Cavity QED with Quantum Gases: New Paradigms in Many-Body Physics
Farokh Mivehvar, Francesco Piazza, Tobias Donner, Helmut Ritsch
RREVIEW ARTICLE
Cavity QED with Quantum Gases: New Paradigms in Many-BodyPhysics
Farokh Mivehvar a and Francesco Piazza b and Tobias Donner c ∗ and Helmut Ritsch a a Institut f¨ur Theoretische Physik, Universit¨at Innsbruck, A-6020 Innsbruck, Austria b Max-Planck-Institut f¨ur Physik komplexer Systeme, D-01187 Dresden, Germany c Institute for Quantum Electronics, ETH Zurich, CH-8093 Zurich, Switzerland ( Compiled February 10, 2021 )We review the recent developments and the current status in the field of quantum-gascavity QED. Since the first experimental demonstration of atomic self-ordering in a systemcomposed of a Bose-Einstein condensate coupled to a quantized electromagnetic mode ofa high- Q optical cavity, the field has rapidly evolved over the past decade. The compositequantum-gas–cavity systems offer the opportunity to implement, simulate, and experi-mentally test fundamental solid-state Hamiltonians, as well as to realize non-equilibriummany-body phenomena beyond conventional condensed-matter scenarios. This hinges onthe unique possibility to design and control in open quantum environments photon-inducedtunable-range interaction potentials for the atoms using tailored pump lasers and dy-namic cavity fields. Notable examples range from Hubbard-like models with long-rangeinteractions exhibiting a lattice-supersolid phase, over emergent magnetic orderings andquasicrystalline symmetries, to the appearance of dynamic gauge potentials and non-equilibrium topological phases. Experiments have managed to load spin-polarized as wellas spinful quantum gases into various cavity geometries and engineer versatile tunable-range atomic interactions. This led to the experimental observation of spontaneous discreteand continuous symmetry breaking with the appearance of soft-modes as well as super-solidity, density and spin self-ordering, dynamic spin-orbit coupling, and non-equilibriumdynamical self-ordered phases among others. In addition, quantum-gas–cavity setups of-fer new platforms for quantum-enhanced measurements. In this review, starting from anintroduction to basic models, we pedagogically summarize a broad range of theoreticaldevelopments and put them in perspective with the current and near future state-of-artexperiments. Keywords: cavity quantum electrodynamics (QED), ultracold quantum gases,Bose-Einstein condensate (BEC), Fermi gases, strong matter-field coupling, Dickesuperradiance, self-organization, cavity-enhanced metrology ∗ Corresponding author. Email: [email protected] a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b ontents1 Introduction 5 P band of the pump lattice . . . . . 523.3.3.2 Structural phase transition . . . . . . . . . . . . . . . 54 U (1) symmetry in ring-cavity geometries . 634.2.2 Collective excitations and dissipationless particle current . . . . 644.3 Supersolidity in a longitudinally pumped ring cavity . . . . . . . . . . 654.4 Supersolidity in other systems . . . . . . . . . . . . . . . . . . . . . . . 65
10 Superradiant self-organization without a steady state 125
11 Cavity-enhanced quantum measurement in quantum gases 135
12 Conclusions, outlook and open challenges 139A1Appendix: The most important and commonly-used symbolsthroughout the paper 143References 145 . Introduction1.1. Historical remarks
Already since the historical conception that matter is built up of indivisible elemen-tary particles, a-tomos , in ancient Greek philosophy, a deeper understanding of thedetailed structure, the formation, and the physical properties of solid matter has beenat the heart of physics and, in particular, material research. Much later with the ad-vent of the periodic table of elements it became finally clear that matter in all of itsuncountable forms and wide ranging properties is composed of only about a hundredatomic elements, which are held together by electromagnetic forces and governed bythe laws of quantum mechanics. While the corresponding Schr¨odinger equation can,at least in principle, readily be written down even in quite general circumstances, itsfull solution can be found solely in very few, exceptionally simple cases.Tremendous progress in experimental techniques have made it possible to nowadaysanalyze the detailed composition, the structure, and essential properties of a wideclass of solids down to the atomic level. Simplified but still very powerful theoreticalmodels and methods as well as a plethora of numerical techniques have been developedin parallel to understand and explain the mechanical, optical, electrical, and thermalproperties of materials. Many fundamental properties like crystal and electronic bandstructures, conductivity, or optical properties nowadays can be numerically predictedand simulated to be at least qualitatively understood. In this context a finite set ofgeneric Hamiltonians, such as lattice spin models in various dimensions [1] or gener-alized Hubbard models [2], have proven to be a viable basis for successful theoreticalmodeling of many observed phenomena. Despite these enormous efforts, however, sev-eral important phenomena, such as high- T c superconductivity, quantum magnetism, orquantum phase transitions, still remain elusive due to the complexity of correspondingmaterials and the hopelessly large Hilbert space needed to study realistic-size quantummany-body problems in full detail.As already pointed out by Feynman decades ago [3], a possible alternative route toinvestigate the physics of more complex and realistic Hamiltonians is to implementthem experimentally by help of other quantum systems. These systems should be cho-sen such that they can be much better controlled, observed in real time in a sufficientlyprecise way, and well understood [4]. While this suggestion of the quantum simula-tion had been considered as a sort of a Gedankenexperiment for decades, breathtakingrecent advances of laser technology and the laser manipulation of atomic gases havemade it feasible to implement a wide class of analog quantum simulations in atomic-molecular-optical (AMO) laboratories [5]. Generic model Hamiltonians developed todescribe key properties of solids can now be tested in extremely well-controlled, tun-able environments magnified by two to three orders of magnitude compared to originalsolid materials. Ultracold quantum gases in tailored optical potentials have so becomea proven workhorse to explore complex multiparticle quantum dynamics in syntheticsolid-state analogs [6,7].A prominent and fundamentally important class of solids exhibits a regular periodiccrystalline structure with a corresponding periodic potential for high energy electrons.The resulting single-particle Bloch energy bands decisively determine electric, mag-netic, optical, and thermal properties of such crystalline solids. A finite set of laserbeams can be utilized to create readily analogous, defect-free, spatially periodic lightintensity distributions mimicking a large class of crystalline potentials at more than ahundred-fold increased scale without any undesired perturbation. Atoms cooled down5o almost absolute zero kinetic temperature can then be used to play the role of elec-trons in crystals and simulate their orbital quantum motion and interactions in theseoptical lattice structures at a directly observable scale [5].By proper choice of laser-light frequencies and atomic species, virtually back-action-free optical potentials spanning over macroscopic distances can be implemented andtuned in space and in real time. Employing more atomic internal states allows also toincorporate spin degrees of freedom and simulate quantum magnetism. Furthermore,geometric phases can be imprinted on charge-neutral atoms via tailored atom-fieldcoupling, in order to mimic minimal coupling and gauge potentials [8,9]. These vastpossibilities have led to a huge number of experimental implementations [6,7], a de-velopment which in turn triggered widespread interest in physics communities beyondthe AMO. This includes, in particular, the solid-state community seeing a unique op-portunity to adapt and implement their paradigmatic model Hamiltonians, and testtheir predictions in a controlled way using quantum-gas setups, which nowadays cananalyze quantum phases with single-site resolution [10–13].Already in the early stage of the field using then available tools for ultracold atoms,a seminal quantum-gas experiment exhibited a reversible quantum phase transitionbetween a superfluid and a Mott-insulator phase exactly as predicted by the Bose-Hubbard Hamiltonian [14]. A huge number of further successful demonstrations ofintriguing condensed-matter phenomena such as superfluidity, quantum magnetism,Abelian and non-Abelian gauge potentials, just to name a few, has followed since [15].Beyond these early, yet very impressive, successful demonstrations, however, sev-eral fundamental open challenges for quantum simulation of solid-state materials withquantum gases have remained. Generally, the underlying lattice structure is externallyprescribed and is independent of particle number and their state. This hinders studiesof self-ordering and crystal formation. Likewise, it is challenging to represent the effectof phonons (i.e., lattice vibrations). More generally, implementation of long- or infinite-range atom-atom interactions proves to be rather demanding. This includes notablypairing interactions in momentum space as in various condensed-matter models for su-perconductivity. Furthermore, fundamental questions concerning crystal growth andmelting, or lattice-defect formation are not so straightforward to implement. The ex-ternally prescribed order generally inhibits structural phase transitions or the appear-ance of new long-range order through dynamical symmetry breaking. As a practicallimitation on the other hand, the observation of any kind of time evolution is gener-ally very slow as measurements are typically destructive in quantum-gas experimentswhich impedes repeated observations within a single measurement sample.The challenge to implement these effects in quantum-gas setups mostly stems fromthe fact that optical potentials are typically required to operate in regimes whereelectromagnetic fields generating the optical potentials are far detuned from any opticalatomic transition, so that most photons propagate almost freely through the atomicgas. Consequently, the back-action of the trapped particles on lattice fields, which is atthe heart of phonon-induced interaction potentials, is extremely small and negligible inmany cases, and sizeable effects only appear in very large setups or closer to resonancewhen dissipation starts to be an issue [16]. To mimic phonon-like long-range couplingin free-space configurations, a diverse set of alternative approaches has been exploited.Amongst others these encompass the creation of hetero-nuclear molecules, the use ofatomic species with strong magnetic dipole moments or the admixture of Rydbergstates giving rise to induced electric dipole moments [17,18] . In all these cases theoriginal lattice remains static, stable, and perfect. As we will see below by help of multi-mode cavity fields and atomic self-ordering this limitation can be largely overcome.6 igure 1. Basic scheme of cavity QED with a quantum gas to realize a self-ordering phase transition. Aquantum gas (blue sphere) is loaded inside a standing-wave cavity and driven by a transverse, coherent pump-laser field. The schematic drawing on the left side illustrates the fundamental photon scattering processesinducing long-range interactions between the atoms, where the grid indicates the standing-wave mode structuresof the cavity and the pump fields: A photon (red wiggly lines) from the pump beam is scattered off a first atominto the cavity mode and then back into the pump off a second atom. Both atoms acquire a momentum kick(blue arrows) due to the photon recoil. These atoms effectively interact with each other via the exchange ofthe cavity photon.
Quantum-gas cavity QED
In this review we concentrate on an alternative approach to implement versatile, tai-lored interatomic interactions and explore novel many-body phenomena using theframework of quantum gases by the help of optical cavities (also referred to as res-onators) composed of high-quality mirrors to enhance atom-photon interaction; seeFigure 1. Any conservative optical force on atoms microscopically originates from thecoherent photon redistribution among various electromagnetic fields acting on theatoms. In principle, any photon scattered by one atom subsequently interacts with theother atoms of the ensemble and can be re-scattered, resulting in non-local photon-induced interactions among the atoms. Hence, non-local interatomic interactions arealready naturally built into optical potentials.Sequential photon scattering processes and strong atom-photon coupling in principlealways lead to atomic back-action on the field, non-local interatomic interactions, andnonlinear effects [16]. However, these sequential photon scattering processes are ex-tremely unlikely and usually negligible for typical free-space configurations operated infar-detuned regimes to reduce dissipation and heating. Luckily the minute back-actionof the atoms on the trapping electromagnetic fields can be dramatically enhanced bythe help of optical resonators. This hinges on the fact that confining the scattered fieldswithin a high- Q cavity allows each photon to multiply pass through the same atomicensemble. Hence resonators significantly enhance the tiny probability of a scatteredphoton to be re-scattered by the same or any other atom in the ensemble, resulting inphoton-induced interatomic interactions—reminiscent of phonon-induced interactionsin natural crystals. Tailored sequential photon scattering in optical resonators thusenables to create a large class of sizable tunable-range interatomic interactions. Inparticular, in multi-mode configurations these can be readily controlled and tuned bythe choice of cavity geometry, the spatial shape, polarization, and frequencies of pumplasers and the internal atomic level structure.Let us note here that for spatially coherent light fields, as they are generally em-7loyed to create optical potentials, all possible scattering amplitudes for a photon bydifferent atoms interfere. Depending on relative phases, they may collectively add up orcancel destructively all together. The collective enhancement is in close analogy withthe well-known Bragg enhancement of coherent wave scattering by periodic structures,which here is further amplified in the presence of an optical resonator.Some intriguing aspects of the collective motional coupling of a quantum gas to acavity mode have been pointed out already two decades ago [19]. It then took severalyears until the first generation of pioneering experiments with cold atomic gases as wellas Bose-Einstein condensates (BECs) in driven optical cavities appeared [20–24]. Asit was shown in several theoretical approaches and first experiments, the collectivelyenhanced atom-field interaction allowed to implement strong coupling of the center-of-mass mode of the atomic ensemble to a single cavity mode. This corresponds tothe implementation of a zero-temperature optomechanic Hamiltonian in the so-calledstrong coupling limit, where single cavity photons create sizable forces and highlynonlinear dynamics [21,24–27].The potential of cavity-enhanced forces and interatomic interactions was, however,fully highlighted experimentally a bit later in a seminal demonstration of superradiantself-ordering of a transversely-driven homogeneous BEC to an ordered crystalline phaseat zero temperature [28]. In detail, quantum fluctuations in the homogeneous atomicdensity stimulate collective scattering of photons from a transversely applied, standing-wave laser into the cavity mode. In turn, photons scattered into the cavity act back non-linearly on the atoms and amplify the BEC density fluctuations. Above a critical pumppower, the system undergoes a reversible zero-temperature quantum-phase transitioninto a perfectly ordered, crystalline state.Mathematically, the underlying dynamics near the self-ordering transition can bewell described by the Dicke Hamiltonian. Hepp and Lieb predicted more than halfa century ago that the Dicke Hamiltonian exhibits a zero-temperature phase transi-tion involving a Z parity-symmetry breaking to a so-called superradiant state in thelimit of very strong coupling [29–31]. Truncating the self-ordering Hamiltonian closeto threshold to low atomic momenta yields exactly the Dicke Hamiltonian with tun-able parameters, allowing thus to emulate the corresponding phase transition [32]. Asthe original Hepp and Lieb prediction based on a very simplified toy model created along-standing and controversial debate on its validity, this first and very clear exper-imental demonstration of the existence of a superradiant phase transition at exactlythe predicted parameter values created huge attention even far beyond the cold-atomcommunity. From a different viewpoint, the experiment can be seen as one of thesuccessful implementations of quantum simulation.Further calculations and a closer look on the detailed underlying dynamics includinglocal atomic interactions revealed even more remarkable properties of the realized self-ordered state. In particular, the self-ordered state has been shown to simultaneouslypossess spatial order associated with the spontaneously broken discrete Z symmetryas well as phase coherence (i.e., off-diagonal long-range order) of the atomic state. Asthe combination of these two properties is usually attributed to lattice supersolids,the self-ordered state constitutes a prominent candidate for such an intriguing stateof matter.After initial theoretical works and these experimental realizations, a plethora ofproposals for generalizations and novel directions to pursue was put forward. A veryprominent example was the introduction of cavity-mediated long-range interatomicinteractions in Hubbard models for bosons [35] as well as for fermions [36]. The phasediagram of the Bose-Hubbard model with cavity-induced global atom-atom interac-8 LSS b Figure 2. Cavity QED with lattice quantum gases. (a) Schematic sketch of a typical setup of cavity QED withquantum gases trapped in a static lattice. The system is described by an extended Hubbard-like Hamiltonianwith cavity-induced global, all-to-all atomic interactions. (b) Phase diagram of an extended Bose-Hubbardmodel with cavity-induced global atomic interactions obtained in the ETH lattice-setup experiment. In additionto the common superfluid (SF) and Mott-insulator (MI) states, the system also exhibits a novel lattice supersolid(LSS) and a charge-density-wave (CDW) state. Figure (a) reprinted from Ref. [33], and Figure (b) adaptedand reprinted with permission from Ref. [34] published at 2016 by the Nature Publishing Group. tions was predicted using various approximate theoretical approaches and subsequentlyalso measured experimentally [34,37], as shown in Figure 2. Here, strong evidence ofan intermediate lattice-supersolid phase between the homogeneous superfluid and thedensity-ordered superradiant phase was found in agreement with theoretical models.In theory, cavity-induced interatomic interactions can be tuned from attractiveto repulsive simply by changing the laser frequencies with respect to cavity reso-nances [38,39]. Already in simple 1D configurations repulsive potentials allow to im-plement intriguing scenarios such as a nonequilibrium Su-Schrieffer-Heeger model withperiodically modulated tunneling amplitude, where symmetry-protected edge statesand other topologically nontrivial effects are expected to appear [40]. Particularly,in 2D configurations interesting phenomena have been predicted, including control-lable dynamical instabilities in conjunction with limit cycles—also referred to as timecrystals in this context—as well as strong indications of chaotic dynamics [38,41–43].Interestingly, cavity-induced interactions can also be tuned to couple various speciesof atoms or even molecules in a single or even distinct traps [44].As atomic species generally possess isotopes with variable nuclear spin, they comewith a large variation of hyperfine level structures. These hyperfine internal states canbe exploited to simulate spin degrees of freedom. Again, the direct spin-spin interactionbetween atoms is quite weak at optical wavelength distances. Nevertheless, it has beensuggested that using specifically designed cavity-enhanced optical Raman transitionsbetween hyperfine states of a multi-component quantum gas allows to implement muchstronger interatomic spin interactions and simulate a wide class of long-range quan-tum spin Hamiltonians, including Heisenberg and Dzyaloshinskii-Moriya interactions,across the self-ordering phase transition [45,46]; see Figure 3. The first fundamentalsteps towards this goal had already been implemented in experiments [47,48], open-ing a new route towards quantum magnetism in atomic quantum gases. The use ofcavity-enhanced Raman transitions between hyperfine states also enabled the first re-alization of a dynamical synthetic spin-orbit coupling [49], paving the way for theimplementation of dynamical artificial gauge potentials.9 ump laser pump laseratomic spinscavity(a) (b)self-ordered atomic spins
Figure 3. Cavity-QED implementation of quantum magnetism. (a) Schematic sketch of the setup. Spinorultracold atoms (light-blue arrows) are driven transversely by coherent pump lasers (red arrows) and stronglycoupled to a cavity mode. By properly designing the spatial forms and frequencies of the lasers and thecavity field, a variety of long-range spin-spin interactions can be induced between atoms by cavity photons.(b) A typical cavity-induced magnetic self-ordering. Here, the cavity mediates Dzyaloshinskii-Moriya-typeinteractions between the atoms, resulting in chiral spin-spiral order. Figure reprinted with permission fromRef. [46] © As a generic, advantageous consequence of the openness of optical resonators, real-time information on dynamics of cavity-QED systems can be obtained simply by non-destructive observation and analysis of cavity output fields [50–52], thus replacing orcomplementing destructive absorption-imaging measurements of atomic density dis-tributions. This, at least in principle, allows novel and improved implementations ofquantum-enhanced sensing [53–57]. Besides, in some cases one can even expect the ap-pearance of genuine consequences of quantum measurement back-action on the system,which can be controlled on a new level of precision [58,59].Mode functions in principle form a basis in the cavity volume. While a single cavitymode covering the whole quantum gas mediates infinite-range interactions among theatoms, by employing a large number of degenerate cavity modes as in confocal orconcentric cavities the range of interatomic interactions can be vastly tuned [60,61].Also by help of a manifold of pump frequencies (as in frequency combs) and modes,range and shape of interatomic interactions can be controlled [62,63].
The present review article
The experimental merger of quantum gases with cavity QED opened a wide field ofpossibilities ranging from cavity optomechanics at zero temperature to the real-timeexploration of quantum phase transitions [64]. The central physical mechanism is thequantum back-action between the dynamic cavity fields and the atomic degrees of free-dom. Combining the excellent control of quantum-gas experiments over external andinternal atomic degrees of freedom with the strong coupling to quantized light fieldsin high-finesse optical resonators provides completely new opportunities specificallyfor the quantum simulation of many-body systems [4,65,66] as well as for quantum-enhanced optimization [62,67], for cavity optomechanics in the bad [22,24,27,68,69]and in the good [70–73] cavity limits, and for quantum sensing [55].The field has rapidly expanded in the last decade. In order for the material to fitinto a single manuscript with a reasonable length, we restrict this review to scenarioswhere the back-action between the quantum gas and the cavity field leads to the emer-gence of correlated many-body phases, some of which go beyond the scope of quantum10imulation in that they offer novel paradigms in many-body physics. In particular, thesystems of interest in the present review are chiefly transversely driven quantum gases inside single- and many-mode cavities. Therefore, we do not include and discuss ex-plicitly thermal gases inside cavities [20,21,74,75], cavity optomechanical systems withquantum gases [24], and Dicke-type cavity models with atomic mediums without quan-tized motion [76–81]. Such scenarios have been extensively considered in the previousreview articles [64,82,83] and we refer interested readers to these excellent reviews.Despite these restrictions in choosing the systems of interest, the scope of the currentreview is quite vast and encompasses plethora of intriguing physical phenomena.
In Section 2 we introduce basic physical notions and models for interacting quantumgases coupled to optical cavities, followed by a short review of typical approximatetheoretical approaches mainly focusing on mean-field treatments and collective ele-mentary excitations for both bosons and fermions. Basic principles and fundamentalexperiments on discrete symmetry breaking in superradiant quantum phase transitionsleading to the spontaneous self-ordering and crystallization are reviewed in Chapter 3,followed by foundational examples for continuous symmetry breaking and the forma-tion of nonequilibrium supersolids in Chapter 4. Section 5 is devoted to quantum gasesstrongly coupled to many modes of optical cavities featuring tunable-range cavity-mediated interatomic interactions as demonstrated recently, while in Chapter 6 wereview the dynamics of multilevel quantum gases in cavities introducing the notion ofcavity-induced long-range spin-spin interactions as a versatile platform for simulatingquantum magnetism. Optical atomic lattice models including cavity-mediated inter-atomic interactions are introduced and discussed in detail in Section 7, while dynamicalsynthetic lattice gauge potentials and nonequilibrium topological states are reviewed inChapter 8. The great versatility of cavity-generated dynamical optical potentials andself-ordering is further highlighted in Chapter 9 discussing quasicrystal formation andemergent symmetries in geometries involving a few cavities. In Chapter 10 we reviewfurther prospects and consequences of the openness of optical resonators in the atomicself-ordering, which feature dissipation forces, strongly nonlinear dynamics, and dy-namical instabilities, followed by a short discussion on measurement back-action andcavity-enhanced sensing on Section 11. Finally in Section 12, we conclude by addingsome remarks on future prospects and challenges.We would like to note here that we have taken a pedagogical approach in composingthis review article in the expense of a longer manuscript. In the view of this, chaptersare, however, almost independent from each other. In particular, after reading Chap-ter 2, readers based on their interest and need can select and jump to any chapterwithout a great discontinuity.
2. Fundamentals: Ultracold atoms dispersively coupled to a single cavitymode
In this chapter, we introduce the framework for studying the many-body physics ofcavity QED with quantum gases. We start our description with the most fundamentalsystem: Ultracold atoms transversely driven by a laser field and dispersively interactingwith a single quantized electromagnetic radiation field of a standing-wave linear cavity.Most of the basic concepts and methods can be captured in this simple model. More-11ver, this system constitutes the first experimental setup where collective phenomenawere observed. In the subsequent chapters, we shall generalize this simple model todifferent scenarios and regimes, including additional atomic and photonic degrees offreedom—most notably spinor gases as well as multiple cavity modes. For the con-venience, we summarize the most important and commonly-used symbols throughoutthe paper in Appendix A1. We note that some of the symbols we use in this reviewcan be different from those used in some literature.
The basic model
Consider ultracold two-level atoms with internal states {| g (cid:105) , | e (cid:105)} and the transitionfrequency ω a inside a single-mode standing-wave linear cavity with the resonance fre-quency ω c . The wave length λ c and the wave number k c of the cavity mode are re-lated to the resonance frequency via ω c = ck c = 2 πc/λ c , with c being the speed oflight. The atoms are driven by a standing-wave classical coherent laser field in thetransverse direction with the position-dependent Rabi frequency Ω( r ) = Ω cos( k c y )and are strongly coupled to the quantized mode of the cavity with the strength G ( r ) = G cos( k c x ); see Figure 4. Ω and G are the maximum Rabi frequency ofthe pump field and the maximum single atom-photon cavity-QED coupling strength,respectively. Here we do not take into account the transverse profile of the cavity mode,which is a Gaussian in the simplest case; the transverse profile of cavity modes will bediscussed in more detail in Section 2.7.1 and will be included in multimode cavitiesin Section 5. Note that the Rabi rate Ω( r ) can have a different spatial form (e.g., viausing a running-wave pump laser) and the pump laser can also impinge the atomiccloud with an angle different than 90 ° with respect to the cavity axis; these scenarioswill be considered in the following chapters.The system in the rotating frame of the pump laser is described by the many-bodyHamiltonian [84],ˆ H = − (cid:126) ∆ c ˆ a † ˆ a + (cid:88) τ = g,e (cid:90) ˆ ψ † τ ( r ) (cid:20) − (cid:126) M ∇ + V ext ( r ) (cid:21) ˆ ψ τ ( r ) d r − (cid:126) ∆ a (cid:90) ˆ ψ † e ( r ) ˆ ψ e ( r ) d r + (cid:126) (cid:90) (cid:110) ˆ ψ † e ( r ) [Ω( r ) + G ( r )ˆ a ] ˆ ψ g ( r ) + H . c . (cid:111) d r + ˆ H int , (1)where ∆ a ≡ ω p − ω a is the relative frequency between the pump-laser frequency ω p and the atomic transition frequency ω a , and ∆ c ≡ ω p − ω c the relative frequencybetween the pump-laser and cavity frequencies. Note that although in general ∆ c (cid:54) = 0,in practice one has | ∆ c | /ω p (cid:28) k p (cid:39) k c , which has been made use ofabove. Here, ˆ ψ τ ( r ) = ˆ ψ τ ( r , t ) and ˆ a = ˆ a ( t ) are the slowly-varying bosonic or fermionicatomic, and bosonic photonic annihilation field operators, respectively. V ext ( r ) is anexternal trapping potential acting identically on both internal atomic states, and ˆ H int accounts for two-body interactions between the atoms. For dilute quantum gases atlow temperatures, the two-body interaction Hamiltonian is well represented by localcontact interactions [85],ˆ H int = (cid:88) τ g ττ (cid:90) ˆ ψ † τ ( r ) ψ † τ ( r ) ˆ ψ τ ( r ) ˆ ψ τ ( r ) d r + g eg (cid:90) ˆ ψ † e ( r ) ψ † g ( r ) ˆ ψ g ( r ) ˆ ψ e ( r ) d r , (2)where the two-body contact-interaction strengths g gg , g ee , and g eg = g ge are related12 igure 4. Schematic sketch of a basic quantum-gas–cavity-QED setup. A cloud of N ultracold atoms (blue)at temperature T is confined by an external potential V ext ( r ), centered with respect to the cavity mode (lightred). The resonance frequency ω c of the cavity is far detuned from the atomic transition frequency ω a . Thecoupling between the atomic cloud and the cavity field is captured by the spatially dependent vacuum Rabirate G ( r ). The atomic cloud is driven by a transverse pump laser (red arrows) with a polarization (cid:15) p at afrequency ω p close to the cavity resonance frequency. The transverse pump laser is retroreflected by a planarmirror and its coupling to the atoms is described by the spatially dependent Rabi rate Ω( r ). The cavity fieldleaking out of the resonator at a rate κ is detected either via a single-photon counter or a heterodyne system.The quantization axis for the atoms is fixed by an applied magnetic field B . to the respective s-wave scattering lengths a sττ (cid:48) by g ττ (cid:48) = 4 πa sττ (cid:48) (cid:126) /m ( τ, τ (cid:48) = g, e ).The strength of the two-body s-wave interactions g ττ (cid:48) can in principle be tuned us-ing Feshbach resonance techniques. In many experiments in cavity QED, the contactinteractions are negligible compared to cavity-mediated interactions (see Section 2.2)and can be omitted. Note that the s-wave interactions proportional to g gg and g ee , i.e.,the first term in Equation (2), are identically zero for fermionic atoms owing to thePauli exclusion principle [86].The dynamics of the system is described by the Heisenberg equations of motion forthe atomic and photonic field operators, i (cid:126) ∂ ˆ ψ e ( r ) ∂t = (cid:20) − (cid:126) M ∇ + V ext ( r ) − (cid:126) ∆ a + g ee ˆ ψ † e ( r ) ˆ ψ e ( r ) + g eg ˆ ψ † g ( r ) ˆ ψ g ( r ) (cid:21) ˆ ψ e ( r )+ (cid:126) [Ω( r ) + G ( r )ˆ a ] ˆ ψ g ( r ) ,i (cid:126) ∂ ˆ ψ g ( r ) ∂t = (cid:20) − (cid:126) M ∇ + V ext ( r ) + g gg ˆ ψ † g ( r ) ˆ ψ g ( r ) + g ge ˆ ψ † e ( r ) ˆ ψ e ( r ) (cid:21) ˆ ψ g ( r )+ (cid:126) (cid:104) Ω( r ) + G ( r )ˆ a † (cid:105) ˆ ψ e ( r ) ,i (cid:126) ∂ ˆ a∂t = − (cid:126) ∆ c ˆ a + (cid:126) (cid:90) G ( r ) ˆ ψ † g ( r ) ˆ ψ e ( r ) d r , (3)where we have assumed, without loss of generality, that { Ω , G } ∈ R . The pump andcavity frequencies are assumed to be far detuned from the atomic transition frequencysuch that 1 / | ∆ a | is the fastest time scale in the system. In this “dispersive” regime, theatomic excited state is not populated significantly—minimizing heating of the atomic13loud due to spontaneous emission—and its dynamics can be eliminated adiabatically:ˆ ψ e, ss ( r ) (cid:39) a [Ω( r ) + G ( r )ˆ a ] ˆ ψ g ( r ) . (4)Kinetic energy, potential, and contact interactions have been omitted as they arenegligible compared to (cid:126) ∆ a . Substituting the steady-state, excited-state field operatorˆ ψ e, ss ( r ) into Equation (3), and ignoring the term ˆ ψ † e ( r ) ˆ ψ e ( r ) ∝ / ∆ a , yields a closedset of two coupled equations for the atomic ground-state and photonic field operators, i (cid:126) ∂ ˆ ψ∂t = (cid:20) − (cid:126) ∇ M + V ext ( r ) + (cid:126) V ( r ) + (cid:126) U ( r )ˆ a † ˆ a + (cid:126) η ( r )(ˆ a † + ˆ a ) + g ˆ n ( r ) (cid:21) ˆ ψ, (5a) i (cid:126) ∂ ˆ a∂t = − (cid:126) (cid:18) ∆ c − (cid:90) U ( r )ˆ n ( r ) d r (cid:19) ˆ a + (cid:126) (cid:90) η ( r )ˆ n ( r ) d r , (5b)where ˆ ψ ( r , t ) ≡ ˆ ψ g ( r , t ) (the explicit position and time dependence of the atomicfield operator has been suppressed above to save space) and g ≡ g gg for the sakeof simplicity of the notation, and ˆ n ( r ) = ˆ ψ † ( r ) ˆ ψ ( r ). Here we have introduced thedefinition of the transverse pump lattice V ( r ) ≡ V cos ( k c y ) with the lattice depth V = Ω / ∆ a , and the intra-cavity lattice per photon U ( r ) ≡ U cos ( k c x ) with thesingle-photon lattice depth U = G / ∆ a . Finally, the checkerboard lattice formedvia the interference of the transverse pump laser with the cavity field is given by η ( r ) ≡ η cos( k c x ) cos( k c y ) with the two-photon Rabi frequency η = Ω G / ∆ a . Westress again that the non-linear contact interaction term g ˆ n ( r ) is absent for fermionicatoms due to the Pauli exclusion principle.The dynamics of the atomic field operator ˆ ψ ( r , t ) depend on the photonic operatorˆ a ( t ), via the dynamical quantum potential (cid:126) ˆ V SR ( r ) ≡ (cid:126) U ( r )ˆ a † ˆ a + (cid:126) η ( r )(ˆ a † + ˆ a ), andvice versa. That is, in addition to the classical optical potential (cid:126) V ( r ) due to theabsorption and emission of pump laser photons, the atoms experience two dynamicalquantum optical potentials: (cid:126) U ( r )ˆ a † ˆ a due to the absorption and emission of cavityphotons, and (cid:126) η ( r )(ˆ a † + ˆ a ) owing to the redistribution of photons between the pumplaser and the cavity field. Note that the atomic field acts back on the photonic field byshifting the cavity resonance frequency by − (cid:82) U ( r )ˆ n ( r ) d r , leading to the dispersively-shifted cavity detuning δ c ≡ ∆ c − (cid:82) U ( r )ˆ n ( r ) d r . It also serves as an indirect effectivepump for the cavity field by Bragg scattering photons from the pump laser into thecavity indicated by the term (cid:82) η ( r )ˆ n ( r ) d r .When the pump field is red detuned with respect to the atomic transition, ∆ a < δ c <
0, the minima of theinterference potential (cid:126) η ( r )(ˆ a † + ˆ a ) are located at the position of the atoms scatteringphotons. This gives rise to an atomic density modulation with maxima at positionswhere constructive photon scattering from the pump into the cavity is supported. Thisleads to a positive runaway feedback mechanism and hence “self-ordering” as it will bediscussed in detail in Section 3.1. If in contrast, the dispersively-shifted cavity detuningis blue, δ c >
0, the phase of the intra-cavity field is shifted by π , such that the emergingcheckerboard potential seen by the atoms is inverted and the atoms are pushed away14 igure 5. Potential V SR ( x ) along the cavity axis given by the sum of the cavity standing-wave lattice andthe interference lattice between pump and cavity fields. In the case of a pump laser field red detuned withrespect to the atomic resonance, ∆ a <
0, the potential is heteropolar and attractive, while it is homopolar andrepulsive for a blue-detuned pump field, ∆ a >
0. The dashed, light-color curves are the respective Z -symmetricpotentials. from their initial positions and constructive photon scattering is suppressed. The caseof atomic blue detuning ∆ a > H eff = (cid:90) ˆ ψ † (cid:20) − (cid:126) ∇ M + V ext ( r ) + (cid:126) V ( r ) + (cid:126) U ( r )ˆ a † ˆ a + (cid:126) η ( r )(ˆ a † + ˆ a ) + g n ( r ) (cid:21) ˆ ψd r − (cid:126) ∆ c ˆ a † ˆ a. (6)The corresponding effective single-particle atomic Hamiltonian density plus the freecavity-field Hamiltonian is then given byˆ H , eff = − (cid:126) M ∇ + V ext ( r ) + (cid:126) V ( r ) + (cid:126) U ( r )ˆ a † ˆ a + (cid:126) η ( r )(ˆ a † + ˆ a ) − (cid:126) ∆ c ˆ a † ˆ a . (7)This effective model has been the basis for the description of many groundbreakingexperiments as well as theoretical works in the field. The generalization of this simplemodel to more sophisticated and other novel scenarios has been the core of the state-of-the-art research, both in theory and experiment. These will be the subject of thefollowing sections.The effective Hamiltonian, Equation (6), is invariant under the common U (1) gaugesymmetry ˆ ψ ( r ) → ˆ ψ (cid:48) ( r ) = e iχ ˆ ψ ( r ), with χ being a position-independent phase shift,ensuring the mass conservation in the system. In addition, in the absence of the exter-nal (harmonic) trap V ext ( r ) = 0, the effective Hamiltonian ˆ H eff possesses a Z paritysymmetry: ˆ a → ˆ a (cid:48) = − ˆ a and r → r (cid:48) = r + ( λ c / e x , where ˆ e x is the unit vector alongthe x direction. That is, the effective Hamiltonian is invariant under the combinedphase rotation of the cavity field ˆ a by π and the λ c / H eff in the absence of the external(harmonic) trap is only λ c -periodic in both x and y directions.Since the pump laser and the cavity field are far detuned from the atomic excitedstate and the atomic excited state is not populated significantly, the spontaneous emis-sion from the atomic excited state is suppressed strongly. However, photon losses outof the cavity constitute another dissipation channel in this intrinsically open system.The photon losses can be accounted for in the master equation for the atom-field15ensity operator ˆ ρ of the system, ddt ˆ ρ = 1 i (cid:126) [ ˆ H eff , ˆ ρ ] + L ˆ ρ, (8)via the Liouvillean terms L ˆ ρ = κ (cid:16) a ˆ ρ ˆ a † − ˆ a † ˆ a ˆ ρ − ˆ ρ ˆ a † ˆ a (cid:17) . (9)Here, 2 κ is the photon decay rate through the cavity mirrors and determines the bandwidth of the cavity.Equivalently, the photon losses introduce a damping term and fluctuations in theHeisenberg equation of motion for the cavity electromagnetic-field operator [87], i (cid:126) ∂ ˆ a∂t = − (cid:126) (cid:18) ∆ c − (cid:90) U ( r )ˆ n ( r ) d r + iκ (cid:19) ˆ a + (cid:126) (cid:90) η ( r )ˆ n ( r ) d r + ˆ ξ, (10)where ˆ ξ ( t ) is a Gaussian noise operator with zero mean value and the correlationfunction (cid:104) ˆ ξ ( t ) ˆ ξ † ( t (cid:48) ) (cid:105) = 2 κδ ( t − t (cid:48) ) for a zero-temperature bath outside the cavity, acondition which is well satisfied in the optical frequency domain. The noise operatorˆ ξ ( t ) induces a Langevin-type stochastic behavior into the field dynamics. The photonlosses have important consequences. As discussed in Section 10, the most noticeableeffect of the photon losses appears in the dynamics, where they can induce new insta-bilities and dynamical phases via a detuning-dependent phase shift of the cavity field.Moreover, in presence of loss even the stationary state of the system is not in generalin thermal equilibrium. Though in many cases it can be approximately described assuch that, the loss-induced noise generates effectively a finite temperature. Cavity-induced long-range interactions between atoms
When the cavity-field dynamics evolve on a much faster time scale compared to thecenter-of-mass motion of the atoms, it follows the latter adiabatically. Consequently,the cavity field can be slaved to the atomic external degrees of freedom in the senseof the Born-Oppenheimer approximation. By setting ∂ ˆ a/∂t = 0 in Equation (10) (andneglecting the loss-induced noise term ˆ ξ ), one obtains the steady-state cavity fieldoperator, ˆ a ss = (cid:82) η ( r )ˆ n ( r ) d r ˜∆ c − (cid:82) U ( r )ˆ n ( r ) d r (cid:39) c (cid:90) η ( r )ˆ n ( r ) d r + O ( 1∆ a ) , (11)by assuming | ˜∆ c | ≡ | ∆ c + iκ | (cid:29) | (cid:82) U ( r )ˆ n ( r ) d r | , up to first order in 1 / ∆ a .Substituting the steady-state photonic field operator ˆ a ss in the Heisenberg equationof motion for the atomic field operator ∂ ˆ ψ ( r , t ) /∂t , Equation (5a), yields an atom-onlydescription of the system with the Hamiltonian,ˆ H eff-at = (cid:90) ˆ ψ † ( r ) (cid:20) − (cid:126) M ∇ + V ext ( r ) + (cid:126) V ( r ) + 12 g ˆ n ( r ) (cid:21) ˆ ψ ( r ) d r + (cid:90) (cid:90) D ( r , r (cid:48) )ˆ n ( r )ˆ n ( r (cid:48) ) d r (cid:48) d r + O ( 1∆ a ) , (12)16p to 1 / ∆ a , where D ( r , r (cid:48) ) = 2 (cid:126) ∆ c | ˜∆ c | η ( r (cid:48) ) η ( r ) = 2 (cid:126) ∆ c Ω G ∆ a | ˜∆ c | cos( k c x (cid:48) ) cos( k c y (cid:48) ) cos( k c x ) cos( k c y ) . (13)Note that in general one has to first symmetrize the atomic Heisenberg equation,Equation (5a), due to the freedom in ordering of the photonic { ˆ a, ˆ a † } and atomic ˆ ψ ( r )field operators as they commute. That is, the term proportional to (ˆ a † + ˆ a ) ˆ ψ ( r ) inEquation (5a) is symmetrized as [(ˆ a † + ˆ a ) ˆ ψ ( r ) + ˆ ψ ( r )(ˆ a † + ˆ a )] / a ss . The potential term (cid:126) U ( r )ˆ a † ˆ a in the atomicHeisenberg equation (5a) is omitted as it yields interactions which scale as ∆ − a .The penultimate term in the effective atom-only Hamiltonian ˆ H eff-at , Equation (12),describes a cavity-mediated infinite-ranged, or “global” density-density interaction. Itstems from the back-action of the quantized cavity field on the atoms. That is, as theatoms are strongly coupled to the dynamic cavity field, a local change in the atomicdistribution affects significantly the cavity field. Since the cavity electromagnetic fieldis global (only restricted by the cavity mirrors), the modification of the cavity fieldis sensed by all the atoms in the cavity, regardless of their distance from the originalatomic perturbation. This induces a density-density interaction among the atoms withthe non-decaying, periodic strength D ( r , r (cid:48) ), Equation (13).This global interaction can also be understood from the microscopic photon scatter-ing processes: A pump-laser photon is scattered by an atom into the cavity mode, andthen later scattered back into the pump mode by a second atom. These two atoms are,therefore, correlated and interact with each other via the exchange of a cavity photon.Since photons are delocalized over the entire cavity mode, the cavity-mediated inter-action is of global range. The range of the cavity-mediated interaction can be tuned byexploiting multiple modes of a cavity, as will be discussed in Section 5. Furthermore, inaddition to the density-density interaction, other forms of global interactions such asa spin-spin interaction can be mediated via cavity photons using more atomic internalstates as described in Section 6.The effective atom-only Hamiltonian can also be obtained by directly substitutingthe steady-state photonic field operator ˆ a ss in the effective Hamiltonian, Equation (6),as well as the Liouville superoperator, Equation (9). The resultant effective atom-onlyHamiltonian will have the same general form as ˆ H eff-at (12), but with a somewhatdifferent cavity-mediated interaction strength. Nevertheless, within the realm of thevalidity of these effective atom-only Hamiltonians, namely, fast cavity-field dynamics,both effective atom-only Hamiltonians yield qualitatively the same results [84].In order to have a constructive photon scattering from the pump laser into thecavity mode, it is usually required to work on the red relative cavity detuning side,∆ c <
0, as mentioned in the previous section. Therefore, for instance for an atom inthe origin (i.e., r (cid:48) = 0) the cavity-mediated interaction D ( r ,
0) is maximally attractive(i.e., negative) for other atoms located at integer multiples of λ c along the x and y directions [ r = ( (cid:96), λ c and r = (0 , m ) λ c , respectively, with (cid:96) and m being integers]as well as at integer multiples of λ c / √ r = ( (cid:96), ± (cid:96) ) λ c / D ( r , r (cid:48) ) can similarly be obtained for any two atomslocated at arbitrary positions r and r (cid:48) .The cavity-mediated interaction (13) favors a λ c -periodic checkerboard density pat-tern. When the energy gain due to the cavity-mediated global density-density in-17eraction in the checkerboard density pattern dominates over other energy penaltiesarising from the local terms in the effective atom-only Hamiltonian ˆ H eff-at , Equa-tion (12), the checkerboard density pattern is then stabilized in the system. Thisso-called “self-ordering” of the atoms into the checkerboard density pattern in theeffective atom-only description of the system coincides with the onset of the “super-radiance” in the cavity—i.e., the appearance of a finite coherent component for thecavity field—as we will discuss in detail in Section 3. That said, in certain regimes thecavity-mediated interaction can induce phase transitions which are not accompaniedby superradiance. Examples include cavity-fluctuation-induced superconductivity dis-cussed in Section 6.3.3, lattice magnetic orders in Section 7.3.2, and cavity-inducedspin glasses with a single spin-1/2 atom per site [77].As discussed intuitively above, the cavity-mediated global atom-atom interactionsfavor an ordered, crystalline density structure. This can also be explained in a math-ematically elegant way through the the atom-only Heisenberg equation of motion forthe atomic field operator [i.e., after substituting the steady-state photonic field oper-ator ˆ a ss (11) in the Heisenberg equation of motion for the atomic field operator (5a),which yields the effective atom-only Hamiltonian (12) as discussed above], i (cid:126) ∂ ˆ ψ ( r , t ) ∂t = (cid:20) − (cid:126) ∇ M + V ext ( r ) + (cid:126) V ( r ) + 2 (cid:126) ∆ c η | ˜∆ c | η ( r ) ˆΘ + g ˆ n ( r ) (cid:21) ˆ ψ ( r , t ) . (14)Here we have kept the terms solely up to 1 / ∆ a (similar to the effective atom-onlyHamiltonian ˆ H eff-at ) and have introduced the operatorˆΘ = 1 η (cid:90) η ( r (cid:48) )ˆ n ( r (cid:48) ) d r (cid:48) . (15)The operator ˆΘ can be envisaged as an order parameter for the system, characterizingthe degree of the density crystallization in the checkerboard potential η ( r ). The groundstate of the system in the limit of fast cavity dynamics is obtained from Equation (14)along with the self-consistent solution for the order parameter (15), as will be discussedin Section 2.4 for a mean-field approach. Note that ˆΘ is directly related to the steady-state cavity field operator (11) asˆ a ss = η ˆΘ∆ c − (cid:82) U ( r )ˆ n ( r )d r + iκ . (16)Through this review paper, we will see how one can engineer various novel self-consistent, emergent orders in ultracold atoms coupled to optical cavities.We note that in the limit of fast cavity-field dynamics considered in this section, agreat deal of interesting physics related to the retardation effects of the electromag-netic field has been already discarded. In Chapter 10, we will encounter examples ofcollective phenomena which can only be described by taking into account the finite-ness of the cavity timescale. One also has to note that such an adiabatically eliminatedatom-only Hamiltonian does not always accurately capture all atomic dynamics [88].18 .3. Applicability of the Dicke model N far-positioned two-level atoms prepared in their excited state decay independentlyfrom one another, incoherently emitting an electromagnetic field with an energy-density maximum proportional to N . However, if the atoms are trapped within afraction of a wavelength, the emitted photons interfere constructively resulting in atransient superradiant pulse with an energy-density maximum proportional to N aspredicted by Dicke in 1954 [89]. Hepp and Lieb in 1973 predicted that a steady-stateform of superradiance can occur in an ensemble of two-level atoms coupled identicallyto a quantized cavity mode [29,90]. However, for atoms with two electronic states thesuperradiance is supposed to take place for atom-photon coupling in optical-frequencyorder, a situation beyond state-of-the-art experimental capabilities. Furthermore, theexistence of the superradiant phase transition for such strong atom-photon couplingshas been under an intense debate due to the ignored A (i.e., vector potential squared)term.Let us now examine the low-energy limit of the effective Hamiltonian ˆ H eff , Equa-tion (6), at zero temperature for bosonic atoms in the absence of the external potentialand the two-body interaction. The quantum potential η ( r ) ∝ cos( k c x ) cos( k c y ) definesthe periodicity of the effective Hamiltonian: The system is λ c periodic along both x and y directions. That is, starting from the zero momentum state, only momentumstates with integer multiples of k c , k = ( k x , k y ) = ( l, m ) k c with l, m ∈ Z , can be oc-cupied via the photon scattering processes. Therefore, the atomic field operator ˆ ψ ( r )can be expanded in the momentum basis in the unit of k c using plan waves as,ˆ ψ ( r ) = 1 √ A (cid:88) l,m ∈ Z e ik c ( lx + my ) ˆ b l,m , (17)where ˆ b l,m is the bosonic annihilation operator for momentum state k = ( l, m ) k c and A is the surface area of the Wigner-Seitz cell spanned by the elementary latticevectors. Therefore, up to an immaterial constant term, the effective Hamiltonian (6)in momentum space takes the formˆ H eff = (cid:88) l,m ∈ Z (cid:20) (cid:126) ω r (cid:0) l + m (cid:1) ˆ b † l,m ˆ b l,m + (cid:126) η a † + ˆ a ) (cid:16) ˆ b † l +1 ,m +1 ˆ b l,m + ˆ b † l +1 ,m − ˆ b l,m + H.c. (cid:17) + (cid:126) V (cid:16) ˆ b † l,m +2 ˆ b l,m + H.c. (cid:17) + (cid:126) U a † ˆ a (cid:16) ˆ b † l +2 ,m ˆ b l,m + H.c. (cid:17) (cid:21) − (cid:126) δ c ˆ a † ˆ a, (18)where ω r = (cid:126) k c / M is the recoil frequency and δ c = ∆ c − N U / N being thetotal atom number .In the low-energy limit, one can restrict the momentum modes solely to l, m = { , ± } . In particular, the momentum states k = (0 ,
0) and ( ± , ±
1) form a closedcoupled set with ˆ b † , ˆ b , + (cid:80) l,m = ± ˆ b † l,m ˆ b l,m = N being a constant of motion. Bydefining the collective atomic raising and lowering ˆ J + = ˆ J †− = ( (cid:80) l,m = ± ˆ b † l,m )ˆ b , andpopulation-imbalance ˆ J z = ( (cid:80) l,m = ± ˆ b † l,m ˆ b l,m − ˆ b † , ˆ b , ) / Note that δ c = ∆ c − NU / δ c ≡ ∆ c − (cid:82) U ( r )ˆ n ( r ) d r defined earlier in Section 2.1. Throughout this review paper, we will use δ c in a ratherliberal way, but all ultimately related to the dispersively shifted cavity detuning. H LE (cid:39) (cid:126) ω ˆ J z − (cid:126) δ c ˆ a † ˆ a + 12 (cid:126) η (ˆ a † + ˆ a )( ˆ J + + ˆ J − ) . (19)This low-energy Hamiltonian has the exact form of the Dicke Hamiltonian [83,91], de-scribing the dynamics of pseudospins with the transition frequency ω = 2 ω r coupledidentically to a quantized bosonic field with the frequency − δ c . The Dicke Hamilto-nian is fully connected and shows a mean-field-type superradiant phase transition atthe critical coupling √ N η c / √− ω δ c / Z parity symmetry of thesystem, ˆ a → − ˆ a and ˆ J ± → − ˆ J ± , both at zero and finite temperature [92]. However,due to photon losses out of the cavity, the transition takes place instead at the criticalcoupling [83], √ N η c = (cid:115) ω ( δ c + κ ) − δ c . (20)Since here the involved atomic states coupled via the radiation field are momentumstates with an energy separation in a much lower recoil-energy regime (cid:126) ω r , the requiredatom-photon coupling for the superradiant phase transition reduces substantially andfalls into the recoil-energy regime. As an important consequence, the no-go theoremsconcerning the superradiant phase transition do not apply in this case as well. Such asuperradiant phase is generically present for both driven bosonic and fermionic atomsinside optical cavities as we will see throughout this review article. Mean-field description
In this section we introduce the commonly-used self-consistent mean-field treatment ofthe system. This approach is justified since the cavity electromagnetic field is macro-scopically occupied in the self-ordered phase and the cavity-mediated interactions insingle-mode cavities are global, as discussed in the preceding section; mean-field ap-proaches generically become accurate in globally interacting systems and fully con-nected models in the thermodynamic limit [29,93–95]. For the photonic field, thismean-field approach amounts to the assumption that the electromagnetic field is ina coherent state, and therefore α = (cid:104) ˆ a (cid:105) is the coherent field amplitude. By takingthe quantum average from the Heisenberg equation of motion of the cavity operator,Equation (10), one obtains a mean-field equation for the field amplitude, i (cid:126) ∂α∂t = − (cid:126) (cid:18) ∆ c − (cid:90) U ( r ) (cid:104) ˆ n ( r ) (cid:105) d r + iκ (cid:19) α + (cid:126) (cid:90) η ( r ) (cid:104) ˆ n ( r ) (cid:105) d r . (21)The crucial approximation involved here is the factorization of the average (cid:104) ˆ n ( r )ˆ a (cid:105) (cid:39)(cid:104) ˆ n ( r ) (cid:105) α . Combined with an equation for the quantum-averaged density (cid:104) ˆ n ( r ) (cid:105) = n ( r ) = n ( r , α ) with α = α ( n ( r )), the set of self-consistent, coupled mean-field equa-tions is closed. Below, we will consider the application of this mean-field approach toboth bosonic and fermionic atoms.We stress that this approximation only regards the atom-light interaction, ignoringany other interaction that might be present among the atoms. This includes intrinsicatom-atom interactions which affect the average density entering the above equations.20n particular, within this mean-field approximation, the two-mode squeezing of thelight-matter state is neglected [92]. However, due to the global cavity-mediated in-teractions, corrections to the above mean-field approximation are suppressed by afactor 1 /V where V is the volume of the atomic cloud [96]. Therefore, the mean-fielddescription becomes exact in the thermodynamic limit N, V → ∞ , with
N/V = const.Before proceeding, let us stress one important consequence of this approximationwhich limits its validity. The mean-field equation for the field amplitude, Equation (21),does not contain the loss-induced noise ξ , which in general leads to violation offluctuation-dissipation relations characterizing the correct balance between dampingand noise in a steady state. While this does not affect the coherent component of thecavity field α , it does affect the atomic density and in general the atomic momentumdistribution. Indeed, coupling atoms to a lossy cavity field is known to lead to the redis-tribution of the atoms and hence an effective temperature for the atoms—referred toas cavity cooling or heating in quantum optics [70]—roughly set by the loss rate. Thecavity-induced redistribution has recently been studied also for quantum gases [97–100] and can even lead to non-thermal steady states [97]. Since the corrections to thenoiseless mean-field approach are suppressed by a factor 1 /V , the characteristic timefor a cavity-induced redistribution to set in scales with the volume V [97]. This isa common feature of long-range interacting systems, where the phase space for col-lisions leading to redistribution of particles is reduced with respect to a short-rangeinteracting system [101]. We will discuss these issues further in Section 3.1.2. Considering zero temperature, weakly-interacting bosons, and neglecting the quantumdepletion of the Bose-Einstein condensate, one can approximate in Equation (21) thequantum-averaged density as n ( r ) = | ψ ( r ) | , where ψ ( r , t ) is the condensate wave-function satisfying the time-dependent Gross-Pitaevskii equation [85,102], i (cid:126) ∂ψ ( r ) ∂t = (cid:20) − (cid:126) M ∇ + V ext ( r ) + (cid:126) V ( r ) + (cid:126) | α | U ( r ) + 2 (cid:126) Re[ α ] η ( r ) + g n ( r ) (cid:21) ψ ( r ) . (22)This equation is derived from the Heisenberg equation of motion of the atomic fieldoperator, Equation (5a), assuming that the many-body atomic wavefunction is a co-herent state such that the field operator ˆ ψ ( r ) can be substituted with its mean-fieldamplitude ψ ( r ) = (cid:104) ˆ ψ ( r ) (cid:105) [19,103]. The above Gross-Pitaevskii equation contains a dy-namical (superradiant) potential (cid:126) V SR ( r ) ≡ (cid:126) | α | U ( r ) + 2 (cid:126) Re[ α ] η ( r ) whose strengthdepends on the cavity field amplitude α , which must be determined self-consistentlyfrom Equation (21).The stationary state of the system can be found using the standard ansatz for thecondensate wavefunction ψ ( r , t ) = ψ ( r ) e − iµt/ (cid:126) , with µ being the atomic chemical po-tential, and correspondingly via the condition ∂α/∂t = 0 for the cavity-field amplitude.The latter yields from Equation (21) the steady-state cavity-field amplitude, α ss = (cid:82) η ( r ) n ( r ) d r ∆ c − (cid:82) U ( r ) n ( r ) d r + iκ = η Θ∆ c − (cid:82) U ( r ) n ( r ) d r + iκ , (23)where Θ = (cid:104) ˆΘ (cid:105) = (1 /η ) (cid:82) η ( r ) n ( r ) d r [see Equations (15) and (16)] is the mean-field checkerboard density order parameter, which is to be determined self-consistently21long with the dispersive cavity shift − (cid:82) U ( r ) n ( r ) d r . In the absence of an external (harmonic) trap V ext ( r ) = 0, as the system is λ c pe-riodic along both x and y directions, one can make a Bloch ansatz for the single-particle states, ψ m, q ( r ) = e i q · r u m, q ( r ), where m is the band index, q = ( q x , q y )is the quasimomentum in the first Brillouin zone [ − π/λ c , π/λ c ] × [ − π/λ c , π/λ c ],and u m, q ( r ) is a periodic function with the same periodicity as the Hamiltonian, u m, q ( r + λ c ˆ e x + λ c ˆ e y ) = u m, q ( r ). Therefore, the fermionic field operator can be ex-panded in the basis of the Bloch functions, ˆ ψ ( r ) = (cid:80) m, q ψ m, q ( r ) ˆ f m, q , where ˆ f m, q is the fermionic annihilation operator for the m th band at quasimomentum q . Theperiodic Bloch functions u m, q ( r ) then satisfy the eigenvalue equation, (cid:20) (cid:126) M ( − i ∇ + q ) + (cid:126) V ( r ) + (cid:126) | α | U ( r ) + 2 (cid:126) Re[ α ] η ( r ) (cid:21) u m, q ( r ) = (cid:15) m, q u m, q ( r ) , (24)with Bloch-band energies (cid:15) m, q . Recall that the local contact interaction is absent dueto the Pauli exclusion principle as here we only consider spin-polarized fermions.Regarding the stationary state of the system, the cavity field amplitude is deter-mined self-consistently via the mean-field equation (23), with the atomic density nowgiven by n ( r ) = (cid:88) m, q | u m, q ( r ) | n FD ( (cid:15) m, q ) , (25)where n FD ( (cid:15) ) = 1 / [1 + e ( (cid:15) − µ ) /k B T ] is the Fermi-Dirac distribution as a function of theenergy (cid:15) at temperature T (with k B being the Boltzmann constant). The chemicalpotential µ is determined self-consistently by fixing the total number of the atoms inthe system, N = (cid:90) n ( r ) d r = (cid:88) m, q (cid:90) | u m, q ( r ) | e ( (cid:15) m, q − µ ) /k B T d r . (26)Note that we have assume here that the steady-state distribution of the atoms is in athermal equilibrium, with T being fixed by the initial energy of the system. This is notstrictly correct since energy is not conserved in this driven-dissipative system and, asdiscussed above, the steady state of the system can be even a non-thermal type [97]. Analytical and numerical methods beyond the mean-field approach
A number of techniques have been employed for the description of the systems un-der consideration, which go beyond the mean-field approximation just introduced.A subgroup of them aims at a better description of correlations induced by the in-trinsic atom-atom interactions, but still treating the atom-cavity correlations at amean-field level and including only the coherent part α of the cavity field operatorˆ a . These methods include dynamical mean-field approaches [104,105], multiconfigura-tional time-dependent Hartree [106], exact diagonalization [107], and quantum Monte22arlo [108]. On the other hand, other beyond-mean-field methods have been devel-oped that also take into account the atom-cavity quantum correlations and thereforethe role of cavity-induced noise in the steady state of the system. These include nu-merically exact methods applicable to small systems [109] and more recently matrix-product-state-based methods allowing to describe larger systems [110], as well as field-theoretical methods for real-time Green’s functions [97,111] or perturbative expansionsaround the mean field [100]. Collective excitations
Ground state and excitations of a coupled system of matter and light are collectivemodes—so-called polaritons—that mix atomic and photonic degrees of freedom owingto the strong coupling between the two constituents. Polaritons govern the linearresponse of the atom-cavity system to perturbations and can be observed either viathe fluctuation spectrum of the cavity field or via cavity probing with a coherent field.In the following we present two complementary approaches to theoretically describethese excitations. Their actual measurement is discussed in Section 3.2.
The polariton spectrum can be computed by starting from the equilibrium mean-field equation (21) for the cavity field amplitude α and expanding it around a self-consistent equilibrium solution. The mean-field equation for the cavity field amplitude,Equation (21), can be rewritten in a generic form as i∂ t α = − (∆ c + iκ ) α + (cid:88) (cid:96) (cid:104) u (cid:96) ( α ) | αU + η | u (cid:96) ( α ) (cid:105) n ( (cid:15) (cid:96) ) , (27)using eigenstates | u (cid:96) ( α ) (cid:105) [and corresponding energies (cid:15) (cid:96) ( α )] of the mean-field single-particle Hamiltonian H , eff ( α ), obtained from Equation (7) by substituting ˆ a with itsaverage α . Note that we have defined here the basis-independent first-quantized opera-tors U and η such that U ( r ) = (cid:104) r | U | r (cid:105) and η ( r ) = (cid:104) r | η | r (cid:105) , with the position eigenstates | r (cid:105) forming a complete basis 1 = (cid:82) d r | r (cid:105)(cid:104) r | . Here we do not yet specify n ( (cid:15) (cid:96) ) to be afermionic or bosonic distribution, nor a choice of the external potential V ext for theatoms. Defining the field-amplitude fluctuation as δ ˆ a ( t ) = ˆ a ( t ) − α and keeping termsin Equation (27) up to linear order in δ ˆ a and δ ˆ a † , which requires computing pertur-bative corrections to the eigenvectors | u (cid:96) ( α ) (cid:105) and the eigenvalues (cid:15) (cid:96) ( α ), we obtain theequation for the fluctuation: i∂ t δ ˆ a = [ − δ c ( α ) − iκ + χ stat ( α )] ( δ ˆ a + δ ˆ a † ) . (28)Here we have defined the dispersively shifted cavity detuning δ c ( α ) = ∆ c − (cid:88) (cid:96) n ( (cid:15) (cid:96) ) (cid:104) u (cid:96) ( α ) | U | u (cid:96) ( α ) (cid:105) , (29)and the static polarization function of the atomic medium χ stat ( α ) = (cid:88) (cid:96),(cid:96) (cid:48) n ( (cid:15) (cid:96) ) − n ( (cid:15) (cid:96) (cid:48) ) (cid:15) (cid:96) − (cid:15) (cid:96) (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) u (cid:96) (cid:48) ( α ) | α ∗ U + η | u (cid:96) ( α ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) . (30)23he static polarization function characterizes the response of the medium to the cavityprobe and can be immediately upgraded to include the energy (cid:126) ω of the probe photon,which gives the dynamical polarization function χ dyn ( ω ; α ) = (cid:88) (cid:96),(cid:96) (cid:48) n ( (cid:15) (cid:96) ) − n ( (cid:15) (cid:96) (cid:48) ) (cid:126) ω + (cid:15) (cid:96) − (cid:15) (cid:96) (cid:48) + i + (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) u (cid:96) (cid:48) ( α ) | α ∗ U + η | u (cid:96) ( α ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) , (31)where the infinitesimally small imaginary part ensures the retarded character of thisresponse function. In the context of condensed-matter physics, this is also known asthe Lindhard function [112], characterizing the density response of a material. Notethat in the dispersive regime considered here, the cavity photons indeed couple to theatomic density.To compute the frequency-resolved response of the system to a cavity probe weFourier-transform Equation (28) and combine it with its complex conjugate to obtainthe following system of equations: D − ( ω ) (cid:20) δ ˆ a ( ω ) δ ˆ a † ( − ω ) (cid:21) = 0 , (32)with D − ( ω ) = (cid:20) ω + δ c + iκ − χ dyn ( ω ) − χ dyn ( ω ) − χ ∗ dyn ( − ω ) − ω + δ c − iκ − χ ∗ dyn ( − ω ) (cid:21) . (33)The matrix D ( ω ) is nothing else than the retarded Green’s function of the cavityelectromagnetic field, including medium-polarization corrections, expressed throughits positive- and negative-frequency components. The complex frequencies ω pol of thecollective polariton modes correspond to the poles of the retarded Green’s function,given by the condition Det[ D − ( ω pol )] = 0. This condition yields the following implicitequation for the polariton frequencies:( ω pol + iκ ) − δ c + 2 δ c χ dyn ( ω pol ) = 0 , (34)where we have exploited the symmetry of the retarded polarization function χ dyn ( ω ) = χ ∗ dyn ( − ω ).The polariton frequencies ω pol obtained from Equation (34) possess an imaginarypart, since they always contain a photonic component. From the point of view of theatoms, this imaginary part arises from the non-adiabaticity of the cavity field. Thatis, the cavity field follows the changes in the atomic state with a delay on the orderof 1 /κ . Therefore, the atomic excitations can be damped out through the cavity losschannel in appropriate parameter regimes, leading to cavity cooling of the atoms.The polariton resonances at ω pol appear as peaks in the cavity optical responsefunction A ( ω ) = − π Im[ D ( ω )]. The resonant peaks have a finite width due to thedecay of cavity photons. A typical cavity optical responce A ( ω ) is shown in Figure 6,both for a fermionic and a bosonic atomic cloud in the absence of an external trap po-tential V ext ( r ) = 0. Generically, two polariton modes appear, corresponding to mixingof the atomic density fluctuation and the single-mode photonic degree of freedom. Byincreasing the atom-cavity coupling η the polariton peaks further separate.In the absence of a coherent cavity field α = 0 [and in the absence of an external trap V ext ( r ) = 0 as considered in Figure 6], the single-particle eigenstates in Equation (31)24 igure 6. Spectrally resolved response of the system to a cavity probe. A fermionic (a) or a bosonic (b) cloudis coupled to a single standing-wave cavity mode of wavevector k c . The cloud is taken to be homogeneous,one-dimensional and oriented along the cavity axis. The cloud average density is n = 0 . k c ( k F = 0 . k c )and T = 0. Fermions are non interacting while the bosons have a contact interaction energy g n = 0 . (cid:126) ω r .The dispersively-shifted cavity detuning is δ c (0) = − . ω r and the loss rate is κ = 0 . ω r . Two polaritoncollective modes are visible as isolated peaks. In the fermionic case, the particle-hole continuum of excitationsis visible between the polaritons. We note that the parameters used here do not correspond to any of thecurrent experiments and are chosen solely for illustrative purposes. are plane waves | u k (cid:105) = e i k · r / √ V with energies (cid:15) k = (cid:126) k / M and the dynamicalpolarization function reads χ dyn , hom ( ω ) = η (cid:88) k n ( (cid:15) k + k c ) − n ( (cid:15) k ) (cid:126) ω + (cid:15) k + k c − (cid:15) k + i + . (35)For (non-interacting) fermionic atoms at thermal equilibrium, n ( (cid:15) ) = n FD ( (cid:15) ) and thedynamical polarization function has an imaginary part, i.e., finite absorption in themedium, corresponding to on-shell particle-hole excitations of the Fermi surface. Thiscontinuum of states is indeed visible in the optical response as a broad feature whichshows sharp edges for low-temperature gas [see Figure 6(a)]. If the polariton energylies within this particle-hole continuum, it does not correspond anymore to a resonantpeak, rather to a locally increased spectral weight [51,97,113,114].On the other hand, in a BEC with a finite two-body contact repulsion, the cavityphoton excites Bogoliubov modes with dispersion (cid:15) k = (cid:113) (cid:15) k ( (cid:15) k + 2 g n ) out of thecondensate n ( (cid:15) k ) = N δ k , . At zero temperature T = 0 and neglecting condensate de-pletion as well as Beliaev damping of phonons [96,115,116], the dynamical polarizationfunction of the BEC has no imaginary part, so that two resonant peaks are alwayspresent in the optical response as shown in Figure 6(b).25 .6.2. Linear stability analysis and Bogoliubov spectrum Here we present an alternative, complementary approach for obtaining low-energycollective excitations of the system with bosonic atoms at zero temperature. It isbased on a linear stability analysis around the mean-field steady-state solutions [19,102,117]. To this end, the Heisenberg equations of motion of the atomic and photonicfield operators, Eqs. (5a) and (10), are linearized around the mean-field steady state ψ ( r ) and α by keeping the quantum fluctuations δ ˆ ψ ( r , t ) = e − iµt/ (cid:126) [ δ ˆ ψ + ( r ) e − iω pol t + δ ˆ ψ †− ( r ) e iω ∗ pol t ] and δ ˆ a ( t ) = δ ˆ a + e − iω pol t + δ ˆ a †− e iω ∗ pol t up to first order, i (cid:126) ∂δ ˆ ψ∂t = (cid:20) − (cid:126) M ∇ + V ext ( r ) + (cid:126) V ( r ) + (cid:126) | α | U ( r ) + 2 (cid:126) Re[ α ] η ( r ) + 2 g n ( r ) − µ (cid:21) δ ˆ ψ + g ψ ( r ) δ ˆ ψ † + (cid:126) ψ ( r ) (cid:110) [ U ( r ) α ∗ + η ( r )] δ ˆ a + [ U ( r ) α + η ( r )] δ ˆ a † (cid:111) , (36a) i ∂δ ˆ a∂t = − (cid:18) ∆ c − (cid:90) U ( r ) n ( r ) d r + iκ (cid:19) δ ˆ a + (cid:90) [ U ( r ) α + η ( r )] (cid:104) ψ ∗ ( r ) δ ˆ ψ + ψ ( r ) δ ˆ ψ † (cid:105) d r . (36b)The linearized equations (36) constitute Bogoliubov-type coupled equations for atomicand photonic quantum fluctuations. The Bogoliubov equations, Equation (36), can berecast in a matrix form, M B F = ω pol F , (37)where F = ( δ ˆ ψ + , δ ˆ ψ − , δ ˆ a + , δ ˆ a − ) (cid:62) and M B is a non-Hermitian matrix obtained fromEquation (36). The eigenvalues ω pol of the Bogoliubov equation (37) yield the collectiveelementary excitation spectrum of the system above the mean-field steady state. Basic experimental scheme and relevant parameters
The basic experimental scheme used to study self-organization of quantum gases inoptical cavities is depicted in Figure 4. Ultracold atoms are trapped in an external po-tential V ext ( r ) whose minimum is centered with respect to the mode waist of an opticalcavity. Typically a near harmonic trapping potential is formed from two orthogonallyintersecting, non-interfering laser beams (not shown) of sufficiently large waist, whichare far red-detuned from atomic resonances [118]. The trapping laser frequencies arechosen to be also far detuned from cavity resonances such that the effect of scatteringtrapping-light photons into the cavity can be neglected.All bosonic experiments to date have used Bose-Einstein condensates of Rubidium-87 atoms [119], with typical atom numbers between a few thousands and a few hundredthousands. Besides the atom number N and the temperature T of the atomic cloud,the most relevant atomic parameters for these experiments are the atomic mass M , theatomic transition frequency ω a , the electric dipole matrix element d ge of the consideredatomic transition and its decay rate γ , as well as the atomic s-wave scattering length a sgg determining atomic two-body interactions. Feshbach resonances of Rubidium-87are mostly very narrow, which makes it practically impossible to modify the scatteringproperties and the interaction strength, and limits the possibilities to realize phenom-26na requiring tunable atomic interactions. However, there is no fundamental difficultyin performing such experiments with atomic species having different properties. Experiments have been carried out using linear cavities (groups at the ETH, Ham-burg, and Stanford) as well as ring cavities (the group in T¨ubingen). Here we focuson the linear cavities and review the role of the most relevant parameters for theirdescription [120–123]. The geometry of a linear cavity is characterized by the radiiof curvature R c , of the two concave mirrors and by their separation, the length l res of the resonator. The g -parameters g i = 1 − l res /R ci are used to describe a resonatorand its stability condition, 0 ≤ g g ≤
1; see also Figure 7. The two limiting and onlymarginally stable cases are the plane parallel Fabry-P´erot resonator ( R c = R c = ∞ )and the concentric configuration ( R c + R c = l res ). A third special, stable case is theconfocal configuration ( R c + R c = 2 l res ).In the paraxial approximation, the so-called transverse electromagnetic modesTEM mn are described by the Hermite-Gaussian wavefunctions, E mn ( r ) = E w w ( x ) H m (cid:18) √ yw ( x ) (cid:19) H n (cid:18) √ zw ( x ) (cid:19) e − ( y + z ) /w ( x ) × cos (cid:20) k c (cid:18) x + y + z R ( x ) (cid:19) − ϕ Gouy mn ( x ) + ϕ offset mn (cid:21) . (38)Here, m and n are the integer transverse mode indices, the wave number k c is fixed bythe mode frequency ν as k c = 2 πν/c , w ( x ) = w (cid:112) λ c x/πw ) is the mode waistwith minimum waist w at longitudinal position x = 0, R ( x ) = x + π w /λ c x is thelocal radius of curvature, and H m is a Hermite polynomial. The wave length λ c is setby k c = 2 π/λ c . Since the radius of curvature of the wavefront of the Gaussian beamis constantly evolving with the propagation distance, the beam acquires an additionalphase shift with respect to the propagation of a plane wave. Indeed, the effective wavelength depends on the distance x from the beam waist. This spatially dependent Gouyphase shift ϕ Gouy mn ( x ) = (1 + m + n ) arctan( λ c x/πw ) depends on the transverse modeindices. Finally, in order to satisfy the boundary conditions at the mirrors, one has toallow for a phase offset ϕ offset mn = ( m + n ) arctan( λ c l res / πw ) [124].In a cavity, the radii of curvature of the wavefronts at the position ofthe mirrors have to match the radii of curvature of the mirror surfaces.This condition determines the minimum mode-waist radius squared, w =( λ c l res /π ) (cid:112) g g (1 − g g ) / ( g + g − g g ) , and therefore the mode volume V = πw l res / with a purely Gaussian transverse pro-file.In order to meet a resonance condition, the phase of the field must be an integermultiple l of 2 π after one resonator round trip, with l thus counting the number ofnodes of a mode in the longitudinal direction. This condition determines the resonancefrequencies of the cavity modes ν lmn = ν FSR (cid:20) l + ( m + n + 1) arccos( ±√ g g ) π (cid:21) , (39)where we have used the fact that the Gouy phase shift across the resonator is given by ϕ Gouy mn ( l res / − ϕ Gouy mn ( − l res /
2) = arccos( ±√ g g ), with the + (-) sign applying to the27pper right g , g > g , g <
0) quadrant. From Equation (39), one candirectly derive the spacing ∆ ν TEM between adjacent transverse modes. The frequencyseparation between two adjacent longitudinal modes, i.e., ( lmn ) and (( l + 1) mn ), isgiven by the free spectral range ν FSR = c/ l res .Equation (39) indicates how the mode frequency separations between the longi-tudinal and transverse modes depend on the g -parameters of the cavity; see Fig-ure 7. For the two extreme cases of a plane parallel ( g = g = +1) and a concentric( g = g = −
1) resonator, all transverse modes TEM mn are degenerate and form modefamilies. In the plane parallel configuration, these mode families share the same lon-gitudinal mode number l , and correspondingly a common longitudinal standing-waveprofile cos( k c x ) with wavevector k c = ω c /c = 2 πν lmn /c (in a ring cavity the standingwave is substituted by two degenerate, counterpropagating plane waves). Moving awayfrom the plane parallel situation, the transverse modes belonging to one such modefamily separate increasingly. Reaching the confocal configuration ( g = g = 0), theinitial mode families mix. At this point, all modes fall into either of two degeneratenew mode families that alternate every half free spectral range, and where the sumof mode indices ( m + n ) is either an even or an odd number. Further decreasing the g -parameters, these sub-families of transverse modes split again until the concentricconfiguration is reached. In this situation, mode families of equal sum of mode indices( l + m + n ) form.The two experimentally most relevant cavity configurations are the near-planar( R ci (cid:29) l res , g i → +1) and the confocal ( R ci (cid:39) l res , g i →
0) geometries, since they arestable resonator configurations. Coupling to a single TEM mode of a near-planarcavity has been considered in the preceding sections (as well as in the large parts ofthis review). Coupling of atoms to confocal (and to some extent also near-concentric)cavities will be discussed in Section 5.Ideal mirror surfaces are assumed to be parabolic and rotationally symmetric. Inpractice, however, spherical mirrors are used and they might have different radii ofcurvature { R cx , R cy } along different axes, resulting in an elliptical cavity waist. In addi-tion, the axes of the mirrors forming the cavity might also be misaligned with respectto each other. The solution of the Helmholtz equation beyond the paraxial approxima-tion shows that the Gouy phase shift—ultimately determining resonance frequenciesof the cavity modes—depends on the orientation of the field polarization with respectto the elliptical focus. To lowest order, this gives rise to a birefringent splitting of thecavity modes along the ordinary and the extraordinary axes by [125],∆ ν = − cλ c π l res R cx − R cy R cx R cy . (40)This geometrically caused birefringence is, however, for macroscopic cavities usuallyvery small and only becomes significant for microscopic cavities with very small radiiof curvature.More importantly for macroscopic cavities is a possible birefringence of dielectriccoatings on the mirror surfaces. This birefringence can be a material property ordue to mechanical stress caused during mounting or machining of the mirrors. Theelectric field penetrates the mirror surfaces typically by several wave lengths and isthus subject to a possible birefringence in the coatings. This leads to a splitting of thecavity resonance frequencies for the orthogonal polarization modes. All these effectslift the theoretically ideal mode degeneracies of multimode resonators to a certaindegree in a realistic experiment. 28 onfocalplaneparallel near planarconcentricnear concentric a b c Figure 7. Stability diagram, typical resonator configurations, and mode spectra. (a) The geometric cavityparameters g i can be used to identify the condition for the resonator stability, 0 ≤ g g ≤
1, shown as greyarea. Special configurations for a symmetric resonator ( g = g , the dashed line) are indicated in the stabilitydiagram and schematically shown in (b). The mode spectra for the near-planar (top), confocal (middle), andnear-concentric (bottom) configurations are shown in (c). In these spectra, the fundamental TEM modes aredepicted above the frequency axes, while the higher transverse modes indicated with the sum ( m + n ) of thetransverse mode indices are shown below. The longitudinal mode index l refers to the TEM modes which arespaced by ν FSR . Moving from the planar to the concentric configuration, all modes move in frequency spaceas indicated by the grey lines. In the confocal configuration, degeneracy of modes with either even or oddsums of transverse mode indices is reached. Note that modes with different longitudinal indices are becomingdegenerate in the confocal and concentric cases, while in the near planar case a (quasi-)degenerate mode familyshares the same longitudinal mode index l . The optical properties of the mirrors are captured by their reflectivity R i , transmis-sivity T i , and losses L i , where R i + T i + L i = 1. The losses can be caused by scattering,diffraction, or absorption. The finesse F of a Fabry-P´erot resonator, F = π √R R − √R R , (41)characterizes the field enhancement inside the cavity. A photon traverses the resonatoron average F /π times before it is lost, resulting in a power enhancement of F /π withrespect to the input coupled power. Finesse and free spectral range also determine thedecay rate κ of the electromagnetic field inside the cavity κ = π ν FSR F , (42)and accordingly the full width at half maximum ∆ ν = κ/π of the transmission peakof a cavity. The coupling between a single atom and a single quantized electromagnetic field insidean optical cavity is given by the vacuum-Rabi frequency G = d ge · (cid:15) c E (cid:126) = d ge (cid:126) (cid:114) (cid:126) ω c (cid:15) V cos( ∠ ( d ge , (cid:15) c )) , (43)29here (cid:15) is the electric permittivity. We used the vector notations d ge for the inducedatomic electric-dipole moment for the transition | g (cid:105) ↔ | e (cid:105) and (cid:15) c for the unit po-larization vector of the intracavity field. Accordingly, ∠ ( d ge , (cid:15) c ) is the angle betweenthe dipole moment and the polarization direction of the electric field. Here, E is themaximum electric field strength of a single photon in the peak intracavity field of theTEM mode volume V . The spatial-dependent Rabi frequency used in the previoussections is then defined as G ( r ) = G E ( r ) E , (44)where E ( r ) is the local electric field strength of a single intra-cavity photon.Cavity-QED systems are often characterized by the cooperativity parameter C = G /κγ (note that κ and γ are the cavity-field and atomic-dipole decay rates, i.e. κ is halfthe cavity linewidth, see Equation (42), and γ is half the atomic linewidth Γ), or by thePurcell factor P F = 24 F /πk c w [126]. However, since self-organization experimentstypically operate in the dispersive regime (∆ a (cid:29) γ ), where the occupation of theatomic excited state is negligible, the more relevant parameter is the collective couplingrate √ N G . The coupling strength G also determines the relevant parameters of theeffective atom-cavity Hamiltonian, Equation (6), i.e., the optical lattice depth (cid:126) U = (cid:126) G / ∆ a created by a single intra-cavity photon and the two-photon Rabi frequency η = G Ω / ∆ a . The energy scales relevant for most of the self-organization physics described in thisreview article are shown in Figure 8 together with the basic coupling scheme. Bosonicatoms are initially in the zero momentum state k = ( k x , k y ) = (0 ,
0) and are coupledvia cavity-assisted two-photon Raman transitions to the symmetric superposition ofstates k = ( k x , k y ) = ( ± k c , ± k c ); see the discussion about the Dicke model in Sec-tion 2.3. The energy difference (cid:126) ω = 2 (cid:126) ω r between these levels is twice the recoilenergy (cid:126) ω r = (cid:126) k c / M . There are two possible excitation paths connecting thesestates. Either first a photon from the pump beam is absorbed by the atoms and thenreemitted into the cavity mode, or these processes take place in the inverse order. Theintermediate states are far detuned by ∆ a = ω p − ω a from the atomic excited state andthus only virtually populated. The cavity resonance is detuned by ∆ c = ω p − ω c fromthe pump frequency ω p . The experiment operates in the sideband resolved regime ifthe cavity-field decay rate κ is on the order of or smaller than the motional excitationfrequency ω .Table 1 summarizes the experimental parameters of the setups which have thus farperformed experiments and published works on self-organization of quantum gases inoptical cavities.
3. Superradiant crystallization breaking a discrete symmetry
In this chapter we explore the superradiant phase transition in laser-driven quantumgases coupled dispersively to single standing-wave modes of linear cavities and someother closely related variants. Fundamental aspects of these coupled systems were dis-cussed in Section 2. Across such a superradiant phase transition, the atomic density30 a r a m e t e r E T H [ ] E T H [ ] H a m b [ ] S t a n [ ] E P F L [ ] T ¨ ub [ ] T ¨ ub [ ] li n e a r s t a nd i n g - w a v ec a v i t y r i n g c a v i t y R c , mm mm . mm . mm mm –– l r e s µ m . mm . mm . mm a d j u s t a b l e ∼ R c , ± (cid:15) . c m mm c m w . µ m . µ m . µ m . µ m µ m . µ m ( × ) µ m µ m ν F S R G H z . G H z . G H z . G H z . G H z . G H z . G H z . G H z ∆ ν T E M . G H z . G H z . G H z . M H z a d j u s t a b l e − M H z . G H z –– F . × . × . × . × . × . × s : . × p : . × . × κ / ( π ) . M H z k H z k H z . k H z k H z k H z s : k H z p : k H z k H z G / ( π ) . M H z . M H z . M H z . M H z . M H z . M H z . k H z . k H z C . . . . . . . . λ n m n m n m n m n m n m n m T a b l e . P a r a m e t e r s o f t h ee x p e r i m e n t a l s e t up s d i s c u ss e d i n t h i s r e v i e w ( E T H a nd E T H a r ee x p e r i m e n t s o p e r a t e d a t E T H Z u r i c h , H a m b i s o p e r a t e d a tt h e U n i v e r s i t y o f H a m bu r g , S t a n a tt h e S t a n f o r d U n i v e r s i t y , E P F L i s o p e r a t e d a t E P F LL a u s a nn e , a nd T ¨ ub a nd T ¨ ub a r e o p e r a t e d a tt h e U n i v e r s i t y o f T ¨ ub i n g e n ) . λ s p ec i fi e s t h e w a v e l e n g t h a t w h i c h t h e p a r a m e t e r s w e r ec h a r a c t e r i z e d a nd m i g h t b e d i ff e r e n t f r o m t h e w a v e l e n g t hu s e d i n a s p ec i fi ce x p e r i m e n t . T h e v a l u e s f o r c o up li n g s t r e n g t h a nd c oo p e r a t i v i t y f o r t h e li n e a r c a v i t i e s h a v e b ee n c a l c u l a t e db a s e d o n t h e g e o m e t r y o f t h e r e s o n a t o r u s i n g t h e m a x i m u m d i p o l e - a ll o w e d t r a n s i t i o n f o r t h e D li n e a nd a ss u m i n g c o up li n g t o t h e T E M m o d e . T h e s e t up E T H p e r a t e s w i t h t w o c r o ss e d c a v i t i e s f o r w h i c h t h e p a r a m e t e r s a r e g i v e n i n c o n s ec u t i v e li n e s . T h e s e t up a t S t a n f o r d o p e r a t e s a c a v i t y w i t h a d j u s t a b l e l e n g t h c l o s e t o t h ec o n f o c a l c o n fi g u r a t i o n . b Figure 8. Basic atom-photon coupling scheme with relevant frequencies and coupling rates in self-orderingphenomena. (a) The relevant energy/frequency scales are given by the atomic kinetic energy (cid:126) ω = 2 (cid:126) ω r , andthe relative atomic ∆ a = ω p − ω a and cavity ∆ c = ω p − ω c detunings with respect to the pump frequency ω p (frequency axis not to scale). The two pairs of red arrows (solid and dashed lines) show the two possibleexcitation paths that couple the zero momentum state k = ( k x , k y ) = (0 ,
0) to the atomic excited momentumstates k = ( k x , k y ) = ( ± k c , ± k c ). (b) The two excitation paths (dashed and solid) corresponding to the twoRaman channels are illustrated schematically in a momentum-space diagram. The operators ˆ a † ˆ J + and ˆ a ˆ J + correspond to the creation and annihilation of a cavity photon while creating atoms in the excited momentumstate, respectively, in the framework of the Dicke model; see Section 2.3. self-orders into a crystalline pattern—spontaneously breaking the discrete Z symme-try of the systems—which constructively scatters the laser light into the cavity suchthat a coherent cavity field is built-up. The self-organization transition occurs whenthe potential-energy gain due to the coupling of the atoms into the cavity field over-comes the kinetic-energy penalty of the density crystallization. Within an equivalentdescription using cavity-mediate atom-atom interactions, presented in Section 2.2, itis these global interactions which favor the self-organization into the crystalline order.This chapter is devoted to the most fundamental self-ordering scenarios. In the follow-ing chapters, we will encounter several other instances of superradiant phase transitionassociated with different types of ordering. Self-organization phase transition
Consider a laser-driven quantum gas coupled dispersively to a single standing-wavemode of a lossy linear cavity, as discussed in detail in Section 2. Such a system gener-ically exhibits a Dicke-type superradiant phase transition where one of the collectivemodes becomes unstable at the onset of the superradiant threshold. In the presentdriven-dissipative system, this corresponds to a low-lying polariton mode switchingfrom being damped to growing in time. That is, the imaginary part of the complexpolariton frequency vanishes at the superradiant threshold; see Section 2.6.At very small atom-photon coupling strengths η (compared to the energy separa-tion between polaritions), the polaritons are almost decoupled from one another andare still mainly either atom-like or photon-like. By increasing the coupling strength η ,32 igure 9. The low-lying (i.e., softening) complex polariton frequencies as a function of the pump strength η for both bosons (black curves) and fermions (blue curves), similar to Figure 6, around the superradiantthreshold η c . After the real part of the frequency vanishes at η = η s , the imaginary branch (associatedwith the damping of the polariton) splits into two parts, where one (shown here) linearly vanishes approachingthe pump-strength threshold η c . Beyond the threshold, this part of the imaginary branch becomes positive,signaling a dynamical instability toward the self-ordered, superradiant phase. the polaritons are mixed more strongly. In the case that the photon-like polariton ini-tially lies at a higher frequency than the atomic counterpart, the atom-like polaritionstarts then to soften—i.e., its real part approaches zero. As can be seen from Fig-ure 6, this happens essentially in the same way for both bosonic and fermionic atoms,apart from the fact that in the latter case the polariton resonant peak becomes visibleonly when it exits the particle-hole continuum. When the real part of the low-lyingpolariton frequency reaches zero at η = η s , the polariton splits into two branchesdiffering only by their imaginary part. The branch with the smallest imaginary partis shown in Figure 9, for both bosonic and fermionic atoms. At the critical pumpstrength η c > η s , one of the imaginary branches crosses zero and becomes positivefor η > η c , signaling that the mean-field solution with α = 0 has become unstabletowards a superradiant state where α (cid:54) = 0. The threshold η c is obtained from thecharacteristic equation (34) by setting the polariton frequency to zero, ω pol = 0. Inthe simple case of a spatially homogeneous 1D, zero-temperature BEC as consideredin Figure 6(b), an explicit analytical expression for the threshold can be obtained, √ N η c = (cid:115) δ c + κ − δ c (cid:114) ω + 2 g n (cid:126) , (45)where δ c = ∆ c − N U /
2. This critical pump strength with g = 0 (i.e., a non-interactingBEC) is consistent with the threshold for the Dicke-superradiance phase transition,Equation (20), except that in the 1D case discussed here there is an extra factor of1 / √ ω = ω r .Equivalently, the superradiant threshold η c , Equation (45), can be obtained fromthe linear stability analysis (see Section 2.6.2) of the trivial mean-field solution, i.e.,a zero-temperature, homogeneous 1D BEC along the cavity axis and a vacuum cav-ity field α = 0. In the low-energy limit in the vicinity of the Dicke superradiancephase transition only the atomic condensate fluctuations δ ˆ ψ ± ( x ) ∝ cos( k c x ) couples33o the field fluctuations δ ˆ a ± and vice versa [102]. By restricting the atomic conden-sate fluctuations to ∝ cos( k c x ) excitations, relevant to the photon scattering from thepump laser into the cavity on the onset of the self-ordeing, the excitations of the sys-tem in the restricted subspace is obtained via the fourth-order characteristic equationDet( M B − ω pol I × ) = 0, (cid:18) ω − (cid:126) (cid:15) k c (cid:19) (cid:2) ( ω pol + iκ ) − δ c (cid:3) + 2 N η ω r δ c = 0 , (46)where (cid:15) k c = (cid:112) (cid:126) ω r ( (cid:126) ω r + 2 g n ) is the energy of the condensate’s phonon mode withmomentum corresponding to the cavity wave number k c . The threshold η c is thesolution of the characteristic equation (46) with zero polarition frequency, ω pol = 0.The solution of the characteristic equation (46) yields the low-energy polaritonspectrum ω pol ( η ) below the threshold η c . For the case of (cid:15) k c (cid:28) (cid:126) ( κ + δ c ), we canobtain simple analytical expressions. In particular, in the region η < η s , the real partof the lowest-lying polariton frequency can be approximated asRe( ω pol ) (cid:39) (cid:15) k c (cid:126) (cid:115) − η η s , (47)and the imaginary part can be approximated asIm( ω pol ) (cid:39) − κ(cid:15) k c (cid:126) ( δ c + κ ) η η c . (48)On the other hand, in the vicinity of the critical point where η /η c (cid:39)
1, the real partis zero and the imaginary part can be approximately written asIm( ω pol ) (cid:39) − δ c + κ κ (cid:18) − η η c (cid:19) . (49)The range of pump strength η s < η < η c where the critical polariton is purelydamped, i.e., Re( ω pol ) = 0 and Im( ω pol ) <
0, is given by, η c − η s = η c (cid:20) κ(cid:15) k c (cid:126) ( δ c + κ ) (cid:21) , (50)which is much smaller than η c given the assumption (cid:126) ( κ + δ c ) (cid:29) (cid:15) k c . If instead κ, δ c ∼ (cid:15) k c / (cid:126) , the overdamped range extends and occupies an appreciable fraction of η c [note that the analytical expression (50) is no longer valid in this regime].By entering the superradiant phase with α (cid:54) = 0, the system spontaneously breaksthe Z parity symmetry of the Hamiltonian, Equation (6). This implies fixing thephase of the cavity field relative to the pump laser and correspondingly λ c -periodicdensity ordering in either even or odd sites of the emergent optical lattice [see alsoexperimental Figure 10(f)]. The resultant λ c -periodic density pattern optimizes indeedthe constructive light scattering from the transverse pump laser into the cavity mode.34 .1.2. Criticality of the self-ordering phase transition From Figure 9 one can see that the critical behavior across the superradiant phasetransition is the same for both bosons and fermions, and is characterized by the linearvanishing of the polariton damping as a function of the effective coupling strength η , asseen by linearizing Equation (49). In this simplest scenario for superradiance featuringa single cavity mode, the critical behaviour coincides with the one of a dissipativeversion of the Dicke model [32,83,91], discussed in Section 2.3. The static universalityclass is of the mean-field, classical Ising type. It is always at finite temperature due tocavity losses (the same holds in the absence of photon losses but at finite temperatureof the atomic cloud). The dynamical universality class is a mean-field version of ModelA in the Hohenberg-Halperin classification, since there is no conserved quantity [132].The mean-field nature of the transition is generic to cases where the atoms are coupledto a finite (in the thermodynamic limit) number of cavity modes, resulting in a zero-dimensional effective theory for the order parameter [96]. In equivalent terms, thecavity-mediated atom-atom interactions in these cases are of infinite-range nature, sothat the interacting models are fully connected.The cavity loss on the one hand makes the superradiant transition to take placealways at a finite temperature and on the other hand shifts the critical light-mattercoupling to a higher value. As discussed above, cavity losses render the superradiantphase transition effectively thermal. The corresponding effective temperature dependson the character of the soft polaritonic mode which becomes undamped at the tran-sition. In the regime where the cavity dynamics is much faster than the atomic one, κ (cid:29) ω r (experiments ETH1, ETH2, Stan, and T¨ub1 in Table 1), the soft polaritoncorresponds to an almost atomic excitation. This picture is supported by the measure-ment of the structure factor in the vicinity of the transition (see Figure 16), wherethe occupation of the soft polariton is consistent with the temperature of the atomicgas ( ∼ kHz) before coupling to the cavity, rather than with the loss rate ( ∼ MHz). Itis important to note, however, that this scenario neglects the effect of cavity-inducedrelaxation (the so-called cavity heating or cooling) of the atomic cloud, whereby theon-shell scattering processes involving cavity photons redistribute the atoms. We havealready touched this issue in discussing the mean-field description in Section 2.4. Notconsidering here the (very interesting) fact that the resulting steady-state atomic dis-tribution is in general not thermal but rather features broad power-law tails [97], wecan restrict to the thermal regime | δ c | (cid:29) ω r . In this regime, the effective temperatureof the atoms becomes on the order of the cavity loss rate κ , i.e., the atoms eventually“thermalize” with the photons. This regime is not reached experimentally in Zurich,the reason being that the cavity-induced relaxation rate [97],Γ rel (cid:39) N η κ δ c ω r κ , (51)is very slow. The relaxation rate is suppressed in two ways: i) By the number of atoms,which is due to the global nature of the light-matter coupling in this single-mode case,ii) by the ratio ω r /κ , which is 10 − in the above mentioned experiments, leadingeventually to a timescale for relaxation of several seconds. The situation is, however,very different in experiments where κ ∼ ω r (experiments Hamb and T¨ub2 in Table 1).Important qualitative differences have indeed already been experimentally observeddue to this fact (see Figure 12). Though cavity cooling has been demonstrated [70],the cavity-induced relaxation and the emergence of non-thermal steady states have not35et been experimentally investigated, though the timescales should be within reach.We described here the superradiant instability in terms of a polariton mode be-coming unstable. The atomic component of this polariton involves a single momentumcorresponding to the cavity wave number k c . This finite-momentum instability is anal-ogous to a roton mode softening. However, in the standard situation of a closed equi-librium system, the softening is associated with an excitation energy approaching zero(the so-called energetic instability), while in the present driven-dissipative open sys-tem what matters is not the excitation energy but its damping which approaches zeroand crosses it at the unstable point. As a consequence, across the transition the modegrows exponentially in time, leading to a so-called dynamical instability. The criticalregime of the self-organization phase transition has been experimentally explored; seeSections 3.2.1 and 3.2.2.The non-equilibrium—or more precisely the driven—nature of this open atom-cavitysystem plays an essential role in the possibility of the experimental realization of theDicke superradiant phase transition. The dispersively driven two-photon transitionsbetween two momentum states lead to the “paramagnetic-like” term proportional toˆ a † + ˆ a in Equation (6) [and correspondingly in Equation (19)], thereby moving thesuperradiant transition from the ultrastrong coupling regime (as in the original Dickemodel) to the strong coupling regime. That is, in the original Dicke model the strengthof the atom-field coupling (i.e., paramagnetic term) has to be comparable to the barecavity and atomic-transition frequencies, which lie in the optical domain of severalhundred THz. In contrast, in the dispersively driven atom-cavity systems the strengthof the atom-field coupling needs to be comparable to the cavity detuning/loss-rateand the recoil frequency, i.e., to be in the several kHz to MHz regime. The validity ofDicke-type models and the fundamental possibility of the superradiant transition inthe ultrastrong coupling regime is still a matter of active debate both in atomic andsolid-state physics [133,134]. The first observation of the superradiant self-organization phase transition of a quan-tum gas made use of an almost pure BEC, such that k B T (cid:28) (cid:126) ω r [28]. The relevantmomentum states can thus be clearly resolved such that the low-energy descriptionand the corresponding mapping to the Dicke model is applicable, as discussed in Sec-tion 2.3. Therefore, the threshold for the superradiant self-ordering phase transitionis set by the competition between the atomic quantum-kinetic energy, i.e., the en-ergetic cost of spatially modulating the wave function, and potential energies. Thatis, the self-ordering is favored when the (negative) potential energy gain outweighsthe kinetic energy penalty due to the density crystallization. This is different fromthe self-organization of thermal classical atoms [20], where the phase transition fromthe normal to the self-organized phase is determined by a competition of thermalfluctuations and potential energy.In typical experiments considered in this section, an atomic BEC is confined byan external trap to the center of a single cavity mode, and illuminated transversallyby a far red-detuned standing-wave pump field [28]; see also Figure 4. The atomicdetuning ∆ a between the pump field and the atomic resonance frequency is on theorder of THz, such that the atoms are not electronically excited but only act as adispersive medium scattering photons from the pump field into the cavity mode andvice versa. At the same time, the cavity detuning ∆ c between the pump field andthe cavity resonance is comparatively small, on the order of ∼
10 MHz (where the36 ormalphase superradiantphase normalphase abcd ef HD Figure 10. Self-organization of a driven BEC inside a single-mode standing-wave cavity, corresponding to thesuperradiant phase transition. (b) The transverse pump power P is ramped up linearly. At the same time,the intensity (c) and the phase (d) of the intra-cavity field are measured. Initially, in the normal phase, thecavity field is close to the vacuum state with no well-defined phase. At the critical pump power, the systementers the superradiant phase with a nonzero cavity field possessing a well-defined phase over time. The processis reversible, i.e., by ramping down the pump field the system enters again the normal phase. (a) Absorptionimages of the BEC after ballistic expansion reveal the occupation of various momentum states at different stagesof the experiment. For increasing pump lattice depth, the peaks at ± (cid:126) k c ˆ e y due to the pump lattice becomevisible. In the self-organized phase, the four momenta at k = ( k x , k y ) = ( ± k c , ± k c ) become macroscopicallypopulated. (e) The heterodyne detection (HD) scheme used to measure amplitude and phase of the cavityfield: a fraction of the pump field is split, frequency shifted, and used as local oscillator. The output of thecavity is interfered with this field and sent to a balanced photo-detector. (f) Symmetry breaking: If the pumppower P is repeatedly ramped across the critical point, the phase of the intra cavity light field takes either 0or π , demonstrating the discrete Z parity symmetry breaking at the phase transition. The blue density plotssymbolize the two possible atomic density configurations where either the even or the odd sites of the emergentcheckerboard lattice are occupied. Figure adapted and reprinted with permission from Ref. [28] published in2010 by the Nature Publishing Group and Ref. [135] © cavity decay rate κ is ∼ (cid:126) ω r . The transverse pump field induces a coupling between the atomic externaldegree of freedom and the cavity field, which leads to a softening of one of the polaritonbranches. This softening reduces the energetic cost of the photon scattering processes,and eventually leads to the superradiant instability described in Section 3.1.1.The basic experimental sequence and main observations of this prototypical experi-ment are summarized in Figure 10. The power P of the transverse pump field inducingthe coupling ( P ∝ η ) is gradually increased. Above a critical pump power, the intracavity light intensity raises sharply, indicating the superradiant phase transition point.At the same time, a superposition of four relevant momentum states k = ( ± k c , ± k c )becomes macroscopically populated, as can be observed in absorption images aftertime of flight expansion of the BEC. In the normal phase, the atomic density is mod-ulated by the transverse pump lattice but homogeneous along the cavity direction,leading to destructive interference of the light scattered off the atoms into the cavitymode. In contrast, in the superradiant phase, the spatial arrangement of the atomic37ystem in the emergent optical lattice corresponds to the formation of a Bragg gratingat which the pump field is efficiently scattered into the cavity mode and back.A good choice as an order parameter operator for the superradiant phase transitionis the overlap integral ˆΘ between the atomic density operator and the checkerboardlattice, introduced in Equation (15). It measures the localization of the atomic densityon either the even ( (cid:104) ˆΘ (cid:105) >
0) or odd ( (cid:104) ˆΘ (cid:105) <
0) sublattice; see also Figure 10. Ifthe system is mapped to the Dicke model as in Section 2.3, the expectation valueof the order parameter is given by the expectation value of the atomic polarization, (cid:104) ˆΘ (cid:105) = (cid:104) ˆ J + + ˆ J − (cid:105) / a ss [see Equa-tion (16)], which can equivalently be used as the order parameter. Since the intra-cavitylight field can be continuously detected in real time via the field leaking through thecavity mirrors, this is a particularly useful and versatile relation. In addition to mon-itoring the steady state of the system, this also enables access to fluctuations of thesystem as we discuss in Section 3.2.2. Furthermore, this provides the fundamental toolfor many cavity-based non-destructive measurement schemes; see Section 11. Whether the even or odd sites of the underlying checkerboard lattice are predominantlyoccupied in the superradiant phase is determined by a spontaneous symmetry breakingprocess. After the phase transition, the phase of the intra-cavity field is locked withrespect to the pump field’s phase. Using a heterodyne detection scheme, this relativephase can be measured. It was experimentally confirmed that the phase locks eitherto 0 or π relative to the pump phase; see Figure 10(f) [135]. This observation oftwo discrete values for the relative cavity-pump phase demonstrates the Z paritysymmetry breaking expected from the basic Hamiltonian (6), and corresponds to theself-organization of the atomic density with maxima at either the even or the odd sitesof the emerging checkerboard lattice; see also Section 3.1.1. An in-situ observation ofthe atomic density distribution has so far not been possible.The symmetry at the phase transition is broken by a symmetry-breaking field ratherthan by a spontaneous process—as probably in all realistic experiments. The breakingis most likely caused by the finite size of the initially homogeneous atomic cloud hav-ing a slightly different overlap with either the even or the odd sites of the emergentoptical lattice. This overlap integral depends λ c -periodically on the relative positionbetween the center of the atomic trap and the checkerboard lattice structure. Thenonzero overlap difference then inhibits the complete destructive interference in thenormal phase and a small coherent field scattered into the cavity acts as a symmetrybreaking light field, such that the order parameter is renormalized by an additive con-stant. Measuring the distribution of the phase of the intra-cavity light field across thephase transition repeatedly allows to analyze the symmetry-breaking field. It is impor-tant to note that an equal distribution of the measurement outcomes does not provethe absence of symmetry-breaking fields, since it can as well be caused by technicalfluctuations of a present symmetry-breaking field. In Ref. [135], a maximal symmetry-breaking field corresponding to ∼
40 atoms distributed in excess on either the evenor odd sites was estimated. When the phase-transition boundary was slowly crossedrepeatedly, the symmetry was found to be broken and indeed one of the two possiblemeasurement results was strongly favored. The symmetry breaking was also studied asa function of the quench time during which the phase-transition boundary was passed.For rapid quenches, the phase-transition boundary was crossed non-adiabatically and38 igure 11. The non-equilibrium phase diagram of a transversely driven BEC dispersively coupled to a singlestanding-wave mode of a linear cavity. (a) The intra-cavity photon number is shown as a function of thepump-cavity detuning ∆ c and the pump power. A sharp phase boundary separating the normal and the Dicke-superradiant states is visible, which shifts for increasing absolute values of the detuning to increasing pumppowers. The dashed curve is the mean-field prediction for the pump threshold, Equation (45), reproducingthe experimental phase boundary. The insets (b) and (c) show time traces of the intra-cavity photon numberwhen entering the self-organized phase. For detunings close to the cavity resonance, an oscillatory behavior isobserved since the dynamic dispersive shift of the cavity pushes the detuning δ c to zero which interrupts thepositive feedback driving the self-organization. Figure reprinted with permission from Ref. [28], published in2010 by the Nature Publishing Group. (thermal or quantum) fluctuations of the order parameter were frozen out analogousto the Kibble-Zurek mechanism [136], such that a more equal distribution of the mea-surement results was found. The superradiant phase in this driven system is a stable steady state with lifetimeson the order of several 100 ms, limited by either atom loss or heating [28,135]. Thisis different from observations of superradiance in free space [137] or in longitudinallypumped ring cavities [26], where the superradiant Rayleigh scattering is a transientphenomenon leading to a pulse of light rather than entering a stable quantum phase ofmatter. By scanning both the pump-cavity detuning ∆ c and the pump power P ( ∝ η ),the steady-state phase diagram of the system can be recorded as shown in Figure 11.With increasing absolute values of the pump-cavity detuning ∆ c , the critical point toenter the superradiant phase shifts to higher values of the pump power. The mean-field prediction for the phase boundary, Equation (45), is in good agreement with theexperimental data. The horizontal phase boundary at small detunings is dispersivelyshifted by − (cid:82) U ( r )ˆ n ( r ) d r with respect to the bare cavity resonance. This shift is adynamic quantity, depending on the overlap between the cavity mode and the atomicdensity distribution, giving rise in addition to nonlinear effects in the system. Fordetunings close to the cavity resonance, the system can enter into an unstable regime,where the optomechanical interaction pushes the dispersively shifted cavity resonance δ c to zero and the positive feedback that drives self-organization is interrupted. As aconsequence, the intra-cavity field shows oscillations; see Figure 11(c).39 igure 12. Quench dynamics accross the non-equilibrium Dicke-superradiant phase transition. (A) The intra-cavity intensity is shown for a quench of the pump lattice from 0 to 4 E rec during 1 . . The experiments considered so far operated in the regime where the time scale 1 /κ for the cavity-field dissipation is much faster than the atomic motion, κ (cid:29) ω r . Inthis regime, the intra-cavity field instantaneously follows the atomic dynamics andcan thus be adiabatically eliminated. In contrast, if the cavity has a very narrowbandwidth, quench experiments allow access to the non-adiabatic regime, where bothphotonic and atomic fields are out of equilibrium and cannot be slaved to each other.Therefore, some of the features predicted for the non-equilibrium phase transition inDicke-type models [138,139] can be observed.This sideband resolved regime has been experimentally realized [37] with κ =2 π × .
45 kHz < ω r = 2 π × . c <
0, theresulting phase diagram is very similar to the phase diagram (Figure 11) of the pre-viously discussed system [28]. However, new features appear in the phase diagram forpositive pump-cavity detunings, ∆ c >
0, as predicted by Refs. [138,139]. Namely, inthis region of the phase diagram short superradiant pulses were emitted into the cavity,during which the atoms were irreversibly scattered into higher momentum states.Features of non-adiabaticity are clearly visible in quench experiments shown in Fig-ure 12. Ramping from the normal phase into the superradiant phase and back each ona time scale of 1 . igure 13. Phase diagram as a function of temperature vs. the driving pump strength for a three-dimensional,non-interacting cloud of driven bosonic atoms coupled dispersively to a single standing-wave mode of an opticalcavity. A superradiant (SR) phase can exist both for thermal and Bose-condensed atoms. There is an optimaltemperature for which the pump-strength threshold for the superradiant phase transition (the blue curve) isthe lowest. Adapted and reprinted with permission from Ref. [96], published in 2013 by the Elsiver. While the criticality of the superradiant phase transition in bosonic atoms is capturedby the effective low-energy Dicke-type model of Equation (19), some important quali-tative effects, even in the simplest geometry considered so far where the driven atomscouple dispersively to a single cavity mode, cannot be reproduced by the low-energyDicke Hamiltonian (19). For instance, the superradiant threshold as a function oftemperature, obtained from the full model of Equation (6) for non-interacting bosonsand shown in Figure 13, behaves differently from the one of the Dicke model. Thesmallest pump-strength threshold appears at an optimal, finite temperature of theatomic cloud, while the Dicke Hamiltonian (19) has the lowest pump-strength thresh-old at zero temperature. This behavior results from the fact that the polarizationfunction χ dyn , Equatiom (35), of the atomic medium at finite photon momentum ini-tially increases if atomic momentum states in the direction perpendicular to the photonmomentum become thermally occupied, simply because more states then participatein the scattering. This, however, holds only until the temperature broadening of theatomic distribution becomes an appreciable fraction of the cavity-photon momentum,which makes the numerator of the polarization function χ dyn smaller as longitudinalmomenta of the order of the photon momentum become occupied.The self-ordering of bosonic atoms can exhibits other interesting features, whichare not shown in Figure 13. For instance, it has been predicted that deep enough inthe superradiant phase the BEC can enter a fragmented phase [142], where multipleatomic modes are macroscopically occupied but with no coherence among one another.The effect of such beyond-mean-field correlations is even stronger if an external (i.e.,not cavity-induced), strong optical lattice is imposed in the system; see Section 7.Even more qualitative differences between the prediction of the full Hamiltonian (6)and the Dicke model (19) appear in the self-ordering of fermionic atoms. Besidesstudies concerning optomechanical effects in Fermi gases inside cavities [143,144], theself-organization of fermionic atoms has been first studied theoretically in Refs. [113,41 igure 14. The superradiant phase transition in a two-dimensional, non-interacting cloud of driven fermionicatoms coupled dispersively to a single standing-wave mode of an optical cavity. (a) Phase diagram for a fixedpump intensity as a function of the dimensionless cavity-pump detuning ˜ ω = − c /U N l vs. the lattice fillingfactor N/N l . The superradiant phase is characterized by a nonzero cavity field. The nesting condition is fulfilledaround the unit filling, visible as a peak around this filling. The empty circle in (a) is a critical point where thefirst order liquid-gas-like transition terminates. Panel (b) shows the effect of temperature, where the nestingeffect is washed out by increasing temperature. Adapted and reprinted with permission from Ref. [145] © k c = 2 k F )in the phase diagram shown in Figure 14(a). The effect of nesting is more dramaticin one dimension [113], where the pump-strength threshold for the self-ordering phasetransition vanishes when the photon momentum is twice the Fermi momentum. Thisphenomenon is known in the condensed-matter context as the Peierls transition inelectron-phonon models [148]. By forming a density wave, a gap is opened in theFermi surface so that the atoms in the superradiant cavity lattice form an insulator.Sufficiently away from the unit filling, the atoms are in a metallic state [113,149,150].Nesting effects require a sharp Fermi surface and are thus washed out with increasingtemperature of the atoms, as illustrated in Figure 14(b). This is especially true in onedimension, where the Peierls instability immediately disappears and the superradiantthreshold is moved to a finite atom-photon coupling strength. The role of disorder inthe self-organization of fermions has been also investigated in Ref. [151]. Excitation spectra and fluctuations
In Section 2.6 we developed the theory for obtaining collective excitations of atom-cavity systems in a general setting. Using this formalism, in Section 3.1.1 we obtainedthe low-energy polaritons of driven bosonic atoms coupled dispersively to a singlestanding wave of a linear cavity. In this section, we provide a summary of the ex-perimental works that measured polaritons and related fluctuation spectra across the42elf-ordering phase transition in bosnic atoms.In the experimental situation of a finite-size atomic BEC with short-range collisionalinteractions and a transverse pump-lattice potential, the energy of the excitations haveto be rescaled [152]. Contact interactions and the optical potentials affect the bareenergy of the excitation mode already without cavity-mediated global interactions.The excited single-particle mode is replaced by a Bogoliubov mode, and the couplingbetween the ground-state wave function and that mode is rescaled by its spatial overlapintegral weighted with the mode structure of the long-range interaction potential. Inthe normal phase, the excited Bogoliubov mode with momentum (respectively quasi-momentum along the transverse pump lattice) ( ± k c , ± k c ) lies in the lowest-energyband of the transverse pump lattice potential, and its energy is shifted with respect tothe bare kinetic energy by the mean-field shift. In the superradiant phase, the emergentcheckerboard lattice halves the Brillouin zone (which is defined initially only along thepump direction, but now along both the pump and cavity directions), and the moststrongly coupled excited Bogoliubov mode at ( ± k c , ± k c ) lies in a higher-energy bandat zero quasi-momentum of the optical checkerboard potential [152]. A nonzero energygap between the ground state and the first excited state is expected to remain evenat the critical point due to the finite size of the system. Experimental access to the polaritonic excitation spectrum can be gained by perform-ing a variant of Bragg spectroscopy [153,154]. The system is prepared at a transversepump power, corresponding to a certain coupling strength, and probed by a weak lightpulse of amplitude √ n pr injected directly into the cavity mode [152]. The frequency ofthe probe light pulse has an adjustable difference δ pr relative to the transverse pumpfrequency, such that the interference between the two gives rise to a time-dependentamplitude-modulated checkerboard lattice potential (cid:126) √ n pr η ( r ) cos( δ pr t + ϕ ) acting onthe atoms, where ϕ is the relative phase between the two fields. If this perturba-tion is resonant with a collective excitation of the system, stimulated scattering ofprobe photons into the pump and vice versa will take place, leading to the occupationof the excited polariton. Subsequent to the probe pulse, all potentials are switchedoff, projecting the created atomic excitations onto the free-space momentum states( ± k c , ± k c ), which can be detected in absorption imaging after time-of-flight expan-sion of the BEC. Instead of measuring the atomic population, the light field leakingfrom the cavity during the probe pulse can also be analyzed. If the atomic excitedmomentum state is populated by the perturbing potential, pump photons will be scat-tered into the cavity mode, leading to an enhanced cavity output field that oscillatesat δ pr . The amplitude of these oscillations is maximal on resonance with a collectiveexcitation of the system. The resulting excitation spectrum is displayed in Figure 15.A clear softening of the lowest-energy polariton mode is observed, consistent withthe square-root vanishing of the real part of the polariton frequency [see Equation (47)].Note that the experimental parameters here do not allow one to resolve the over-damped critical regime η s < η < η c [see Equation (50)], where the polariton oscilla-tion frequency (i.e., the real part of the polariton frequency) is zero but the damping(i.e., the imaginary part of the polariton frequency) is finite. We will return to thepolariton damping in the next section. We note that if such a measurement would beperformed in the Hamburg setup where κ ∼ ω r [37], the overdamped critical regimeshould become accessible experimentally. 43 cr )0246810 E x c i t a t i onene r g y ( h × k H z ) atom dataphoton data Figure 15. Measurement of the lowest excitation energy as a function of the transverse pump power across theself-ordering phase transition of a driven BEC inside a single-mode linear cavity. The excitation spectrum wasmeasured via either the atomic population of the excited momentum state (blue), or via the light field leakingfrom the cavity (red). The observed mode softening (filled circles) for effective attractive cavity-mediated atom-atom interactions (corresponding to δ c <
0) culminates around the critical pump power P cr . For the oppositeblue detuning between pump and cavity ( δ c > The atom-photon coupling η ( r ) induces checkerboard density correlations in theatomic gas with a λ c spatial periodicity in the x - y plane. These correlated fluctua-tions, which correspond to the atomic component of the lowest polariton mode, arealready present in the normal phase below the self-ordering critical point and are theprecursors of the mean density modulation emerging in the superradiant phase. In amicroscopic picture, these fluctuations correspond to the creation and annihilation ofcorrelated atoms in an excited Bogoliubov mode. They are induced by photons beingscattered by the atoms from the pump into the cavity mode and vice versa. Thesephoton scatterings impart momentum kicks onto the atoms, and accordingly lead toa frequency shift of the involved photons, observable in the power spectral density S aa ( ω ) of the light field leaking from the cavity. When an atom is scattered from theBEC into the excited momentum state via a photon scattering process, the involvedphoton will be red shifted as it deposits energy in the system. For the opposite processof annihilating a mechanical excitation of the atomic gas, the photon is accordinglyblue shifted. The energy shift depends on the distance to the critical point and is givenby the real part of the lowest polariton energy (cid:126) ω pol [see Equation (47)].The density correlations are directly connected to the dynamic structure factor S ( k , ω ) of the atomic system, which is the spatial and temporal Fourier transform ofthe density-density correlations. The experimental setting allows direct access to thisobservable. While the cavity-enhanced Bragg spectroscopy described above is based onthe stimulated scattering of photons from one coherent field to another, the measure-ment of fluctuations requires the analysis of spontaneously and inelastically scatteredphotons. With respect to free space, the probability to detect spontaneously scatteredphotons leaking from the cavity mode is increased by several orders of magnitude dueto the enhanced vacuum field in the resonator. The density of states for scatteringphotons has to be high but flat in the energy range of interest, which requires the44 .7 0.8 0.9 1.0 1.1 − − − ( k H z ) − ( − / H z ) − − ( d B m / H z ) Figure 16. Measured power spectral density of the cavity field across the self-ordering phase transition of adriven BEC inside a single-mode linear cavity. The power spectral density S aa ( ω ) of the light leaking fromthe cavity is shown as function of transverse pump power P/P cr and frequency difference ω with respect tothe transverse pump frequency. The two sidebands originate from the density fluctuations of the atomic gas,while the signal at ω = 0 stems from elastic scattering of pump photons at a density modulation. The cutsbelow the main figure display the extracted dynamic structure factor S ( k , ω ) at the wave vector given by k = ( ± k c , ± k c ). Figure adapted and reprinted with permission from Ref. [52] published in 2015 by the NaturePublishing Group. cavity to operate in the bad cavity limit with respect to the energy of the mechanicalexcitation, κ (cid:29) ω r .A heterodyne detection scheme can be used to obtain a frequency-resolved measure-ment of the light field leaking from the cavity [52]. Figure 16 displays a measurementof the power spectral density S aa ( ω ) of the cavity field together with the extracteddynamic structure factor S ( k , ω ) as a function of the relative transverse pump power P/P cr and the frequency ω relative to the frequency ω p of the transverse pump. Thegeometric setup of pump and cavity fields determines the momentum k = ( ± k c , ± k c )at which the dynamic structure factor is measured. Power in the frequency bin at ω = 0 corresponds to the elastic (Rayleigh) scattering of pump photons at an existingdensity modulation, while power at nonzero frequencies corresponds to inelastic (Ra-man) scattering of pump photons creating or annihilating density fluctuations. Thesignal at ω = 0 raises by 5 orders of magnitude, which is a measure of the densitymodulation increasing dramatically, above the critical point at P/P cr = 1. The red andblue detuned sidebands are a measure of the density fluctuations. Their frequency isdecreasing towards the critical point in accordance with the expected mode-softeningmeasurement of Figure 15.The width of the sidebands is a measure of the damping of the polaritonic mode.Since this mode has a dominant atomic character, the damping is mainly attributedto the decay of atomic momentum excitations, and the contribution of cavity decay tothe damping is negligible (see the discussion at the beginning of Section 3.1.3). The45ehavior of the damping as a function of the atom-photon coupling strength has beeninvestigated theoretically in detail and originates from Beliaev damping of the checker-board density modulation and from finite temperature effects [115,116,155]. However,in the overdamped critical regime, where the polariton mode frequency is zero, thecontribution to damping from Beliaev decay (or other collision-induced decays) alsovanishes. This leaves the cavity loss as the only source of damping which cannot any-more be neglected in principle. As noted above, this regime was not accessible in theETH-Zurich experiments, but could be within reach in the Hamburg setup.The lower panels in Figure 16 clearly show a sideband asymmetry, where the lowerenergetic red sideband has a larger amplitude than the blue sideband. Similar tothermometry in trapped ion experiments, the occupation of the relevant quasi-particlemode can be extracted from this data and was found to be on the order of a fewquanta. Analyzing the system in terms of thermodynamic quantities further allowedto extract the irreversible entropy production rate of the system across the phasetransition [156].The integrated power in the sidebands is a measure of the strength of the densityfluctuations. In accordance with an earlier measurement [157], the density fluctuations,which are the fluctuations of the order parameter of this phase transition, divergewhen approaching the critical point. An analysis of the scaling of the order parame-ter fluctuations as a function of the distance from the critical point revealed criticalexponents of 0 . . Variants and extensions
The experimental schemes introduced in the previous sections have been extendedto realize a number of variants of the self-ordering phase transition breaking discretesymmetries, which we summarize in this section.
Atomic self-organization can also take place in ring cavities, which support a pair ofdegenerate, counterpropagating running-wave modes. First experiments investigatedthe phenomenon of collective atomic recoil lasing (CARL), which is an instability anal-ogous to a free electron laser and has a similar origin as the Dicke phase transition.The atomic gas is trapped at the position of the cavity modes where one of them ispumped through one of the cavity mirrors; see Figure 17(a). Photons injected into theforward-propagating cavity mode can be scattered back via the atoms into the un-pumped, counterpropagating cavity mode, while the atoms recoil. Exponential gain ofthis unpumped, counterpropagating field then triggers the so-called CARL instability.Specifically, the backscattering process leads to the buildup of an interference patternof the two counterpropagating light fields which acts as a dynamic potential on theatoms. Correspondingly, the atoms form a matter-wave pattern further enhancing thephoton scattering in a runaway process. This density wave is accelerated in space, syn-chronized with the co-moving optical potential generated from the interference of thetwo cavity fields. As a consequence, the frequency of the light scattered at the atomsis increasingly red-shifted with respect to the probe light due to the Doppler effect.In the bad cavity limit, this leads to the occupation of increasingly higher momen-46 b Figure 17. Self-organization of a longitudinally driven BEC in a ring cavity with two degenerate, counter-propagating running-wave modes. (a) Atoms are placed within a ring cavity that is pumped on the TEM mode (green) through one of the cavity mirrors. Photons scattered off the atoms populate the counterprop-agating, degenerate mode (blue), leading to the emergence of a dipole potential from the interference of thetwo counterpropagating fields. (b) Stability diagram of the system. The white circles indicate the critical pumppower above which the condensate zero-momentum state is significantly depopulated due to photon scattering.The background color shows the population of the condensate mode from a numerical simulation, and thered line indicates the numerical phase boundary obtained using a low-energy model without cavity damping.The inset shows an example from which a threshold can be deduced. Depending on the detuning between thecavity and pump field ∆ c , the system either enters the recoil lasing regime or the momentum correlated regimeseparated by the white dashed line. Figure adapted and reprinted with permission from Ref. [130] © tum states. For narrow band resonators in the good cavity limit, processes occupyingincreasingly higher momentum states are, however, suppressed, effectively stabilizingthe system. The CARL instability, originally predicted by Bonifacio et al. [159], hasbeen demonstrated in both the thermal [160] and the ultracold regime [26].The CARL instability sets in for positive, dispersively-shifted pump-cavity detuning, δ c >
0, and is a runaway process. In an intuitive classical picture [161], the atoms arelocated close to the maxima of the periodic dipole potential building up from thepumped and back-scattered fields, leading to the acceleration of the atomic system.A pulse of light is emitted while the atoms are transferred from the zero momentumstate to higher momentum states in the direction of the pump mode. In contrast,for negative pump-cavity detuning δ c <
0, the phase of the field inside the resonatoracquires a phase shift of π with respect to the situation for δ c >
0. The atoms thusare located close to the minima of the emerging dipole potential and can enhancescattering via constructive interference by building up a stationary density gratingwith a λ c / δ c < So far, we did not consider the spin degree of freedom of the atoms in the self-organization process. Tilting the quantization axis of the atoms with respect to thepolarization of the driving electric fields, and considering scattering of light into bothpolarization modes of the cavity allows to extend the single-mode self-organization toschemes involving two photonic modes, as discussed in Ref. [47,166].To this end, we need to rephrase the atom-field interaction Hamiltonian in termsof the atomic polarizability, taking into account that the atom has a more complexinternal structure than the two-level scheme assumed in Section 2. The Hamiltoniandescribing the interaction between an atomic dipole operator ˆ d and an electric field ˆ E in the dipole approximation ˆ H A − L = − ˆ d · ˆ E can be rewritten in the dispersive limitas [47,167,168],ˆ H A − L = − α s ˆ E † · ˆ E + iα v ( ˆ E † × ˆ E ) · ˆ F F − α t (cid:88) i,j =1
3( ˆ F i ˆ F j + ˆ F j ˆ F i ) − F δ ij F (2 F −
1) ˆ E i · ˆ E j , (52)where ˆ F is the total angular momentum operator with maximum eigenvalue F , and α s , α v , α t are the scalar, vector, and tensor polarizabilities of the atom, respectively [168].They depend on the electronic structure of the atom and the frequency of the drivingelectric field. The scalar part captures the induced atomic dipole oscillating in phaseand in the direction of the polarization of the driving electric field, which we assumeto be linear for the moment. Light scattered by the scalar part thus maintains thepolarization of the driving light field. In contrast, the vector part is imaginary and canchange the polarization of the scattered light field through the Farady effect. Lightemitted due to the vector polarizability has a polarization direction orthogonal toboth the atomic spin and the pump polarization, and oscillates π/ E at the positionof the atoms contains contributions from both the (classical) pump fields E p and the(quantum) cavity fields ˆ E c , ˆ E = (cid:88) pumpfields E p + (cid:88) cavitymodes ˆ E c , (53)48nd each electric field is described by its unit polarization vector, amplitude, spatialmode profile, and time dependence. Inserting the pump and cavity modes assumed inSection 2.1 into Equation (52), one recovers the potentials derived in Equation (5),now in term of the scalar polarizability. Specifically, one finds the pump lattice depth V = Ω / ∆ a = − α s E , the cavity lattice depth per photon U = G / ∆ a = − α s E ,and the two-photon scattering rate η = Ω G / ∆ a = − α s E E . Here, E and E denote the electric field amplitudes of the pump and the cavity fields, respectively. The term proportional to the spin-dependent vec-tor polarizability α v becomes important if the cross product ( ˆ E † × ˆ E ) in Equation (52)becomes non-zero. This can be achieved for example by rotating the polarization ofthe pump field (illuminating the atoms in the z direction) with respect to the polar-ization of the cavity field by an angle ϕ , as shown in Figure 18(a) [47]. That is, thepolarization of the pump field is rotated to make an angle π/ − ϕ with respect to thecavity axis, since the polarization of the cavity mode is parallel to cavity mirrors. Theinduced atomic dipole can then be decomposed into a scalar and a vector componentˆ d = ˆ d s + i ˆ d v = − α s ˆ E / iα v ˆ F × ˆ E / F [where the orientation of ˆ F is fixed by the biasmagnetic field B ; see Figure 18(a) and also Equation (52)], such that light scatteredinto the cavity mode will be phase shifted with respect to the phase of the pump field.This can also be seen from the many-body Hamiltonian for N atoms—mapped intoa Dicke model with macroscopic spin operators as introduced in Section 2.3—in theZeeman state m F [47],ˆ H m F = − (cid:126) δ c ˆ a † ˆ a + 2 (cid:126) ω r ˆ J z,m F + (cid:126) (cid:104) η s (ˆ a † + ˆ a ) cos ϕ + iη v (ˆ a † − ˆ a ) sin ϕ (cid:105) ˆ J x,m F . (54)Here, cos ϕ (sin ϕ ) appears due to the projection of the scalar ˆ d s (vector ˆ d v ) dipolecomponent along the polarization of the cavity field in the y direction; see Figure 18(a).The polarization angle ϕ of the pump electric field controls the ratio between the scalarcoupling η s ∝ α s and the vectorial coupling η v ∝ α v . Since the scalar and vector po-larizabilities have different strengths, the critical point for the self-organization phasetransition depends on the angle ϕ . Figures 18(b) and (c) display the threshold for theself-organization and the phase shift of the light scattered into the cavity as a functionof the polarization angle of the pump field, respectively. The behavior depends on theZeeman state the atoms are prepared in. Note that we assume here the Zeeman split-ting of the atomic levels to be much larger than both the coupling strength and thedetuning δ c , such that spin changing processes are suppressed; spin changing processesare considered and discussed in Section 6. For m F = 0, only the scalar polarizabilityis present. Accordingly, the time phase of the cavity field is independent of ϕ , andthe threshold for self-organization diverges if the polarization of the pump field be-comes parallel to the cavity axis (i.e., ϕ = π/ m F = ± ϕ = π/
2; the field scattered into the cavity experiences either a positive or anegative phase shift.In a more general case of a balanced spin mixture of atoms prepared in the Zeemanstates m F = +1 and m F = −
1, the light fields scattered by the individual spin com-ponents interfere, leading to competition between the different self-organized states.Depending on the ratio of real and imaginary components of the field scattered intothe cavity, it can become energetically more favorable for the system to arrange ina spin modulated texture than in a checkerboard density modulation. Such a phase49
30 60 90 120 150010203040 V c T P ( E R ) y - po l . x - po l . ϕ (deg) − π − π π π φ (r ad ) y - po l . x - po l . V TP ( E R ) ¯ n ph V c T P (a)(b) a bc Figure 18. Spin-dependent self-organization of a transversely driven BEC in a single-mode optical cavity.(a) The BEC is illuminated by a pump field along the z direction whose linear polarization can be rotated inthe x - y plane by an angle ϕ . The induced atomic dipole has a scalar ˆ d s and a vector ˆ d v component whichoscillate π/ ϕ . (b) Critical value of the pump-lattice depth V c ∝ η c for self-organization as a function of the polarizationangle ϕ for the different Zeeman states (circles: m F = 0, triangles and squares: m F = ± © transition was observed in Ref. [47], where the phase of the light field scattered intothe cavity with respect to the phase of the pump field jumped from 0 to π/ λ c -periodic checkerboard modulation of the atomic spins. The phase transition be-tween the normal and either of the self-organized states (spin modulated or densitymodulated) has the same behavior as the Dicke model. However, for sufficiently smalldetuning with respect to the cavity resonance, an intriguing dynamical instability canbe induced driving the system to oscillate between the different atomic patterns (seeSection 10.1.2). So far, we have considered super-radiance of atoms in only one of the two (theoretically) degenerate modes of a linearcavity with orthogonal polarizations. Coupling to the other orthogonally polarizedmode of the cavity is, however, also possible if the atom has a non-zero vector polariz-ability at the frequency of the transverse pump. Self-organization can thus take placeinvolving both polarization modes, as has been experimentally demonstrated [166].Figure 19(a) shows the basic experimental scheme. The polarization of the pumpelectric field E p is oriented along the z axis, and an externally applied bias magneticfield defining the atomic quantization axis allows to rotate the orientation of the atomicpseudospin ˆ F by an angle ϕ with respect to the cavity axis. Therefore, the induceddipole moment ˆ d = ˆ d s + i ˆ d v = − α s ˆ E / iα v ˆ F × ˆ E / F can have components along twoorthogonally polarized cavity modes designated by the annihilation operators ˆ a (cid:107) (forthe mode with a parallel polarization with respect to the pump polarization) and ˆ a ⊥ b cdef Figure 19. Two-mode self-organization of a driven BEC inside a linear cavity via competing of two mutuallyorthogonal, polarized cavity modes. (a) The electric field E p of the pump laser is polarized along z while theatomic spin ˆ F can be rotated by an angle ϕ in a plane spanned by the cavity axis and the z axis. The vectorpolarizability can lead to scattering of photons from the pump into both polarization modes of the cavity.(b) The population of both polarization modes is simultaneously measured and shown in blue and orange,respectively, for an angle of ϕ = 0(4) ◦ . Two regions of self-organization are visible in the resulting phasediagram. The self-organized phase SO (cid:107) is completely suppressed for detunings close to the cavity resonance ofthe orthogonally polarized mode, as indicated by the grey region. For detunings effectively red with respect tothe orthogonal polarized pump, the self-organized phase SO ⊥ emerges. The dashed lines show the theoreticallypredicted phase boundaries. Panels (c)-(f) show neasurements of this phase diagram for different angles ϕ =30(4) ◦ , ◦ , ◦ , ◦ . Figure adapted and reprinted with permission from Ref. [166] © (for the mode with an orthogonal polarization with respect to the pump polarization).In particular, ˆ d s is aligned along the polarization of the ˆ a (cid:107) mode, while ˆ d v is parallelto the polarization of the ˆ a ⊥ mode. The degeneracy of the two mutually orthogonalcavity modes is lifted in the experiment due to birefringence (see Section 2.7.1), andhence they have different resonance frequencies. Depending on the detunings ∆ (cid:107) and∆ ⊥ between the pump frequency and the respective cavity-mode frequency, the pumpfield might thus be red-detuned to one cavity mode while it is blue-detuned withrespect to the other cavity mode.Using the general atom-field interaction formalism described above [see Equa-tion (52)] and the mapping of Section 2.3, the low-energy Hamiltonian of the systemfor atoms in the Zeeman substate m F is [166],ˆ H = − (cid:126) ∆ (cid:107) ˆ a †(cid:107) ˆ a (cid:107) − (cid:126) ∆ ⊥ ˆ a †⊥ ˆ a ⊥ + (cid:126) ω ˆ b † ˆ b + α s E E √ (cid:16) ˆ a †(cid:107) + ˆ a (cid:107) (cid:17) (cid:16) ˆ b † ˆ b + ˆ b † ˆ b (cid:17) + iα v (cid:20) E E √ (cid:16) ˆ a †⊥ − ˆ a ⊥ (cid:17) (cid:16) ˆ b † ˆ b + ˆ b † ˆ b (cid:17) + E (cid:16) ˆ a †(cid:107) ˆ a ⊥ − ˆ a †⊥ ˆ a (cid:107) (cid:17) ˆ b † ˆ b (cid:21) m F F cos ϕ, (55)51here ˆ b (ˆ b † ) is the operator annihilating (creating) an atom in the momentum su-perposition state with energy ω resulting from the scattering of photons between thepump and the cavity, and ˆ b (ˆ b † ) is the operator annihilating (creating) an atom in thezero-momentum BEC. The term proportional to the scalar polarizability α s describescoupling of the BEC to the real quadrature (ˆ a (cid:107) + ˆ a †(cid:107) ) of the parallel polarized cavitymode. The first term proportional to the vector polarizability α v captures the couplingof the BEC to the imaginary quadrature i (ˆ a †⊥ − ˆ a ⊥ ) of the orthogonal polarized cavitymode, while the second term describes a direct scattering of photons between the twomutually orthogonal cavity modes via the atoms in the BEC.Since for certain pump frequencies the signs of the detunings ∆ (cid:107) and ∆ ⊥ can beopposite, the coupling to the two polarization modes competes, and self-organizationcan be fully suppressed in a certain parameter range, as can be seen in Figure 19(b).Specifically, if the pump frequency is blue detuned with respect to the resonancefrequency ω ⊥ of the orthogonal polarized mode but red detuned with respect to theresonance frequency ω (cid:107) of the parallel polarized mode, the phase diagram shows a sliverwithout self-organization also for large pump fields. In this region, a mode hardeningdue to the effective blue detuning competes with a mode softening due to the effectivered detuning. Accordingly, the critical point in the grey shaded area is pushed toinfinity. Self-organization can also take place for low-field seeking atoms with ∆ a >
0, i.e.,in the repulsive regime of blue-detuned pump and cavity lattices. This was predictedtheoretically early on in Ref. [138]. In particular, the possible existence of dynamicalinstabilities in this parameter regime has attracted more interest recently [38,43,139,169]; see Section 10 for discussions regarding dynamical aspects.At first sight self-organization in repulsive fields might seem counterintuitive, sincethe atoms will localize in the intensity minima of the pump field, suppressing thescattering of photons into the cavity and thus inhibiting self-organization. However,this purely classical picture of point-like particles does not take into account the finiteextent of the BEC wavefunction around the potential minima and that the localizationof particles comes at the cost of kinetic energy. As a result, the atomic gas still hasa finite overlap with the blue detuned pump field. Since the induced atomic dipolesare driven above their resonance, the light field scattered by the atoms into the cavityoscillates out of phase with respect to the pump field. At the position of the atoms,these two fields thus interfere destructively, again creating potential minima for theatoms; see also Figure 5. This process allows the atoms to lower their potential energyvia increased scattering of photons into the resonator, and hence self-organization ina repulsive potential takes place. P band of the pump lattice. Although the basic process forself-ordering of low-field seeking atoms is very similar to self-organization of atomsin red-detuned, attractive light fields, significant differences exist. In a blue-detunedpump lattice, the atoms in the ground state localize at the nodes of the pump latticepotential. The maxima of the atomic density are thus shifted by λ c / η ( r ) in Equation (7) describing the coupling betweenthe pump and the cavity thus changes parity and becomes ∝ sin( k c y ) instead of52 ormal phaseself-organized phase ∆ a − 1 0 1q/k024 SP Pump Lattice Depth V p ( E rec ) − − C a v i t y D e t un i n g ∆ c / ( π ) ( M H z ) bc (a) − C a v i t y L a tt i ce D e p t h ( E r ec ) (cid:31) k p (cid:31) k c (b) (cid:31) ( k c (cid:31) k p ) (c) c1 c1c2 c2 cba Figure 20. Self-organization of a driven BEC in a single-mode linear cavity in the repulsive lattice regime.(a) The BEC (red) is exposed to a repulsive, blue-detuned optical pump lattice. In the self-organized phase, thefield scattered by the atoms into the cavity interferes at the position of the atoms destructively with the pumpfield. This way the atoms can lower their potential energy by self-organization. (b) The two-photon processfor the blue atomic detuning (grey arrows) couples the BEC in the ground S band to a higher momentumstate in the P band of the pump lattice. With increasing pump-lattice depth the coupling term leads to amode softening at the according wave vector (pale blue lines) and finally to self-organization when this modetouches zero relative to the ground state energy. At the same time, the energy of the P band increases withthe pump-lattice depth (blue arrows). For deeper lattices, this effect becomes dominant and suppresses self-organization. (c) Phase diagram showing the intra-cavity lattice depth as a function of the cavity detuningand the pump-lattice depth. The self-orderd phase has only limited extent for increasing pump lattice depth.Dotted lines are results from a numerical mean-field calculation showing the phase boundaries. The insets showabsorption images of the atomic cloud after ballistic expansion for parameters indicated by the crosses in themain plot. Figure adapted and reprinted with permission from Ref. [39] © ∝ cos( k c y ). Accordingly, while for atomic red detuning the BEC is coupled from thezero-momentum state of the S band to finite quasi-momenta within the S band, forblue detuned lattices the BEC is coupled to quasi-momenta in the P band, localizedat the maxima of the pump potential. The coupling leads to a mode softening atthe wave vector of the scattering process [see Figure 20(b)], eventually enabling self-organization for a sufficiently strong pump field. However, for a further increasedpump-lattice depth, the energy gap between the S and P bands increases, and theself-organization is finally terminated at some point. As a result, the extent of theself-organized phase as a function of the pump-lattice depth is limited, as can be seenfrom the experimentally recorded phase diagram in Figure 20(c) [39]. This is in starkcontrast to the self-organization in the red-detuned regime, cf. Figure 11.The self-organized atomic density favours the emergence of stripes [see Figure 20(a)]rather than a checkerboard pattern for two reasons. First, in the experiment the pumpfield is tilted with respect to the cavity mode by 60 ◦ , such that two atomic momentummodes with different energies are addressed by the two-photon scattering. The lowerenergetic one is predominantly occupied. Second, the structure of the coupling disfavorsfor ∆ a > a < c betweenthe pump and the cavity since also the dispersive shift, − (cid:82) U ( r ) n ( r ) d r , has a flippedsign with respect to the case for ∆ a <
0. For increasing pump-lattice depth at positivecavity detunings, cyclic spiking of the cavity field and population of very high atomicmomenta was observed. Uniquely for ∆ a , ∆ c >
0, self-organization can become ametastable state, and the system eventually lowers its energy by letting the photonnumber in the cavity diverge. 53 .3.3.2. Structural phase transition.
Applying a pump field with a standing-waveand a running-wave component allows to realize for low-field seeking atoms, ∆ a >
0, asituation where a simultaneous coupling to the P band and to the S band of the pumplattice can be brought to competition [170]. Since the self-organized density patternsof these two possibilities are not compatible with each other, a first order structuralphase transition between them can be induced.Experimentally this was implemented by applying imbalanced pump fields withwave vectors ± k p and electric field amplitudes E ± to a BEC inside a cavity withthe mode wave vector k c ; see Figure 21(a). The according pump electric field is thengiven by E p = ( E + + E − ) / E ± = E ± e ± i k p r e z , which results in a pump-latticedepth V p = − α s E + E − . The interference between the pump electric field and the cavityelectric field generates two potential energy terms with different symmetry. This can beseen from inserting the total electric field ˆ E = E p + ˆ E c into the atom-field interactionHamiltonian (52), where ˆ E c = E (cid:15) c ˆ a cos( k c · r ) is the positive frequency componentof the cavity field. Ignoring the term proportional to the vectorial polarization inEquation (52), this results inˆ H A − L ( r ) = (cid:126) V cos ( k p · r ) + (cid:126) U ˆ a † ˆ a cos ( k c · r ) + (cid:126) η (ˆ a + ˆ a † ) cos( k p · r ) cos( k c · r )+ i (cid:126) η (ˆ a − ˆ a † ) sin( k p · r ) cos( k c · r ) , (56)where the potential depths (cid:126) η , (cid:126) η depend on the tunable imbalance between thepump fields E ± .From the resultant atom-field interaction Hamiltonian, Equation (56), it becomesobvious that the two different spatial structures involving cos( k p · r ) and sin( k p · r )couple to the real and the imaginary quadrature of the cavity field, respectively. Thisallows to distinguish them experimentally by measuring the phase φ of the cavityfield, as shown in Figures 21(b) and (c). Two superradiant phases, termed SR andSR , are observed which differ in amplitude and phase of the cavity field. In addition,absorption images of the atomic cloud after ballistic expansion reveal that also thespatial structure of the two self-ordered states is different. SR shows momentumcomponents indicating a dominantly striped density modulation, while SR showsmomentum components corresponding to a checkerboard density pattern.When the system is driven from SR to SR by increasing the pump-lattice depth V , a structural phase transition takes place. This transition is of first order as wasshown theoretically by investigating the energy landscape. In addition, the real-timeobservation of the phase of the cavity field shows an abrupt jump by π/
2, followedby a damped oscillation. The oscillation frequency is in agreement with the curvatureof the free energy landscape minimum, as calculated numerically from a mean-fieldmodel [170].
4. Superradiant crystallization breaking a continuous symmetry:supersolids
In the preceding section, the self-ordering of ultracold atoms inside single-mode,standing-wave linear cavities was discussed. We saw that the crystallization in thesesetups breaks spontaneously the discrete Z parity symmetry of the Hamiltonian,leading to a two-fold degenerate ground state with gapped collective excitations. Inthis section, we consider scenarios where the superradiant crystallization of a super-54 yz b cad i i iiii iiiiii iiiiii Figure 21. Self-organization of a driven BEC inside a single-mode cavity with imbalanced pump fields. (a) ABEC inside a linear cavity is illuminated by imbalanced pump fields E + , E − , effectively leading to the couplingof the quantum gas to a standing-wave and a running-wave pump field. The interference of these two componentswith the cavity field gives rise to two incompatible patterns the atoms might self-organize to. Phase diagramshowing the intra-cavity lattice depth V c = (cid:126) U (cid:104) ˆ a † ˆ a (cid:105) (b) and the phase mapped to the quadrant φ ∈ [0 , π/
2] ofthe cavity field (c) as a function of the pump-lattice depth (cid:126) V and the pump-cavity detuning ∆ c . Besides thenormal superfluid phase (SF), two self-organized superradiant phases SR and SR can be distinguished. Thecavity field has different amplitude as well as a π/ i - iii ). (d) A first order structural phase transition between the SR and SR phases canbe induced by ramping up the pump-lattice depth to (cid:126) V = 25 E r at a detuning of ∆ c / π = − .
75 MHz. Thephase of the cavity field shows a sudden jump of π/
2, followed by a damped oscillation. Figure adapted andreprinted with permission from Ref. [170]. fluid BEC breaks spontaneously an external continuous U (1) symmetry [in additionto the spontaneously broken U (1) superfluid gauge symmetry of the BEC], giving riseto an infinitely degenerate ground state with a gapless Goldstone mode (in additionto the gapless condensate phonon mode associated with the broken superfluid gaugesymmetry). This state fulfills the criteria for the minimal version of a “supersolid”, anenigmatic state of matter characterized by the coexistence of crystalline and super-fluid orders which thus supports frictionless flow of particles in a crystal. Already thepreviously discussed self-organized phases of superfluid BECs breaking the discrete Z symmetry [28,152] can be regarded as a variant of supersolids, often termed “latticesupersolids” [171]. However, the underlying discrete Z symmetry does not allow forthe continuous ground-state degeneracy envisioned in the original discussions of su-persolidity. As a consequence, lattice supersolids do not sustain dissipationless particlecurrents.More precisely, during the formation of a supersolid two continuous symmetriesmust be spontaneously broken, namely, a continuous spatial translational symmetryand an internal U (1) superfluid gauge invariance. This paradoxical state of matter waspredicted almost 50 years ago to exist in solid helium-4 [172–174]. Despite intensiveexperimental efforts, there has been so far no conclusive evidence for the observa-tion of supersolidity in helium. Dilute bosonic quantum gases had been proposed aspossible candidates for the realization of a supersolid state of matter, if appropriatelong-range interactions which favor the superfluid gas to form a crystal could be en-55ineered [175,176]. These interactions might stem from induced or permanent electricdipole moments of the atoms, or can exist for atoms with strong magnetic dipole mo-ments. Cavity-mediated global interactions turn out to be a suitable, alternative routeto realize a supersolid, if the discrete Z symmetry of composite atom-cavity systemscan be replaced by a continuous U (1) symmetry [60]. This was achieved via symmetryenhancement using two crossed standing-wave linear cavities [127], as we detail belowin Section 4.1. Supersolids have since then also been experimentally demonstrated inquantum gases making use of spin-orbit coupling [177], of magnetic dipolar interac-tions [178–180], and of ring cavities [181]. Supersolids in systems with finite-rangeinteractions exhibit a continuous dispersion relation and thus have finite rigidity. Incontrast, systems with exclusively global-range interactions show (in the thermody-namic limit) a mode softening at isolated points in momentum space and thus formdefect-free and perfectly rigid crystals. The resulting supersolid in the latter systemsthus can be seen as the minimal possible version of a supersolid. Supersolidity in two-crossed linear cavities
A continuous symmetry can be engineered through symmetry enhancement with com-peting order parameters, as has been discussed in the context of high-temperaturesuperconductors, high-energy physics, and cosmology [182]. The concept is based onfine-tuning Hamiltonian parameters in order to combine the symmetry groups of mul-tiple order parameters into a single group with a higher symmetry. Several schemesenhancing symmetries based on generalized Dicke models exploiting the atomic inter-nal degrees of freedom have been discussed [183–185]. The idea can also be applied tothe external degree of freedom of a driven quantum gas coupled dispersively to twostanding-wave linear cavities. This allows to realize a supersolid in the self-organizedsuperradiant phase, if the two systems each breaking the discrete Z parity symmetryindividually are combined to break a continuous U (1) symmetry in real space andcavity field space.In the experimental realization of a supersolid at the ETH group, a single BEC wascoupled dispersively to two linear optical cavities crossed at a 60 ° angle, and subjectto a standing-wave transverse pump beam [127]; see Figure 22(a). Self-organizationof the atoms can take place in either of the cavities, each breaking a discrete paritysymmetry. However, if the two cavity modes are tuned to be degenerate, photonsfrom the pump field can be scattered into either of the two cavity modes withoutan energetic bias (neglecting scattering between the cavities for the moment), andsimultaneous self-organization populating both modes in the two cavities can takeplace. The optical potential acting on the atoms is formed from the interference of allthree involved fields [186]. As a consequence, the x position of the emergent latticethe atoms arrange into depends on the ratio of the field amplitudes α and α inthe two cavities. Conversely, the position of the atomic ordering along the x axisdetermines the cavity-field amplitudes α and α . Since the cavities are degenerate,the energy of the system is independent of the ratio α /α , which can take any value.Thus the self-ordered atomic crystal can be continuously translated along x while themode occupations are accordingly redistributed without energetic cost. The light-fieldamplitudes α , are the individual scalar order parameters characterizing the BECcrystallization according to either cavity-mode field; see Section 3.1.3. They can becombined into a vector order parameter α = α + iα = | α | exp( iφ ) that exhibits a56 ba α α α α z p z p x p y xy ħ k d Figure 22. Self-organization of a driven BEC in two crossed standing-wave cavities, giving rise to a supersolidstate. (a) A BEC (blue) is symmetrically coupled to two linear cavities that cross at a 60 ° angle. The atomic gasis dispersively illuminated by a standing-wave transverse pump in the plane of the two cavities. (b) Momentumspace representation of the involved two-photon scattering processes. Processes scattering photons from thepump field into either cavity are colored in red and yellow, respectively. Since pump and cavity fields are notorthogonal, a higher and a lower energetic momentum state (ˆ c j ± ) can be occupied. Scattering processes betweenthe two cavities shown in grey (ˆ c ± ) are negligible for large atomic detunings. (c) A vector order parameter α = α + iα = | α | exp( iφ ) can be constructed from the two scalar field-amplitude order parameters α and α .Different ratios α /α are energetically degenerate but lead to a spatial translation of the crystallized atomicdensity along the x axis. Figure adapted and reprinted with permission from Ref. [127] published at 2017 bythe Nature Publishing Group. continous rotational U (1) symmetry; see Figure 22(c). This idea is in close analogy toconstructing the XY model from two Ising models with equal couplings.The basic model introduced in Section 2.1 can be extended to two cavities ( j = 1 , a j , wave vectors k cj , detunings ∆ cj between pumpfrequency and cavity resonances, and maximum two-photon coupling rates η j . Theresultant many-body Hamiltonian of the system readsˆ H eff = − (cid:126) (cid:88) j =1 , ∆ cj ˆ a † j ˆ a j + (cid:90) ˆ ψ † ( r ) (cid:26) − (cid:126) ∇ M + (cid:126) V cos ( k p · r + ϕ )+ (cid:88) j =1 , (cid:104) (cid:126) U j cos ( k cj · r )ˆ a † j ˆ a j + (cid:126) η j cos( k p · r + ϕ ) cos( k cj · r )(ˆ a † j + ˆ a j ) (cid:105) + (cid:126) (cid:112) U U cos( k c · r ) cos( k c · r )(ˆ a † ˆ a + ˆ a † ˆ a ) (cid:27) ˆ ψ ( r ) d r . (57)Here, the wave vector of the pump is k p , and ϕ is the experimentally tunable spatialphase of the pump lattice with respect to the crossing point of the two cavity modes.The last line in Equation (57) describes the scattering of photons between the twocavities.Following the procedure of Section 2.3, we expand the atomic field operator ˆ ψ ( r ) inthe momentum basis. The angle of 60 ° between the different modes implies that two-photon scattering processes with a bare energy of (cid:126) ω − = 1 (cid:126) ω r and of (cid:126) ω + = 3 (cid:126) ω r arepossible as depicted in Figure 22(b). The resulting annihilation operators are labeledˆ c j − , ˆ c j + for the momenta involved in photon scattering processes between the pumpfield and cavity j , and ˆ c − , ˆ c for the momenta involved in photon scattering processbetween the two cavities. The annihilation operator for the zero-momentum state ofthe BEC is designated by ˆ c . Neglecting terms that do not involve the macroscopicoccupation of the BEC zero-momentum state ( (cid:104) ˆ c (cid:105) (cid:39) √ N ), the low-energy limit of57he Hamiltonian (57) takes the form,ˆ H LE = − (cid:126) (cid:88) j =1 , δ cj ˆ a † j ˆ a j + (cid:88) j =1 , (cid:88) s = − , + (cid:20) (cid:126) ω s ˆ c † js ˆ c js + (cid:126) η j √ (cid:16) ˆ c † js ˆ c + H . c . (cid:17) (cid:16) ˆ a † j + ˆ a j (cid:17)(cid:21) + (cid:126) √ U U √ (cid:16) ˆ c † − ˆ c + ˆ c † ˆ c + H . c . (cid:17) (cid:16) ˆ a † ˆ a + ˆ a † ˆ a (cid:17) , (58)where the dispersive shifts are incorporated in the detunings δ cj . The two-photon scat-tering rate between the pump and either cavity is proportional to η j = Ω G ,j / ∆ a ,while the two-photon scattering between the two cavities is proportional to √ U U = G , G , / ∆ a . The pump Rabi frequency Ω and the atomic detuning ∆ a are freely tun-able experimental parameters. This allows to go to a regime of large detunings, wherephoton-scattering processes from one cavity to the another cavity become negligible.In this case, the Hamiltonian of the system reduces to the first line of Equation (58).Besides the U (1) superfluid gauge symmetry of the BEC, the system in the regime ofnegligible inter-cavity photon scattering exhibits a continuous U (1) symmetry for thefine-tuned situation of equal couplings ( η = η ) and equal detunings ( δ c = δ c ):The Hamiltonian [i.e., the first line of Equation (58)] remains unchanged under asimultaneous rotation by an arbitrary angle θ , R = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) , (59)in the space of photonic and atomic-momentum field operators, (cid:18) ˆ a ˆ a (cid:19) → R (cid:18) ˆ a ˆ a (cid:19) , (cid:18) ˆ c ± ˆ c ± (cid:19) → R (cid:18) ˆ c ± ˆ c ± (cid:19) . (60)This transformation shifts photon occupations between the two cavities and simul-taneously redistributes the momentum excitations. The according continuous U (1)symmetry ˆ U ( θ ) = e iθ ˆ C has the generatorˆ C = − i (cid:34) ˆ a † ˆ a − ˆ a † ˆ a + (cid:88) s = ± (cid:16) ˆ c † s ˆ c s − ˆ c † s ˆ c s (cid:17)(cid:35) . (61) In the experiment [127], a BEC was symmetrically coupled to the two crossed cavitymodes, and the spatial phase ϕ of the pump field was set to π/
2. An atom-pumpdetuning ∆ a = − π × . cj of thetwo cavities to construct a phase diagram of the system. Three extended phases wereobserved: a normal phase without self-organization and with no coherent field in eithercavity for large enough detunings ∆ cj , and two self-organized phases where either thecavity 1 or the cavity 2 was populated. In addition, self-organization where bothcavities were simultaneous populated was only observed for fine-tuned parameterswhere the effective cavity detunings were identical, δ c = δ c .58 cavity 1 field amplitude b n Transverse pump lattice depth ( ħω rec ) a c ħ k c a v i t y fi e l d a m p l i t u d e p h o t o n n u m b e r
00 10 20 3050100
Figure 23. Observation of supersolid properties across self-organization of a driven BEC in two crossedstanding-wave cavities. (a) For degenerate cavity modes, the mean intra-cavity photon numbers were recordedwhile the transverse pump-lattice depth was increased. Both cavity modes are populated, however, with dif-ferent amplitudes. (b) Absorption image after ballistic expansion of the atomic cloud. The observation of clearmomentum peaks shows the phase coherence of the BEC and is a strong indication of the superfluid character.The observed momentum peaks also indicate a crystallized atomic density in accord with the pump and bothcavity fields. (c) Field amplitudes in both cavities for repeated experimental runs under identical conditionssignal the breaking of a continuous U (1) symmetry. Along with the phase coherence of the BEC in (b), thisis an indication of the supersolidity of the system. The data point highlighted in red corresponds to the traceshown in (a). Figure adapted and reprinted with permission from Ref. [127] published at 2017 by the NaturePublishing Group. An experimental trace of both mean intra-cavity photon numbers as a function oftransverse pump power is shown in Figure 23(a) for the case of degenerate cavities (i.e., δ c = δ c ). Light was detected in both cavities, however, with different amplitudes. Theabsorption image of the atomic cloud taken after ballistic expansion indeed showedmomentum peaks corresponding to simultaneous self-organization in both cavities;see Figure 23(b). The clear observation of this matter-wave interference is a strongindication of the superfluid nature of the BEC.In order to proof that the system across the double-superradiance crystallizationbreaks spontaneously the fine-tuned continuous U (1) symmetry [see Equation (60)]of the system, the experiment was repeated multiple times. Each repetition showed adifferent ratio of the field amplitudes in the two cavities, while all parameters were keptconstant; see Figure 23(c). From this observation one can conclude that the superfluidcrystallizes at a different location in each realization, thereby breaking a continuousspatial symmetry. The observed combination of a broken continuous spatial symmetrywith the superfluid phase coherence of the atomic gas is a sufficient requirement forrealizing a spontaneously formed crystal with dissipation-free flow, i.e. the minimalversion of a supersolid. The presence of an infinitely degenerate ground state has a direct consequence on theexcitation spectrum of a system. In the Landau picture, the free energy of a super-solid resembles a sombrero hat. Accordingly, the fluctuations of the order parameter α = α + iα = | α | exp( iφ ) reveal two different excitations in the supersolid phase: Anamplitude mode (or Higgs or massive Goldstone mode) originating from amplitudefluctuations δ | α | at constant phase φ with a finite excitation energy due to the radialcurvature of the Landau potential, and a phase mode (or Goldstone mode) with van-ishing excitation energy which stems from phase fluctuations δφ at constant amplitude | α | .The excitation spectrum was probed in the two-crossed cavity system described59 E xc i t a t i on f r equen cy ( k H z ) Transverse pump lattice depth ( ħω rec )10 15 20 25 30 35 40 == ⎪α⎪⎪α ⎪, ⎪α ⎪ φ ⎪α⎪ φ ⎪α ⎪, ⎪α ⎪ M ean i n t r a c a v i t y pho t on nu m be r n π /20505 0 π /2 a bc normal phase supersolid phase Figure 24. Excitations of the supersolid state realized in the two crossed cavity setup. (a) Excitation spec-trum recorded using cavity-enhanced Bragg spectroscopy shows a mode softening twoards the critical point atthe normal-to-supersolid phase transition. Above the critical point, two excitation branches appear: the lowerpolariton remains (almost) gapless while the higher polariton hardens. Investigating real-time dynamics after ashort excitation pulse at higher (b) and lower (c) polaritons reveals that these excitation branches can be iden-tified as a gapped amplitude (Higgs) and a gapless phase (Goldstone) mode, respectively. Figure adapted andreprinted with permission from Ref. [187] published in 2017 by the American Association for the Advancementof Science. above using cavity-enhanced Bragg spectroscopy (see Section 3.2.1) measuring the re-sponse of the system to a weak probe field injected into either cavity. Figure 24(a)shows the resulting excitation spectrum as a function of transverse pump-lattice depth.The second order phase transition from the normal phase to the supersolid is accompa-nied by the softening of the lowest polariton that vanishes at the critical point. In thesupersolid phase, two excitation branches emerge. One polariton mode hardens again,similar to the raising excitation branch of the lattice supersolid, cf. Figure 15. Thesecond polariton, however, remains at low energies. The observed non-zero excitationgap of the lower exitation branch in the supersolid phase is likely caused by imper-fect fine-tuning of the degeneracy of the two cavities, and/or by residual cavity-cavityscattering as discussed below. In principle, photon dissipation from the cavities mightalso have a contribution to this non-zero excitation gap, as it mixes the amplitude andphase modes.This separation of excitation frequencies inside the ordered phase suggests an inter-pretation as Goldstone and Higgs modes. The nature of these modes can, however, alsobe directly tested by observing real-time dynamics of the order parameter after excit-ing the system with a strong pulse. Since the supersolid order parameter is composedof the two cavity field amplitudes, α = α + iα , both amplitude and phase responseof the system can be directly reconstructed. Figure 24(b) and (c) show real-time dy-namics after a 1-ms-long excitation pulse quasi-resonant with the amplitude mode andthe phase mode, respectively. While the amplitude (or Higgs) mode captures changesof the strength of the atomic density modulation, the phase (or Goldstone) mode isassociated to changes in the position of the otherwise perfectly rigid crystal. The re-constructed phase and amplitude in the insets clearly confirm the expected characterof the two modes. 60 b c d Figure 25. Mutual enhancement and competition of self-organization in two crossed cavity modes. Experimen-tal phase diagrams as a function of the two cavity detunigns ∆ cj for ∆ a / (2 π ) = −
73 GHz (a), −
400 GHz (b), − − . The system with two crossed cavities was theoretically analyzed in Ref. [188], show-ing that the U (1) symmetry is an approximate symmetry that holds in the limit ofvanishing cavity fields. The atom-mediated scattering of photons between the two cav-ities favors a state with equal cavity populations for degenerate cavities and reducesthe U (1) symmetry to a Z ⊗ Z symmetry. An analysis of the system in terms of aGinzburg-Landau picture was carried out [188] including finite temperature effects andfinite cavity-decay rates. The estimated non-zero effective mass of the Goldstone modedue to higher order scattering processes slightly breaks the U (1) symmetry (which isexactly restored only at the critical point [189]), giving rise to shallow minima in theLandau free-energy landscape. However, the high probability to escape these shallowminima due to the added noise from cavity losses restores the continuous symmetry.The rate of photon scattering processes between the two cavities can be tunedrelative to the rate of photon scattering processes from the pump to either cavityby changing the atom-pump detuning ∆ a ; see Section 4.1.1. The inter-cavity photonscattering processes involve coupling of the zero-momentum BEC to additional highermomentum states. If one cavity is populated with photons, this cavity field can thenact as an additional pump field for the second cavity and influence the critical pointfor self-organization [190]. Such inter-cavity photon scattering processes break thedegeneracy and either favor a superposition of both cavity modes or a symmetrybreaking between them.Figure 25 shows experimental phase diagrams as a function of the two cavity de-tunings ∆ cj for different atomic detunings ∆ a . For small absolute atomic detunings,inter-cavity photon scattering processes are strongly enhanced, and self-organizationin one cavity favors self-organization in the other cavity. Thus, for a wide range ofparameters, the system organizes simultaneously in both cavity modes, giving rise toa broken Z ⊗ Z symmetry; see Figure 25(a). The opposite limit of large absoluteatomic detunings has already been discussed in detail in Section 4.1.2. In this case,the simultaneous superradiance in both cavities is suppressed due to the absence ofphoton scattering between the two cavities and only possible for degenerate cavitymodes [i.e., along the diagonal in Figure 25(d)], resulting in a broken U (1) symmetry.In the limit of very strong inter-cavity photon scattering, it was theoretically dis-cussed that an exotic vestigially ordered phase, located between the normal and the61 b Figure 26. Self-ordering of a transversely-driven BEC in three crossed cavities. (a) A BEC (blue circle) isplaced at the intersection of three single-mode linear cavities, which are all located in the x - z plane and alignedat the same angle of 60 ◦ from each other. An external, circularly polarized laser dispersively drives the atomsfrom the transverse ( y ) direction, resulting in two-photon scattering processes between the pump and the cavitymodes. (b) Mean-field ground-state phase diagram of the system as a function of the effective couplings η ,j to the individual cavities. The orange cube indicates the normal phase with no photon in any cavity. PhasesS with a single superradiant cavity breaking a Z symmetry, phases S with double superradiance breakinga U (1) symmetry, and the phase S with triple superradiance breaking a continuous SO (3) symmetry areindicated. Figure adapted and reprinted with permission from Ref. [192] © superradiant phases, can be entered [191]. In this vestigially ordered phase, the atomiccloud acquires a density modulation, but neither cavity mode is macroscopically pop-ulated. In order to reach this phase, its kinetic energy cost has to be lower than thekinetic energy cost of the density wave from the standard self-ordered phase. This canbe achieved for small angles between the two cavities (or in multi-mode cavities), sincethe atomic kinetic energy cost is given by the momentum difference between the twocavity-mode wave vectors. The double crossed linear-cavity setup can be extended to more cavities with highergeometrical complexity, in order to realize higher continuous symmetries with moreintriguing phase diagrams. By arranging three cavities in a plane symmetrically suchthat they make a 60 ° angle with one another and placing a transversely-driven BECin their common intersection [see Figure 26(a)], one can obtain a much richer phasediagram as shown in Figure 26(b) [192]. In particular, a continuous SO (3) rotationalsymmetry is realized in the low-energy physics for symmetric coupling of the BEC toall cavities [the red dashed line denoted by S in Figure 26(b)]. Here the continuous SO (3) symmetry corresponds to the redistribution of photons among the three cavitiesat a given fixed total intensity. The continuous SO (3) rotational symmetry is spon-taneously broken at the onset of the triple superradiance, where all the cavity modesare populated. Correspondingly, two gapless Goldstone modes appear in the collectiveexcitations of the system. This can be understood based on the fact that SO (3) isa higher dimensional symmetry, related to three different continuous translations ofthe atomic positions along the directions perpendicular to each one of the cavity wavevectors. Therefore, the emergent superradiant potential can be located anywhere inthe plane of the cavities. 62 a) (b)
Another scheme to engineer a supersolid has been proposed based on a BEC withina ring cavity with a pair of degenerate counterpropagating running electromagneticmodes e ± ik c x with the corresponding annihilation operators ˆ a ± [60,111,193]. Thishinges quite intuitively on the ring geometry of the cavity [194], which does not im-pose any hard-wall boundary on the electromagnetic fields, thereby respecting thecontinuous translational symmetry of the space. U (1) symmetry in ring-cavity geometries A one-dimensional BEC is dispersively coupled to two cavity modes ˆ a ± with strengths G ± ( x ) = G e ± ik c x and illuminated coherently by a standing-wave pump laser withthe Rabi frequency Ω in the transverse direction as depicted in Figure 27(a) [193].Following a procedure similar to Section 2.1 [195], in the rotating frame of the pumplaser the system is described by the effective Hamiltonian ˆ H eff = (cid:82) ˆ ψ † ( x ) ˆ H , eff ˆ ψ ( x ) dx − (cid:126) ∆ c (ˆ a † + ˆ a + + ˆ a †− ˆ a − ) , with the effective single-particle atomic Hamiltonian density:ˆ H , eff = − (cid:126) m ∂ ∂x + (cid:126) U (cid:16) ˆ a † + ˆ a + + ˆ a †− ˆ a − + ˆ a † + ˆ a − e − ik c x + ˆ a †− ˆ a + e ik c x (cid:17) + (cid:126) η (cid:16) ˆ a + e ik c x + ˆ a − e − ik c x + H.c. (cid:17) . (62)The photon losses are accounted for by the master equation (8), where the Liouvilleannow includes both cavity modes, L ˆ ρ = κ (cid:80) (cid:96) =+ , − (2ˆ a (cid:96) ˆ ρ ˆ a † (cid:96) − { ˆ a † (cid:96) ˆ a (cid:96) , ˆ ρ } ) . In addition to the continuous U (1) gauge symmetry, the system possesses another U (1) symmetry, as the effective Hamiltonian ˆ H eff and the Liouvillean are both invariantunder a simultaneous spatial translation x → T X x = x + X and cavity-phase rotations63 a ± → U X ˆ a ± = ˆ a ± e ∓ ik c X . This continuous U (1) symmetry is ultimately related tothe ring geometry of the cavity which does not impose a hard wall boundary on theelectromagnetic fields and respects the continuous spatial translational symmetry. Incontrast to the two crossed linear cavities where the external U (1) symmetry—i.e.,the invariance under simultaneous continuous spatial translation and field rotation—is merely an approximate and fine-tuned symmetry as discussed in Section 4.1, theexternal U (1) symmetry in the ring-cavity setup is an exact symmetry, independentof parameter regimes.The external U (1) symmetry is spontaneously broken at the onset of the super-radiant phase transition, where the field amplitudes α ± = (cid:104) ˆ a ± (cid:105) = | α ± | e iφ ± ac-quire non-zero values with equal absolute values | α | = | α + | = | α − | and arbitraryphases φ ± . That is, in the superradiant state the relative phase ∆ φ ≡ ( φ + − φ − ) / π , sponta-neously breaking the the continuous external U (1) symmetry. A spontaneously chosenvalue of ∆ φ fixes the position of the minima of the emergent superradiant potential V SR ( x ) = 2 U | α | cos(2 k c x +2∆ φ )+4 η | α | cos( k c x +∆ φ ) cos(Φ), with Φ ≡ ( φ + + φ − ) / e ± ik c x correlated with the corre-sponding density fluctuations peaked at x emerges, forming a highly entangled atom-field state: | Ψ (cid:105) = (cid:82) λ c dx | ψ x (cid:105) ⊗ (cid:12)(cid:12) αe ik c x (cid:11) ⊗ (cid:12)(cid:12) αe − ik c x (cid:11) [55]. This highly entangled state isvery fragile, susceptible to quantum fluctuations and noises. This state subsequentlycollapses to a state with a certain random relative phase via quantum jumps induced bycavity photon losses, forming the supersolid state: | Ψ (cid:105) = | ψ x (cid:105)⊗ (cid:12)(cid:12) αe ik c x (cid:11) ⊗ (cid:12)(cid:12) αe − ik c x (cid:11) . The collective excitations of the system shown in Figure 27(b) confirm the supersolid-ity of the system [193]. In particular, a gapless Goldstone mode (the solid blue curve)appears beyond the pump strength η s (cf. Figure 9), corresponding to the sponta-neously broken continuous U (1) symmetry. The Goldstone mode corresponds to thecenter-of-mass motion of the entire modulated BEC along the cavity axis dragging thesuperradiant optical lattice V SR ( x ) with itself. The lowest gapped polariton branch(the dashed red curve) instead corresponds to a Higgs amplitude mode. These arereminiscent of the observed Goldstone and Higgs modes in the crossed-cavity experi-ment, cf. Figure 24. Note that the self-ordering in the ring cavity (and the two crossedlinear cavities) is in sharp contrast to the self-organization in a single linear cavity,where only a discrete Z symmetry is spontaneously broken and the first excitationgap closes at the critical pump strength but then re-opens again; cf. Figure 15.The imaginary parts of the polaritons exhibit a peculiar behavior. Above the criticalpump strength η c , all the collective excitations except the gapless Goldstone modeacquire imaginary parts. The imaginary part of the lowest polariton mode as a functionof √ N η /ω r is shown in the inset of Figure 27(b). We note that although this modeis damped for small pump strengths, the imaginary part vanishes at the critical pumpstrength η c (note that the corresponding real part vanishes at the slightly lower pumpstrength η s , where the damping reaches its maximum value; see also Figure 9). This64eans that the center of mass of the entire modulated BEC can move freely along thecavity axis dragging the superradiant optical lattice with itself without experiencingany friction. The fact that supersolidity survives even in presence of dissipation is dueto the fact that the corresponding Lindblad operators also respect the U (1) symmetryof the system. Such a dissipationless supersolid particle flow in a ring cavity—an openquantum system—can be used as a very sensitive tool for a precise, non-destructivemeasurement of various forces including the gravitational force [55]; see Section 11.3for more details. Supersolidity in a longitudinally pumped ring cavity
In a related ring-cavity–BEC geometry, signatures of supersolidity have been experi-mentally observed in a BEC of Rb atoms [181]. This scheme exploited two pairs ofrunning-wave transverse modes (TEM and TEM ), where the modes in each pairare degenerate (i.e., propagating in opposite directions) and do not interfere with themodes in the other pair because of a large frequency offset between the pairs. Twocounterpropagating modes, one mode from each pair, were longitudinally pumped bycoherent laser fields through a cavity mirror; see Figure 28(a). For symmetric andstrong enough pumping, the uniform BEC was self-ordered into a stationary crys-talline state, breaking the continuous spatial symmetry of the system. The crystallinestate maintains the long-range phase coherence of the BEC; hence, it is a supersolidstate. Superfluidity of the atomic crystal was inferred from the observation of distinctmomentum peaks after ballistic expansion. The breaking of the continuous spatialsymmetry was, however, not proved in the experiment.The system also remains in the stable self-ordered phase when the pump fields arenot fully symmetric. The atomic system adapts to a stable momentum distributionsuch that the net force on its center of mass vanishes. This stabilizing mechanismcompensating asymmetric pumping is related to cavity cooling, and makes this systemmore robust. For strongly asymmetric pumping, however, there is no stationary state.In particular, above a certain pump asymmetry the system enters the collective atomicrecoil laser (CARL) regime; see Section 3.3.1. Figure 28(b) shows the phase diagramof the system as a function of the sum of the two pump strengths, S , and the relativepump asymmetry, | A | /S . Three phases are visible: the normal superfluid phase, thestable supersolid phase, and a runaway CARL-like state. Supersolidity in other systems
The supersolids induced by the interaction with dynamical light fields as described inthis section crystallize with a lattice constant that is determined by the wave lengthof the light. As discussed in Section 2.2, the light fields induce an effective interatomicinteraction of global range with a particular spatial structure stemming from the in-terfering light fields. The crystallization is thus also expected to be defect-free andhomogeneous, because all atoms couple equally to the cavity modes. This results in aperfectly rigid crystalline structure, which inhibits the presence of phonons at non-zerowave numbers. Similar arguments also apply for supersolid states induced by spin-orbitcoupling [177].More recently, supersolidity has also been engineered using quantum gases withstrong magntic dipole moments [178–180], exploiting the interplay between trap shapesand collisional and dipolar interactions. In contrast to the light-induced supersolids65 b Figure 28. Supersolid state in a longitudinally pumped ring-cavity–BEC geometry. (a) A BEC placed insidea ring cavity dispersively couples to two pairs of transverse running-wave modes { ˆ a ± , ˆ b ± } , from which thecounterpropagating modes ˆ a + and ˆ b − are pumped at rates η + and η − . (b) Phase diagram of the system as afunction of the sum S of the two pump-field strengths and their normalized asymmetry | A | /S . Shown as colorscale is the kinetic energy transferred to the atomic system within an interaction time of 1 ms. A superfluid,a supersolid, and an instable phase of collective atomic recoil lasing (CARL) are visible. Figure adapted andreprinted with permission from Ref. [181] © in cavities, dipolar supersolids can have phononic excitations. System sizes have sofar, however, been limited to a handful of lattice sites. Light-induced supersolids can,in principle, also show phononic excitations, provided a continuum of electromagneticmodes is available. This is for instance possible in confocal cavities, as discussed inSection 5, and even in free space [196], though experiments have shown that largeatomic clouds are needed to achieve stability.
5. Multimode cavities: supermode polaritons, quantum crystallinephases, and droplets
In this section, we shall consider the case where atoms are coupled to a large number ofmodes of an optical cavity. We will discuss novel phenomena emerging in this case withrespect to the few-mode cases discussed in the preceding sections, and illustrate conse-quences for the superradiant self-organization and corresponding crystalline phases. Inabsence of a multifrequency or a Floquet drive [197–199], coupling to a large number ofcavity modes is achieved by tuning the cavity to a degenerate point, where several tensor hundreds of modes are almost resonant with the driving laser (see Figure 7). In thissituation, the collective polaritonic excitation, whose coherent occupation correspondsto superradiance, is a superposition of a large number of cavity modes. The superpo-sition of many cavity modes generically leads to destructive interference which makesthe scattered light spot more local. As an important consequence, cavity-mediatedatom-atom interactions become finite range, offering a promising route to observeinteraction-induced fluctuations and beyond-mean-field physics. As the largest con-structive interference is obtained by a localized atomic cloud, the cavity-mediatedinteraction is attractive and strongest at short distances. Unless countracted by suf-ficiently strong Van-der-Waals interatomic repulsion, this leads to the formation ofsuperradiant, self-bound crystalline droplets instead of extended crystals.66 .1.
Cavity-mode structure and stable degenerate points
The generalization of the theoretical model of Equation (6) to the multimode case isstraightforward and readsˆ H eff = (cid:90) ˆ ψ † ( r ) (cid:26) − (cid:126) M ∇ + V ext ( r ) + (cid:126) ∆ a (cid:104) | Ω( r ) | + (cid:88) ν,ν (cid:48) G ∗ ν (cid:48) ( r ) G ν ( r )ˆ a † ν (cid:48) ˆ a ν + (cid:88) ν (cid:0) Ω ∗ ( r ) G ν ( r )ˆ a ν + H.c. (cid:1)(cid:105) + g n ( r ) (cid:27) ˆ ψ ( r ) d r − (cid:126) (cid:88) ν ∆ µ ˆ a † ν ˆ a ν , (63)where we have allowed for complex electromagnetic mode functions. The functions G ν ( r ) are defined as in Equation (44). Note that the contact interaction term pro-portional to g is identically zero for fermions due to the Pauli exclusion principle.We assume a single frequency ω p for the pump laser, defining the relative detuningof cavity modes ∆ ν = ω p − ω ν . Here ν = ( l, m, n ) is a multi-index composed of thelongitudinal mode index l and the transverse mode indices m, n of a given TEM mn cavity mode (see Figure 7). In addition, we define a loss rate κ ν for each mode inducingthe Lindblad dynamics [cf. Equation (9)] L ˆ ρ = (cid:88) ν κ ν (cid:16) a ν ˆ ρ ˆ a † ν − ˆ a † ν ˆ a ν ˆ ρ − ˆ ρ ˆ a † ν ˆ a ν (cid:17) . (64)As discussed in Section 2.7.1, in the near-planar case all the modes of a given family[see Figure 7(c)] share the same longitudinal index l as well as the same longitudi-nal standing-wave modulation cos( k c x ). That is, the whole family possesses the samenumber of nodes in the longitudinal direction. The higher-frequency modes within thefamily are then characterized by an increasing transverse index m + n .On the other side of the stability diagram in the concentric case (see Figure 7),modes belonging to the same family instead share the same total number of nodes l + m + n . These degenerate modes thus can have a different number of longitudinalnodes, captured by the index l . Sufficiently away from the cavity center x (cid:29) λ c , allthe modes in a given family still show a longitudinal periodic modulation with wavenumber k c . Indeed, due to the small beam waist w ∼ λ c , the Gouy phase ϕ Gouy mn ( x )[see Equation (38)] quickly saturates to a spatially independent constant away fromthe center. Differently from the near-planar case, however, in the near-concentric casethe Gouy phase is crucial in determining the mode frequency as well as in inducing aphase offset ϕ offset mn which differs for modes with different total transverse index m + n .Requiring that the electric field satisfies the boundary conditions at the mirrors x = ± (cid:96) res / ϕ offset mn . Equation (38) then implies that modeswith different m + n are offset with respect to each other by a phase ϕ offset mn =( m + n ) arctan( λ c (cid:96) res / πw ). Let us consider the oscillating part (i.e., without theGouy phase shift) of the longitudinal modulation in combination with the offset justdiscussed. In the concentric limit, arctan( λ c (cid:96) res / πw ) = π/
2, so that different trans-verse modes come with a periodic longitudinal modulation cos( k c x ) for m + n = 0,sin( k c x ) for m + n = 1, − cos( k c x ) for m + n = 2, . . . . In the confocal case instead,arctan( λ c (cid:96) res / πw ) = π/
4, so that different transverse modes of a family come with amodulation cos( k c x ) for m + n = 0, sin( k c x ) for m + n = 2, − cos( k c x ) for m + n = 4, . . . , and similarly for an odd family m + n = 1 , , , . . . .67pproaching both the plane-parallel and the concentric limit in principle allows toreach full mode degeneracy. The concentric degeneracy is preferable over the plane-parallel one since the latter is characterized by a diverging mode volume, hence avanishing light-matter coupling [see Equation (43)]. We will return to the concentriccase in Section 5.4. However, the extent to which degeneracy can be achieved in bothlimits is affected by resonator stability. Experiments realizing multimode cavity QEDwith ultracold atoms have for this reason so far utilized the confocal degeneracy [61,128,200,201]. As shown in Figure 7, the latter lies indeed in the middle of the stabilitydiagram and can successfully be reached, as we will discuss in the next two sections.Before proceeding, however, let us stress one difference between confocal and con-centric degeneracies which is especially relevant for the purpose of studying many-bodyphysics. In the concentric case, a degenerate family includes modes with all possiblenumbers of nodes, so that the full destructive interference resulting from their su-perposition allows for light fields which can be local over the characteristic distancebetween atoms. On the other hand, in the confocal case, a degenerate family excludesevery second mode, so that the destructive interference is only partial and revivals ofcavity fields at large distances appear (see Section 5.3). Multimode superradiance
A few years after the first observation of steady-state superradiance with a single-mode cavity discussed in Section 3, the Stanford group observed the superradianttransition with Bose-condensed atoms also in the multimode regime, by tuning a cavityaround the confocal point [200]. The main novelty in this respect is the emergence ofcavity supermode polaritons. This mixes the bare cavity modes by the intra-cavityatomic medium. If this system becomes superradiant, it forms a supermode polaritoncondensate.To understand this concept let us revisit the computation of collective polaritonicexcitations of Section 2.6 in the present multimode case described by Equations (63)and (64). The linearized equation of motion for the field amplitude of mode ν reads[cf. Equation (28)], i∂ t δ ˆ a ν = (cid:88) ν (cid:48) [ − (∆ ν + iκ ν ) δ ν,ν (cid:48) + χ stat ,νν (cid:48) ] ( δ ˆ a ν (cid:48) + δ ˆ a † ν (cid:48) ) , (65)with the static polarization function given by χ stat ,νν (cid:48) = (cid:88) (cid:96),(cid:96) (cid:48) n ( (cid:15) (cid:96) ) − n ( (cid:15) (cid:96) (cid:48) ) (cid:15) (cid:96) − (cid:15) (cid:96) (cid:48) (cid:104) u (cid:96) | η ν | u (cid:96) (cid:48) (cid:105)(cid:104) u (cid:96) (cid:48) | η ν (cid:48) | u (cid:96) (cid:105) . (66)Here | u (cid:96) (cid:105) are the single-particle eigenstates of the mean-field atomic Hamiltonian withthe corresponding eigenvalues (cid:15) (cid:96) , and η ν ( r ) = Ω( r ) G ν ( r ) / ∆ a = (cid:104) r | η ν | r (cid:105) . Furthermore,we have for simplicity assumed to be in the non-superradiant phase, α ν, = 0, neglectedthe terms quadratic in the photon operators, and also assumed real laser and cavity-mode functions, { Ω( r ) , G ν ( r ) } ∈ R .The main observation is that the polarization function is in general a non-diagonalmatrix in the cavity-mode space. This means that the presence of the atoms leads toscattering of photons between the cavity modes, thus mixing them to form a supermodepolariton. An additional source of coupling between different modes besides the atomic68 igure 29. Observation of supermode-polariton superradiance with a BEC inside a near-confocal cavity. (a) Ex-perimental configuration with the BEC at the center of the cavity. (b,c) Cavity transmission spectrum showingthe quasi-degenerate mode families, even in (b) and odd in (c), in a near-confocal cavity. (d-e) Superradiantemission above the superradiant-threshold coupling for a laser frequency indicated by the corresponding arrowsin panel (b). A varying admixture from different transverse TEM mn modes is observed by changing the laserfrequency. Figure adapted and reprinted with permission from Ref. [200] published at 2017 by the NaturePublishing Group. cloud is mirror aberrations [200].As discussed in relation to Equations (32) and (34), the polariton excitations areobtained as poles of the Green’s function of the electromagnetic field. Since the po-larization function is not diagonal, the original two-by-two matrix structure resultingfrom the positive and negative frequency components of the Green’s function in Equa-tion (33) acquires now a block structure, where each of the four entries is a matrixin mode space with the non-diagonal part coming from χ stat ,νν (cid:48) . It is clear that eachpolaritonic pole acquires in general contributions from all cavity-mode sectors, i.e.,becomes a supermode polariton.At the superradiant instability point where one of the complex polariton frequenciesvanishes, the admixture of each bare cavity mode to the unstable supermode polaritoncan be read out from the eigenvectors of the inverse electromagnetic Green’s functionat ω = 0. This phenomenology has been experimentally analyzed in Ref. [200] fora BEC trapped inside a near-confocal cavity such that the atomic-cloud’s extensionwas significantly smaller than the mode waist, leading to an almost mode-independentpolarization function as discussed above. The experimental results are summarizedin Figure 29. The resonator length is tuned to a near-confocal point with the corre-sponding mode frequencies visible in the transmission spectrum of Figure 29(b), wherethe even transverse-mode families are still separated by few tens of MHz. Referringto Section 2.7.1, we recall that a given family can be labelled by just m + n , as thelongitudinal index l is fixed in the near-confocal condition by l + ( m + n ) / m + n = 6sub-family, a superradiant emission containing a different admixture of cavity modesis observed.When the atomic cloud is coupled to a single mode of the cavity, the superradiant69hase transition breaks spontaneously the discrete Z symmetry of the system, byfixing the phase of the cavity field relative to the pump laser (see Section 3). Thisis the case since the longitudinal modulation has a single possible phase offset, andso does the corresponding density modulation ∝ cos( k c x ) of the atomic cloud in thesuperradiant phase. As we have seen in Section 5.1, the situation changes at theconfocal degeneracy, where an additional longitudinal modulation ∝ sin( k c x ) is alsopossible. This can effectively lead to a continuous U (1) symmetry [124], as we willdiscuss in the next section. For now, we just note that in the near-confocal case ofFigure 29 the distance from confocality is large enough to allow for only a singlepossible longitudinal modulation and thus realizes only the standard Z symmetrybreaking. Cavity-tuning of the atom-atom interaction in the confocal regime
Following the procedure of Section 2.2 for many cavity modes, the cavity-mediatedinteractions between two atoms in the multimode case can be obtained as, D ( r , r (cid:48) ) = 2 G Ω( r )Ω( r (cid:48) )∆ a (cid:88) m,n ∆ mn E mn ( r ) E mn ( r (cid:48) ) E (∆ mn + κ mn ) ≡ G Ω( r )Ω( r (cid:48) )∆ a | ∆ | ˜ D ( r , r (cid:48) ) , (67)where the last equality defines the dimensionless interaction potential ˜ D solely due tothe cavity field. The Hermite-Gaussian functions E mn ( r ) are given in Equation (38).Having restricted to a single almost-degenerate family, the sum over modes in Equa-tion (67) runs only over the transverse indices { m, n } , since the longitudinal index l for given transverse indices m and n is determined by the resonance condition (39).As discussed already previously, having chosen a confocal degeneracy, only half of thetransverse modes contribute, either with even or odd m + n .The dimensionless interaction potential is shown in Figure 30 for a cavity tunedslightly away from the confocal point, so that a large number of modes contributes.This is measured by the mode-spacing parameter (cid:15) = cδl res /l with δl res = l res − R c ,which enters the mode detuning: ∆ mn = ∆ − ( m + n ) (cid:15) . Following the notation ofRef. [61], we quantify the amount of confocality by the parameter M ∗ = | ∆ | /(cid:15) ,whose square can be loosely associated with the number of modes participating. Inthe numerically computed potential ˜ D ( r , r (cid:48) ) of Figure 30, one can observe all the mainqualitative features of cavity-mediated interactions around a confocal point. In thenext section, an approximate analytical form will be derived [61].We first note that for a laser which is red-detuned from the lowest mode of the fam-ily, the interaction potential has an attractive minimum at zero interatomic distance.This is a generic feature of cavity-mediated interactions in the multimode case, result-ing from the fact that the largest constructive interference in the scattering of laserphotons into the cavity happens always when the atoms are at the same spot. Thisattractive minimum has important consequences for the many-body physics, whichwill be discussed in Section 5.4.The width of the attractive minimum as a function of the interatomic distancealong the transverse direction is roughly given by ξ = w / √ M ∗ , i.e., it is set bythe amount of confocality M ∗ . In the ideal confocal case M ∗ → ∞ , this width goesto zero. However, this does not happen in reality mainly due to the finite number of70 igure 30. Dimensionless interaction potential ˜ D ( r , r ) [defined in Equation (67)] between two atoms at po-sitions r and r mediated by a cavity close to the confocal degeneracy point. (a) The dimensionless interactionpotential as a function of the relative interatomic distance y along the transverse direction on the mid-planeof the cavity, for different values of the center-of-mass position Y along the transverse direction. Even for theatoms off the cavity center, Y (cid:54) = 0, a residual long-range oscillating interaction remains as a function of y .(b) ˜ D ( r , r ) as a function of the relative interatomic distance x along the longitudinal direction. Inset: Thesame quantity shown over a smaller range and for two different values of the center-of-mass position Y alongthe transverse direction. (c) ˜ D ( r , r ) as a function of the center-of-mass position X along the longitudinaldirection, for different values of the center-of-mass position Y along the transverse direction. The distance fromthe confocal point is δl res = l res − R c = 0 . µ m, i.e., (cid:15) = 1 . = −
30 MHz from the lowestmode of an even family. The curvature of both mirrors is R c = 1 cm, with the loss rate κ = 1 MHz. The modewaist is w = 35 µ m and the number of supported transverse modes is such that n + m < modes supported by the cavity, which limits the destructive interference between themodes. In the following, we argue that the length ξ plays the role of the interactionrange. This is, however, not strictly true since in the confocal case only either even orodd transverse modes contribute to a given degenerate family. One consequence of thisis that the interaction potential in general does not fall to zero at large interatomicdistances, as shown in Figure 30(a), which can also be explained as resulting fromthe presence of a mirror image for each atom. However, moving the center of massof the atoms transversally away from the cavity axis, the image-induced backgroundinteraction decays to zero over a length set again by ξ , as shown by the green and redcurves in Figure 30(a). A further consequence of having only half of the modes involvedis the presence of residual oscillations around zero, also shown in Figure 30(a). In thelimit of infinitely many modes the ratio between the amplitude of the oscillationsand the global minimum approaches zero. The oscillations have a period set by thebeam waist w and essentially no decay (the much faster wiggles visible in the figureare instead a spurious effect of using a hard cutoff in the cavity-mode sum instead ofthe more physical smooth cutoff). In the equivalent atom-image picture, these residualoscillations correspond to the interference fringes generated by the atom and its image.As discussed in Ref. [124], this long-range oscillating part can, however, be eliminatedby bringing the neighboring odd family into play using a second driving laser tunedone free spectral range away.Figure 30(b) shows instead the behavior of the interaction potential in the longitu-dinal direction as a function of the interatomic distance. Here the potential oscillateswith a period set by the wavelength λ c . In the confocal case the dependence of theGouy phase ϕ Gouy mn ( x ) on the longitudinal coordinate is slow since w (cid:29) λ c , so thatevery transverse mode has essentially the same longitudinal spatial dependence overseveral periodic oscillations. As discussed in Section 5.1, different transverse modescan still have a different phase offset of the longitudinal oscillations. The consequenceof this on the interaction potential is observable in its dependence on the center-of-mass position along the longitudinal direction, shown in Figure 30(c). The longitudinal71scillations quickly change from a cosinusoidal to a sinusoidal modulation by movingthe center of mass transversally beyond a distance w from the cavity center. Alongthe longitudinal direction the interaction potential does not decay as a function ofinteratomic distance on experimentally relevant length scales [see Figure 30(b)]. The features of the cavity-mediated interaction potential around the confocal degen-eracy point have been explored in a series of recent experiments performed by theStanford group [61,201]. The experimental approach consisted in using movable BECsof a small size compared to the beam waist, and measuring the interaction strengthas a function of position. Based on a theoretical model, which we will discuss it inthe following, the interaction strength as well as the length scales characterizing thespatial form of the interaction have been inferred from the experimental value of thesuperradiant threshold as a function of the BEC position.The theoretical model describing the experiments is discussed in Refs. [61,124]. Itis based on a multimode version of the mean-field approach introduced in Section 2.4.In the particular case of a BEC, the mean-field approximation corresponds to the self-consistent solution of the two coupled equations (21) and (22), which can be directlygeneralized to the present multimode case starting from the Hamiltonian (63). Weare interested in the regime close to the superradiant threshold. We thus consider thelinearized version of the coupled equations about the homogeneous, non-superradiantphase with α mn = 0. The equation of motion for the cavity fields is thus given byEquation (65), specified to the BEC case.In the single-mode self-ordering discussed in Section 3.1, it was sufficient to con-sider the following ansatz for the BEC wavefunction ψ ( r ) = ψ el ( x ) ψ etr ( y, z )[ ψ + √
2Θ cos( k c x ) cos( k c y )], where ψ el , ψ etr are fixed envelope functions describing thedensity modulation along the longitudinal and transverse directions from an exter-nal trap, and Θ is the Z order parameter of the superradiant phase transition[see Equation (15)]. The modulation cos( k c y ) with period λ c = 2 π/k c along thetransverse direction y accounts for the imposed transverse standing-wave pump laserΩ( r ) = Ω cos( k c y ), while the similar modulation cos( k c x ) along the longitudinal di-rection x accounts for the spatial dependence of the cavity mode function in the lon-gitudinal direction for a near-planar cavity. The combination of the two modulationscorresponds to the total chequerboard interaction potential of Equation (13).The analysis of the confocal interaction potential of the previous section implies thefollowing generalization of the superradiant ansatz for the BEC wavefunction: ψ ( r ) = ψ el ( x ) ψ etr ( y, z ) (cid:110) ψ + √ c cos( k c x + γ ) + Θ s sin( k c x + γ )] cos( k c y ) (cid:111) . (68)Apart from the common phase shift γ that has been introduced for later conve-nience [124], the only difference with respect to the ansatz for the single-mode case isthe introduction of a second component Θ s of the order parameter. This accounts forthe fact discussed in the previous section that the longitudinal modulation of modesbelonging to the same confocal degenerate family can be either cosinusoidal or si-nusoidal [see also Figure 30(c)]. These two components of the order parameter are,however, only independent under fine-tuned conditions, which will be discussed later.The experimental characterization focused on the behavior of the interaction poten-tial along the transverse direction r ⊥ = ( y, z ). The linearized equations for α mn andΘ c , s must, therefore, be integrated along the longitudinal direction x . Since a tight72weezer-trap confines the BEC such that both envelopes ψ el , etr are centered aroundthe position R = ( X , Y , Z ) and have characteristic size w BEC ∼ µ m (cid:28) w /λ c ,one can neglect the non-periodic longitudinal dependence of the cavity mode func-tions. At the same time, since w BEC (cid:29) λ c , one can set all oscillating terms to zeroin the integrated equations. The cavity fields α mn can be adiabatically eliminated bysubstituting their steady-state solution [see Equation (23) for the single-mode case]into the equations of motion for the atomic order parameters Θ c , s . This then leads toeffective coupled equations of motion for the atomic order parameters, i∂ t Θ τ = ( µ + 2 ω r ) ψ avgetr Θ τ + g ψ avgetr ψ Θ ∗ τ + G Ω N ∆ a | ∆ |× (cid:90) d r ⊥ d r (cid:48)⊥ (cid:88) τ (cid:48) =c , s ˜ D eff ττ (cid:48) ( r ⊥ , r (cid:48)⊥ ; X ) ψ etr ( r ⊥ ) ψ etr ( r (cid:48)⊥ ) | ψ | (Θ τ (cid:48) + Θ ∗ τ (cid:48) ) , (69)where τ, τ (cid:48) = { c , s } , g is the short-range atom-atom interaction strength, ψ avgetr = (cid:82) d r ⊥ ψ etr ( r ⊥ ), and˜ D eff ττ (cid:48) ( r ⊥ , r (cid:48)⊥ ; X ) = | ∆ | Re (cid:88) m,n Ξ mn ( r ⊥ ; X )Ξ mn ( r (cid:48)⊥ ; X )∆ mn + iκ O τmn ( X ) O τ (cid:48) mn ( X ) , (70)is the dimensionless, effective cavity-mediated interaction potential [cf. Equation (67)].Here the dimensionless transverse Hermite-Gaussian functions are defined asΞ mn ( r ⊥ ; X ) = w w ( X ) H m (cid:18) √ yw ( X ) (cid:19) H n (cid:18) √ zw ( X ) (cid:19) e − r ⊥ /w ( X ) , (71)and the overlap integrals (resulting from the integration along the longitudinal direc-tion) as O cmn ( X ) = cos[( m + n ) φ ( X )] and O smn ( X ) = sin[( m + n ) φ ( X )], with φ ( X ) = π λ c X πw . (72)An analytical expression for the effective cavity-mediated interaction potential canbe written in the following matrix form in the order-parameter space, specified forsimplicity to an even degenerate family at perfect confocality (cid:15) = 0 and at the cavitymidplane X = 0, neglecting losses κ = 0 [124]:sgn(∆ ) ˜ D eff ( r ⊥ , r (cid:48)⊥ ; X ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) = κ = X =0 = I ˜ D effloc ( r ⊥ , r (cid:48)⊥ ) + σ z ˜ D effnonloc ( r ⊥ , r (cid:48)⊥ ) , (73)where I and σ i are the identity and Pauli matrices in the order-parameter space. Thepotential is split into a local part,˜ D effloc ( r ⊥ , r (cid:48)⊥ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = 14 (cid:20) δ (cid:18) r ⊥ − r (cid:48)⊥ w (cid:19) + δ (cid:18) r ⊥ + r (cid:48)⊥ w (cid:19)(cid:21) , (74)73nd a nonlocal part,˜ D effnonloc ( r ⊥ , r (cid:48)⊥ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) = κ = X =0 = 1 π cos (cid:18) r ⊥ · r (cid:48)⊥ w (cid:19) . (75)The former originates from the peak of size ξ visible in the actual interatomic potentialin Figure 30(a). The Dirac-delta form of ˜ D effloc ( r ⊥ , r (cid:48)⊥ ) is a peculiarity of the idealconfocal case where an infinite number of modes contributes to the cavity-inducedinteratomic potential. Slightly away from confocality, one has instead [61],˜ D effloc ( r ⊥ , r (cid:48)⊥ ) = M ∗ π (cid:88) s = ± K (cid:32) r ⊥ − s r (cid:48)⊥ ξ (cid:115) r ⊥ + s r (cid:48)⊥ ) M ∗ w (cid:33) , (76)with K being the modified Bessel function of the second kind. In both Equations (74)and (76), there is a contribution to the local interaction potential from the mirror imagelocated at − r (cid:48)⊥ . The nonlocal term (75) instead oscillates without decay, with a periodset by the beam waist w . This corresponds to the residual long-range oscillations inthe interatomic interaction potential in Figure 30(a).Let us now turn to the actual experimental characterization of the interaction po-tential. As mentioned, this relies on inferring its strength from the value of the superra-diant threshold. For a BEC of small transverse size w BEC compared to the beam waist w , we see from Equation (69) that close to the confocal point the local part of theinteraction potential is much more important than the nonlocal part in determiningthe threshold, which would depend on the self-interaction ˜ D eff ( R ⊥ , R ⊥ ; X ) aroundthe center of the small atomic cloud R = ( X , R ⊥ ) = ( X , Y , Z ). The contributionof the nonlocal part to the self-interaction is indeed of order one, while the contribu-tion from the local part is of order M ∗ (cid:29) c , s can be thus considered approximately decoupled. This allows to obtaina simple relation between the critical value Ω of the laser strength and the strengthof the local part of the effective self-interaction potential [61]:˜ D effloc ( R ⊥ , R ⊥ ) (cid:39) − ω r ∆ a | ∆ | N G Ω . (77)It is evident that the self-interaction has to be negative (i.e., the short-range part ofthe interatomic interaction potential must be attractive) in order for the superradiantthreshold to be reachable. We note at this point the different sign convention betweenthe present review article and Ref. [61].The relation of Equation (77) has been used in the experiment [61] to reconstructthe spatial dependence of the local effective interaction (76). More precisely, due tothe divergence of the Bessel function at zero argument, Equation (77), which assumesthe transverse envelope of the BEC to be infinitely localized ψ etr ∝ δ ( r ⊥ − R ⊥ ), can-not be directly used. The transverse spatial average in Equation (69) needs insteadto be performed over a finite BEC size. The experimental results are presented inFigure 31 for different cavity lengths. As clear from Equation (76), even though theexperimental data of Figure 31(b) only measure the self-interaction as a function ofthe center-of-mass position Y , they already show the relevant scales characterizingthe local interaction potential: the width ξ of the (attractive) peak and a background74 igure 31. Experimental characterization of the cavity-mediated interatomic potential close to and away fromconfocality. (a) Cavity transmission spectrum for five different cavity lengths, from confocal δl res = l res − R c = 0to a regime similar to the one presented in Figure 29, where the even mode families are well separated. Forall the different cavity lengths in (a), panel (b) shows the absolute value of the strength of the effective self-interaction potential for a small-sized BEC located at different positions on the cavity midplane: X = 0,measured using the theoretical model of Equation (77). The change in the width of the peak at the cavitycenter as a function of the degree of confocality is appreciable. The inset shows a zoom of the data close to thecavity center. Figure adapted and reprinted with permission from Ref. [61] © value decaying on a scale much slower than w . Those features of the effective potentialare also found in the microscopic interatomic interaction potential shown in Figure 30.Measurements involving two separate BECs have also been performed, which allowedto access the finite-distance behavior of the local part of the effective interaction po-tential. This first Stanford experiment [61] did focus only on the local part of theinteraction potential. The nonlocal part was the object of a second experiment [201],to which we turn next.While the nonlocal part does not appreciably modify the superradiant threshold, itis crucial in determining the associated density modulation of the BEC. This is alreadyevident from Equation (73), where the nonlocal interaction favors one of the two com-ponents Θ c , s of the order parameter depending on the transverse position. This meansthat, depending on the BEC location the superradiant longitudinal density modula-tion would be either cosinusoidal or sinusoidal. In fact, as discussed in Ref. [124], awayfrom the cavity midplane the nonlocal part of the effective interaction is not a realquantity anymore, which results in a coupling between the two order parameters viaan additional term in Equation (73) proportional to σ x Im ˜ D effnonloc . This implies thatin general the superradiant density modulation along the longitudinal direction is noteither cosinusoidal or sinusoidal, but rather a superposition of the two with a phasedependent on the position of the BEC. The behavior of this relative phase has alsobeen characterized experimentally by means of the cavity output field [201]. The prin-ciple of such measurement can be understood in the adiabatic cavity limit, which weused to derive Equation (69). In this limit, the cavity field components α mn are lockedto the order-parameter components Θ c , s , i.e., the cavity fields are fully determinedby the instantaneous value of the atomic order parameters. The total cavity field inreal space is then given by a sum over the Hermite-Gaussian modes, weighted by thecoefficients α mn . The measured cavity output corresponds to the forward-travelling75 igure 32. Experimental characterization of the superradiant density wave via the cavity output field at theconfocal point. For two different positions of a single BEC in the transverse direction, the light intensity, (a)and (c), and phase, (b) and (d), are shown. Panel (e) shows the theoretical prediction (see text) for the relativephase between the fringes and the bright spot, with circles marking the values used in panels (b) and (d).Figure adapted and reprinted with permission from Ref. [201] © component of the total cavity field passing through one of the mirrors. Choosing theparameters as in Equation (73), it reads [124]: α F ( r ⊥ ) ∝ (cid:90) d r (cid:48)⊥ ψ etr ( r (cid:48)⊥ ) ˜ D effloc ( r ⊥ , r (cid:48)⊥ ) + e − i arg(Θ c + i Θ s ) (cid:90) d r (cid:48)⊥ ψ etr ( r (cid:48)⊥ ) ˜ D effnonloc ( r ⊥ , r (cid:48)⊥ ) , (78)where we have assumed the BEC wavefunction to be real. In the cavity output field,the two terms in Equation (78) can be clearly separated, as shown in Figure 32.In panels (a) and (c), for two different positions of the BEC the light intensity ispresented, showing two bright spots corresponding to the BEC and its mirror image,together with interference fringes. In accordance with the discussion above, the brightspots localized around the BEC and its image originate from the local part of theeffective interaction potential, while the periodic fringes originate from the nonlocalpart. Having separated the two contributions, a measurement of the local phase ofthe output field, as shown in panels (b) and (d), allows then to extract the relativephase between the local and nonlocal parts of the interaction potential, and thus therelative weight of the cosinusoidal and sinusoidal modulations of the superradiant BECdensity. Panel (e) shows the theoretical prediction for the relative phase as a functionof the BEC position, with the two circles marking the values used for the experimentaloutput of panels (b) and (d). The measured sign flip is consistent with the theoreticalprediction.We conclude this section by discussing the symmetries of the effective model ofEquation (69). In the generic case just discussed, where the nonlocal part of theinteraction potential shifts and also couples the two components Θ c , s of the orderparameter, one is left with the usual single-mode scenario where the superradianttransition corresponds to the spontaneous breaking of a discrete Z symmetry [seeEquation (15)]. In the present notation, this symmetry corresponds to the choice ofthe global shift γ = 0 , π in Equation (68), while Θ s , c are not independent so that asuperposition of fixed relative phase is chosen, as shown in Figure 32. However, undercertain conditions the two components of the order parameter can become indepen-76ent and equivalent, so that a continuous U (1) symmetry emerges in the system [124].Assuming perfect confocality and choosing the cavity midplane X = 0, the effectiveinteraction is given by Equation (73). Taking the transverse BEC size w BEC muchsmaller than λ c , one can approximately consider only the self-interaction r ⊥ = r (cid:48)⊥ inEquation (69) for the order-parameter components. By choosing the position of theBEC such that ˜ D effnonloc ( R ⊥ , R ⊥ ) = 0, the equations for Θ c , s are not only decoupledbut also equivalent, so that their relative weight can be chosen arbitrarily, thus a U (1)symmetry emerges. Such special positions exist also out of the cavity midplane, andthe U (1) symmetry can exist approximately also for small deviations from confocalityor finite BEC size [124]. Superradiant quantum crystals versus self-bound crystalline droplets
Having discussed the characteristics of the cavity-mediated interaction potentialaround the confocal degeneracy, let us now explore the consequences for the many-bodyphysics. In particular, we focus on the types of crystalline phases and phase transitionsthat can appear in this case. As already observed in Section 5.3.1, the main featureof the cavity-mediated interaction in or around the confocal point is the presence ofan attractive minimum at zero interatomic distance [see Figure 30(a) and (b)]. Thisfeature is quite generic to laser-driven atoms coupled to multiple cavity modes, as thestrongest constructive light scattering occurs typically when all the atoms are locatedat the same position. For a laser which is red detuned from a degenerate mode family,the resulting potential is attractive. As indicated by Equation (77), red detuning isnecessary to observe superradiance and the accompanying crystallization of the atoms.Moreover, for a blue-detuned laser with respect to the cavity modes, heating of theatomic cloud might become an issue.The consequence of an attractive interaction minimum at zero interatomic distancefor an atomic bosonic cloud is the instability of the latter towards collapse. This col-lapse is counteracted by the finite temperature or the quantum kinetic energy, as wellas by the short-range atomic repulsion, therefore leading to the formation of a stable,self-bound droplet state. Considering that the cavity-mediated interaction genericallyfeatures also an oscillatory part, depending on the period of such oscillations com-pared to the droplet size, the latter may exhibit a density modulation associated withsuperradiance. The density crystallization breaks spontaneously a (typically discrete)translation symmetry of the system, accompanied with a phase-locking of the superra-diant cavity field with respect to the pump laser. Noting Equation (67), we stress thatthe oscillatory part of the interaction might stem from the cavity modes, i.e., from ˜ D ,and/or from the spatial profile of the driving laser Ω( r ).Such superradiant droplets have been predicted to appear for multimode cavi-ties [202,203] and also for a related geometry involving a retroreflected beam pass-ing through the atomic cloud [204]. The generic zero-temperature phase diagram, as afunction of the strength of the atom-cavity coupling G versus the interatomic-repulsionstrength g , is shown in Figure 33. As expected, a strong enough interatomic repulsionis necessary to stabilize an extended crystalline phase against the droplet. Otherwise,the superradiant instability of the homogeneous phase always leads to the formation ofa self-bound, crystalline droplet state through a first-order phase transition [202,203].Taking further into account the static optical lattice created by the pump laser, astrong enough repulsion g suppresses number fluctuations and brings in the Mott-insulator physics, as discussed in Section 7 for extended superradiant states. As shown77 igure 33. Qualitative phase diagram for a BEC of atoms interacting via a cavity-mediated interaction po-tential with an attractive minimum of finite width ξ at zero distance, as is the case in confocal cavities (seeFigure 30). The axes correspond to the strength of the atom-cavity coupling G and the strength of the short-range interatomic repulsion g . The color scale indicates the value of the droplet order parameter or the inverseparticipation ratio (cid:82) d r ρ ( r ) / (cid:2)(cid:82) d r ρ ( r ) (cid:3) . Adapted from Ref. [202]. in Ref. [202], this can lead to different types of quantum liquid-like droplets. In Sec-tion 9, we will see another instance of transition between a superradiant localized stateand a superradiant extended quasicrystalline phase due to the competition betweenthe atom-cavity coupling (i.e., the cavity-mediated attractive atomic interaction) andthe interatomic repulsion in a different geometry involving few cavity modes.The qualitative phase diagram of Figure 33 is generic to systems where the attrac-tive minimum of the cavity-mediated interaction at zero distance has a finite width ξ , as is the case close to a confocal degeneracy (see Figure 30). The attractive min-imum of size ξ has been indeed observed from the superradiant light emission usinga BEC inside a confocal cavity as discussed in the previous section. Due to the tighttweezer trap, the BEC had a characteristic size w BEC ∼ µ m ∼ min ξ ∼ w / ξ (depending on the interatomic repulsion), the cavity-output spatial resolution avail-able in Refs. [61,201] would not have allowed to identify the droplet. However, the useof a shallower trap creating a BEC whose transverse extension is above ξ should allowto distinguish the superradiant droplet from the extended crystalline phase.In the phase diagram of Figure 33, the nature of the transition between the homoge-neous and the crystalline phase has not been specified. The reason is that this dependson the actual range of the interaction and not strictly on the width ξ of the attractiveminimum at zero distance. A clear example is the confocal interaction potential ofFigure 30, where the actual range of the interaction is infinite (compared to availableatom-cloud sizes) while ξ can be small. As already discussed, this is a consequenceof the imperfect destructive interference due to the missing odd or even modes in aconfocal family. In this situation, the crystallization transition is of second-order andof the mean-field type, i.e., the same as in the single-mode case (see Section 3.1.1).78 igure 34. Brazovskii-type superradiant crystallization. (a) Proposed setup involving a quasi-two-dimensionalcloud of bosons trapped on a longitudinal plane around the cavity center. (b) Nodal structure of one of the modesof the concentric cavity (TEM with 5 longitudinal nodes in the picture). (c) Sketch of the free energy landscapeas a function of the superradiant order parameter. The right column corresponds to a fluctuation-induced first-order transition of the Brazovskii-type. Panels (a) and (b) have been adapted and reprinted with permissionfrom Ref. [60] published at 2009 by the Nature Publishing Group, and panel (c) from Ref. [111] © However, as already mentioned above, by keeping the cloud away from the cavity axisand eliminating the long-range oscillating background via an additional laser, a confo-cal cavity can be used to engineer a genuine short-range interaction with a scale givenby ξ , in which case beyond-mean-field effects would become accessible.A further option is to approach the concentric limit. As discussed in Sections 2.7.1and 5.1, a degenerate family in the concentric case possesses both even and odd trans-verse modes, which allows for completely destructive interference at long distances andthus a finite cavity-mediated interaction range. As already mentioned before, the per-fect concentric limit can never be achieved as it corresponds to an unstable point, sothat only a near-concentric regime is accessible. As shown in Figure 7(c), this regimehas the peculiarity that higher-order transverse modes lie lower in frequency, whichmeans that for a red-detuned pump laser the highest-order transverse modes are theclosest ones to resonance and thus mostly coupled with the atoms. Therefore, the resul-tant cavity-mediated interaction potential strongly depends on the mode cutoff, i.e., onthe details of the cavity mirrors beyond the simple curvature and reflectivity. A sharpmode cutoff is certainly not a good approximation. While proper mode-dependentloss rates κ ν in Equation (64) can work well in concentric cavities, where among oth-ers diffraction losses due to finite mirror sizes need to be accurately modeled [205].Therefore, qualitative features of the photon-mediated interaction in near-concentriccavities will depend on the actual cavity employed. While a large portion of the phasediagram should in general feature a droplet phase due to the attractive minimum atzero distance expected also in near-concentric cavities, the properties of the transitionbetween the homogeneous and the extended crystalline phase have to be discussedcase by case.Here we limit ourselves to briefly discuss a particular example investigated theoret-ically in Ref. [60]. Though considering an idealized case, it well illustrates the promise79f multimode cavities for realizing interesting many-body physics of crystallization.The proposed setup is illustrated in Figure 34. The basic idea is that (thermal orquantum) fluctuations, which become important for a short-enough interaction range,can modify the order of the transition as illustrated by the free energy sketched inFigure 34(c). In Refs. [60,111], starting from the microscopic description of the atom-cavity system, an effective free energy for the superradiant order parameter is derived.We will not present the calculation here and limit ourselves to describe the qualita-tive picture instead. Assuming one can approximately realize a radial symmetry onthe atomic plane for a perfectly concentric cavity, the superradiant order parametercorresponds to a density modulation with a wave vector whose modulus is set by thelaser wavelength but its direction is arbitrary chosen in the plane. This plane-wave-like picture relies on an approximate momentum conservation valid in the limit of verylarge clouds with respect to the wavelength, and also on excluding (via a sharp cut-off) modes which have a more radial than longitudinal nodes. Within this description,getting closer to the superradiant threshold the order parameter fluctuations grow.Since they can have an arbitrary direction in the plane, they form a quasi-continuumwhich impacts the critical behavior very strongly, so that the order of the transitionis changed from the second-order mean-field type to a fluctuation-induced weak firstorder. This mechansim is known as the Brazovskii mechanism. Besides relying on aquasi-continuum of order-parameter components with radial symmetry, it also requiresthe free energy not to contain cubic terms, so that at the mean-field level the crys-tallization transition can remain of second order (which is actually not the typicalcase in most materials [206]). In the specific geometry of Figure 34(a) and within theapproximations employed in Ref. [60], one can show that the cubic terms are indeedabsent from the effective free energy for the superradiant order parameter [60,111].We note that the instability towards the self-bound droplets discussed above, whichcorresponds to a strongly first-order transition, is automatically excluded if the cubicterms are absent.
6. Multi-component quantum gases inside cavities: density and spinself-ordering
So far, we have mainly focused on many-body cavity QED systems where the internaldynamics of atoms are not relevant (i.e., they evolve much faster than the externaldegrees of freedom) and are omitted. That is, ultracold atomic gases are approxi-mated as polarizable quantum media. Nonetheless, it is natural to wonder what willhappen if the internal dynamics of the atoms also become important. This resultsin a new degree of freedom—“pseudospin”—which follow the same algebra as a gen-uine “spin”, and allows in turn for studying combined density and spin self-orderingdynamics [45,46,48,107,207–210]. In particular, the exchange of cavity photons bythe atoms in this case can also result in long-range interactions between atomic spinsindependent of the temperature of the atomic cloud, similar to dipolar interactionsbetween polar molecules. This is in contrast to spin exchange interactions stemmingfrom the Hubbard contact interactions in atomic gases trapped in free-space opticallattices, which only come into play at very low temperatures. The cavity-mediated spininteractions can be modified and tuned experimentally, allowing one to study differentmagnetic orders. Therefore, many-body cavity-QED systems provide a promising, ver- We will just use spin instead of pseudospin to refer to internal states and internal dynamics of the atomthroughout this review paper.
Multi-component ultracold atoms coupled to cavities: Fundamentals
The concept of combined, stationary density and spin self-ordering was introduced inRef. [45], where they studied the Dicke superradiance phase transition for a generalizedatomic system with both internal and external quantized degrees of freedom. Thenonequilibrium dynamics of a similar, related setup were also studied in Ref. [207]. Theconsidered setup consists of four-level ultracold atoms with the ground-state manifold {↓ , ↑} and excited states { , } with energies { (cid:126) ω ↓ = 0 , (cid:126) ω ↑ , (cid:126) ω , (cid:126) ω } , tightly confinedinto the x - y plane containing the axis of a standing-wave linear cavity [46,48,76]. Theatoms are pumped in the transverse direction using two sufficiently red-detuned, non-interfering pump lasers and also strongly coupled to an off-resonance empty modeof the cavity. Namely, the first (second) laser with the frequency ω p ( ω p ) drives thetransition ↑↔ ↓↔
2) with the position-dependent Rabi rate Ω ( r ) [Ω ( r )], while thetransitions ↓ ↔ ↑ ↔ ω c and the same coupling strength G ( x ) = G cos( k c x ) as depicted in Figure 35(a). Forinstance, for an atomic species with Zeeman sublevels with corresponding magneticquantum numbers m ↓ = m and m ↑ = m = m ↓ + 1, the polarization of the cavityfield is chosen to be linear along the quantization axis z , while the pump lasers havein-plane polarizations. The scheme constitutes a double Λ configuration with largeatomic detunings ∆ ≡ ( ω p + ω p ) / − ω and ∆ ≡ ω p − ω , where the ground states {↓ , ↑} are the relevant spin states and form a spin-1/2.Adiabatic elimination of the atomic excited states yields an effective Hamiltonianˆ H eff = (cid:82) ˆΨ † ( r ) ˆ H , eff ˆΨ( r ) d r − (cid:126) ∆ c ˆ a † ˆ a , where ∆ c ≡ ( ω p + ω p ) / − ω c , and ˆ a andˆΨ = ( ˆ ψ ↑ , ˆ ψ ↓ ) (cid:62) are the photonic and two-component (spinor) atomic annihilation fieldoperators, respectively. The single-particle atomic Hamiltonian density reads [46],ˆ H , eff = (cid:126) (cid:18) − (cid:126) M ∇ + δ + V ↑ ( r ) + U ↑ ( r )ˆ a † ˆ a η ∗ ( r )ˆ a + η ( r )ˆ a † η ( r )ˆ a † + η ∗ ( r )ˆ a − (cid:126) M ∇ + V ↓ ( r ) + U ↓ ( r )ˆ a † ˆ a (cid:19) , (79)with the classical pump potentials V ↑ ( ↓ ) ( r ) = | Ω ( r ) | / ∆ , the quantum cavity po-tentials U ↑ ( ↓ ) ( r ) = |G ( x ) | / ∆ , and the two-photon Raman-Rabi coupling strength η j ( r ) = G ∗ ( x )Ω j ( r ) / ∆ j with j = 1 ,
2. Here, δ ≡ ω ↑ − ( ω p − ω p ) / δ Zeeman is theeffective frequency difference between the two spin states, where an external magneticfield B ext has been applied to fix the quantization axis (along z ) and tune the effec-tive two-photon detuning via the Zeeman shift δ Zeeman ( B ext ). The two-body contactinteractions are assumed to be small and are omitted. Note that for bosonic atoms byrestricting to an appropriate low-energy manifold including both internal and externalstates, one can map the effective Hamiltonian ˆ H eff , corresponding to Equation (79),into the Dicke Hamiltonian [48,211]. Alternatively, the pump lasers Ω , ( r ) can be aligned in the z direction with left/right circular polarizations,where now the physics in the x - z plane is identical to the one presented in the text for given choices. ( x ) x y z BEC ⌦ ( r )⌦ ( r ) G ( x ) G ( x ) | i | i |"i | ⌦ ( r )⌦ ( r ) R FM domain-wall spin spiralchiral spin spiraldomain-wall X-AFM p N ⌘ /! r | ↵ ss | / p N (a) (b) Figure 35. Scheme and phase diagram for the cavity-induced density and spin self-ordering. (a) Four-levelbosonic atoms are coupled to a single standing-wave mode of a linear cavity and pumped in the transversedirection by two pump lasers. As can be seen from the inset, the atom-photon couplings comprise a doubleΛ scheme, with far off-resonant, red-detuned transitions. (b) The phase diagram of the system in the R - η plane for Re(˜ δ c ) < η c ( R ), the ground state of the system is a spin-polarized ferromagnetic (FM) normal state.Increasing the pump-laser strength beyond the threshold, the system undergoes the Dicke superradiant phasetransition and exhibits an emergent magnetic order: domain-wall X -antiferromagnetic ( X -AFM), domain-wallspin-spiral, or chiral spin-spiral order depending on the competition among different cavity-induced long-rangespin-spin interactions. The structure of the density self-ordering is different in each phase. Figure adapted andreprinted with permission from Ref. [46] © From the steady-state photonic field operator,ˆ a ss = (cid:82) [ η ( r )ˆ s − ( r ) + η ( r )ˆ s + ( r )] d r ∆ c + iκ − (cid:82) [ U ↓ ( r )ˆ n ↓ ( r ) + U ↑ ( r )ˆ n ↑ ( r )] d r , (80)where ˆ n τ ( r ) = ˆ ψ † τ ( r ) ˆ ψ τ ( r ) and ˆ s ( r ) = ˆΨ † ( r ) σσσ ˆΨ( r ) (with σσσ being the vector of Paulimatrices) are the local density and spin operators, respectively, one can see that thecavity field is derived by pump-photon scattering processes involving the atomic spinoperators ˆ s + ( r ) = ˆ s †− ( r ) = ˆ ψ †↑ ( r ) ˆ ψ ↓ ( r ), rather than the atomic density as in the normalself-ordering of single component atoms; cf. for instance Equation (11). That is, thescattering of pump-laser photons into the cavity mode is accompanied by the atomicspin flips ↑↔↓ and corresponding momentum kicks to the atom depending on thepump lasers’ spatial profiles Ω , ( r ). For the cavity-field dynamics evolving on a much faster time scale compared to theinternal (i.e., spin) and external (i.e., center of mass) atomic dynamics, the formeris locked to the latter ones in the sense of the Born-Oppenheimer approximation. Tothis end, the steady-state photonic field operator (80) can be adiabatically eliminatedin close analogy to the single-component atoms described in Section 2.2. This leadsto an effective atom-only Hamiltonian consisting of a (local) spin-independent non-interacting part for the center-of-mass motion, plus a long-range interaction part for82he spin degree of freedom [46]—cf. the penultimate term in the single-componenteffective atom-only Hamiltonian (12),ˆ H spin = (cid:90) (cid:90) (cid:110) (cid:88) β = x,y J β Heis ( r (cid:48) , r )ˆ s β ( r (cid:48) )ˆ s β ( r ) + J z DM ( r (cid:48) , r ) (cid:2) ˆ s x ( r (cid:48) )ˆ s y ( r ) − ˆ s y ( r (cid:48) )ˆ s x ( r ) (cid:3) + J xy cc ( r (cid:48) , r ) (cid:2) ˆ s x ( r (cid:48) )ˆ s y ( r ) + ˆ s y ( r (cid:48) )ˆ s x ( r ) (cid:3) (cid:111) d r d r (cid:48) + (cid:90) B z ( r )ˆ s z ( r ) d r . (81)The first term in the effective spin Hamiltonian ˆ H spin , Equation (81), correspondsto the x and y components of a Heisenberg-type interaction ˆ s ( r (cid:48) ) · ˆ s ( r ). The secondterm corresponds to the z component of a Dzyaloshinskii-Moriya-type (DM) inter-action ˆ s ( r (cid:48) ) × ˆ s ( r ). The third term is a long-range coupling between the x and y components of the spins, hence referred to as the cross-component spin interaction.Finally, the last term serves as a local magnetic bias field, fixing the spin quantizationaxis. Depending on the sign of J β Heis , the Heisenberg interaction favors ferromagnetic(FM) or antiferromagnetic (AFM) ordering. On the other hand, the DM and cross-component interactions favor chiral spin states such as spin spiral and skyrmion. Itis the competition among these cavity-induced spin-spin interactions as well as thelocal magnetic bias field that determines the nature of the emergent magnetic or-der in the superradiant phase. The magnetic field and the spin-coupling coefficientsare position dependent and are related to the cavity mode function and the pump-field spatial profiles as [cf. the density-density interaction strength D ( r (cid:48) , r ) in Equa-tion (13)]: B z ( r ) = (cid:126) δ/ (cid:126) | Ω ( r ) | / − (cid:126) | Ω ( r ) | / , J x/y Heis = Re( c ) ± Re( c ), J z DM = − Im( c ), and J xy cc = − Im( c ) with c ( r (cid:48) , r ) = 2 (cid:126) G ( x (cid:48) ) G ( x ) (cid:34) ˜ δ c Ω ( r (cid:48) )Ω ∗ ( r ) + 1∆ ˜ δ ∗ c Ω ∗ ( r (cid:48) )Ω ( r ) (cid:35) ,c ( r (cid:48) , r ) = 2 (cid:126) G ( x (cid:48) ) G ( x )∆ ∆ (cid:20) δ c Ω ( r (cid:48) )Ω ∗ ( r ) + 1˜ δ ∗ c Ω ∗ ( r (cid:48) )Ω ( r ) (cid:21) . (82)The dispersively-shifted complex cavity detuning ˜ δ c ≡ δ c + iκ = ∆ c − (cid:82) [ U ↓ ( r )ˆ n ↓ ( r ) + U ↑ ( r )ˆ n ↑ ( r )] d r + iκ has been introduced for the shorthand. We note that thecoupling coefficients are periodic and sign changing, reminiscent of the Ruder-man–Kittel–Kasuya–Yosida interaction in some heavy-fermion materials and metallicspin glasses [77].One can implement other spin models by utilizing different atom-photon couplingschemes. For instance, in a single Λ scheme where one leg of the transition is drivenby a pump laser and the second leg is coupled to another pump laser and a linear-cavity mode, in addition to the long-range XXZ
Heisenberg interactions and thecross coupling between the x and z components of the spins, cavity photons alsoinduce long-range density-density and density-spin interactions [210]. Therefore, thiscavity-QED system offers an alternative approach for simulating t - J - V - W -like modelsimplemented via polar molecules in optical lattices [212]. We also note that the rangeof the cavity-mediated spin-spin (and density-spin) interactions can be tuned andbecome finite range by utilizing multimode cavities similar to the tunable density-density interactions in the single-component atoms coupled to multimode cavities asdiscussed in Section 5.3 [77]. 83 .2. Spinor Bose gases
In this section, we focus on bosonic atoms with the effective Hamiltonian (79) [46].The steady state of the system can be obtained for any given laser profiles Ω , ( r ).An instructive and interesting scenario is to consider the spatial profiles Ω , ( y ) =Ω , ( e ± ik c y + Re ∓ ik c y ) / (1 + R ), where 0 (cid:54) R (cid:54) R = 0 and 1 correspond to pure running-wave and purestanding-wave pump lasers, respectively, while 0 < R < , and G are taken to be real, with the bal-anced Raman condition η ≡ G Ω / ∆ = G Ω / ∆ . Once again we note that thepump lasers do not interfere with each other, and the pump lattices V ↑ / ↓ ( r ) do notadd up or cancel one another as they act on different internal atomic spin states. Thetwo extreme limits of the above considered laser profiles corresponding to R = 1 and0 have been experimentally realized and will be discussed in Sections 6.4 and 8.1.3,respectively.The mean-field approach for spinor bosonic atoms is identical to the single-component bosonic atoms discussed in Section 2.4.1, saving that here Ψ( r ) = (cid:104) ˆΨ( r ) (cid:105) =( ψ ↑ , ψ ↓ ) (cid:62) is a two-component condensate wave function. The mean-field phase diagramof the system in the R - η plane is shown in Figure 35(b) for Re(˜ δ c ) <
0, to ensurethe possibility of the superradiance phase transition. Below the pump-laser strengththreshold η c ( R ), the ground state of the system is a spin-polarized FM normal statewith no photon in the cavity. At the threshold η c ( R ), the system undergoes a Dickesuperradiant phase transition into a magnetically ordered state. The nature of theemergent magnetic order depends on the spatial profiles of the lasers through R .For pure standing-wave pumps Ω , ( y ) = Ω , cos( k c y ) corresponding to R = 1,a spin wave is stabilized in the superradiant phase; see Figure 36(b) which showsthe projection of the normalized local spin ˜ s ( r ) ≡ s ( r ) /s n ( r ), with s ( r ) = (cid:104) ˆ s ( r ) (cid:105) and s n ( r ) = (cid:113) s x ( r ) + s y ( r ) + s z ( r ), in the ˜ s x -˜ s z plane as a function of r . Note thatdifferent spin domains are separated by domain-wall lines. In strong pump strengths,the optical potentials V τ ( r ) + U τ ( r ) | α | (with τ = {↓ , ↑} ) and the Raman coupling η ∗ ( r ) α + η ( r ) α ∗ = 2 η Re( α ) cos( k c x ) cos( k c y ) localize the atoms in a λ c / λ c -periodic checkerboard X -AFMspin lattice order in a more conventional sense.In contrast, for pure running-wave lasers Ω , ( y ) = Ω , e ± ik c y corresponding to R = 0, transverse conical, chiral spin-spiral order is favored. The spirals appear in the˜ s x -˜ s y plane as the spin performs a full 2 π rotation in this plane over one wave length λ c along the y direction; see Figure 36(a). For 0 < R <
1, the system exhibits “domain-wall spin-spiral” texture; see Figure 36(c). In this phase, although the projection ofthe spin in the ˜ s x -˜ s y plane still sweeps a full 2 π angle over one wave length λ c alongthe y direction, the y component of the spin is considerably suppressed compared tothe chiral spin-spiral order in the R = 0 case.We note that in the case of two pure running-wave pumps Ω , ( y ) = Ω , e ± ik c y ,a dynamical spin-orbit coupling is induced for the BEC on the superradiant phaseand the emergent spin-spiral magnetic order is indeed intimately connected to thisdynamical spin-orbit coupling; this will be discussed in more detail in Section 8.1.2.By choosing more general pump configurations Ω , ( r ), more exotic spin textures canbe obtained. 84 ˜ s x , ˜ s z ) (˜ s y , ˜ s z ) ˜ s ( r ) x/ c y / c (˜ s x , ˜ s y ) (a) (b)(c) ' ( r ) y / c (˜ s x , ˜ s y ) x/ c (˜ s x , ˜ s z ) x/ c y / c (a-i)(a-ii) (a-iii)(a-iv) Figure 36. Typical cavity-induced chiral spin-spiral, domain-wall X -antiferromagnetic ( X -AFM), and domain-wall spin-spiral textures. (a) Shown is a chiral spin-spiral order (a-i) and its projections in different mutualspin planes (a-ii)-(a-iv). The spin does a full 2 π rotation in the ˜ s x -˜ s y plane along the y direction over one λ c , as can be seen clearly in the projection of ˜ s ( r ) into the ˜ s x -˜ s y spin plane (a-ii). The projection of thenormalized spin texture into the (b) ˜ s x -˜ s z spin plane for the domain-wall X -AFM order and (c) ˜ s x -˜ s y spinplane for the domain-wall spin-spiral texture. The color code in each figure indicates the respective spin angle,e.g., ϕ = tan − (˜ s y / ˜ s x ) in (a-ii). Figure adapted and reprinted with permission from Ref. [46] © The various emergent magnetic orders in the superradiant phase can be describedand accounted for in an elegant way based on cavity-mediated spin-spin interac-tions and the spin Hamiltonian ˆ H spin , Equation (81). For the pure standing-wavepumps Ω , ( y ) = Ω , cos( k c y ) corresponding to R = 1, the coefficients c and c are identical (i.e., c = c ) and real. Therefore, all the long-range spin-spin in-teractions vanish except the x component of the Heisenberg Hamiltonian [see Equa-tion (81) and the paragraph below it]. The spin Hamiltonian ˆ H spin then reduces to a X -type long-range Heisenberg model (cid:82)(cid:82) J x Heis ( r (cid:48) , r )ˆ s x ( r (cid:48) )ˆ s x ( r ) d r d r (cid:48) with the periodicallymodulated coupling strength J x Heis ∝ Re(˜ δ c ) cos( k c x (cid:48) ) cos( k c x ) cos( k c y (cid:48) ) cos( k c y ). Forthe red-detuned dispersively shifted cavity frequency Re(˜ δ c ) <
0, the cavity-inducedHeisenberg interaction, therefore, leads to the checkerboard domain-wall X -AFM spintexture shown in Figure 36(b). On the other hand for the pure running-wave pumpsΩ , ( y ) = Ω , e ± ik c y corresponding to R = 0, the coefficients c and c are both com-plex and different from each other, c (cid:54) = c . As a consequence, all the long-range spin-spin interactions in the effective spin Hamiltonian ˆ H spin are nonzero. They all have thesame Re(˜ δ c ) cos( k c x (cid:48) ) cos( k c x ) position dependence along the x direction, but differentmodulations along the y direction: J x Heis ∝ cos( k c y (cid:48) ) cos( k c y ), J y Heis ∝ sin( k c y (cid:48) ) sin( k c y ), J z DM ∝ − sin k c ( y (cid:48) − y ), and J xy cc ∝ sin k c ( y (cid:48) + y ). The DM and cross-component spininteractions stabilize transverse, conical spin-spiral order, while the magnetic domainsare favored by the Heisenberg interactions as before; see Figure 36(a). However, thesecavity-mediated interactions are compatible with one another and do not lead to afrustration for this chosen geometry of the pump lasers and the cavity. The spirals85ppear solely in the ˜ s x -˜ s y plane as the DM interaction has only the z componentand the cross-component spin interaction only couples the x and y components of thespins. Note also that the λ c / s x -˜ s z and ˜ s y -˜ s z planes along the y axis is accurately accounted for by the y dependence of J x/y Heis givenabove. For 0 < R <
1, all the cavity-mediated spin-spin interactions are again nonzero.However, the coupling coefficients J y Heis and { J z DM , J xy cc } approach zero, respectively,as (1 − R ) and 1 − R when R →
1. Whereas, the coupling coefficient J x Heis is indepen-dent of R and remains the dominant coupling coefficient. This is indeed the reason forthe suppression of the y component of the spin in the domain-wall spin-spiral phasecompared to the spin-spiral state where all the spin-spin interactions are in the sameorder of magnitude. For chosen pump lasers Ω , ( y ) = Ω , ( e ± ik c y + Re ∓ ik c y ) / (1 + R ), the system pos-sesses a Z ⊗ U (1) symmetry, where Z is the parity symmetry ˆ a → ˆ a (cid:48) = − ˆ a and r → r (cid:48) = r + ( λ c / e x , and the continuous U (1) symmetry represents the freedomof the total phase of the two bosonic atomic fields ˆ ψ τ ( r ). The U (1) symmetry as-sociated with the freedom of the relative phase of the two atomic fields is explic-itly broken on the onset of the superradiant phase transition by the cavity-inducedposition-dependent Raman coupling term with the zero quantum average below thesuperradiant threshold (cid:104) η ( r )ˆ a † + η ∗ ( r )ˆ a (cid:105) = 0 and with the always zero spatial average (cid:82) [ η ( r )ˆ a † + η ∗ ( r )ˆ a ] d r = 0; see Equation (79). As a consequence, in the self-orderedphase the relative condensate phase varies in space to minimize the Raman couplingenergy, leading to the spin orders. For instance, for the pure standing-wave pump laserswith R = 1, the Raman energy is 2 (cid:126) η ( α + α ∗ ) (cid:82) √ n ↑ n ↓ cos( k c x ) cos( k c y ) cos(∆ φ ) d r ,where n τ ( r ) = (cid:104) ˆ n τ ( r ) (cid:105) are the condensate densities and ∆ φ ( r ) is the relative phase be-tween the two condensate wave functions. As cos( k c x ) and cos( k c y ) change sign alongthe x and y directions, respectively, the relative phase ∆ φ ( r ) then jumps (discontinu-ously) along both x and y directions to minimize the energy, resulting in domain-wallAFM spin order. On the other hand, for the pure running-wave pump lasers with R = 0, the Raman energy is 2 (cid:126) η ( α + α ∗ ) (cid:82) √ n ↑ n ↓ cos( k c x ) cos( k c y − ∆ φ ) d r . As be-fore the relative phase ∆ φ ( r ) jumps (discontinuously) along the x direction to min-imize the energy. However, to minimize the energy along the y direction, it changessmoothly in a screw-like manner due to the term cos( k c y − ∆ φ ), resulting in the chiralspin-spiral texture along the y direction. Owing to the explicit breaking of the U (1)relative-phase symmetry by the zero quantum- (below the superradiant threshold) andspatial-averaged Raman field, one may envisage the self-organization and the magneticordering in this generalized Dicke model (with both internal and external quantizeddegrees of freedom of the atoms included) as an order-by-disorder process [45], withdisorder stemming from strong fluctuations of the cavity field around the superradiantthreshold. This can be considered reminiscent of the spontaneous ordering in a two-dimensional classical ferromagnetic XY spin model with a uniaxial random magneticfield [213–216]. The relative condensate phase is related to the spin angle, and theposition-dependent Raman field with zero quantum (below the superradiant thresh-old) and spatial average plays the role of the random magnetic field. However, one hasto note that quantum fluctuations here might not necessarily lower the free energy asin the original concept of order-by-disorder processes [217,218].86 .3. Spinful Fermi gases
Closely related schemes to the model discussed in Section 6.1 have been consideredwith ultracold fermionic atoms [107,208,219]. Reference [219] has considered a similardouble-Λ atom-photon-coupling scheme as in the inset of Figure 35(a) with a 2D Fermigas perpendicular to the cavity axis (i.e., in the y - z plane), where the pumps are alsoaligned along the cavity axis. In the mean-field limit, above a certain critical pumpstrength the Fermi gas undergoes a phase transition from a normal state into a super-radiant state with partial polarization. Depending on the effective energy separationbetween the two spin states, the Dicke phase transition is either first order or secondorder. A variant of the scheme discussed in Section 6.1 has been considered for a 1DFermi gas within a ring cavity [208]. In particular, the transitions ↓ ↔ ↑ ↔ G ± ( x ) = G e ± ik c x . Below the Dicke superradiant phase transition,the system is in a normal mixed or polarized Fermi-gas state depending on the effec-tive energy separation between the two spin states. A spin wave is stabilized in thesuperradiant phase, which evolves into an X -AFM lattice order in the strong pumplimit. Due to the ring geometry of the cavity the system possesses a U (1) × Z sym-metry, where the former is spontaneously broken by the emergent density order andthe latter by the X -AFM spin order at the onset of the superradiant phase transition.Furthermore, in moderate pump strengths the self-ordered states exhibit unexpectedpositive momentum pair correlations between fermions with opposite spin. The recent experimental realization of a tunable Fermi gas inside an optical cavity [129]has opened up interesting avenues of research with interacting spinful Fermi gases. Inthe experiments performed at the EPF Lausanne the scattering length governing theinteractions between fermions in different spin states can be tuned using a standardFeshbach resonance. This allows to control the pairing between the atoms within thecavity, interpolating between the BEC limit (small positive scattering length), wheretightly-bound pairs are well described as a condensate of localized bosons, and theBCS limit (small negative scattering length), where weakly-bound pairs are formed.Between these two limits the so-called unitary point is crossed, at which the scatteringlength diverges. The behavior of the cavity-induced self-organization transition acrossthe BEC-BCS crossover has been studied theoretically in Ref. [220]. The main finding,shown in Figure 37, is that self-organization is maximally enhanced in the unitaryregime for low densities, in the BCS regime for moderate densities close to Fermisurface nesting, or in the BEC regime for high densities.
In the scenario just discussed in the preceding section, the pairing is induced by theshort-range atom-atom interactions, while the cavity-mediated interactions favor theself-organization of the atomic cloud into a density and spin pattern. Another interest-ing question is whether and how pairing can be instead induced by the cavity-mediatedinteractions. This question has been addressed in Ref. [107], where Λ-type fermionicatoms are coupled to a far red-detuned transverse pump laser with the Rabi rateΩ( r ) and a standing-wave cavity mode with strength G cos( k c x ). In the bad cavity87 igure 37. Behavior of the density response function χ stat at the cavity wave vector, for interacting fermions inthe BEC-BCS crossover. The latter is parametrized by the inter-species scattering length a s ↑↓ . The images belowthe plot illustrate pictorially how the size of the Cooper pairs changes in the BEC-BCS crossover. As discussedbelow Equation (30), χ stat quantifies the response of the atoms to a cavity photon and thus the tendency ofthe system to self-organize into the superradiant phase. Figure adapted and reprinted with permission fromRef. [220] © limit, cavity photons mediate global-range interaction between the atomic ground spinstates, (cid:90) (cid:90) D ( r , r (cid:48) ) ˆ ψ †↓ ( r ) ˆ ψ ↑ ( r ) ˆ ψ †↑ ( r (cid:48) ) ˆ ψ ↓ ( r (cid:48) ) d r d r (cid:48) , (83)where D ( r , r (cid:48) ) = ( D / Ω )Ω( r )Ω( r (cid:48) ) cos( k c x ) cos( k c x (cid:48) ) with D = (cid:126) ∆ c Ω G / ∆ a | ˜∆ c | ;cf. Equation (13). Restricting to 1D along the cavity axis, i.e., Ω( r ) = Ω , for D < X -AFM spin texture discussed above and shown inFigure 36(b). One the other hand, for D > D > c >
0, superradiance is strongly suppressed. The superfluid paired phase isindeed not of the superradiant type, i.e., coherent macroscopic occupation of the cav-ity mode does not occur in any pump strength. In this case, superradiant and pairingprocesses, therefore, strongly exclude each other, and in particular for D < c < Experimental realizations of spin self-organization
Spin self-organization of atomic systems has been experimentally explored usingschemes similar to the one discussed in Sections 6.1 and 6.2. While first experimentswere based on thermal atoms [78,222], we here discuss a more recent realization using88 bc x x yy yx Figure 38. Observation of the density and spin self-organization of a spinor BEC. (a) Experimental schemeusing two hyperfine states of Rb as the effective spin-1/2 system. These states are coupled via two separatecavity-assisted Raman transitions, where each transition involves the cavity mode and an additional coherentpump field. (b) The Raman scattering processes flip the spin of the atoms while transferring them to a su-perposition of momentum states (bottom). The system undergoes a transition from the normal phase with nospin ordering to a spin-decorated checkerboard superradiant state breaking a Z symmetry. (c) Spin-selectiveabsorption images of the atomic cloud after ballistic expansion in the normal (left) and the superradiant phase(right). The images reveal that the excited momentum states in the superradiant phase are only populatedby spin flipped atoms. Figure adapted and reprinted with permission from Ref. [48] © a BEC [48]. Operating in the quantum degenerate regime implies that experimentstake place at effectively zero temperature, that all atoms couple identically to lightfields as they share the same spatial wave function, and that the self-organization ofthe matter wave can be clearly observed in the appearance of sharp momentum peaks.In the spin-changing superradiant phases as discussed in Section 6.2, the atomic spinsare flipped due to the cavity-assisted Raman scattering processes [see Figure 35(a)].Correspondingly, recoil momenta associated with the Raman processes are impartedto the atoms. As a result, the spinor superradiant phases are spin-decorated patterns(see Figure 36).The experimental scheme used in Ref. [48] is shown in Figure 38(a), which corre-sponds to the R = 1 case discussed in Section 6.2. Two different hyperfine groundstates of Rb are chosen as the effective spin-1/2 system ( |↑(cid:105) , |↓(cid:105) ) and coupled viatwo different Raman processes, each involving a coherent pump field and the TEM cavity mode. The amplitudes of the Raman processes are balanced, such that an ef-fective Dicke model is realized. The standing-wave pump beams illuminate the atomsfrom the side and are arranged such that the resulting pump lattices are in phase atthe location of the atoms. The frequency of the light field scattered into the cavity byeither Raman process is the mean frequency of the two coherent Raman pump fieldsilluminating the atoms. Since the frequencies of the pump fields and the cavity fieldare different, the potentials for the densities are square lattices while the spin potential89as a checkerboard shape [see Equation (79)]. During self-organization, an accordingspin-decorated checkerboard pattern forms.Figure 38(b) shows the underlying scattering processes together with illustrations ofthe normal and possible superradiant phases breaking a Z symmetry. In the organizedphase, coherent Raman scattering creates a superposition of atoms in the initial zero-momentum state with the spin state |↓(cid:105) and of atoms in the recoiled momentumstate with the spin state |↑(cid:105) . Spin-selective absorption imaging of the atomic cloudafter ballistic expansion allows to distinguish both the momentum states and the spinstates. According images in the normal and the superradiant phases are shown inFigure 38(c).The experiment was repeated by utilizing the TEM mode instead of the TEM mode. The node in the field of the TEM mode is equivalent to a π phase shift in theplane perpendicular to the cavity axis. This spatial phase shift of the cavity field leadsto the creation of a spinor domain wall across the superradiant self-organization. Inthe experiment, the phase of the field leaking from the cavity was analyzed spatiallyby interfering it with a local oscillator in a camera, thus realizing a spatially resolvedhomodyne detection. Indeed, a π phase shift across the nodal line was observed in thecavity field. This nodal line acts as a domain wall in the atomic systems. Accordingly,also the momentum distribution of the atomic cloud—derived from absorption imagesafter ballistic expansion—shows a node in the Bragg peaks of the self-organized atomiccloud.
7. Lattice superradiance: generalized extended Hubbard models
So far for single cavity setups, we have considered scenarios where the atoms areinitially free along the cavity axis below the superradiance phase transition. Onlybeyond the superradiant phase transition, photons scattered from the pump laser intothe cavity create an optical lattice along the cavity axis for the atoms. The atoms thenself-order in this emergent potential into a Bragg density grating which maximizes thephoton scattering from the pump into the cavity. We now consider lattice scenarioswhere the atoms are already initially trapped in a strong 2D “external”, static opticallattice below the superradiant phase transition. As before, photons scattered by theatoms from a transverse pump field into the cavity result in cavity-mediated long-rangeinteractions, competing directly with the kinetic energy and the local interactions ofthe strongly correlated atoms [75,84,98,99,104,105,223–237]. Here, for instance, thecavity-mediated long-range interactions can be incommensurate with respect to theexternal static lattice spacing, leading to frustration.For ultracold atoms in a static optical lattice, the competition between the kineticenergy and the local contact interaction leads to the quantum phase transition betweena superfluid (SF) state and a Mott-insulator (MI) phase [14,238]. The Mott-insulatorstate is favored in the strongly interacting regime where the atoms are localized inpotential minima and the coherence between the atoms is lost. Adding the cavity-mediated long-range interactions into this system enriches significantly the physicsresulting, in addition to the superfluid and Mott-insulator phases, in new states in thesuperradiant phase including lattice-supersolid (LSS), charge-density-wave (CDW),and Bose-glass (BG) states. The lattice supersolid is the lattice variant of the su-persolid discussed in Section 4, where here the crystalline order is stabilized througha discrete spatial symmetry breaking (cf. Section 4). The charge-density wave is aninsulating state with a modulated periodic density. The Bose glass is a compressible90tate, containing disjoint superfluid islands in the Mott-insulator background. It is acharacteristic state of disordered Bose-Hubbard models.
One-component extended Bose-Hubbard model
Let us now consider a generic lattice model inside a linear cavity [84,98,99,104,105,225–230,233]. Ultracold bosonic atoms are trapped in a three dimensional external static lattice, generated by three mutually orthogonal standing-wave laser fields notinterfering with each other. The BEC is split into a stack of identical 2D pancakesalong the z direction by the lattice, and we focus our description into one of thesepancakes. In the x - y plane, the atoms experience the external, static square opticallattice V ext ( r ) = V (0)latt (cid:2) cos ( k latt x ) + cos ( k latt y ) (cid:3) , (84)with the lattice spacing λ latt / π/k latt and lattice depth V (0)latt . The polarizationsand frequencies of the lasers generating this lattice potential are chosen such that thephoton scattering from these lasers into the cavity by the atoms are either not possibleor strongly suppressed. As depicted in Figure 39, the atoms are further driven in the y direction by a standing-wave pump laser with wavenumber k c = 2 π/λ c and stronglycoupled to an empty mode of the cavity with the mode function cos( k c x ), similar to thebasic model described in Section 2.1. The initially empty cavity mode is populatedby photons scattered off the atoms from the transverse pump field. This results inan additional, dynamical potential on top of the classical external static lattice. Inthe 2D x - y plane, the system is still described by the Hamiltonian of Equation (6),with V ext ( r ) given by the lattice potential of Equation (84). The external static latticeand the dynamical potentials can be commensurate or incommensurate with eachother, depending on the ratio λ c /λ latt . These two cases will be considered separatelyin Sections 7.1.1.1 and 7.1.1.2.For a very strong external classical lattice V ext ( r ), as a lowest-order approximationone can ignore the modification of V ext ( r ) by the transverse pump and dynamicalpotentials and expand the atomic field operator in the basis of Wannier functions ofthe lowest Bloch band of the classical lattice as ˆ ψ ( r ) = (cid:80) j W j ( r )ˆ b j , where W j ( r ) is thelowest-band Wannier function localized at lattice site r j = j λ latt / j x , j y ) λ latt / { j x , j y } ∈ Z and ˆ b j is the corresponding atomic annihilation operator satisfyingthe canonical bosonic commutation relation. By doing so, the transverse pump and thedynamical potentials are basically treated as perturbations; see Section 7.1.2 for a briefdiscussion regarding beyond perturbative treatments. Therefore, in the lowest-orderapproximation the system is described by the lattice Hamiltonian,ˆ H latt = − J (cid:88) (cid:104) j , j (cid:48) (cid:105) (ˆ b † j ˆ b j (cid:48) + H.c.) + 12 U s (cid:88) j ˆ n j (ˆ n j −
1) + (cid:88) j (cid:16) (cid:126) V B ( y ) j − µ (cid:17) ˆ n j + (cid:126) (cid:88) j (cid:104) U ˆ a † ˆ aB ( x ) j + η (ˆ a † + ˆ a ) A j (cid:105) ˆ n j − (cid:126) ∆ c ˆ a † ˆ a, (85)where J is the hopping amplitude, U s the strength of the short-range contact interac-tion, ˆ n j = ˆ b † j ˆ b j the local particle number operator at site j , µ is the chemical potentialincluded explicitly to fix the number of the particles, and { V , U , η } are defined as91 y z Figure 39. Schematic sketch of a setup for realizing a generic lattice model with cavity-mediated global in-teractions. Bosonic atoms in a 2D pancake in the x - y plane are trapped by a strong external, statice squarelattice generated by two standing-wave fields (indicated in blue) inside a cavity. The cavity mode is populatedvia photon scattering of the pump laser (indicated in red) off the atoms. The cavity photons mediate global,all-to-all interactions among the atoms, described by an extended Bose-Hubbard-type Hamiltonian. Likewise,an extended Hubbard-type Hamiltonian can be realized for fermionic atoms. Figure adapted and reprintedwith permission from Ref. [105] © before (see Section 2.1). Like { J, U s } , the coefficients A j and B ( x/y ) j are related tooverlap integrals of the Wannier functions (i.e., matrix elements of the spatial part ofthe pump and dynamic cavity potentials): A j ≡ (cid:90) | W j ( r ) | cos( k c x ) cos( k c y ) d r ,B ( x ) j ≡ (cid:90) | W j ( r ) | cos ( k c x ) d r ,B ( y ) j ≡ (cid:90) | W j ( r ) | cos ( k c y ) d r . (86)Note that off-site matrix elements of the potentials have been ignored here, as thesecontributions are normally much smaller than on-site matrix elements, Equation (86).The contribution of these higher-order corrections will be discussed briefly in Sec-tion 7.1.2. We note that in the absence of the cavity field [the second line in Equa-tion (85)], the lattice Hamiltonian ˆ H latt reduces to the common Bose-Hubbard Hamil-tonian describing the dynamics of locally interacting bosonic atoms in the static lattice V ext ( r ) with a site-dependent chemical potential µ − (cid:126) V B ( y ) j due to the transversepump-field lattice potential (cid:126) V ( r ) that can have a different wave vector than theoriginal static lattice.We focus on the large cavity decay-rate limit, where the cavity field reaches veryfast its steady state, ˆ a ss = η (cid:80) j A j ˆ n j ∆ c + iκ − U (cid:80) j B ( x ) j ˆ n j . (87)The cavity field acquires a nonzero value only if the “weighted” average density oper-92tor [cf. Equation (15)], ˆΘ ≡ N Λ (cid:88) j A j ˆ n j , (88)with N Λ being the total number of lattice sites, is nonzero. Adiabatically eliminatingthe steady-state photonic field operator ˆ a ss (87) under the assumption that ∆ c , κ (cid:29) U (cid:80) j B ( x ) j ˆ n j and retaining terms up to 1 / ∆ a yields a generalized, extended Bose-Hubbard model,ˆ H eBH = − J (cid:88) (cid:104) j , j (cid:48) (cid:105) (cid:16) ˆ b † j ˆ b j (cid:48) + H.c. (cid:17) + 12 U s (cid:88) j ˆ n j (ˆ n j −
1) + (cid:88) j (cid:16) (cid:126) V B ( y ) j − µ (cid:17) ˆ n j + N Λ U l (cid:16) N Λ (cid:88) j A j ˆ n j (cid:17) , (89)where the last term represents the cavity-mediated global interaction with the strength U l = 2 (cid:126) N Λ ∆ c η / (∆ c + κ ) [225]; cf. Equations (12) and (13). A negative cavity-mediated long-range interaction U l < (cid:104) ˆΘ (cid:105) . The parameter Θ is zero in the superfluid and Mott-insulator stateswith average uniform densities. The onset of nonzero Θ signals a quantum phase transi-tion into other ordered states with modulated densities, driven by the cavity-mediatedinteraction N Λ U l ˆΘ . The nonzero Θ also corresponds to the superradiant phase withnonzero cavity field amplitude α ss = (cid:104) ˆ a ss (cid:105) (cid:54) = 0; see Equation (87). Therefore, Θ = (cid:104) ˆΘ (cid:105) can be considered as an order parameter, the lattice analog of the continuum orderparameter introduced in Equation (15). The nature of the states with Θ (cid:54) = 0 dependscrucially on the ratio λ c /λ latt , as we will see shortly in Sections 7.1.1.1 and 7.1.1.2.The extended Bose-Hubbard Hamiltonian (89) can be diagonalized exactly for smallsystem sizes. Other numerical techniques such as quantum Monte Carlo [228] andmulti-configurational time-dependent Hartree method [234] have also been utilized tofind the ground state of this system. Due to the fully-connected, long-range natureof cavity-mediated interactions, the mean-field approach provides, however, a faithfuldescription of the system in a wide range of parameters in the thermodynamics limitwhich we focus on in the following section (see also Section 2.4). Before discussingthe mean-field approach, we finally also note that such extended Bose-Hubbard mod-els can also be obtained in the deep superradiant phase without an external opticallattice [223,234] or even in longitudinally pumped cavities due to the optomechanicalinteraction [75,239,240]. The extended Bose-Hubbard Hamiltonian, Equation (89), can be studied most readilyusing a mean-field method. This mean-field approach decouples the off-site terms inthe Hamiltonian ˆ H eBH [241]:ˆ b † j ˆ b j (cid:48) + ˆ b † j (cid:48) ˆ b j (cid:39) β j (cid:48) ˆ b † j + β ∗ j (cid:48) ˆ b j + β j ˆ b † j (cid:48) + β ∗ j ˆ b j (cid:48) − (cid:0) β ∗ j β j (cid:48) + β j β ∗ j (cid:48) (cid:1) , ˆΘ (cid:39)
2Θ ˆΘ − Θ , (90)93here β j = (cid:104) ˆ b j (cid:105) is the superfluid order parameter and Θ = (cid:104) ˆΘ (cid:105) is the “Bragg density”order parameter, assumed to be independent of the superfluid order parameter. Thisassumption is used frequently in the literature, but its validity has not yet been checkedthoroughly. We first consider the simplest case corresponding to λ c = λ latt .One then has B = B ( x ) j = B ( y ) j independent of site j and A j = ( − j x + j y A (with A ≡ | A j | ) alternating its sign between (cid:96) ≡ j x + j y ∈ even (e) and odd (o) sites. TheHamiltonian (89) can then be recast as,ˆ H ceBH = − J (cid:88) (cid:104) e,o (cid:105) (cid:16) ˆ b † e ˆ b o + ˆ b † o ˆ b e (cid:17) + 12 U s (cid:88) (cid:96) ∈ e,o ˆ n (cid:96) (ˆ n (cid:96) − − ( µ − (cid:126) V B ) (cid:88) (cid:96) ∈ e,o ˆ n (cid:96) + A N Λ U l (cid:16) (cid:88) e ˆ n e − (cid:88) o ˆ n o (cid:17) , (91)where the total number of atoms N = (cid:104) ˆ N (cid:105) = (cid:80) j (cid:104) ˆ n j (cid:105) has to be determined self-consistently. Let us first heuristically discuss the extended Bose-Hubbard Hamilto-nian (91) and consider in particular the effect of the cavity-mediated long-range in-teractions N Λ U l ˆΘ ∝ U l ( (cid:80) e ˆ n e − (cid:80) o ˆ n o ) . In the absence of the cavity-mediated in-teraction, U l = 0, the Hamiltonian (91) reduces to the common Bose-Hubbard model,where for J/U s > J c ( µ/U s ) /U s with J c ( µ/U s ) being the critical hopping amplitudethe system is in the superfluid state. For J/U s < J c ( µ/U s ) /U s , the system enters intothe Mott-insulator state, where the atoms are localized on lattice sites with an equal,integer number of atoms per site and the quantum coherence between the atoms is lost.Both states respect the symmetry between even and odd sites. However, a negativecavity-mediated long-range interaction U l < Z symmetry of even and odd sites [originatingfrom the independence of the long-range interaction energy ∝ ( (cid:80) e ˆ n e − (cid:80) o ˆ n o ) onthe sign of the population imbalance]. For large enough cavity-mediated interactionstrength, this results in a λ c -periodic checkerboard density structure which acts asa Bragg grating and maximizes the photon scattering from the pump lattice to thecavity field, signaling the superradiant phase transition. The resultant state in thesuperradiant phase with a population imbalance is either a charge-density wave or alattice supersolid, depending on if the quantum coherence between the atoms is lostor not.The phase diagram of the system can be obtained using the decoupling mean-field approach, Equation (90). Application of the mean-field decoupling (90) to theHamiltonian ˆ H ceBH (91) leads to an effective two-site mean-field Hamiltonianˆ H MFceBH (cid:39) (cid:88) (cid:96) ∈ e,o (cid:20) − J ˜ β (cid:96) (cid:16) ˆ b † (cid:96) + ˆ b (cid:96) − β (cid:96) (cid:17) + U s n (cid:96) (ˆ n (cid:96) − − ( µ − (cid:126) V B ) ˆ n (cid:96) (cid:21) + 2 AU l Θ (cid:16) (cid:88) e ˆ n e − (cid:88) o ˆ n o (cid:17) − N Λ U l Θ , (92)where, without loss of generality, we have assumed that the superfluid order parametersare real β e/o ∈ R (as there is no gauge potential), ˜ β e = (cid:80) n.n.o β o = 4 β o (assuming an94 .750.50.2500 1 2 300.250.50 (a) (b) Figure 40. Phase diagram of a cavity-induced commensurate extended Bose-Hubbard model with λ c = λ latt .(a) Phase diagram as function of rescaled parameters µ/U s , zJ/U s , and − A U l /U s with coordination number z . The colorbar refers to zJ/U s = 0 and indicates the average imbalance Θ between the number of atoms onan even and an odd site. The transparent colors show the different phases for zJ/U s >
0: MI (green), CDW(red), and LSS (blue). The SF phase is not indicated, but fills the remaining space. The labels correspond tothe site populations ( n e , n o ). (b) Evolution of the different order parameters as a function of tunneling zJ/U s for filling 1/2, corresponding to a cut through the CDW(1,0) lobe at U l /U s = 0 . zJ/U s , the system evolves from CDW over LSS to SF. Here, all order parameters change continuously acrossthe phase transitions. Figure and caption adapted and reprinted with permission from Ref. [225] © isotropic situation in this 2D model each even site has four identical, nearest-neighborodd sites and vice versa), ˜ β o = 4 β e , and Θ = (cid:104) ˆΘ (cid:105) = A ( (cid:80) e n e − (cid:80) o n o ) /N Λ with n e/o = (cid:104) ˆ n e/o (cid:105) . The Bose-Hubbard model with the cavity-mediated infinite-range inter-action in the mean-field level, represented by the Hamiltonian ˆ H MFceBH in Equation (92),is equivalent to the extended Bose-Hubbard Hamiltonian with the nearest-neighbordensity-density interaction ∝ (cid:80) (cid:104) j , j (cid:48) (cid:105) ˆ n j ˆ n j (cid:48) [225]. The mean-field Hamiltonian ˆ H MFceBH can be diagonalized in a unit cell consisting of an even and an odd lattice site, withthe order parameters β e/o and Θ to be determined self-consistently.All phases of the system can be identified using the order parameters β e/o and Θ.Zero superfluid order parameters β e = β o = 0 indicate an incompressible insulatingstate ∂ ¯ n/∂µ = 0, with ¯ n = (cid:80) j (cid:104) ˆ n j (cid:105) /N Λ = N/N Λ being the average density. Thisincompressible state is a Mott insulator if Θ = 0, otherwise a charge-density-wavestate. On the other hand β e = β o (cid:54) = 0 and Θ = 0 corresponds to a superfluid sate,while β e/o (cid:54) = 0 with β e (cid:54) = β o and Θ (cid:54) = 0 signals the existence of a so-called latticesupersolid. The phase diagram of the system is shown in Figure 40. The system canbe studied in a similar way for other integer ratios of λ c /λ latt . We now consider the case where theratio λ c /λ latt is not a rational number [108,237,242,243]. Therefore, the external clas-sical lattice is not commensurate with the pump and dynamical potentials, resultingin a non-periodic configuration. One may envisage the cavity-mediated interactionas a disorder in this case, since the amplitude of the interaction proportional to A j oscillates at the cavity wavelength λ c , which is incommensurate with respect to theexternal lattice. The phase diagrams of the system in 1D (along the cavity axis x )and 2D are shown in Figures 41(a) and (b), obtained from quantum Monte Carloand mean-field simulations, respectively [108]. In addition to the superfluid and Mott-insulator states, the system exhibits another phase with interesting properties. It com-prises a compressible state ∂ ¯ n/∂µ (cid:54) = 0 with vanishing superfluid density. The atomicdensity in this phase exhibits a checkerboard density-wave pattern with an envelope95 /U s µ / U s (a) (b) µ / U s J/U s Figure 41. Phase diagram of a cavity-induced incommensurate extended Bose-Hubbard model in the µ - J planefor the irrational ratio of λ c /λ latt = 830 / © at the beating wavenumber | k latt − k c | , such that the Bragg density order parame-ter Θ = (cid:80) j A j (cid:104) ˆ n j (cid:105) /N Λ acquires a nonzero value. The envelope arises due to the factthat the system tries to maximize Θ by convoluting the (checkerboard) density as A j oscillates at the cavity wavelength λ c and becomes out of phase with respect to theexternal lattice with the incommensurate wavelength λ latt . Therefore, the atoms canconstructively scatter photons from the pump laser into the cavity. This state has thecharacteristic of a Bose-glass (BG) phase typical in disordered systems.The effect of the cavity-mediated interactions is particularly evident in the 1D case,where the pump term is a constant shift in the chemical potential [cf. Equation (89)]:At small hopping amplitudes the sizes of the Mott lobes reduce and they give way forthe Bose-glass states (compare it with the dashed curves for the phase diagram withoutcavity-mediated interactions U l = 0). By increasing the hopping amplitude, the Boseglass melts and gives way to the superfluid state. Note that in 1D the Bose-glass statecorresponds exactly to the superradiant phase, where the cavity field is populatedmacroscopically. However, in 2D a Bose-glass state can also arise due to the pumppotential which is also incommensurate with respect to the external optical lattice.The color map in Figure 41(b) represents the average superfluid order parameter (cid:80) j β j /N Λ and the dashed line delineates the superradiant phase. In the single-particlecase, the incommensurate cavity potential can lead to the Anderson-type localizationof the atom and mobility edges—energy eigenstates that separate coexisting extendedand localized states—even in the purely optomechanical regime [244–246].In Ref. [247], a closely related scenario to the schemes discussed in this section hasbeen studied: it has been assumed that non-interacting bosonic atoms are alreadytrapped in a 1D static, external quasiperiodic potential—composed of two incommen-surate lattices and described by the single-band Aubry-Andr´e model—along the cavityaxis and pumped by a laser in the transverse direction. If the strength of disorder inthe external Aubry-Andr´e lattice exceeds a critical value, it leads to the Andersonlocalization of the atoms. As a consequence, the atoms scatter pump photons into the96avity mode completely in phase and constructively, driving the Dicke-superradiantinstabilty even in vanishingly small pump strengths. Such superradiant instabilty withvanishing pump strengths has also been predicted for fermionic atoms, but due to acompletely different mechanism—Fermi surface nesting [113]. In deriving the lattice Hamiltonian (85), we have assumed that the external classiclattice V ext ( r ) and its lowest-band Wannier functions are not modified by the trans-verse pump and dynamical potentials. However, this is not strictly true, specially inthe strong pump and the deep superradiant regime. Therefore, one has to find self-consistently the Wannier functions of the lowest Bloch band of the total lattice poten-tial, V ext ( r ) + (cid:126) V ( r ) + (cid:126) U ( r )ˆ a † ˆ a + (cid:126) η ( r )(ˆ a † + ˆ a ). That is, the steady-state cavity fieldˆ a ss , Equation (87), depends on the matrix elements (86) calculated using the Wannierfunctions, which in turn depend on the cavity field through the total lattice potential.Correspondingly, the hopping amplitude J and the strength of the short-range contactinteraction U s must be also determined self-consistently. This is a genuine many-bodycavity-QED nonlinear effect, where the coupled matter-field dynamics depend cruciallyon the atomic density. Such a nonlinear effect is most prominent when the dynamiccavity potential is the sole lattice potential in the system, e.g., by longitudinally pump-ing a cavity [75] where the existence of overlapping, competing Mott-insulator statesas well as bistable behavior has been predicted [239,240].As mentioned already, in deriving the lattice Hamiltonian (85) only the on-sitematrix elements of the spatial part of the pump and dynamic cavity potentials, Equa-tion (86), have been kept. Although much smaller compared to the on-site matrix ele-ments, retaining the off-site matrix elements of the potentials alters the lattice Hamil-tonian (85) crucially. Namely, after adiabatic elimination of the steady-state cavity fieldˆ a ss these off-site contributions modify the bare hopping amplitude J and result also incavity-induced correlated two-particle hoppings proportional to [ (cid:80) (cid:104) j , j (cid:48) (cid:105) (ˆ b † j ˆ b j (cid:48) + H.c.)] in the extended Bose-Hubbard Hamiltonian. As a consequence, phase boundaries be-tween different phases are modified [75,84]. In particular, in the vicinity of phaseboundaries different photon-number states are associated with different atomic states,so that the ground state of the system in the proximity of phase boundaries con-tains atomic states of different nature correlated with different photon numbers dueto photon-number fluctuations. The experimental realiza-tion of the above described commensurate extended Bose-Hubbard model with cavity-mediated global-range interactions (Section 7.1.1.1) requires to create a static buttunable 3D optical lattice inside an optical cavity. The depth of this lattice controlsthe strength of the on-site short-range collisional interactions between the atoms, whilethe cavity is used to induce competing global-range interactions. For a sufficiently deep3D optical lattice, the BEC will—for small enough global-range interactions—undergoa quantum phase transition from a superfluid phase to a Mott-insulating phase featur-ing an integer number of particles with equal occupations of even and odd sites [14].In contrast, cavity-mediated global-range interactions favor superradiant states withan occupation imbalance between even and odd sites. For small enough short-rangeinteractions, the system will undergo a transition from the SF phase to a lattice su-97 igure 42. Experimental setup to realize the commensurate extended Bose-Hubbard model. A BEC is trappedat the center of an optical cavity and loaded into a deep optical lattice along the z axis, which slices the cloudinto 2D layers in the x - y plane. A TEM mode of the cavity is pumped to create an intra-cavity lattice alongthe x direction, which forms together with a free space lattice along the y direction a square optical lattice oftunable depth V (0)latt . The lattice along the y direction simultaneously acts as a transverse pump with latticedepth V = V (0)latt for the orthogonally-polarized TEM cavity mode inducing cavity-mediated global-rangeinteractions. The figure on the right displays the fundamental processes determining the physics within a 2Dlayer: atoms are subject to global interactions with strength ∝ U l , favoring a density modulation, and torepulsive on-site interactions with strength U s , favoring localization of the atomic wave function on individuallattice sites. Finally, tunneling between neighboring lattice sites with amplitude J favors delocalization of theatomic wavefunction. Figure adapted and reprinted with permission from Ref. [34] published at 2016 by theNature Publishing Group. persolid phase with a modulated superfluid state. If both short-range and global-rangeinteractions are strong, the system will enter a superradiant, insulating charge-densitywave state with a periodically modulated integer site occupation.Figure 42 displays the basic experimental setup together with the relevant atomicprocesses. A BEC trapped at the center of an optical cavity by external fields isloaded into a deep optical lattice along the z axis, effectively freezing out tunnelingin this direction. This slices the cloud into 2D pancakes in the x - y plane. These 2Dlayers are further exposed to a classical optical lattice in the x - y plane, formed byone free space lattice along the y direction and one intra-cavity optical standing wavealong the x direction. By a choice of perpendicular polarizations and a large enoughrelative detuning, these two fields do not interfere such that the resulting potentialis a square lattice. This corresponds to the external optical lattice V ext ( r ) introducedin Equation (84) with adjustable lattice depth V (0)latt , allowing to tune the strengthof the short-range interactions as well as the hopping amplitude. At the same time,the standing-wave potential along the y axis acts additionally as a transverse pumplattice with lattice depth V = V (0)latt [cf. Equation (85)] for another cavity mode with thesimilar polarization, inducing the cavity-mediated global interactions. The resonancefrequency of the cavity is chosen with an adjustable detuning ∆ c with respect tothe frequency of this pump field. As a result, the strength of the cavity-mediatedglobal-range interactions ∝ A U l can be tuned by changing ∆ c or by varying V (0)latt .Most importantly, this arrangement allows to change the ratio of short-range andglobal-range interactions which can thus be brought into competition. The physicsof this system is captured in a wide range of parameters by a Hamiltonian of theform of Equation (91). It is, however, important to note that due to the twofold roleof the transverse pump as the external lattice and the pump field simultaneously,the transverse pump is not a perturbative term anymore. This requires to calculatethe Wannier functions, J , U s , and other coefficients consistently using V (0)latt = V . Thefundamental processes underlying the different terms in the Hamiltonian are visualizedin Figure 42.In order to detect different quantum phases reached by tuning the parameters, two98 igure 43. Experimental observables of the cavity-induced extended Bose-Hubbard model as a function ofthe cavity-pump detuning ∆ c and the lattice depth V (0)latt . Panel (b) shows the BEC fraction extracted frombimodal fits to absorption images after ballistic expansion of the atomic cloud [examples are given in panel(a)]. Panel (d) shows the imbalance Θ between even and odd lattice sites, calculated from the measured meanintra-cavity photon number n ph [example traces of n ph are shown in panel (c)]. The white and black data pointsindicate the threshold for transitions between superfluid and insulating phases [panel (b)] and between balancedand imbalanced site occupations [panel (d)], respectively. Figure adapted and reprinted with permission fromRef. [34] published at 2016 by the Nature Publishing Group. observables are necessary. First, the presence or absence of an intra-cavity light field isa measure whether the atomic system has an imbalanced occupation of the even andodd lattice sites or not [see Equations (87) and (88)], allowing to distinguish the super-fluid and Mott-insulating phases from the lattice-supersolid and the charge-density-wave phases. Second, absorption images of the atomic cloud after ballistic expansionare used to probe the coherence of the gas. Observation of narrow momentum peaksand a high condensate fraction indicate the presence of coherence typical for the su-perfluid and lattice-supersolid phases. In contrast, broadened peaks and a vanishingcondensate fraction indicate the loss of coherence and can be associated with the lossof superfluidity and the formation of an insulating phase (i.e., the Mott-insulator andcharge-density-wave phases). Combining both observables—related to β e/o and Θ inthe mean-field description of Section 7.1.1.1—thus allows to distinguish all the possiblephases of the system.This scheme was experimentally implemented in Ref. [34], and Figure 43 displays themeasurement of both observables as a function of the lattice depth V (0)latt and the relativecavity-pump detuning ∆ c . The BEC fraction vanishes for increasing lattice depth V (0)latt and a threshold for the transition from the superfluid to insulating phases can bedefined, as indicated by the white data points; see Figure 43(a) and (b). Figure 43(c)and (d) show respectively the simultaneously measured intra-cavity photon numberand the imbalance Θ = (cid:104) ˆΘ (cid:105) between even and odd sites as defined in Equation (88), butassuming completely localized atoms on even or odd sites. This imbalance is directlycalculated from the mean intra-cavity photon number using Equation (87). Also forthis observable, a threshold can be defined that indicates the transition from a normalphase with balanced occupation of even and odd sites to a superradiant state with animbalanced occupation. 99 SS LSS a b
Figure 44. Phase diagram of the cavity-induced extended Bose-Hubbard model constructed from the thresholdsidentified in Figure 43. All four theoretically expected phases can be identified: superfluid (SF), lattice supersolid(LSS), Mott insulator (MI), and charge-density wave (CDW). Panel (a) shows the phase diagram as a functionof the experimental parameters ∆ c and V (0)latt , while panel (b) shows the same phase diagram as a function ofthe Hamiltonian parameters U s /J and − A U l /J , calculated from the experimental parameters. Figure adaptedand reprinted with permission from Ref. [34] published at 2016 by the Nature Publishing Group. Combing the measurement results a phase diagram can be constructed, as shown inFigure 44. All four expected phases [i.e., superfluid (SF), lattice supersolid (LSS), Mottinsulator (MI), and charge-density wave (CDW)] can be identified. The transition linebetween the superfluid and the lattice supersolid shifts to smaller values of V (0)latt whenapproaching cavity resonance, as expected from self-organization without the addi-tional lattices (see Section 3). The transition line between the lattice supersolid andthe charge-density wave follows also this trend, indicating that the insulating phaseis entered for smaller lattice depths if a density modulated state is favored. This hasbeen attributed to a reduced nearest-neighbor tunneling due to an increasing energyoffset between even and odd lattice sites in the superradiant phases. The transitionbetween the charge-density wave and the Mott insulator phase is especially interest-ing, since the system has to transition between two different incompressible densitypatterns while tunneling is strongly suppressed. An experimental exploration of thisfirst-order phase transition is discussed below in Section 7.1.3.2.The experimental setup also included a shallow harmonic potential trapping theatomic cloud, leading to an inhomogeneous density profile. As a consequence, it isenergetically more costly for the atoms to be away from the trap center and theinsulating phases will form a wedding cake structure, also allowing for the coexistenceof phases. Furthermore, since the global character of the cavity-mediated interactionsleads to a break-down of the local-density approximation, a direct comparison of theexperimental results with theoretically calculated phase diagrams is challenging. Therole of the trapping potential in this system has, however, been theoretically discussedin Refs. [104,227].As already mentioned, even without an external optical lattice important aspectsof the competition between short-range and cavity-induced global-range interactionscan be explored in the deep superradiant phase [223,234]. Indeed, such a setup wasthe first experimental scheme to explore the competition between short-range andglobal-range interactions [248]. Figure 45(a) shows the experimental scheme of theHamburg group where the atomic cloud was sliced into 2D pancakes, however, noadditional lattice was applied along the cavity axis. This scheme relies on the fact that100 b Figure 45. Extended Bose-Hubbard model realized in the emergent checkerboard lattice of the superradiantphase. (a) Schematic representation of the experimental setup. A BEC is magnetically trapped at the centerof an optical cavity (oriented vertically) and sliced into 2D layers by an external lattice. A second lattice, notinterfering with the first one, is applied in the perpendicular direction and also acts simultaneously as a pumpbeam for the cavity mode inducing global-range interactions. Photons leaking from the cavity are recordedwith a photon counter and measure the density modulation of the atomic system. (b) The measured meanintra-cavity photon number n ph is shown as a function of the dispersively shifted pump-cavity detuning δ c and the pump lattice depth V . Combining this measurement with absorption images taken after time of flightexpansion of the atomic cloud allows to identify the different phases of the system. The homogeneous superfluid(HSF) phase undergoes a phase transition with increasing pump lattice depth into a self-organized superfluid(SSF) phase and then finally into a self-organized Mott-insulating (SMI) state. Figure adapted and reprintedwith permission from Ref. [248] © the emergent superradiant checkerboard lattice can become sufficiently deep such thatthe self-organized superfluid (SSF) phase undergoes eventually a transition into a self-organized Mott-insulator (SMI) phase. Since there is no underlying 3D optical lattice,this system can not enter a Mott insulating phase without imbalanced occupationsbetween even and odd sites.The phase diagram can again be constructed by combining measurements of theintra-cavity field with the interpretation of absorption images after ballistic expansionof the atomic cloud. The resultant phases are shown in Figure 45(b) on top of themeasured mean intra-cavity photon number as a function of pump-cavity detuningand pump lattice depth. Besides the homogeneous superfluid phase (HSF), the self-organized superfluid and the self-organized Mott insulator could be identified. The phase transition between theMott-insulator state and the charge-density-wave phase described above is a first orderphase transition with an according parameter region of metastability [see hatchedregion in Figure 44(a)]. Metastability, hysteresis, and a self-induced switching betweenthe Mott insulator and the charge-density wave have been observed experimentallyusing the real-time access typical to cavity-based quantum gas experiments [249].Figure 46(a) shows the experimental sequence for this study. After the system isprepared in the superfluid phase at a detuning ∆ c corresponding to vanishing global-range interactions, the depth V (0)latt of the square lattice is ramped up such that thesystem undergoes a phase transition to the Mott-insulator state. This constitutes thestarting point for the experiments exploring the first-order phase transition betweenthe Mott-insulating and charge-density-wave states. In the next step, the absolutevalue of the detuning is first reduced (ramp I) and then increased again (ramp II)101 attice depth D e t u n i n g ∆ c SFLSS MICDW E n e r g y a b c Figure 46. (a) Schematic phase diagram of the system with a superfluid (SF, gray), a lattice supersolid (LSS,blue), a Mott insulator (MI, orange), and a charge-density wave (CDW, green) phase. The shaded regionbetween the MI and CDW indicates a region of hysteresis between the phases. The black arrow illustrates theexperimental sequence: The atoms are prepared in the SF phase and the 3D optical lattice is ramped up toincrease U s , which brings the system into the MI phase. Subsequently, a detuning ramp toward cavity resonanceis carried out which increases the strength of the global-range interaction. (b) Non-normalized imbalance, asderived from the intra-cavity light field. The imbalance created during ramp I is shown in orange and theimbalance during ramp II is shown in green. Arrows indicate the ramp directions. (c) Mean-field results fromthe toy model assuming unit filling. In the presence of short- and global-range interactions, atoms placed ina lattice potential can show metastable behavior. States (indicated by circles) can be protected by an energybarrier and the present state of the system depends on its history, leading to hysteresis. The Mott insulator(orange line) and the charge-density wave (green lines) are stable (solid), metastable (dashed), or unstable.Figure and caption adapted and reprinted with permission from Ref. [249] published in 2018 by the UnitedStates National Academy of Science. over time, which brings the global-range interactions ∝ U l into competition with theshort-range interactions ∝ U s and back, driving the system from the Mott insulatorto the charge-density-wave state and back. The light field leaking out of the cavity isa measure for the spatial ordering of the atoms, captured by the imbalance Θ. WhileΘ ≈ H ceBH (91) in deep in-sulating phases, the system can be characterized by the competition of short-range andglobal-range interactions captured by the Hamiltonian ˆ H insul = ( U s / (cid:80) (cid:96) ∈ e,o ˆ n (cid:96) (ˆ n (cid:96) − A U l /N Λ ) ( (cid:80) e ˆ n e − (cid:80) o ˆ n o ) . Figure 46(c) shows the average ground-state energyper particle as a function of the imbalance Θ and the ratio of the interactions U l /U s .In the limit of small global-range interactions, a global energy minimum at zero imbal-ance stabilizes the system in the Mott-insulator phase. With increasing global-rangeinteractions, Z -symmetric local minima at finite imbalance emerge. By further in-creasing global-range interactions, the Mott-insulator state becomes a local minimumin the energy landscape, separated from the charge-density-wave states by an energybarrier resulting from competing short- and global-range interactions.Not captured by this simple model is the observed jump in the imbalance [bluesymbol in Figure 46(b)], where the atomic system is strongly re-organized via tunnelingbetween neighboring sites in a dynamic process. Time-resolved studies of this regimeindicate that the process is a collective tunneling of several thousand atoms within102he expected tunneling time of a single double well. The microscopic description ofthe observed dynamics relies on the harmonic trapping potential experienced by theatoms. The atomic system accordingly forms a wedding-cake structure with a Mott-insuolator core and a superfluid surface. Particles in the surface possess a high mobilityand can locally organize into a density-modulated state. Local self-organization leadsto a global energy offset between even and odd sites. Depending on the position inthe trap, this offset can bring the tunneling between neighboring sites into resonance,which leads to an avalanche dynamics and very fast collective tunneling. Two-component extended Bose-Hubbard model
As discussed in Section 6, when the internal states of atoms become relevant [250],the exchange of cavity photons by the atoms can result in global-range spin-spin andspin-density interactions among the atoms in addition to the density-density interac-tions [46]. When the atoms are trapped in an external lattice, the cavity-induced long-range spin interactions can then compete directly with contact-interaction-inducedshort-range spin interactions in the deep insulating phases, opening the possibility forrealizing intriguing lattice spin models including topological spin orders and frustratedspin states.Cavity-induced lattice spin models can be obtained by imposing an external opticallattice potential in the spinor models discussed in Section 6. The simplest model ofthese kinds is the lattice version of a transversely-driven spinor BEC with standing-wave pump lasers Ω , ( y ) = Ω , cos( k c y ) inside a single-mode linear cavity. Namely,a state-independent external, static optical lattice potential V ext ( r ), Equation (84), isadded to the effective two-component Hamiltonian ˆ H , eff , Equation (79). Followingexactly the same procedure as discussed in Section 7.1, for a strong external opticallattice and after adiabatic elimination of the cavity field one obtains up to 1 / ∆ a atwo-component extended Bose-Hubbard model,ˆ H = − J (cid:88) τ, (cid:104) j , j (cid:48) (cid:105) (cid:16) ˆ b † τ j ˆ b τ j (cid:48) + H.c. (cid:17) + 12 (cid:88) τ, j U s,τ ˆ n τ j (ˆ n τ j −
1) + U s, ↑↓ (cid:88) j ˆ n ↑ j ˆ n ↓ j + (cid:88) τ, j (cid:16) (cid:126) V B ( y ) j − µ (cid:17) ˆ n τ j + N Λ U l (cid:16) N Λ (cid:88) j A j ˆ s x, j (cid:17) , (93)where ˆ b τ j is the bosonic annihilation operator destroying a boson at spin state τ andsite j , and satisfies the bosonic commutation relation [ˆ b τ j , ˆ b † τ (cid:48) j (cid:48) ] = δ τ,τ (cid:48) δ j , j (cid:48) . Here we haveassumed the balanced Raman condition as well as V ↑ ( r ) = V ↓ ( r ) = V cos ( k c y ). Thelast term in the Hamiltonian ˆ H is the cavity-mediated global spin-spin interaction.It is indeed nothing but the lattice version of the continuum spin Hamiltonian ˆ H spin ,Equation (81), in a special case where only the x component of the Heisenberg termis present (recall that for standing-wave pump lasers other spin-spin interactions areabsent; see Section 6.2.1).For the case of the commensurate lattice with λ c = λ latt , the global spin-spin inter-action in Equation (93) reduces to the imbalance of the x component of the spin oneven and odd sites, ∝ U l ( (cid:80) e ˆ s x,e − (cid:80) o ˆ s x,o ) . Therefore, for large enough (negative)cavity-mediated interaction strength U l , a ground state with spin imbalance betweeneven and odd sites (cid:80) e (cid:104) ˆ s x,e (cid:105) (cid:54) = (cid:80) o (cid:104) ˆ s x,o (cid:105) is favored. The ground state in this regimeis either a spin-density wave or a lattice supersolid depending the superfluid order103arameter is zero or finite [251].More interesting lattice spin models can be obtained by considering different pump-laser configurations as discussed for the continuum models in Section 6. Therefore, avariety of emergent magnetic orders can be explored in the framework of lattice cavityQED with multi-component quantum gases; see also the next section for the fermionicvariants. Extended Fermi-Hubbard models
Similar to the bosonic atoms discussed in the previous section, ultracold fermionicatoms can also be loaded into the external lattice (84) inside the cavity, yield-ing extended Fermi-Hubbard-like Hamiltonians with global, all-to-all atomic inter-actions [252]. The only difference with respect to the extended Bose-Hubbard models,Equations (89) and (93), is that the only nonzero onsite atomic interaction would bethe inter-species interaction U s, ↑↓ (cid:80) j ˆ n ↑ j ˆ n ↓ j for two-component fermionic atoms dueto the Pauli exclusion principle. The effect of the cavity-mediated global, all-to-all commensurate interactions ∝ U l [ (cid:80) (cid:96) ( − (cid:96) ˆ n (cid:96) ] on the many-body localization of spin-polarized fermionic atomsin a 1D external, static optical lattice with nearest-neighbor atomic interactions U n.n. (cid:80) (cid:96) ˆ n (cid:96) ˆ n (cid:96) +1 and random onsite disorder (cid:80) (cid:96) (cid:15) (cid:96) ˆ n (cid:96) , with (cid:15) (cid:96) being uncorrelated ran-dom variables uniformly distributed in [ − W, W ], has been explored via an exact diag-onalization for a small system size. It has been found that the many-body localizationpersists in the presence of the cavity-mediated global interactions, but occurs at largerdisorder values [253]; see Figure 47(a). Similar results have also been predicted for one-component bosonic atoms with on-site contact and random interactions [253].
The steady-state phase diagram of spin-1/2 (i.e., two-component) fermionic atoms sub-ject to the cavity-mediated global, commensurate spin interaction ∝ U l [ (cid:80) (cid:96) ( − (cid:96) ˆ s x,(cid:96) ] in a 1D lattice at half filling has been mapped out using a numerical density-matrix-renormalization-group method [254]; see Figure 47(b). The phase diagram exhibits arich structure containing anisotropic magnetic orders: FM (AFM) indicates a statewhere both ˆ s x and ˆ s z exhibit ferromagnetic (antiferromagnetic) correlations, whileFM z -IAFM x (AFM z -IAFM x ) is a state with a ferromagnetic (antiferromagnetic) cor-relation for ˆ s z and an incommensurate—with respect to the static optical lattice—antiferromagnetic correlation for ˆ s x . In the absence of the cavity-mediated global in-teraction, U l = 0, a repulsive onsite interaction U s, ↑↓ > U l < U l > U l ∝ ∆ c > α = (cid:104) ˆ a (cid:105) ≈
0. Nonetheless, cavity-field fluctuations (cid:104) δ ˆ a † δ ˆ a (cid:105) / (cid:104) ˆ a † ˆ a (cid:105) is considerably large(in the order of one) in this regime, signaling that the FM, FM z -IAFM x , and AFM z -IAFM x magnetic orders are induced by quantum fluctuations of the photonic field.104 a) (b)
Figure 47. Steady-state phase diagram of cavity-induced commensurate, 1D extended Fermi-Hubbard modelfor spin-polarized (a) and spin-1/2 (b) fermions. (a) In addition to the cavity-mediated global interactions,fermions experience nearest-neighbor interactions with strength U n.n. and random onsite disorder character-ized by W . While the disorder drives the system into the many-body localized phase (the blue region charac-terized by the mean gap ratio shown as the background color map), the global interactions favor eigenvector-thermalization-hypothesis phase (the yellow region). The simulation has been done at half-filling for a lattice ofsixteen sites. The solid yellow line indicates the phase boundary between the two phases, obtained via finite-sizescaling analysis. (b) The cavity-mediated global spin interaction with strength U l induces different magneticorders: ferromagnetic (FM), antiferromagnetic (AFM), and incommensurate antiferromagnetic (IAFM) orders.The subscripts in the figure indicate the direction of the magnetic order in the internal spin space. The magneticorders in the repulsive cavity-mediated interaction regime, U l ∝ ∆ c >
0, are induced by quantum fluctuationsof the photonic field. Panel (a) is adapted and reprinted with permission from Ref. [253] published in 2019 bythe SciPost and panel (b) from Ref. [254] © In Section 6.3.3 we described a scenario for superfluid pairing mediated via cavityfluctuations in the blue-detuned regime, ∆ c >
0. Here we consider a lattice scenario forcavity-induced pairing in the red-detuned regime ∆ c < D <
0. In particular, two ring cavities oriented alongaxes of a 2D external optical lattice (84) mediate global interactionsˆ H int = D (cid:88) k , k (cid:48) , k c (cid:88) τ,τ (cid:48) ˆ f † τ, k ± k c ˆ f τ, k ˆ f † τ (cid:48) , k (cid:48) ∓ k c ˆ f τ (cid:48) , k (cid:48) , (94)among spin-1/2 fermionic atoms in the external optical lattice, where k c = k c ˆ e i with i = x, y . In Ref. [252], such cavity-mediated interactions have been considered in aregime where | k c | ∼ k latt , which is characterized by the strong competition betweena cavity-induced superfluid paired phase and other types of non-paired, charge- orspin-density phases. On the other hand, in the regime where | k c | (cid:28) k latt considered inRef. [255], the superfluid pairing instability dominates. Furthermore, different types ofpairing compete with one another, and can give rise to interesting exotic topologicalsuperfluid states. This phenomenon is a consequence of an accidental degeneracy ofcompeting paired phases, shown in Figure 48, due to the global range of the cavity-mediated interaction, leaving the system in a frustrated state. The perturbative ad-dition of local interactions splits this degeneracy and creates topological p + id wavesuperfluid-paired state featuring Majorana fermions. The resulting phase diagram isshown in Figure 48(e).These results highlight the promise of cavities for inducing novel mechanisms forpairing as well as stabilizing more exotic superconducting states. This direction ofinvestigation is currently being received a lot of attention also in the context of solid-105 latexit sha1_base64="y9B2VgfKMmxikfqBgsIW3yPUKDk=">AAACA3icbVDNSgMxGMzWv1r/Vr3pJVgED1J3RdFj0Yt4quC2hXZZsmnahmaTJclaylLw4qt48aCIV1/Cm29jut2Dtg4Ehplv+PJNGDOqtON8W4WFxaXlleJqaW19Y3PL3t6pK5FITDwsmJDNECnCKCeeppqRZiwJikJGGuHgeuI3HohUVPB7PYqJH6Eep12KkTZSYO95QaqO20mMpBTDdkcMecbGJ7eBXXYqTgY4T9yclEGOWmB/mTxOIsI1ZkipluvE2k+R1BQzMi61E0VihAeoR1qGchQR5afZDWN4aJQO7AppHtcwU38nUhQpNYpCMxkh3Vez3kT8z2slunvpp5THiSYcTxd1Ewa1gJNCYIdKgjUbGYKwpOavEPeRRFib2kqmBHf25HlSP6245xXn7qxcvcrrKIJ9cACOgAsuQBXcgBrwAAaP4Bm8gjfryXqx3q2P6WjByjO74A+szx+a/Jgi U s, " /J
8. Cavity-induced synthetic gauge potentials and topological states
Gauge potentials and quantum gauge theories play central roles in our understandingof Nature. The simplest example is electromagnetism with its Abelian vector and scalerpotentials describing the coupling between charged matter and electromagnetic fields.Quantum electrodynamics (QED), a relativistic gauge theory describing the couplingof a relativistic charged matter field to the Abelian electromagnetic gauge potentials,is among the most successful and most accurate theories of physics. In the StandardModel of elementary particles, gauge fields are the mediators of fundamental interac-tions among the elementary particles. Gauge potentials are also of great significancein condensed-matter physics. In fact, they are the essential ingredients of topologicalstates of matter [260,261]. In addition to their fundamental significance, topologicalinsulators are also of great importance from a technological point of view: they possessconducting edge or surface states which are topologically protected and robust againstexternal perturbations. These topological edge states can have exotic properties. Inthe case of quantum Hall insulators, such edge states carry precisely one quantum ofconductance. They can also host anyonic excitations, quasiparticles that do not obeyBose or Fermi statistics.Since ultracold atoms are charge neutral, they do not couple to gauge potentialsthe same way charged particles (such as electrons) do. However, a variety of gaugepotentials and minimal couplings can be engineered via the light-matter interactionfor neutral atoms, to mimic phenomena encountered not only in condensed-matterphysics but also in high-energy physics [8,9,262–265]. The induction of such synthetic106auge potentials generally relies on geometric phases acquired through adiabatic mo-tion of quantum particles with internal structures. That is, the light-matter interactioncouples parametrically the atomic internal dynamics to its external degree of freedom(i.e., position), resulting in the appearance of a geometric vector potential under spe-cial conditions. The laser-induced gauge potentials are normally static , in the sensethat the back-action of the quantum matter on these gauge fields is negligible. Theyare solely background gauge potentials, unaffected by the matter dynamics (thoughthey can still have an externally imposed time dependence) and described by extrafixed terms in the Hamiltonian of the system. For instance, the synthetic electric andmagnetic fields induced by the light-matter interaction for quantum gases need notto obey the Maxwell’s equations. However, gauge potentials encountered in quantumgauge theories such as QED, quantum chromodynamics, etc., and strongly correlatedmany-body condensed-matter systems are normally dynamic . That is, the dynamicsof these gauge potentials are governed by their own Hamiltonians and affected bythe back-action of the matter. Motivated by these dynamical gauge potentials, someproposals have been put forward to simulate dynamical gauge potentials for quan-tum gases [266]. A simplest interacting gauge potential toward full dynamical gaugepotentials is perhaps a density-dependent gauge potential [267,268].An alternative, natural route to implement dynamic gauge potentials for neutralatoms is the framework of cavity QED. This hinges on the non-linear, dynamicalnature of cavity fields which can induce gauge potentials. Unlike lasers interacting withatoms in free space, cavity fields are dynamical and affected by the atomic back-action.Therefore, cavity-induced gauge potentials inherit the dynamical nature of the cavityfields and respond naturally to the atomic back-action. Furthermore, as radiation fieldsdecay out of a cavity, quantum-gas–cavity systems with induced gauge potentials arenaturally out of equilibrium. This leads to nonequilibrium topological phases in theframework of cavity QED, a scenario beyond common condensed-matter and free-space quantum-gas systems. Guided by this nonlinear coupled dynamics of matter andradiation in cavity QED, proposals have been put forward to induce dynamical gaugefields [46,269–280] and topological phases [40,281,282] for quantum gases in cavities,with the experimental realization of a dynamical spin-orbit coupling [49]. Anotherpeculiarity of cavity-induced gauge potentials and topological states, in addition totheir nonequilibrium dynamical nature, is the appearance of them on the onset ofthe superradiance phase transition. Since the threshold of the superradiant quantumphase transition depends crucially on atomic properties, the onset of a cavity-inducedgauge potential and topological phase is also sensitive to atomic structures and isself-consistent.Here we will review some of the cavity-based schemes for inducing gauge potentialsfor quantum gases. We start with the cavity-QED implementation of a synthetic mag-netic field corresponding to a position-dependent vector potential, the simplest gaugepotential with the U (1) gauge symmetry. Our focus shall be mainly on lattice modelswhere the synthetic magnetic field is induced via a cavity-assisted Peierls phase. Wethen study rich physics arising from this dynamical synthetic magnetic field includingtopological effects, chiral currents, fractal energy bands, and Meissner-like states. Wethen consider cavity-assisted two-photon Raman schemes to engineer dynamical spin-orbit coupling for neutral atoms, where the interplay between the dynamical spin-orbitcoupling, contact interactions, and cavity-mediated long-range interactions results inrich physics. Finally, we discuss nonequilibrium topological phases induced in quantumgases via the cavity field dynamics. 107 .1. Cavity-induced synthetic gauge potentials
Following Ref. [274], consider spin-polarized fermionic atoms trapped in a strong ex-ternal classical 2D lattice in the x - y plane with the superlattice structure along the x direction. The optical lattice is located inside an initially empty linear cavity andcreated using standing-wave lasers with wavelength λ ( y )latt along the y direction, andwith wavelengths λ ( x )latt and 2 λ ( x )latt along the x direction (which is the cavity axis). Thephase difference of the two lasers along the x direction is chosen such that to create po-tential offsets ± (cid:126) ∆ latt between adjacent sites of the lattice, resulting in an imbalancedsuperlattice structure along the cavity (i.e., x ) axis. Therefore, the external lattice isseparated into a set of decoupled ladders as depicted schematically in Figure 49(a).The atoms can tunnel along the legs of the ladders (i.e., the y direction) due to theirkinetic energy, while the tunneling along the rungs (i.e., the x direction) is stronglysuppressed due to the potential offset (cid:126) ∆ latt . The atoms are dispersively pumped inthe transverse direction by two running-wave pump lasers with space-dependent Rabicouplings Ω , ( y ) = Ω , e ± ik c y and frequencies ω p ,p such that ω p − ω p = 2∆ latt .The atoms are strongly coupled to a standing-wave mode of the cavity with the barefrequency ω c such that two-photon Raman transitions involving the pump lasers andthe cavity mode are close to resonant, ω p − ω c ≈ ω c − ω p ≈ ∆ latt ; see Figure 49(b).The double Λ scheme ensures that the system is not optically pumped into one leg dueto the cavity photon losses. The cavity mode is initially in the vacuum state; however,it can be populated due to the constructive, collective Raman scattering from thepump lasers by the atoms. This in turn restores the hopping along the rungs (i.e., thecavity axis along x ), imprinting a phase into the wavefunction of the atoms due to therunning-wave pump lasers.In the tight-binding limit, one ladder of the system is described in the rotatingframe of the lasers by the Hamiltonian,ˆ H latt = − J (cid:107) (cid:88) (cid:96) =1 , (cid:88) m (cid:16) ˆ f † (cid:96),m +1 ˆ f (cid:96),m + H.c. (cid:17) + (cid:126) η (ˆ a † + ˆ a ) (cid:88) m (cid:16) Ae − imϕ ˆ f † ,m ˆ f ,m + H.c. (cid:17) − (cid:126) δ c ˆ a † ˆ a, (95)where ˆ f (cid:96),m is the fermionic operator annihilating a particle at leg (cid:96) and rung m , J (cid:107) the kinetic-energy hopping amplitude along the legs, δ c = ( ω p + ω p ) / − ω c − U (cid:80) (cid:96),m B ( x ) (cid:96),m ˆ n (cid:96),m the dispersively shifted cavity detuning with respect to the pumpswith U = G / ∆ a being the dispersive shift per atom [where ∆ a = ( ω p + ω p ) / − ω a is the atomic detuning with respect to the pumps], and η = Ω G / ∆ a = Ω G / ∆ a the balanced two-photon Raman coupling. The coefficient B ( x ) (cid:96),m is the same as the onedefined in Equation (86), while the coefficient A has now the form A = (cid:90) W ∗ ,m ( r ) cos( k c x ) exp (cid:20) − ik c (cid:18) y − mλ ( y )latt (cid:19)(cid:21) W ,m ( r ) d r . (96)The phase ϕ = k c λ ( y )latt / πλ ( y )latt /λ c arises due to the photon-assisted tunneling.In particular, the hopping amplitude ∝ η (ˆ a † + ˆ a ) along the rungs is an operatordepending on the cavity field operator ˆ a , which in turn depends on atomic properties108
4, from left to right. The solid red curves mark the onset of the superradiant phasetransition, above which a synthetic magnetic field appears for the fermionic atoms. As a consequence of thelatter, a chiral current is induced in the system. Depending on the filling n , the atomic ground state correspondsto either a chiral insulator (CI) or a chiral liquid (CL). Figure adapted and reprinted with permission fromRefs. [274,275] © highlighting the highly nonlinear nature of the system. In the absence of the cavityphoton, the system consists of decoupled 1D legs along the y direction. The nonzerocavity field restores the hopping along the rungs, where the atoms collect the totalphase ϕ while traversing a plaquette counterclockwise: − mϕ +( m +1) ϕ = ϕ . This phasesimulates an electromagnetic vector potential through the Peierls substitution, wherethe vector potential corresponds to a synthetic transverse magnetic field perpendicularto the x - y plane. The phase and correspondingly the synthetic magnetic field can betuned by changing the ratio between the wavelengths λ ( y )latt /λ c of the external latticealong the y axis and the cavity.The steady state of the system can be obtained in the thermodynamic limit using amean-field approach; see also Section 2.4.2. To this end, the cavity field operator ˆ a isreplaced with its steady-state expectation value α = η (cid:104) A ˆ K ⊥ + A ∗ ˆ K †⊥ (cid:105) / ( δ c + iκ ), whereˆ K ⊥ = (cid:80) m e − imϕ ˆ f † ,m ˆ f ,m is the directed atomic tunneling in all rungs. Substitutingthe steady-state field amplitude α in Equation (95) results in an effective atom-onlyHamiltonian for a single ladderˆ H ladd = − J (cid:107) (cid:88) (cid:96) =1 , (cid:88) m (cid:16) ˆ f † (cid:96),m +1 ˆ f (cid:96),m + H.c. (cid:17) − (cid:88) m (cid:16) J ⊥ e − imϕ ˆ f † ,m ˆ f ,m + H.c. (cid:17) , (97)with the self-consistent hopping amplitude J ⊥ = − U l A (cid:104) A ˆ K ⊥ + A ∗ ˆ K †⊥ (cid:105) / , (98)along the cavity axis, where U l = 4 (cid:126) η δ c / ( δ c + κ ) [cf. definition of U l in Equation (89)].The self-consistency condition (98) can be solved, for instance, graphically and has a109ontrivial solution only for δ c <
0. Furthermore, the existence of a nontrivial solutiondepends also crucially on the system’s parameters including the filling n = N/ L with N being the total number of the atoms and L the size of the system (i.e., thenumber of the rungs), the flux ϕ , and the pump strength η ( ∝ √ U l ), which in turnthey determine the energy bands and the Fermi surface (i.e., Fermi points). Therefore,for a given parameter set including a fixed η , the transverse hopping J ⊥ is variedcontinuously and correspondingly (cid:104) A ˆ K ⊥ + A ∗ ˆ K †⊥ (cid:105) is calculated. The system has anontrivial self-consistent solution if the left- and right-hand sides of Equation (98)intersect in a nonzero value.Recall that a nontrivial solution of the self-consistency equation implies a nonzerocavity field amplitude α = − J ⊥ ( δ c + iκ ) / (cid:126) η Aδ c . Thanks to the restored hopping J ⊥ along the rungs of the ladder for a nontrivial solution of Equation (98), fermionicatoms become subject to a synthetic magnetic field which induces a chiral current j c = (cid:80) m (cid:104) ˆ j ,m − ˆ j ,m (cid:105) / ( L −
1) with ˆ j (cid:96),m = − iJ (cid:107) ( ˆ f † (cid:96),m +1 ˆ f (cid:96),m − H.c.) being the localcurrent operator along the leg (cid:96) .Figures 49(c) and (d) show, respectively, the real part of the cavity-field amplitude α and the chiral current j c as a function of the flux ϕ and − U l A L/J (cid:107) ∝ η for threerepresentative filling n = 1 / , / , / η c ( ϕ ), indication of the appearanceof the cavity-induced synthetic magnetic field beyond the nonlinear pump-strengththreshold η c ( ϕ ). This is in sharp contrast to the free-space implementation of thisscheme (in square lattices) [283,284]. For a given filling n , however, the pump-strengththreshold η c is strongly suppressed at the critical flux ϕ c = 2 πn (and 2 π − ϕ c ) andthe hopping is restored even for an infinitesimal pump strength. This behavior isintimately related to the band structure and the Fermi points. For critical fluxes ϕ c ,the cavity field amplitude increases slowly above the zero pump strength. On theother hand, for ϕ < ϕ c ≤ π and ϕ > π − ϕ c ( ϕ c < ϕ < π − ϕ c ) the cavity-fieldamplitude acquires a nonzero value only beyond a nonzero pump-strength threshold η c , exhibiting a first (second) order phase transition into the self-ordered superradiantphase. Depending on the filling n , the atomic ground state corresponds to a bandinsulator or a liquid, carrying a chiral edge current [indicated respectively by CI andCL in Figures 49(c) and (d)]. In some parameter regimes, there exists a second solution.This second solution vanishes at hight pump strengths, signaling the unstable nature ofit. Furthermore, the exact diagonalization of the corresponding master equation for asmall-size system reproduces solely the first stable solution (above the pump-strengththreshold η c ). The field amplitude and the chiral current exhibit similar behavior asfunctions of the filling n and U l A L/J (cid:107) ∝ η for representative fluxes. In particular, thecavity field and the chiral current appear again only above a pump-strength threshold η c ( n ), where the threshold approaches zero at the critical filling n = ϕ c / π .This cavity-based scheme to implement a synthetic magnetic field in a ladder can bestraightforwardly generalized into a square/rectangle optical lattice [280,282], corre-sponding to the quantum Hall effect for electrons. This allows one to study the Harper-Hofstadter model with fractal energy bands—known as the Hofstadter’s butterfly—arose from a dynamical synthetic magnetic field. Indeed, the cavity-induced syntheticmagnetic field in a finite-size system leads to topological edge states [282]. In an infi-nite system, the dynamic nature of the magnetic field induces a nontrivial deformationof the Hofstadter butterfly in the superradiant phase due to the delicate interplay be-tween the collective superradiant scattering inducing the synthetic magnetic field andthe emerging underlying fractal band structure [280]. This is in sharp contrast to the110ree-space implementation of this scheme as first proposed in Ref. [285], where theenergy bands show the conventional Hofstadter-butterfly structure.In the above discussed schemes [274,275,280,282], although the cavity-induced syn-thetic magnetic field emerges dynamically owing to the backaction of the atoms, thespatial profile of the synthetic magnetic field itself is fixed by the cavity wave number k c . The latter can also be made dynamical as proposed in Refs. [276,277], provid-ing a local dynamical coupling between the matter and the gauge potential as in theGinzburg-Landau theory. A variant of the scheme discussed in detail above was consid-ered in Ref. [276], where an external 1D lattice with an energy offset (cid:126) ∆ latt betweenadjacent sites is located perpendicular (i.e., along the y direction) to the axis of asingle-mode linear cavity. The suppressed hopping is restored by a cavity-assisted “di-rectional” hopping containing a Peierls phase, which is proportional to the phase ofthe cavity field and is dynamic and must be determined self-consistently. In an infinitelattice under certain initial conditions, a dynamical vector gauge potential appears inthe system, inducing a net, steady-state directed atomic current. In contrast, for afinite system the atomic current shows a transient pulse behavior due to a superra-diant pulse (similar to those observed by illuminating degenerate quantum gases infree space) and a nonequilibrium gauge potential. In a different approach, coupling aBEC to many nearly degenerate transverse modes of a near-confocal cavity in a par-ticular configuration induces a synthetic magnetic field B x , which its spatial profile,proportional to the spatial derivative of the transverse intensity profile of the cavitylight field, can evolve freely in response to the atomic state [277]. In particular, thecavity-induced synthetic magnetic field is expelled from the bulk of the condensatedue to the diamagnetic response of the system, reminiscent of the Meissner effect insuperconductors; see Figure 50(d). Correspondingly, vortices in the BEC density alsodisappear as shown in Figure 50(b). As the name suggest, the spin-orbit (SO) coupling refers to the coupling between thespin and the (linear or angular) momentum of a particle and it is a relativistic effect.A charged particle moving with velocity v in an electric field E experiences a magneticfield B = E × v /c in its own rest frame according to the Lorentz transformation upto the first order in ( v/c ) (with c being the speed of the light). Consequently, themagnetic moment ˆ µ of the particle proportional to its charge q and its spin ˆ s couplesto this magnetic field ˆ H SO = − ˆ µ · B ∝ q ˆ s · ( E × ˆ p ), resulting in the SO coupling.Depending on the spatial form of the electric field, different forms of SO coupling canarise.Such a coupling between the spin and the external degree of freedom is absent foran atom, as atoms are charge neutral. The SO coupling can be simulated for neutralatoms via different methods such as dressing the atoms by laser lights, where in thesimplest case two atomic (ground) pseudospin states are needed to encode a spin-1/2 degree of freedom. As for the simulation of synthetic dynamical magnetic fieldsfor the atoms using cavity fields described in Section 8.1.1, dynamical SO couplingcan also be engineered for neutral atoms based on cavity fields. We will review herea few concrete schemes based on cavity-assisted two-photon Raman processes. TheRaman technique using (non-copropagating) plane-wave lasers has so far been theonly method to experimentally engineer an equal contribution of the Rashba andDresselhaus dynamical SO couplings for quantum gases inside cavities [49]—it was alsothe first method to implement static SO coupling for quantum gases in free space [286,111 latexit sha1_base64="rru0s2CMRHPiQ91S4wYdK/fIBN8=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU01E0WPRi8cKxhbaUCbbTbt0swm7GyGU/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmKPNpIhLVDlEzwSXzDTeCtVPFMA4Fa4Wj26nfemJK80Q+mDxlQYwDySNO0VjJz8+w5/aqNbfuzkCWiVeQGhRo9qpf3X5Cs5hJQwVq3fHc1ARjVIZTwSaVbqZZinSEA9axVGLMdDCeHTshJ1bpkyhRtqQhM/X3xBhjrfM4tJ0xmqFe9Kbif14nM9F1MOYyzQyTdL4oygQxCZl+TvpcMWpEbglSxe2thA5RITU2n4oNwVt8eZk8nte9y7p7f1Fr3BRxlOEIjuEUPLiCBtxBE3ygwOEZXuHNkc6L8+58zFtLTjFzCH/gfP4AM2aOSA== y/a
1, the atomic excited state can beadiabatically eliminated. This results in an effective Hamiltonian for the ground spin112 latexit sha1_base64="n8IJ9o41SD3d3ay8VSMA92RTG2g=">AAAB8nicbVDLSgNBEJyNrxhfUY9eBoPgKeyKosegF48RzAM2S5id9CZD5rHMzAphyWd48aCIV7/Gm3/jJNmDJhY0FFXddHfFKWfG+v63V1pb39jcKm9Xdnb39g+qh0dtozJNoUUVV7obEwOcSWhZZjl0Uw1ExBw68fhu5neeQBum5KOdpBAJMpQsYZRYJ4U9JWBI+nkaTPvVml/358CrJChIDRVo9qtfvYGimQBpKSfGhIGf2ign2jLKYVrpZQZSQsdkCKGjkggwUT4/eYrPnDLAidKupMVz9fdEToQxExG7TkHsyCx7M/E/L8xschPlTKaZBUkXi5KMY6vw7H88YBqo5RNHCNXM3YrpiGhCrUup4kIIll9eJe2LenBV9x8ua43bIo4yOkGn6BwF6Bo10D1qohaiSKFn9IrePOu9eO/ex6K15BUzx+gPvM8fVlCRSQ== ! p
1) at even and odd sites. (d) The mean-field phase diagramof the system in the parameter plane of η -Ω p , exhibiting checkerboard, stripe, and LVAP phases. Thesephases are characterized by the cavity field amplitude α and the order parameters { Θ , Ξ } , shown respectivelyin panels (e) and (f) for a vertical cut through the phase diagram at (cid:126) η = 0 . E r . The LVAP phase arises dueto the competition between the static and dynamic SO coupling. Figure adapted and reprinted with permissionfrom Ref. [269] © states,ˆ H SO = (cid:90) ˆΨ † ( r ) ˆ H , eff ˆΨ( r ) d r + 12 (cid:88) τ,τ (cid:48) = ↓ , ↑ g ττ (cid:48) (cid:90) ˆ ψ † τ ( r ) ˆ ψ † τ (cid:48) ( r ) ˆ ψ τ (cid:48) ( r ) ˆ ψ τ ( r ) d r − (cid:126) ∆ c ˆ a † ˆ a, (99)where ∆ c ≡ ω p − ω c , and ˆ a and ˆΨ = ( ˆ ψ ↑ , ˆ ψ ↓ ) (cid:62) are the photonic and two-componentatomic bosonic annihilation field operators, respectively. The single-particle atomicHamiltonian density reads,ˆ H , eff = − (cid:126) M ∇ + V ext ( r ) + (cid:126) (cid:20) δ Ω p ( r ) + Ω c ( r )ˆ a Ω ∗ p ( r ) + Ω ∗ c ( r )ˆ a † V ( r ) + U ( r )ˆ a † ˆ a + η ( r )(ˆ a † + ˆ a ) (cid:21) , (100)with the state-independent harmonic trap V ext ( r ), and the classical potential V ( r ) =Ω ( r ) / ∆ a = V cos ( k c y ), the cavity potential U ( r ) = G ( r ) / ∆ a = U cos ( k c x ) perphoton, and the interference potential η ( r ) = G ( r )Ω ( r ) / ∆ a = η cos( k c x ) cos( k c y )for the spin- ↓ component. The two spin states are coupled via the classical Ω p ( r ) =Ω ( r )Ω ∗ ( r ) / ∆ a = Ω p cos( k c y ) e − ik c y and cavity-induced Ω c ( r ) = G ( r )Ω ∗ ( r ) / ∆ a =Ω c cos( k c x ) e − ik c y two-photon Raman-Rabi processes. Here it has been assumed with-out loss of generality { Ω , Ω , G } ∈ R . The maximum strength of the potentials andRaman-Rabi couplings are defined as V = Ω / ∆ a , U = G / ∆ a , η = G Ω / ∆ a ,Ω p = Ω Ω / ∆ a , Ω c = G Ω / ∆ a . Here, δ ≡ ω Z + ∆ ω p + Ω / ∆ a is the effectivetwo-photon detuning and can be modified independently via the external bias mag-netic field tuning ω Z . Both classical and quantum Raman-Rabi interactions Ω p ( r ) andΩ c ( r )ˆ a induce coupling between internal and external degrees of freedom of the atom,113esulting in a dynamical SO coupling. The interplay of this dynamical SO couplingwith the classical and quantum optical potentials leads to interesting phenomena asdiscussed in the following.The mean-field approach ˆ ψ τ ( r ) → ψ τ ( r ) = (cid:104) ˆ ψ τ ( r ) (cid:105) and ˆ a → α = (cid:104) ˆ a (cid:105) can be usedto describe the system; see Section 2.4.1. The corresponding coupled Gross-Pitaevskiiequations for the condensate wavefunctions are solved self-consistently with the steady-state field amplitude α ss = η Θ + Ω c Ξ∆ c − U B + iκ , (101)where B = (cid:82) | ψ ↓ ( r ) | cos ( k c x ) d r , Θ = (cid:82) | ψ ↓ ( r ) | cos( k c x ) cos( k c y ) d r [cf. Equa-tion (15)], and Ξ = (cid:82) ψ ∗↓ ( r ) ψ ↑ ( r ) cos( k c x ) e ik c y d r . Here it has been assumed the cavitydecay κ is large so that the cavity field follows the atomic dynamics instantly. Thecavity field has two contributions: i) from collective photon scattering from the firstpump with spin-down atoms without changing their internal state characterized by Θ,and ii) from collective photon scattering from the second pump via the spin-flippingprocess quantified by Ξ. The first scattering process favors a checkerboard densitypattern, while the second process results in the cavity-induced SO coupling favoringa lattice of vortex-antivortex pairs (LVAP). Therefore, Θ and Ξ can be identified asorder parameters, characterizing different phases of the system.The mean-field phase diagram of the system in the parameter plane of η -Ω p fora fixed δ is presented in Figure 51(d). In order to reduce the number of the free pa-rameters, the ratio Ω / G has been fixed at 10, implying Ω c = 0 . p . In relativelysmall Ω p the checkerboard phase arises beyond a critical threshold η c , similar to theone-component checkerboard self-ordering discussed in Section 3. For chosen immis-cible two-body contact interaction strengths g ↓↑ > g ↓↓ g ↑↑ , in this phase the densitystructure of the spin- ↓ component is determined by the potentials V ( r ), U ( r ), and η ( r ), while the density structure of the spin- ↑ component is mainly fixed by the con-tact interactions. In moderate Ω p , the system undergoes a phase transition into thedensity-modulated stripe phase due to the classical potential V ( r ) and the “static” SOcoupling induced by the classical pump Raman-Rabi coupling Ω p ( r ). The cavity fieldamplitude is zero in this phase, signaling that it is a normal (i.e., non-superradiant)state. Note that the stripe phase here does not possess supersolid characteristics asthe translational symmetry along the transverse ( y ) direction is broken explicitly bythe classical Raman-Rabi coupling Ω p ( r ) and the classical potential V ( r ). The rel-ative phase ∆ φ ( r ) of the two condensate wave functions is periodically modulatedalong the y direction to minimize the Raman-coupling energy; see also Section 6.2.2for more details. The LVAP phase is, however, favored in the strong Ω p limit wherethe cavity-induced “dynamical” SO coupling by the Raman-Rabi coupling α Ω c ( r ) iscomparable with the static SO coupling owing to the large number of photons in thesystem. In this phase, the relative phase of the two condensate wave functions exhibitsa lattice of vortex-antivortex pairs, hence the name LVAP of this phase. Consequently,the transverse spin s ⊥ ( r ) = ( s x , s y ) forms a hedgehog (quadrupole) texture with thewinding number ± Z symmetry of the system; seeFigure 51(c). The cavity field amplitude α and the order parameters { Θ , Ξ } are shownin Figures 51(e) and (f), respectively, for a vertical cut through the phase diagramshown in Figure 51(d). Note that | Θ / Ξ | (cid:29) | Ξ / Θ | (cid:29)
1) in the checkerboard (LVAP)phase, while both order parameters as well as the cavity field are zero in the stripephase. 114e now revisit the model described in Sections 6.2 and 6.4 for realizing density andspin self-ordering. As it was shown, the Hamiltonian of Equation (79) for two purerunning-wave pump lasers leads to a transverse conical, chiral spin-spiral texture. Infact, this emergent spin order is due to the cavity-induced dynamical SO coupling inthe system. This has been studied in detail in Ref. [288], by also including the two-body contact interactions between the atom in the model. This is the first and onlycavity-based scheme where a dynamical SO coupling has been realized for ultracoldatoms so far [49], as will be discussed in more details in Section 8.1.3. This cavity-induced SO-coupled BEC system has also been studied in 1D along the direction of thepumps using a finite-size density matrix renormalization group (DMRG) algorithm inthe matrix product state form [279]. By adiabatically eliminating the cavity field anddiscretizing the space, the system amounts to an interacting Bose-Hubbard ladderpierced with a cavity-induced synthetic magnetic field (similar to the ladder modelfor the fermionic atoms with the cavity-induced synthetic magnetic field discussed inSection 8.1.1). In the continuum limit, a Meissner superfluid with a net chiral (i.e.,spin) current is dynamically stabilized in some parameter regimes.
Another approach for inducing a dynamical SOcoupling in the framework of cavity QED is to utilize ring cavities [270–273]. Since thenatural modes of ring cavities are plane waves, this provides the important ingredientof a SO coupling based on the Raman scheme. Reference [271] has considered a scenariowhere non-interacting Λ-type bosonic atoms are placed inside one arm of a ring cavity.The atoms are Raman-dressed by one running-wave laser and one counterpropagatingplane-wave mode of the ring cavity pumped longitudinally through a cavity mirror.The single-particle energy dispersion of the system has two branches, similar to thefree-space SO-coupled atoms. However, the energy dispersion develops a loop struc-ture in some parameter regimes due to the nonlinearity of the system. The nonlinearityalso leads to dynamical instabilities, which are more pronounced in the limit of lowcavity-photon number. An interacting Λ-type BEC has also been considered insidea ring cavity, where the atoms are Raman-dressed instead by two distinct counter-propagating plane-wave modes of the cavity pumped longitudinally through a cavitymirror [272]. The cavity fields induce a synthetic SO coupling as well as long-range in-teractions between the atoms, where the latter stems from the dynamical nature of thephotonic fields and their backaction on the atomic fields (see Section 2.2). The strengthof the cavity-mediated interactions can be tuned readily by changing the pump am-plitudes and frequencies as well as the decay rate of the cavity modes. The interplaybetween the single-particle energy dispersion on one hand, and the two-body contactand cavity-induced global interactions on the other hand determines the many-bodyground state of the system. In a wide range of parameters, the stripe phase is favoredover the plane-wave phase owing to the cavity-induced interactions. Notably, the stripephase is energetically stabilized even for condensates with attractive intraspecies andinterspecies contact interactions for sufficiently large cavity-induced interactions. Fur-thermore, for a symmetric double-well energy dispersion it is also possible by tuningthe cavity-mediated interactions to obtain a stripe phase with an arbitrary superpo-sition of atoms in the left and right wells, in sharp contrast to a stripe phase in afree-space symmetric double-well energy dispersion consisting of an equal number ofatoms in the left and right wells. Both Refs. [271,272] have ignored the unpumpeddegenerate counterpropagating mode(s) corresponding to the pumped mode(s) of thering cavities. This is a good first approximation for strongly pumped cavities in shorttime scales, as the probably of scattering a photon into a strongly pumped mode is115uch higher than the probability of scattering a photon into the degenerate emptymode due to the Bose-enhancement factor. By building on Ref. [272], the effect ofthe unpumped degenerate counterpropagating modes in this model has been includedand studied in Ref. [209]; though the two-body interactions were omitted. These un-pumped modes are populated solely by the coherent backscattering of photons fromthe pumped modes via the atoms and play an important role in the dynamics of thesystem in long times. The system exhibits three fundamentally different steady-statephases, as the pump strengths and pump frequencies are varied, characterized bydistinct spatial density distributions and spin textures: a combined density and spinwave, a continuous spin-spiral texture with a homogeneous atomic density, and a spinspiral with a modulated density. The spin-spiral states are topologically nontrivial,as discussed in Section 6.2, and are intimately related to cavity-induced SO couplingappearing beyond a critical pump power. The topologically trivial density-wave–spin-wave state spontaneously breaks two continuous symmetries, the U (1) freedom of thetotal phase of the two condensate wave functions and a U (1) screw-like symmetry cor-responding to a continuous spatial translation accompanied by phase rotations of theunpumped cavity-field amplitudes and condensate wave functions, a characteristic of asupersolid state as discussed in Section 4. The transitions between different phases areeither topological and first order, or non-topological and second order. Remarkably,all the phase transitions can be detected nondestructively at the cavity output viamonitoring the unpumped cavity modes. The first experiment demonstrating cavity-assisted dynamical spin-orbit couplingmade use of the experimental scheme shown in Figure 52(a) and (b) [49]. It is a variantof the scheme presented in Section 6.4, however, employing running-wave fields for thepumps instead of the standing-wave fields. More precisely, it is a special case of theHamiltonian of Equation (79) for two pure running-wave pump lasers correspondingto R = 0 in Section 6.2. A BEC is loaded into a single-mode cavity and transversallyilluminated with two balanced counterpropagating running-wave laser fields. The fre-quencies of these laser fields are chosen such that they, in combination with vacuumfluctuations of the cavity mode, each induce a Raman transition between two atomichyperfine states, labeled as |↓(cid:105) and |↑(cid:105) . For pump rates below threshold, the atomicspins and positions are disordered, and the Raman scattering rates are low. Above acritical pump strength, spin and density of the gas self-order and the cavity mode ispopulated with a coherent field due to superradiance. The frequency of the emergingcavity field is the center frequency between the two coherent pump-field frequencies.Above threshold, SO coupling emerges in this scheme. In contrast to a situationwith standing-wave pump fields where no net momentum is transferred to a cloud (cf.Section 6.4), during the self-organization with the running-wave pumps momentum isimparted to the system while the atomic spins are flipped. This can also be under-stood from the momentum-space representation shown in Figure 52(c) and (d). Whilethe cloud is spin polarized in |↓(cid:105) below the critical pump strength and only occupiesthe zero momentum state, above threshold additional momentum states are occupied.Since the Raman scattering process is spin selective, the spin state |↑(cid:105) is only pop-ulated if a net momentum along the positive y axis is transferred to the atoms. Incontrast along the x axis, the accompanying momentum change averages to zero dueto the standing-wave mode of the cavity. Going into an according co-moving referenceframe, opposite spin states thus move in opposite directions, which thus realizes a SO116 b c de f yx, -g-z, B Figure 52. Schematic of the cavity-induced dynamical spin-orbit coupling. Two Raman pump beams (red andblue arrows), polarized along the cavity axis, counterpropagate through a BEC of Rb atoms (purple) insidea TEM cavity mode. The cavity emission (green arrow) is detected by a single-photon counter, and theatoms are imaged after ballistic expansion. (b) Level diagram illustrating the cavity-assisted Raman couplingbetween two hyperfine levels of the Rb atoms acting as the spin states. δ (cid:48) is the Raman detuning. (c),(d)Momentum-space depiction of the emergence of SO coupling. (c) Initially atoms are in a spin-polarized state |↓(cid:105) . (d) If the transverse pump strength is sufficiently strong, SO coupling emerges and the spin componentsare in different momentum states. The ± sign of the |↑(cid:105) spin component indicates the Z symmetry-brokenphase freedom. (e),(f) Energy-momentum dispersion relation of each spin state, transitioning from free (e) toSO-coupled (f) dispersion bands. The coupling strength ˆΩ SOC is proportional to ˆ a and ˆ a † and therefore arisesdynamically as the atoms scatter pump photons into the cavity. The zero of the momentum has been shiftedwith respect to the lab frame by − k c / © coupling. Since this process depends self-consistently on the evolution of the atomiccloud and the cavity field in order to maximize the superradiant collective photonscattering, the SO coupling has a dynamical character.The relevant physics is captured—neglecting dispersive shifts—by a Hamiltonianobtained from Equation (79) for two pure counterpropagating running-wave pumplasers Ω , ( y ) = Ω ± e ± ik c y (corresponding to R = 0) after transferring to the co-movingframe,ˆ H = − (cid:126) ∆ c ˆ a † ˆ a + (cid:90) ˆΨ † ( r ) (cid:32) M (cid:0) ˆ p + (cid:126) k c e y (cid:1) − (cid:126) ˜ δ (cid:126) ˆΩ SOC cos k c x (cid:126) ˆΩ † SOC cos k c x M (cid:0) ˆ p − (cid:126) k c e y (cid:1) (cid:33) ˆΨ( r ) d r , (102)where (cid:126) ˜ δ is the effective two-level spin splitting and ˆΨ = ( ˆ ψ ↑ , ˆ ψ ↓ ) (cid:62) is a spinor contain-ing the atomic annihilation operators ˆ ψ ↑ , ↓ . The strength of the two-photon Raman-Rabi rate giving rise to the SO coupling is an operator, given byˆΩ SOC = G ( y, z )Ω + √ a + ˆ a † + G ( y, z )Ω − √ a − ˆ a, (103)where ∆ a ± are the atomic detunings with respect to the two pump fields. Considering117 x yy Figure 53. Experimental realization of dynamical spin-orbit coupling. (a) Cavity emission detected by single-photon counters (solid blue curve) and optical power in the Raman beams (dashed black line), both as afunction of time. Steady-state SO coupling persists up to a few milliseconds. (b),(c) Spin-resolved momentumdistribution in time of flight, taken at the points labeled in (a). All atoms are in the |↓(cid:105) state (blue) just belowthreshold (b). Above threshold (c), spin-up atoms (red) have acquired a net momentum in the y direction, asshown by the spin-colored Bragg peaks at nonzero momenta. Also shown are second-order diffraction peaksalong the cavity direction due to the reverse Raman process. Figure adapted and reprinted with permissionfrom Ref. [49] © the dispersion relation of both spin states in presence of the SO coupling, one findsthat the energy bands are displaced from one another due to the opposite momentumrecoils and coupled via the Raman-Rabi term. This leads to a double-minima lowestenergy-band structure and a band gap given by ˆΩ SOC proportional to ˆ a † and ˆ a , shownschematically in Figure 52(e) and (f).The superradiant state displaying SO coupling corresponds in real space to anatomic spinor state that exhibits a helical pattern (i.e., spin spiral) along the pumpaxis according to |↓(cid:105)± e ik c y cos( k c x ) |↑(cid:105) . This state corresponds to the chiral spin-spiralphase in Figure 35(b) and Figure 36(a). The experimental observables of this stateare the intra-cavity light field that rises to a finite value above threshold and thespin-selective occupation of momentum states according to the expectations describedabove; see Figure 53. The lifetime of the superradiant state is only on the order of afew milliseconds, most likely limited by a dephasing of the two pump beams. Cavity-induced topological states
The landscape of the superradiant optical lattice (cid:126) V SR ( x ) = (cid:126) U | α | cos ( k c x ) + 2 (cid:126) η ( α + α ∗ ) cos( k c x ) , (104)depends crucially on the sign of U ∝ ∆ − a = ( ω p − ω a ) − as illustrated in Figure 5.The lattice potential (cid:126) V SR ( x ) is λ c periodic due to the interference term ∝ cos( k c x ).For a conventional red-detuned pump laser (and cavity field) with respect to the rel-evant atomic transition U ∝ ∆ − a <
0, the superradiant lattice has a single globalminimum in every unit cell of length λ c . More precisely, the unit cell contains twolattice sites with an energy offset between them. For the blue-detuned repulsive case U ∝ ∆ − a >
0, in stark contrast (cid:126) V SR ( x ) is homopolar and its unit cell possessestwo lattice sites with no energy offset between them. In the deep lattice limit, thislatter case maps to the Su-Schrieffer-Heeger (SSH) model, a non-interacting 1D tight-binding fermionic model with staggered (i.e., different intracell and intercell) hoppingamplitudes [289]. The SSH model has two distinct dimerizations: i) the intracell hop-118ing amplitude is larger than the intercell hopping amplitude, ii) the intracell hoppingamplitude is smaller than the intercell hopping amplitude. In the former case, thesystem is a trivial insulator at half filling. Whereas in the latter case, the Bloch bandshave a nontrivial topological structure, characterized by a non-zero bulk topologicalinvariant—the winding number—and topologically protected edge states.The self-ordering of spin-polarized, low-field-seeking (i.e., blue detuned ∆ − a > U ∝ ∆ − a >
0; see also Section 2.4.2. Note that the two-body contact interaction is identically zero owing to the Pauli exclusion principle and V ext ( x ) = (cid:126) V ( x ) = 0.At low pump strengths, the system is in the normal Fermi-gas state with no pho-ton inside the cavity. By increasing the pump strength, the system becomes unstabletoward the superradiant phase α (cid:54) = 0, where the Fermi gas crystallizes under the emer-gent superradiant potential (cid:126) V SR ( x ), Equation (104), in turn enhancing constructivephoton scattering from the pump laser into the cavity mode. Across the superradiantphase transition, the relative phase of the cavity field with respect to the pump laser islocked at either 0 or π (under the assumption ∆ c (cid:29) κ ), spontaneously breaking the Z symmetry (i.e., the invariance under the transformation α → − α and x → x + λ c /
2) ofthe system. That is, the system chooses spontaneously one of the two possible superra-diant lattice configurations shifted by λ c / η versus the Fermi momentum k F at a finite temperature is shown in Figure 54(a) [40].The emergent lattice (cid:126) V SR ( x ) opens up gaps in the quadratic energy dispersion of thefree Fermi gas across the superradiant phase transition. For atomic densities aroundhalf filling k F = k c /
2, the superradiant state is an insulator, while for filling suffi-ciently away from half filling the system corresponds to a superradiant metal (SRM).The transition to the insulating state with a dimerized structure at half filling occursthrough the same mechanism underlying the Peierls instability in 1D electron-phononmodels [150]. Depending on the spontaneously chosen dimerization via the superradi-ant Z symmetry breaking, the superradiant insulating state is either a trivial bandinsulator (SRBI) or a topological insulator (SRTI), characterized by the Zak phase.For this system the Zak phase is Z quantized and serves as a topological invariant,with the value 0 ( π ) corresponding to the SRBI (SRTI). Remarkably, the value of theZak phase coincides exactly with the relative phase of the cavity field with respect tothe pump laser, providing a nondestructive way to read out the topological Zak phasevia the leaked cavity field. Note that the superradiant phase terminates at large pumpstrengths, indicated by the gray area in the phase diagram of Figure 54(a); see alsoFigure 20 for a similar behavior in self-ordering of low-field-seeking bosonic atoms. Wealso note that the critical pump strength at half filling becomes infinitesimally smallat zero temperature due to the Fermi nesting [113].The SRTI state possesses a pair of edge states within the superradiant-opened en-ergy band gap in a finite size system, another indication of the nontrivial topology ofthe state, represented in the inset of Figure 54(b). These edge states can be detectedindirectly via the spectral properties of the cavity output. Figure 54(b) shows for afinite-size system the imaginary part of the optical polarizability χ dyn ( ω ) [see Equa-tion (31)], corresponding to the atomic absorption with respect to the propagationof cavity photons. The edge states modify the polarization function significantly andopen an additional absorption channel in a well-defined frequency range as shown in119 latexit sha1_base64="o75iwHQ92Sc12Q04YMBddsQXHto=">AAACAHicbVBNS8NAEN3Ur1q/oh48eAkWwVNNRNFjUQRPUsF+QBPCZrtpl242cXcilJCLf8WLB0W8+jO8+W/ctjlo64OBx3szzMwLEs4U2Pa3UVpYXFpeKa9W1tY3NrfM7Z2WilNJaJPEPJadACvKmaBNYMBpJ5EURwGn7WB4Nfbbj1QqFot7GCXUi3BfsJARDFryzT13EGDpUsC+7aoHCdltfnztS9+s2jV7AmueOAWpogIN3/xyezFJIyqAcKxU17ET8DIsgRFO84qbKppgMsR92tVU4IgqL5s8kFuHWulZYSx1CbAm6u+JDEdKjaJAd0YYBmrWG4v/ed0UwgsvYyJJgQoyXRSm3ILYGqdh9ZikBPhIE0wk07daZIAlJqAzq+gQnNmX50nrpOac1ey702r9soijjPbRATpCDjpHdXSDGqiJCMrRM3pFb8aT8WK8Gx/T1pJRzOyiPzA+fwCUWpZk ~ ⌘ p N / E r
0. (a) The phase diagram of the system in the η - k F parameter space at the finite temperature k B T = 0 . E r . Above the critical pump strength (the blue curve),the Fermi gas crystallizes as a consequence of the superradiant phase transition. The superradiant crystallinephase is either a metal (SRM) or an insulator. Depending on the spontaneously chosen dimerization (i.e., thecavity field phase) via the Z symmetry breaking process, the superradiant insulating state is either a bandinsulator (SRBI) or a topological insulator (SRTI), characterized by their Zak phases. (b) The imaginary partof the optical polarizability χ dyn ( ω ) for a finite-size system, corresponding to the atomic absorption of thecavity photons. The edge states within the energy gap open an additional absorption channel in a well-definedfrequency range (the red dashed curve). The inset shows the energy spectrum of a finite-size system, with apair of edge states within the superradiant-opned energy gap. Figure adapted and reprinted with permissionfrom Ref. [40] © Figure 54(b).
A closely related scheme to the spinor bosonic self-ordering discussed in Section 6has also been considered for 1D spin-1/2 fermionic atoms in the repulsive regime,i.e., where the pump laser and the cavity field are blue detuned with respect to rel-evant atomic transitions [281]; see Figure 55(a) and (b). The transverse pump laserand the cavity mode A couple the two ground spin states {|↓(cid:105) , |↑(cid:105)} via a double-Λ scheme (i.e., two independent Raman processes), while the longitudinally pumpedcavity mode B provides a background spin-independent “classical” optical lattice po-tential V ext ( x ) = V (0)latt cos ( k c x ) for the atoms. The system is described by a specialcase of the Hamiltonian density (79),ˆ H , eff = (cid:32) p x M + V ext ( x ) + (cid:126) U ( x )ˆ a † ˆ a + (cid:126) δ (cid:126) η ( x )(ˆ a + ˆ a † ) (cid:126) η ( x )(ˆ a + ˆ a † ) p x M + V ext ( x ) + (cid:126) U ( x )ˆ a † ˆ a (cid:33) , (105)where ˆ a is the annihilation operator of the mode A, U ( x ) = U cos ( k c x ), and η ( x ) = η cos( k c x ), with U and η defined as before. However, note that here U ∝ ∆ − a > (cid:126) δ between the two spin states can be tuned via an external bias magneticfield, where the latter also fixes the quantization axis along the z direction.At small pump strengths η , there is no constructive photon scattering from thepump laser into the cavity mode, (cid:104) ˆ a (cid:105) = 0. Therefore, there is no (Raman) coupling120 e)
0. The schematic representation of the setup (a) and the atom-photon couplingscheme (b). The Bloch energy bands before (solid) and after (dashed) the superradiant phase transition for δ <δ c (c) and δ > δ c (d). For δ < δ c , the superradiance-induced gap opening in finite momenta marked by circlesresults in the band inversion. (e) The phase diagram of the system at half filling and finite temperature k B T =0 . E r is represented in the η - δ parameter plane, exhibiting metallic (M), insulating (I), superradiant (SR),and topological superradiant (TSR) states. The various boundaries merge at a tetracritical point (the purpledot). The insets show the evolution of the energy bands (solid-dotted-dashed) across the phase boundariesindicated by arrows. Note, in particular, that the band gap closes and re-opens across the TSR-SR phaseboundary. Figure adapted and reprinted with permission from Ref. [281] © between the two spin states and the atoms only experience the background optical lat-tice potential V ext ( x ) and the Zeeman field (cid:126) δ , with Bloch energy bands correspondingto different spins being independent from each other. The normal Fermi gas at halffilling is either a gapless metal (M) or a gapped insulator (I) depending on δ ; seeFigures 55(c) and (d).By increasing the pump strength above the critical value η c ( δ ), the system entersthe superradiant phase with (cid:104) ˆ a (cid:105) (cid:54) = 0. As a consequence, the cavity-assisted space-dependent Raman processes (with periodicity twice as the background lattice poten-tial) induce a spin-orbit coupling which mixes different spin Bloch bands and may opengaps in the bulk energy spectrum within the first Brillouin zone; see Figures 55(c) and(d). Depending on δ , this interband mixing in the superradiant phase can result in aband inversion, leading to topologically nontrivial states.The steady-state phase diagram of the system at half filling in the η - δ parameterplane is illustrated in Figure 55(e). In particular, for δ < δ c the system is in the metallicstate below the superradiant threshold and a gap opens in the bulk energy spectrumacross the superradiant phase transition. Intriguingly, a pair of topological edge modeswith localized wave functions appears within the superradiant-opened energy gap in afinite-size system. Furthermore, the topologically nontrivial nature of this superradiantstate (TSR) is confirmed by a nonzero bulk winding number, a topological invariantcounting the number of times the momentum-space spin texture sweeps the circle S (i.e., does a full 2 π rotation) across the first Brillouin zone. In contrast, for δ >δ c the system is in the insulating state below the superradiant threshold and thetrivial insulating gap persists across the superradiant phase transition, resulting in atopologically trivial superradiant (SR) stat. However, by further increasing the thepump strength the trivial superradiant state crosses into the topological superradiantphase. This is due to the increased interband mixing, which closes and then re-opens121he lowest band gap, resulting in a band inversion; see the inset of Figure 55(e-ii).The topological phase transition across the SR-TSR boundary can be detected inthe abrupt change of the momentum-space density distribution of the spin-up andspin-down components via a spin-selective time-of-flight imaging technique. However,since the cavity field here is driven by the self-ordering of the atomic spin, this providesa nondestructive tool to monitor the atomic spin texture and the SR-TSR phase tran-sition. In particular, the derivative of the cavity photon number exhibits a pronouncedpeak across the SR-TSR phase boundary.
9. Superradiant self-ordering with emergent discrete rotationalsymmetry: quasicrystals
The preceding sections, specially Sections 3, 4, and 5, were focused on cavity-QEDsystems with the discrete ( Z ) and continuous [ U (1) and SO (3)] symmetries, andhow the self-ordering of the atoms into “crystalline” structures in the superradiantphase spontaneously breaks these symmetries. We noted that these hybrid atom-cavitysystems exhibit fundamentally different steady-state phase diagrams, depending onunderlying symmetries and their spontaneous breaking . This is the common paradigmof condensed-matter physics, described by Landau theory. That said, in rare exoticsituations in some condensed-matter models, global [290–293] and local gauge [294–298] symmetries emerge in the proximity of certain quantum phase transitions and/orin some quantum phases.Recently, a novel many-body cavity-QED setup has been proposed in Ref. [299],where a discrete eightfold rotational symmetry C (i.e., a rotation by 45 ° ) emerges inthe low-energy states across the superradiance quantum phase transition. Since theeightfold rotational symmetry C is not consistent with any translational symmetry,this results in an emergent “quasicrystalline” superradiant potential for the atomsin analogy to natural quasicrystalline materials. Quasicrystals are quasiordered (i.e.,orientationally ordered) materials with no translational symmetry, rather with crystal-lographically forbidden rotational symmetries such as five-, seven-, eightfold rotationalsymmetries, as discovered from their diffraction patterns first by Shechtman et al. in1984 [300]. Four crossed linear cavities
In the the setup proposed in Ref. [299], four identical linear cavities are arranged sym-metrically in a plane with a common center such that they make a 45 ° angle withone another. A BEC is tightly confined in the direction perpendicular to the planeat the intersection of these initially empty four cavities, and is strongly coupled toan in-plane polarized mode of each cavity ˆ a j . The BEC is also driven with the Rabifrequency Ω by a spatially uniform, right circularly polarized pump laser propagat-ing perpendicular to the cavity-BEC plane as depicted in Figure 56(a). The involvedatomic internal states satisfy the magnetic selection rule ∆ m = m e − m g = 1. Gen-eralizing the procedure of Section 2.1 (see also Section 5.1), in the dispersive regimeof a large atom-pump detuning ∆ a the system is described by the effective many-body Hamiltonian ˆ H eff = (cid:82) ˆ ψ † ( r ) ˆ H , eff ˆ ψ ( r ) d r + ˆ H int − (cid:126) ∆ c (cid:80) j ˆ a † j ˆ a j with the effective122 ˆ e x ˆ e y ˆ ✏ + ˆ ✏ ˆ + BEC x y z NH SRQCSRL p N ⌘ /! r | ↵ j | / p N c r o ss o v e r
In the preceding section, we discussed a scenario where a quasicrystalline superra-diant optical potential with an eightfold rotational symmetry emerges for ultracoldatoms coupled to four-crossed linear cavities, where the atoms in turn self-order in thisemergent quasicrystalline potential. This proposal is an extension of the ETH group’stwo-crossed cavity setup (see Section 4.1). Other superradiant quasicrystalline opticalpotentials with different rotational symmetries can also be implemented by employingdifferent cavity setups, although they might become progressively more challenging124xperimentally. However, they can provide a platform to study many fundamentalopen questions concerning the formation and nature of quasicrystals. For instance,it is still not completely clear whether quasicrystals are only entropy-stabilized high-temperature states or can also be thermodynamically stable at low temperatures. Theconditions and nature of quasicrystal growth are also under debate with a lack ofa generally accepted model. Further enriching the physics in cavity-based compositeatom-light quasicrystals compared to solid-state quasicrystalline materials is the in-terplay between two-body contact and cavity-mediated long-range interactions, bothtunable in principle.
10. Superradiant self-organization without a steady state
So far we mostly considered situations, where atomic and photonic degrees of freedomreach a steady state. In the present section we will instead discuss some interestingquantum-gas–cavity scenarios with non-steady states. In order to put the results pre-sented in the following into the general context of quantum many-body dynamics,some remarks are in order. First and foremost, a hybrid atom-cavity setup is intrin-sically an open quantum system due to the photon leakage out of the cavity. Thismeans that the dynamics always leads to an attractor (which needs not to be unique),defined by the eigenvalue of the master equation (8) with vanishing real part, i.e., withonly damping. This scenario has to be distinguished from the case of isolated systems,where the dynamics are classified in terms of ergodic versus non-ergodic behavior. Inthe context of quantum many-body systems, this characterization is still an activefield of research, with the phenomenon of many-body localization being a prominentexample [301].Besides being dissipative, the atom-cavity systems we consider are also driven, asexternal lasers in conjunction with cavity fields are used to generate the dispersivelight-matter coupling discussed throughout this review. In this case, the long-timeattractor does not necessarily need to be a steady state, even though the Liouville op-erator governing the dynamics has no explicit time dependence (in the frame rotatingwith the driving laser). This situation will be considered in Section 10.1, while thecase of explicitly time-dependent Liouvillians will be discussed in Section 10.2.Some of the non-steady states discussed below can be classified, in a broader sense, asso-called time crystals. Such phases, which spontaneously break some time-translationsymmetry and show robust periodic oscillations in time, have recently received con-siderable amount of attention, with examples from different physical platforms [302].We will also encounter in the following non-steady phases which do not oscillate peri-odically in time but rather exhibit irregular behavior, similar to the ergodic dynamicsof isolated systems. In the context of open quantum systems, the definition and char-acterization of ergodic or chaotic behavior is still an open issue, with very recentinteresting developments [303–310]. In this context, atom-cavity setups seem a verypromising platform for further investigation.Before proceeding, let us note that in a driven-dissipative setting like the onesconsidered here, even a steady attractor can be very nontrival, since it is not in generalof the thermal Gibbs type [97]. We refer here to the discussion given in Section 3.1.2.125
Non-steady-state phases with time-independent driving
In this section, we discuss two examples, where a stable, non-steady behavior emergeswithout an explicit time-dependent external driving. This type of phenomena sharesstrong similarities with limit cycles, quantum synchronization, and some types of dis-sipative time crystals.
A possible scenario through which stable non-steady attractors can emerge is via thecompetition between different steady attractors. This can be achieved in the simplestsingle-mode superradiant self-ordering scenario by driving the atoms with a laser blue-detuned with respect to the atomic resonance, ∆ a >
0, as first pointed out in Ref. [38].Blue-detuned laser fields lead to repulsive optical potentials but still allow for superra-diant self-ordering in some limited parameter regimes, as discussed in Section 3.3.3. Tobetter understand the phenomenon, instead of considering the two-dimensional config-uration of that section, we re-examine here the simpler one-dimensional blue-detunedmodel introduced in Section 8.2.1, this time for bosonic atoms instead of fermionicatoms.The phase diagram of this 1D system is shown in Figure 57(a). The competi-tion which destabilizes the steady states can be immediately understood from Equa-tion (104), expressing the optical potential felt by the atoms in the superradiant phase.For a laser which is blue detuned from the atomic transition, one has U ∝ ∆ − a > ( k c x ) part of the potential pushes the atoms out of the self-ordered pattern induced by the cos( k c x ) part, thereby favoring a homogenous densityand thus the non-superradiant (normal) phase. At very small photon numbers, thecos( k c x ) term dominates so that a steady-state superradiant phase is stabilized. How-ever, for sufficiently strong pumps the cos ( k c x ) term becomes equally important, sothat the system cannot decide between the normal and the superradiant phases. Thisleads to self-sustained oscillations between these two states which are stabilized by thecavity loss. These limit-cycle oscillations are shown in the middle panel of Figure 57(b).By further increasing the pump strength the number of harmonics increases by theknown mechanism of period doubling, eventually crossing over into irregular behaviorexemplified in the lower panel of Figure 57(b). Within a (semi-)classical descriptionof the system based on coupled nonlinear equations for the cavity coherent field andthe BEC wavefunction (see Section 2.4.1) this irregular behavior can be associatedwith a chaotic attractor [311]. Indeed, the dissipative, i.e., non-Hamiltonian, nature ofthe equations of motion leads to the contraction of the phase-space volume to withina finite region. Still, within the attractor region the dynamics is chaotic, that is, itfeatures at least one positive Lyapunov exponent.What this classically known scenario implies for the full quantum state of the systemis an open question. Differently from the closed-system case, the study of quantumdissipative chaos is still in its beginning and matter of active research [304,305,309,310,312]. It is thus also an open question whether this type of classical dynamics underliesome kind of ergodic or thermal-like behavior of our quantum many-body system. Thisis a matter of future research, for which the atom-cavity systems considered in thisreview article seem an ideal platform. From the point of view of the atoms, the chaoticattractor corresponds to a fast increase of the kinetic energy [38,169], which reachesa value much larger than that in the limit-cycle attractor. The chaotic attractor isthus characterized by a more effective redistribution, i.e., a behavior which is “more126 igure 57. Emergence of stable, non-steady-state behavior for a BEC inside a cavity, driven transversely bya blue-detuned laser. A quasi-one-dimensional BEC is trapped along the axis of a linear cavity and coupledto a single standing-wave mode of it. The transverse, driving laser is blue detuned from the atomic transition,∆ a >
0. (a) Phase diagram of the system. While the bare cavity detuning ∆ c is positive in the graph, thedispersively-shifted detuning is still negative: δ c <
0. The phase diagram features the usual steady-state normal(N) and superradiant self-organized (S-O) states, as well as non-steady-state superradiant self-organized limit-cycle (S-O-L-C) and chaotic (S-O-Chaos) phases. (b) Time evolution of the coherent cavity field in the threedifferent self-organized phases: S-O, S-O-L-C, and S-O-Chaos in order. (c) Time evolution of the BEC density inthe chaotic-like phase, where the white circles indicate the space-time coordinates where solitons are nucleated.The parameters are U N = 12 . ω r and κ = 10 ω r . Panel (c) also features a finite short-range atom-atomrepulsion ng = (cid:126) ω r . Figure adapted and reprinted with permission from Ref. [38] © ergodic” than for the limit-cycle attractor. Still, the increase in the kinetic energy iscounteracted by the cavity loss via the mechanism of cavity cooling, which eventuallystops the heating [38].A further interesting aspect connected with the chaotic attractor is the phenomenonof phase slippage and related soliton nucleation in the BEC [38], as shown in Fig-ure 57(c). This behavior is peculiar to superfluids excited out of equilibrium, wherebyspatial gradients in the condensate phase are converted into localized excitations carry-ing a phase slip. In the present case the phase gradients are caused by the modulationof the cavity field and have a characteristic length scale fixed by λ c . On the otherhand, the characteristic size of a soliton is set by the healing length, whose value(2 M g n ) − / depends on the interatomic repulsion and is set to 0 . λ c in Figure 57(c).The main contribution to the BEC kinetic energy can be attributed to soliton prolif-eration. The latter is counteracted by cavity cooling, in a situation which resemblesturbulence. This interesting analogy should be further investigated in the future.The Gross-Pitaevskii study of Ref. [38] has been extended to include non-mean-fieldcorrelations between the bosonic atoms via a multiconfigurational time-dependentHartree method for indistinguishable particles (MCTDH-X) [169], as well as to in-clude the effect of harmonic trapping of the atomic cloud. The phase diagram for theone-dimensional case is shown in Figure 58(a). A non-mean-field, steady-state phasepeculiar to this blue-detuned case is found. In this so-called self-organized second-127 igure 58. (a) Extension of the mean-field phase diagram of Figure 57(a) by including non-mean-field atomiccorrelations as well as an harmonic trap for the atoms. The dimensionless potential depth is defined as A = Nη (cid:112) U /ω r / (cid:112) (∆ c − NU B ) + κ , with the overlap B = (cid:82) dxn ( x ) cos ( k c x ). Besides the normal (NP) andself-organized superfluid (SSF) phases present in Figure 57(a), a further mean-field-type phase appears asa self-organized dimerized superfluid (SDSF). Moreover, beyond-mean-field phases are also stabilized: a self-organized Mott insulator (SMI) phase [also present in the phase diagram of Figure 45(b)] and self-organizedsecond-order superfluid (2-SSF) phase. Additionally, non-steady-state phases are also present: A quasi-periodicattractor (QA) related to the limit-cycle phase of Figure 57(a) and a chaotic attractor corresponding to thechaotic phase of Figure 57(a). (b) Pictorial representation of the different steady-state phases. Figure adaptedand reprinted with permission from Ref. [169] © order superfluid phase, the phase coherence is limited to within each double well, i.e.,the unit cell of the dimerized superradiant lattice. Within this beyond-mean-field de-scription a limit-cycle phase is also identified, with the only difference with respectto the mean-field version discussed above being the quasi-periodic nature of the os-cillations, which has been attributed to the combination of the external trap and theshort-range atomic repulsion [169]. The chaotic attractor is also present within thisbeyond-mean-field description, indicating that this phenomenon is not an artifact ofthe (semi-)classical description. This opens the interesting perspective for further in-vestigating of this type of chaotic phases in a fully quantum setting.Further insight into the limit-cycle phase and its relation to time crystals is providedby the analysis of the temporal coherence presented in Figure 59. These results [43]are obtained using the truncated Wigner approximation, which goes beyond the mean-field approximation by including the noise induced by cavity loss [see Equation (10)],as well as the quantum noise in the initial state. In Ref. [43], the latter has beenincluded by occupying low-lying BEC and cavity modes with vacuum fluctuations.Observables are then computed by averaging over a set of stochastically-generatedtrajectories. Even in the presence of these forms of noise, the temporal coherence ofthe limit-cycle phase is clearly visible in Figure 59(a) as finite-amplitude harmonicoscillations in the quantity C ( t ) = (cid:104) α ∗ ( t ) α ( t ) (cid:105) , where the averaging is performed overan ensemble of trajectories. The emergence of temporal coherence in the limit-cyclephase is a characteristic of time crystals. On the contrary, the chaotic phase does notshow appreciable temporal coherence. The amplitude of the oscillations of temporal128 igure 59. Time crystals in non-steady-state self-organized phases in the repulsive, blue-detuned regime. (a)Temporal coherence C ( t ) (defined in the text) for the self-organized phases shown in the phase diagram ofFigure 57(a) [here the DW corresponds to the S-O phase in Figure 57(a)]. The emergence of temporal coherencein the limit-cycle (LC) phase is a characteristic of time crystals. (b) The amplitude of the oscillations of C ( t ) asa function of the pump strength across the three phases (the LC phase extends roughly between 0.7 and 1.5).Here the simulations are done for a two-dimensional BEC inside a standing-wave cavity pumped transversallywith a standing-wave laser. Adapted and reprinted with permission from Ref. [43] © coherence is shown in Figure 59(b) as a function of the ramping of the pump strengthfrom the self-organized steady-state phase, across the limit-cycle phase over to thechaotic phase.We conclude this section by noting that the above described limit-cycle phase hasnot been yet experimentally demonstrated, though experiments at the ETH grouphave been carried out in the blue-detuned regime, as discussed in Section 3.3.3. Onepossible explanation is the two-dimensional geometry of the ETH experiment, whichis different from the two-dimensional geometry studied in Ref. [43] and presented inFigure 59, where the pump laser is orthogonal to the cavity axis. Another possibleexplanation is the large value of the cavity loss rate in the experiment, reducing thesize of the limit-cycle region in the phase diagram. Addressing this issue requiresfurther investigation. Non-steady-state long-time dynamic behavior can also be induced by a competitionof coherent and dissipative couplings, as has been experimentally demonstrated inRef. [313] and theoretically discussed in Refs. [314,315]. We consider the model al-ready introduced in Section 3.3.2.1, where a spinor BEC with equal populations N ofthe Zeeman states m F = +1 and m F = − V can be adjusted with an angle ϕ with respect to the y axis [see Figure 18(a)],which changes the ratio of scalar and vector polarizability of the illuminated atoms.Generalizing the effective Dicke Hamiltonian of Equation (54) for a single-componentBEC to both Zeeman states, the many-body Hamiltonian of the system can be ex-129ressed asˆ H = − (cid:126) δ c ˆ a † ˆ a + 2 (cid:126) ω r (cid:16) ˆ J z, + + ˆ J z, − (cid:17) + (cid:126) (cid:104) η D (ˆ a † + ˆ a ) (cid:16) ˆ J x, + + ˆ J x, − (cid:17) − iη S (ˆ a † − ˆ a ) (cid:16) ˆ J x, + − ˆ J x, − (cid:17)(cid:105) . (107)The subscripts ± of the effective angular momentum operators ˆ J x, ± , ˆ J y, ± , andˆ J z, ± indicate the Zeeman states m F = ±
1. If the two spin components occupythe same checkerboard pattern, the expectation value for the density modulation x D = ( (cid:104) ˆ J x, + (cid:105) + (cid:104) ˆ J x, − (cid:105) ) /N becomes non-zero and the system self-organizes its den-sity. If on the other hand the two spin-components occupy the opposite checkerboardpatterns, the expectation value for the spin modulation x S = ( (cid:104) ˆ J x, + (cid:105) − (cid:104) ˆ J x, − (cid:105) ) /N is non-zero. The parameters η D = η s cos ϕ and η S = η v sin ϕ [cf. Equation (54)]describe the coherent coupling strengths to the density and spin modulated states,respectively. The coupling with the density modulated state is mediated via the realquadrature (ˆ a † + ˆ a ) of the cavity field, which is thus simultaneously occupied with afinite value of x D . While the imaginary quadrature i (ˆ a † − ˆ a ) is occupied with the emer-gence of the spin modulated state, x S (cid:54) = 0. Without dissipation, this system showsstable self-organization with either a density or a spin modulation above a criticalcoupling strength, when either the eigenfrequency ω D of the density modulated stateor ω S of the spin modulated state have softened to zero. Density and spin modulatedstates are distinguishable via the phase of the cavity field; see Section 3.3.2.1.A non-zero cavity-field dissipation rate κ leads to a phase shift φ κ = tan − ( − κ/δ c )of the light field scattered from the pump field into the cavity mode. As a consequence,a pure density modulation, for instance, thus leads to an occupation of both the realand the imaginary quadratures of the cavity field. The light field in the imaginaryquadrature then drives self-organization of the spin mode. This process exerts aneffective force onto the system and triggers an instability constantly driving it betweendensity and spin modulation already far below the threshold for self-organization. Inthe experiment, this was observed via a constantly evolving phase of the intra-cavitylight field and the associated emergence of sidebands in the spectrum of the cavityfield; see Figure 60.Adiabatically eliminating the cavity field, linearized equations of motion for theamplitudes x D and x S can be derived [313]: d dt (cid:18) x D x S (cid:19) = (cid:18) − ω D − K K − ω S (cid:19) (cid:18) x D x S (cid:19) , (108)where ( ω D − ω S ) ∝ sin δφ with δφ = 2 tan − ( − η S /η D ) − π/ K ∝ V sin φ κ cos δφ which can be increasedby increasing the cavity-induced phase shift (via the detuning δ c ) or by making the twomodes more degenerate (via the polarization angle ϕ affecting the ratio η D /η S ). Ex-pressing the state of the system by a vector in a plane spanned by the two amplitudes x D , x S , this dissipative coupling induces a force which is always orthogonal to theinstantaneous position vector of the system. Accordingly, solutions to the equations ofmotion (108) show that the system undergoes amplified rotations with fixed chiralityin the x D - x S plane. This mean-field description does not fully cover the experimentalobservations. Specifically, the theoretical model predicts an amplified motion, while in130 V ( E R ) V ( E R ) V ( E R ) V ( E R ) − − ( k H z ) (E R ) 0 10 20 30 40 50Time (ms)0 6 13 19 26 32V (E R ) 0 10 20 30 40 50Time (ms) 10 − − (E R ) x D x S x D x S x D x S π π φ (r a d ) π π φ (r a d ) A BC D E
Figure 60. Dissipation-induced dynamical behavior in a two-component BEC coupled to a single mode ofa linear cavity via both scalar and vector polarizabilities. (A and B) Time evolution of the mean numberof photons (black) in the cavity and the corresponding phase of the light field modulo 2 π (yellow), whilethe transverse pump-lattice depth V is ramped up over time (dashed line). Panel (A) displays data for apolarization angle and detuning where stable self-organization takes place, while panel (B) shows data forparameters where a transition to a dissipation-induced instability happens. Panels (C) to (E) show spectrogramsof the mean intra-cavity photon number as a function of frequency while the pump-lattice depth is increasedover time. The corresponding insets show the evolution of the state of the system in the x D - x S plane. Panel(C) displays the situation for stable self-organization, and panel (D) for the dissipation-induced instabilitywith degenerate mode frequencies ω S ≈ ω D . The observation of a single red sideband corresponds to a linearlyrunning phase of the light field, or a circular motion in the x D - x S plane. Panel (E) shows the situation fornon-degenerate modes, which leads to the appearance of also a blue sideband and a corresponding ellipticmotion in the x D - x S plane. Figure adapted and reprinted with permission from Ref. [313] published in 2019by the American Association for the Advancement of Science. the experiment an evolution similar to limit-cycles without increasing amplitude wasobserved. This behavior has been attributed to collisional interactions of the atoms.An analysis of the eigenspectrum reveals that the emergence of this dynamical in-stability is connected to the presence of exceptional points with the according levelattraction as shown in Figure 61(b). This behavior is well known for systems describedby non-Hermitian matrices such as Equation (108). After the instability has been in-duced, the two eigenmodes of the system synchronize and oscillate at a mean frequency.It was shown that the dissipative processes in the system generate a nonreciprocalcoupling between the two collective spins that eventually induces this instability ina certain parameter regime, shown in Figure 61(a). Going beyond adiabatic elimina-tion of the cavity field, it was though shown that the instability should occur in thelong-time limit for basically all parameters due to additional anti-damping from thecavity field fluctuations. In the bad-cavity limit that applies to the experiment, theinstability is, however, confined to a finite parameter region [314].An analysis studying the full quantum model of this system confirmed that the131 igure 61. Dissipation-induced dynamical behavior in a two-component BEC coupled to a single mode ofa linear cavity via both scalar and vector polarizabilities. Panel (A) shows the boundary of the dissipation-induced structural instability (DSI) regime as a function of dispersively shifted cavity detuning δ c and angleof the pump field polarization ϕ . Data points indicate where the sidebands from Figure 60 become dominant,the colored background is the result of a mean-field calculation. Panel (B) shows the frequencies of the twoeigenmodes in presence (solid lines) and absence (dashed lines) of dissipation. Without dissipation the twomodes are the spin modulation (SM, blue) and the density modulation (DM, green). Dissipation leads to levelattraction and finally synchronization in the instability dominated region (orange). The corresponding gains ofthe amplified and damped modes are shown in black and grey, respectively. Around the critical angle ( ϕ ≈ ° ),the eigenmodes become degenerate, and the DSI extends to large absolute values of δ c . Figure adapted andreprinted with permission from Ref. [313] published in 2019 by the American Association for the Advancementof Science. inclusion of cavity field fluctuations renders the system unstable indeed also in regionswhere the mean-field model predicts stability [315]. Using the framework of Lindbladmaster equation, the eigenspectrum of the system was studied. The interpretationwas that the presence of equally spaced imaginary parts of the eigenvalues prevents adephasing of the dynamics and thus leads to a long-time non-steady-state dynamics.This can be contrasted to steady-state situations, where the eigenfrequencies are denseand incommensurate and thus lead to dephasing as it would be the case for a closedsystem. An analysis of the correlation functions of the system revealed indicationsof beyond mean-field behavior such as entanglement induced by dissipation. The de-scribed behavior can be connected to dissipative time crystals breaking a continuoussymmetry, where the description in a rotating frame is given by a time-independentmaster equation [315]. Non-steady-state phases with time-dependent driving
Non-trivial dynamics without reaching a steady state can also be induced by applyinga time-dependent transverse pump lattice, modulated either in amplitude or in phase;see Figure 62(a). Restricting the discussion to the self-organization of the atomicdensity in the low energy sector, the modulation results in a Dicke model with anexplicitly time-dependent coupling rate, even in the frame rotating with the naturalpump frequency.
First investigations of a parametrically driven Dicke model showed that additionalminima in the energy landscape of the system appear already for infinitesimally smallmodulation if the modulation frequency is close to an eigenfrequency of the undrivensystem [31]. For strong enough modulation, the predicted phase diagram features132 b Figure 62. Parametrically driven Dicke model. (a) Sketch of the setup. A BEC inside a dissipative single-modecavity is driven transversely by a pump laser with an oscillating intensity V ( t ). The atoms are coupled to theenvironment with a dissipation rate γ , and photons with a rate κ . (b) Exemplary trajectories of the atomicorder parameter on a Bloch sphere as a function of time in the steady state for different driving parameters.Figure adapted and reprinted with permission from Ref. [316] © metastable phases and according nonequilibrium first-order phase transitions. Includ-ing dissipation both via the cavity and via the atoms stabilizes the system againstthe instability seen on resonance for the closed system, such that a finite modula-tion strength is required to enter the non-steady-state phases [316]. This work alsoinvestigated possible trajectories of the Bloch vector in the non-steady-state phases.Examples of such trajectories are shown in Figure 62(b). For non-steady states in theso-called “dynamical normal phase”, the phase of the cavity field rotates in more or lesscomplicated trajectories and the field amplitude averages to zero. For the atomic de-gree of freedom this corresponds to a quasi-resonant switching between the two ordereddensity patterns allowed by the Z symmetry. Likewise, the superradiant phase dis-plays oscillations, however, around a mean non-zero photon number. The simulationsalso showed the existence of chaotic behavior: While the normal and the self-organizedphases seem to be robust against parametric heating also in presence of the modu-lated drive, the above introduced dynamical normal phase shows ergodic-like behaviorsimilar to the one characterizing the chaotic phases discussed in Section 10.1. Numeri-cal simulations also taking into account atomic collisional interactions and a trappingpotential confirm the existence of the parametrically driven phases [317].The response of a BEC to a modulated field in the case of a cavity in the sideband-resolved limit has been numerically studied using an open system truncated Wignerapproximation [42]. Also in this work, the pump field amplitude was assumed to bemodulated periodically. In presence of this modulation, which effectively imprints twosidebands at the modulation frequency onto the pump field, the system was found toeither relax to a steady state or enter dynamical states. Depending on the modulationfrequency, density waves involving different higher atomic momentum modes withmultiple recoils being transferred along the pump or the cavity direction can be excited.Another theoretical proposal for a periodically driven atom-cavity system is basedon a phase modulated rather than an amplitude modulated pump field [318]. Sucha phase modulation results in a periodically shaken optical pump lattice which theatoms are exposed to. This configuration is expected to lead to self-organization inspecific density waves, depending on the modulation frequency. Different to the caseof an amplitude modulated pump field, here the atoms can also order in a state withdensity maxima at the nodes of the emergent superradiant optical lattice, dynmaicallyswitching between the symmetry broken states. The induced oscillations can have avery low frequency, set by the beating between the mode eigenfrequency and the modu-133ation frequency. This beating leads to a subharmonic response also at incommensuratefrequency ratios, and has been interpreted as incommensurate time crystal.Periodic phase modulation of the cavity-induced optical lattice can also modify theproperties of the superradiant phase transition. For instance, periodic phase modu-lation of a cavity field has been predicted to yield a first-order superradiant phasetransition with a hysteresis loop, provided the modulation frequency is chosen ap-propriately and the bosonic gas is sufficiently-weakly interacting [319]. The periodicphase modulation of a cavity field can also result in cavity-assisted long-range hop-pings for fermionic atoms in an external lattice inside the cavity, leading to topologicalsuperradiant states [320]. The first experimental realization of a modulated atom-cavity system employed thebeating between two different pump-field frequencies [321]. One field is red detunedwith respect to the cavity resonance and drives the self-organization phase transitionin the system for sufficiently strong amplitudes. The amplitude of the second pumpfield is increased from zero only after self-organization has set in. The frequency ofthis second field is closer to, and blue detuned with respect to the cavity resonance,such that it counteracts atomic self-organization. Indeed, a suppression of the self-organized density-wave state was observed with increasing amplitude of the secondpump field. This behavior was present for a wide range of modulation frequencies.It was argued [42,321] that this mechanism of suppressing the self-organized densitywave and enhancing atomic coherence in this setup bears analogy with light-inducedrestoration of superconductivity in hight- T c cuprates. From dynamical point of view,this suppression of the self-organization can be understood from (reduced) rescalingof the atom-field coupling η in the corresponding time-independent effective MagnusHamiltonian [42]. Accordingly, the self-organization threshold is pushed to strongeratom-field couplings. Resonant features as reported in Ref. [316] were not observedemploying this scheme, except for rapid excitations if the modulation frequency hitmultiples of the vibrational frequency of the pump lattice.When an amplitude modulation was implemented experimentally by imprinting in-stead two sidebands onto the pump field as suggested in Ref. [42], a resonant featurewas observed in addition to the suppression of superradiance described above [322].The experimental scheme and observations are displayed in Figure 63. The pump in-tensity is modulated at a frequency ω d according to V ( t ) = V [1 + f cos( ω d t )], whichimprints frequency sidebands at ω p ± = ω p ± ω d . The higher frequency sideband isclose to and blue detuned with respect to the dispersively-shifted cavity resonance.Operating in the sideband-resolved regime is important to select specific atomic ex-citations [323]: In a certain range of modulation frequencies ω d , a different set ofmomentum states is occupied than for usual self-organization induced by an unmodu-lated pump field. For the given parameters, this state correponds to a transfer of threerecoil momenta along the pump axis and one recoil momentum along the cavity axis.Such resonant occupation of specific momentum states, corresponding to a nonequilib-rium density wave, has also been theoretically predicted (see above) if the modulationfrequency is close to multiples of the atomic momentum-excitation energy [42,322].Interestingly, the observed density wave is a subradiant non-steady state where thecavity occupation is suppressed while higher order momentum states are occupied. Asshown in Figure 63(b), the cavity field decays rapidly once the amplitude modulationis switched on. At the end of the sequence absorption images are taken after ballistic134 c b (b)(c)(d)30 35 40 45 500.00.20.40.6 Modulation frequency d /2 [kHz] F - F (a) dcab ef Figure 63. Dynamical self-organization of a BEC in a standing-wave cavity driven transversely by anamplitude-modulated pump lattice. (a) The carrier of the pump field ω p is red detuned with respect to thedispersively shifted cavity resonance at ω c, eff . The amplitude modulation leads to sidebands at frequencies ω p ± . (b) The driving protocol is shown in grey, where the pump lattice depth is first increased linearly suchthat the atomic system self-organizes. Deep in the self-organized phase the amplitude modulation is rampedup. The blue line indicates the mean intra-cavity photon number. (c) Dynamical phase diagram showing thedifference of the occupations in a state induced by the modulation F , (a density wave with three recoil mo-menta along the pump direction and one recoil momentum along the cavity direction) and in the state inducedby standard self-organization F , (one recoil momentum along both pump and cavity directions) as a functionof modulation strength and modulation frequency. (d-f) Absorption images of the atomic cloud after ballisticexpansion for a modulation frequency of 2 π ×
40 kHz and modulation strength (d) f = 0 .
60, (e) f = 0 .
47, and(f) f = 0 .
10. Figure adapted and reprinted with permission from Ref. [322]. expansion; examples are displayed in Figures 63(d)-(f). Away from the resonance, mo-mentum peaks signaling standard self-organization are observed, while on resonancethe occupation of higher momentum states is visible.Theoretical analysis provided together with the experiment shows that the self-organization in this nonequilibrium density wave takes place at unstable positions, andthat the atomic pattern is expected to periodically switch between symmetry brokenstates, similar to the discussion in Ref. [316]. The theoretically predicted transition tochaotic behavior for increasing modulation depth could not be observed experimentallydue to driving-induced heating and accordingly shorter lifetimes.A recent experimental work [324] on time-modulated pump fields employed a het-erodyne detection scheme to gain access to the phase of the light field leaking fromthe cavity. The observed period doubling of the atomic response to the modulationwas interpreted as a dissipative time crystal.
11. Cavity-enhanced quantum measurement in quantum gases
In the previous sections we demonstrated that ultracold quantum gases trapped to-gether with light within high-Q optical cavities constitute an incredibly rich field ofmodern AMO physics with growing connections and application prospects towardssolid-state physics. In many respects the specific quantum properties or quantum fluc-tuations of driving laser light fields and cavity modes can be neglected and a mean-field description in terms of coherent fields yields a valid and reliable description (seeSection 2.4 for more details). However, from a wider perspective quantum-gas cavityQED, i.e., the combination of cavity quantum electrodynamics with ultracold quan-tum gas physics, goes well beyond the mean-field picture of the coherent cavity-fieldmode evolution. In its full blossom the field can address and study novel phenomena,135 igure 64. Cavity-based non-destructive measurement scheme: Light scattered by the quantum gas from apump laser is collected and builds up in a cavity mode. The field leaking through the cavity mirrors can beanalyzed by various detection schemes and lead to quantum measurement back-action on the atomic dynamics.Figure adapted and reprinted with permission from Ref. [325] © where the quantum statistical nature of both light and ultracold matter play equallyimportant roles. While the light fields create forces and dynamical potentials withquantum properties, the atoms constitute a dynamical refractive index with genuinequantum properties as well. This wider view has several intriguing consequences as wewill discuss in the following sections. Projective measurements
Beyond the mean-field approximation, a quantum description of the electromagneticmodes allows for dynamic field-matter entanglement beyond classical correlations.Hence, a quantized cavity mode can still induce forces and long-range interactionvia photons even for zero average field-mode amplitude. Consequently, this opens acompletely new venue for projective measurement-based preparation of special many-body states. In particular, a wide class of emerging atomic states can be probabilis-tically prepared via the choice of optical geometry. As generic basic examples, atomnumber-squeezed as well as Schr¨odinger-cat states can be prepared in close analogy tocavity-based spin squeezing [54].As we generally deal with open systems with respect to the cavity-field dynam-ics, the output light continuously provides information about the intrinsic dynamicswith minimal perturbation [325]; see Figure 64. Ultimately it can serve as a quan-tum non-demolition (QND) probe of the spatial distribution, phase diagram, or thephase-transition dynamics of the ultracold quantum gas. As an example, a Mott-insulator state in an optical lattice inside a cavity exhibits clearly distinct light scat-tering properties compared to a superfluid state. In this way the corresponding phasetransition between the two states can be directly monitored from angle resolved scat-tering [50,325] or cavity transmission measurements [326] as shown in Figure 65. Inaddition to being able to monitor non-destructively the orbital state (i.e., Mott in-136
10 15 20 250.00.30.60.9 SFMISFMI (a)0 10 20 30 40 50 60 700.00.10.2 P ho t on nu m be r , a . u . K =68 K =35SF K =10 (c) Probe detuning D p / U (b) Figure 65. Transmission spectra of a cavity filled with a BEC in an external lattice. The transmission peaksdirectly map out the full atom-number distribution of the lattice sites within the cavity mode as shown inFigure 64. (a) A single Lorentzian for a Mott-insulating (MI) phase reflects the non-fluctuating atom number.Many Lorentzians for a superfluid (SF) phase indicate atom-number fluctuations, which are imprinted on thepositions of narrow resonances in the spectrum. For a very narrow bandwidth cavity, i.e. when the cavity decayrate κ is smaller than the cavity light shift per atom U , all cavity transmission are resolved. Here we seethe example of a superfluid of N = 30 atoms in a lattive with M = 30 sites, where K = 15 sites within thecavity are illuminated. (b) The same as in panel (a) but for a worse cavity with κ = U , which gives a smoothbroadened contour for SF. Although the individual peaks are not resolved, the spectra for SF and MI states arestill very different. (c) Same spectra as above for SF with N = M = 70 ( κ = 0 . U ) and different number ofsites illuminated K = 10 , ,
68. The transmission spectra have different forms, since different atom distributionfunctions correspond to different K . Adapted and reprinted with permission from Ref. [326] published in 2006by the Nature Publishing Group. sulator or superfluid) of one-component atoms in an optical lattice inside a cavitythrough cavity output fields, it is also feasible to probe non-destructively the spinorialstate of multi-component atoms through cavity transmission spectrum. This providesa novel approach for non-destructive detection of various magnetic orders in quantumgases [327]. The QND property also allows for intriguing and promising applicationsfor quantum enhanced sensing as in, e.g., a supersolid-based accelerometer in a ringcavity [55]; see Section 11.3 for more details.From a fundamental physics point of view such systems allow detailed studies ofquantum measurement theory for many-body quantum systems ranging from the quan-tum Zeno effect [328,329] to quantum measurement-induced ordering [330]. Implement-ing weak measurement schemes on the field modes even allows controlled steering ofthe atom-field quantum evolution [331] or generation and observation of multi-partiteentanglement via nonlocal cavity-enhanced measurements [58].By trapping atoms inside an optical cavity, one creates optical potentials and forces,which are not externally prescribed but are quantum dynamical variables themselves.Ultimately, cavity QED with quantum gases requires a self-consistent solution forlight and particles, which enriches the picture of quantum many-body states of atomstrapped in quantum potentials. A proper quantum treatment of interaction and back-action turns out to be particularly important to implement quantum simulations or toexploit quantum-enhanced optimization using cavity-mediated tailored and externallycontrolled interactions [62]. Interestingly, a mean-field treatment of the fields here leads137 igure 66. Probability distribution of the population imbalance between odd and even sites for single quantumtrajectories (column1) and averages over 200 trajectories (column 2) for different values of the feedback gain.Column 3 shows the power spectrum averaged over 200 trajectories. In the absence of feedback [panel (a)]the oscillations of the population between even and odd sites are visible only in a single trajectory. For anincreased feedback strength z beyond a creitical value z > z c [panel (b)] the imbalance between odd and evensites soon becomes frozen for each individual quantum trajectory . Below the critical feedback strength z < z c the frequency of the oscillations can be tuned above [panel (c)] or below [panel (d)] the frequency defined bythe tunneling amplitude J . Again, the phase varies from run to run and the oscillatory dynamics is visible onlyin a single quantum trajectory. Adapted and reprinted with permission from Ref. [333] published in 2016 bythe Optical Society. to a significantly reduced success probability hinting for a real quantum advantage [33]. Measurement-induced dynamics and quantum feedback
One can make use of the continuously measured output-field properties for a feedbackon the coupled atom-field system. Applying weak measurements and feedback to aquantum system induces phase transitions driven by fundamental quantum fluctua-tions due to measurements. Non-Markovianity and nonlinearity of feedback enablessimulating special spin-bath problems and Floquet time crystals with long-range andmemory time interactions [59,332]. As an example we show in Figure 66 the simulationof quantum trajectories for a continuous measurement on an ultracold gas in a latticewith continuous monitoring of the cavity outputs [333]. The interplay between mea-surement projection and nonlinear dynamics creates an intriguing dynamical patternfor the atom number distribution on even or odd sites along the cavity axis.Classical feedback derived from the output of a cavity field has been experimen-tally demonstrated [334]. A BEC trapped inside a linear cavity was brought to self-organization by applying a transverse pump field. Using the field leaking out fromthe cavity as a signal, the intensity of the transverse pump laser was regulated by anelectronic circuit such that the mean intra-cavity photon number was stabilized to agiven value. While this photon number decreases without feedback due to atom losses,applying the feedback scheme allowed to stabilize the system over multiple seconds.
Quantum sensing with ultracold gases in cavities
Measurement-induced back-action and feedback are intriguing tools to study genuinequantum physics in action and the measurement outcomes provide in-depth informa-138 - - - - - - - - t [ s ] Δ g / g (a) (b) Figure 67. Supersolid-based gravimeter in a ring cavity. (a) Schematic sketch of the setup. A transverselydriven 1D BEC is coupled to two degenerate counterpropagating modes ˆ a ± of a ring cavity. The cavity istilted with respect to the horizontal direction by an angle θ such that the BEC experiences the gravitationalpotential V ext ( x ) = Mgx sin θ (cf. Section 4.2 and Figure 27). The relative phase φ of the two cavity modesis measured in the cavity output, in order to monitor the motion of the BEC along the cavity axis under thegravitational force. (b) Relative sensitivity for state-of-the-art experimental parameters as a function of time.The red dashed curve presents the sensitivity for κ = 0 (for the sake of comparison) while the blue solid curvetakes into account the effective friction arising from photon loss κ (cid:54) = 0. Figure reprinted with permission fromRef. [55] © tion on the quantum gas dynamics [51]. Furthermore, as quantum-gas–cavity systemsare extremely sensitive to external perturbations, these hybrid systems constitute anew class of quantum sensors [57,335–337]. A remarkable example is a recently pro-posed supersolid-based gravimeter in a ring cavity, which exhibits Heisenberg-like-scaling sensitivity with respect to the atom number [55]. For state-of-the-art experi-mental parameters, the relative sensitivity ∆ g/g of such a gravimeter has been pre-dicted to be of the order of 10 − –10 − for a condensate of a half a million atomswithin a few seconds; see Figure 67.Also the real-time monitoring of Bloch-Zener oscillations via the light field leak-ing from a cavity has been theoretically discussed [53,338,339] and experimentallyexplored [72,340]. Such a method holds the promise to help to increase the data ac-quisition speed in precision measurements of forces.
12. Conclusions, outlook and open challenges
Starting from a short historical overview and an introduction of the basic underlyingphysical models and mechanisms, the central goal of this review has been to presentthe state of the art in the field of quantum-gas–cavity QED (QG-CQED). As somecomprehensive and nice reviews on the classical gas aspects [82], simple cavity-latticemodels [35], and the early quantum-gas developments [64] already exist, for the sake ofcompactness here we concentrated on more recent important developments and out-standing achievements in the field. Nevertheless, the overwhelming recent growth intheory and experiment has rendered the goal of being comprehensive, detailed enough,and complete rather futile. Even well exceeding the initially planned length of about100 pages did not allow us to include everything, which would deserve to be men-139ioned here in sufficient detail. Even while writing this review article intriguing newdevelopments continuously surfaced, we finally had to strongly restrict ourselves andleave some of the most recent developments for future reviews. Hence, we here haveto apologize for some important omissions. Nevertheless, we hope that on the onehand this review gives a good and solid starting point to new researchers interested inentering the field, and on the other hand fosters new and closer collaborations amongthe existing diverse community.Let us wrap things up shortly here: starting from early theoretical ideas and ab-stract proposals somewhat in the style of
Gedankenexperiments at the turn of thecentury [341–343], the field of QG-CQED has enormously grown and expanded in var-ious directions. First aims were driven by theoretical visions of cavity-induced ground-state cooling towards implementing a continuous BEC and a continuous-wave atomlasers [344,345], or creating macroscopic superposition states of larger particle num-bers [346]. Nevertheless, it soon became clear that using a simpler implementationstarting from a zero-temperature degenerate quantum gas prepared by conventionalmeans within an optical cavity, one already realizes a very versatile and well control-lable test ground for many-body physics in an extreme quantum limit. Here cavity-induced thermalization limits the available measurement time, but this process is inmany cases slow enough such that it does not hinder typical experimental sequences.As a particular example, it proved straightforward to emulate a wide class of optome-chanical Hamiltonians at zero temperature [22,24,27], a goal far out of reach of otheroptomechanical systems at that time [347].Almost a decade after theoretical suggestions, a series of cutting edge founda-tional experiments trapping a BEC within a cavity mode has been implemented inZ¨urich [348], T¨ubingen [26], Berkeley [22], and Paris [23]. These allowed transforming
Gedankenexperiments into real setups in laboratories and thus sparked a great deal ofinterest well beyond the ultracold-atom community, in particular, in the solid-state andquantum-information communities. The subsequent multitude of theoretical models,predictions, and suggested applications initiated another wave of experimental efforts,encompassing BECs in recoil-resolved linear and ring cavities [70,131], in multi-modecavities [128], in crossed linear cavities [127], and in folded ring cavities [349], as wellas most recently also of degenerate Fermi gases coupled to a linear cavity [129].A second decisive boost to the field came from the discovery that cavity-mediatedglobal interactions can induce self-ordering of trapped atomic clouds [20,74]. Most in-terestingly, in the zero-temperature regime this corresponds to a reversible quantumphase transition. In a seminal experiment, this self-ordering phase transition was ex-plored in real time [28]. Even more interestingly, a direct mapping of the correspondingHamiltonian to the Dicke model allowed to confirm an almost 50 year old prediction ofthe Dicke superradiant phase transition [28,32], a prediction that was controversiallydisputed in literature for decades with its actual direct realization being still disputedand sought for in solid-state platforms [133,134]. Nevertheless, this proof-of-principledemonstration highlighted the remarkable possibilities for QG-CQED as a versatileplatform for quantum simulation of many-body Hamiltonians with all-to-all interac-tions. With these possibilities at hand a new wave of theoretical work was triggered.With the subsequent extension and refinement of experimental apparatus, experi-ments went well beyond the existing theoretical capabilities. Soon it became clear thatexisting 1D mean-field models, which allow quite general understanding of the emerg-ing physical phenomena, could not adequately account for all observations. This be-came particularly obvious for the case of 3D intra-cavity lattices or setups with severalcavity modes dynamically involved [34,248]. As a notable example, the clear experi-140ental observation of signatures of a lattice-supersolid phase required new theoreticalapproaches such as dynamical mean-field theories [104], Keldysh-type approaches [96],or multi-configuration time-dependent Hartree methods [142]. As experiments becomemore complex but also more precise, these theoretical efforts strongly grow in variousdirections such as, e.g., a thrust to make use of powerful DMRG methods [110], whichturns out to be rather delicate due to the competition of local and cavity-mediatedglobal interactions.In a next phase theory and experiments pushed the field ahead further by exploitingZeeman sub-levels of atomic ground-state manifolds which added an additional spindegree of freedom to the dynamics and paved the way for exploring emergent magneticorders in real time [45–48]. Here the dynamical nature of the cavity fields allowed toextend models for synthetic gauge fields to the dynamical regime [49] previously inac-cessible in free-space configuration. New realms of physics also became accessible byintroducing repulsive optical potentials in the blue-detuned regime allowing to emulatea nonequilibrium form of the Su-Schrieffer-Heeger model [40] as well as entering the do-main of nonlinear dynamical instabilities, deterministic chaos, and a new arising areaof time-crystal formation [38,41]. Using complex resonator geometries even allowed tostudy quasicrystal formation beyond naturally existing crystal symmetries [299].Beyond studying fundamental aspects of quantum physics, the application perspec-tive of QG-CQED has been theoretically discussed already since the early stage ofthe field as the extreme sensitivity of the systems to external perturbations and thereal-time observation capabilities hint for high measurement precision and accuracy.In this context several new proposals for acceleration, force, and magnetic-field sen-sors have recently been put forward [55,57] and are currently refined. In a paralleldevelopment the large versatility of cavity-induced interactions, i.e., in principle al-most arbitrary forms of all-to-all couplings, allowed to envisage novel types of quantumsimulators [65,77] or annealers [62,67]. In some special cases it could be shown thatthese implementations, which typically require a number of laser frequencies, onlyrequire a minimal q-bit overhead so that implementations beyond the capabilities ofclassical computers are within reach of current experimental technology [33]. Recently,in an another direction QG-CQED–based setups for active atomic clocks in the formof so-called superradiant lasers were theoretically proposed [350,351] and many effortsstarted for their experimental implementation and technical refinement [352,353]. Herethe cavity-induced all-to-all coupling among clock atoms is centrally responsible forthe synchronization of the atomic dipoles at very low photon numbers in the spirit ofthe successful micro-maser–based clock devices.Let us also mention that cavity QED with quantum matter recently became possiblealso in solid-state platforms [256,257,259,354–362], potentially attracting the interestof an even larger community. Those platforms partially differ from their ultracold-atomic counterparts both in terms of models and parameter regimes. Nevertheless, alarge number of phenomenological features are still common, so that we believe thatthis review article should be useful also for readers interested in solid-state many-bodycavity QED.Although it is very difficult to make predictions about future diections, we fi-nally point out some recent promising developments and comment on a possible roadmap. As a recent central advancement, first cavity-QED experiments with fermionicatoms have recently been presented [129]. This will bring the field even closer toemulate genuine solid-state Hamiltonians and condensed-matter phenomena, such asmomentum-space pairing in high- T c superconductors [107,208,255]. With more andmore quantum-gas species trapped within resonators, one can also envisage multi-141pecies configurations soon [44], which will pave the way to molecule formation viadark-state-induced collective photoassociation [363] and subsequently superradiantquantum chemistry [364]. Important new possibilities will arise from more complexresonator geometries covering two and three dimensions, as already demonstrated in2D [127], and by the help of higher-order transverse modes [128]. Similarly, the wholemanifold of longitudinal modes available in any optical resonator opens the possibilityto emulate inter-particle forces of virtually arbitrary form. Here the natural frequency-periodic identical nature of cavity and frequency-comb spectra can certainly help toreduce the complexity of corresponding setups [63]. In addition, the combination ofhigh-resolution optical access with QG-CQED experiments may make new schemespossible for the shaping of cavity-induced interactions, but also for the simultane-ous preparation of multiple atomic systems, or for the in-situ imaging of self-orderedatomic structures. Finally, the open system physics of non-steady states and in partic-ular dynamic periodic solutions of many-body systems, often called “time crystals,”has drawn strongly increased recent attention. Here the nonlinear nature of the cou-pled atom-field dynamics and real-time observation capabilities of the cavity fieldstogether with the excellent control over the system make QG-CQED setups a primecandidate for corresponding experimental studies and tests [43,324]. Acknowledgements
We are grateful to Nishant Dogra, Yudan Gao, Sarang Gopalakrishnan, Andreas Hem-merich, Jonathan Keeling, Ronen Kroeze, Benjamin Lev, Brendan Marsh, and ClausZimmermann, for critical reading and valuable feedback on our manuscript. We thankJean-Philippe Brantut, R. Chitra, Tilman Esslinger, Petr Karpov, Corinna Kollath,Giovanna Morigi, and Oded Zilberberg for stimulating discussions. We also thankElvia Colella for the help to create Figure 4.
Funding
F. M. is supported by the Lise-Meitner Fellowship M2438-NBL of the Austrian ScienceFund FWF. F. M. and H. R. acknowledge the International Joint Project No. I3964-N27 of the the Austrian Science Fund FWF and the National Agency for Research ofFrance ANR. T. D. acknowledges funding from the Swiss National Science FoundationSNF: NCCR QSIT and the project “Cavity-assisted pattern recognition” (Project No.IZBRZ2 186312). T. D. and H. R. acknowledge funding from EU Horizon 2020: ITNgrant ColOpt (Project No. 721465). 142
1. Appendix: The most important and commonly-used symbolsthroughout the paperSymbol Definition
Fundamental constants c speed of light (cid:15) vacuum permittivity h Planck’s constant, (cid:126) = h/ π Properties concerning the atoms M atomic mass d ge matrix element of the induced atomic electric dipole moment for thetransition | g (cid:105) ↔ | e (cid:105) ω a atomic transition frequency∆ a atomic detuning, relative frequency between the pump field and theatomic resonance: ∆ a = ω p − ω a γ decay rate of the atomic dipoleΓ natural linewidth of the atomic transition: Γ = 2 γa sττ (cid:48) atomic s-wave scattering length between internal atomic states τ, τ (cid:48) g ττ (cid:48) strength of the two-body contact interactions: g ττ (cid:48) = 4 πa sττ (cid:48) (cid:126) /mN number of atoms T temperature of the atomic cloud V ext ( r ) external trapping potentialˆ ψ τ ( r , t ) atomic annihilation field operator for internal state τ ˆ b j , ˆ c j ( ˆ f j ) bosonic (fermionic) atomic annihilation operator for momentumstate/lattice site jψ τ ( r , t ) condensate wave function for internal state τ Properties concerning the pump field ω p frequency of the pump field λ p wave length of the pump field: λ p (cid:39) λ c k p wave number of the pump field: k p = 2 π/λ p (cid:39) k c (cid:15) p polarization vector of the pump field ω r recoil frequency: ω r = (cid:126) k c / MP optical power of the pump fieldΩ maximum Rabi frequency of the pump fieldΩ( r ) position-dependent Rabi frequency of the pump field (cid:126) V maximum pump-lattice depth: (cid:126) V = (cid:126) Ω / ∆ a (cid:126) V ( r ) position-dependent pump lattice: (cid:126) V ( r ) = (cid:126) Ω ( r ) / ∆ a η maximum two-photon Rabi frequency: η = Ω G / ∆ a η ( r ) position-dependent two-photon Rabi frequency: η ( r ) = Ω( r ) G ( r ) / ∆ a Properties concerning the cavity R ci radius of curvature of mirror il res resonator length g i g -parameter of the cavity: g i = 1 − l res /R ci w beam waist radius of the cavity mode: w = ( λl res /π ) (cid:113) g g (1 − g g )( g + g − g g ) V mode volume of the cavity: V = πw l res / R i , T i , L i reflectivity, transmissivity, and losses of mirror i : R i + T i + L i = 1 ν FSR free spectral range of the cavity: ν FSR = c/ l res Continued on next page
Continued from previous page
Symbol Definition ν lmn resonance frequency of the cavity mode with longitudinal index l andtransverse mode indices ( mn ) ω c cavity resonance frequency: ω c = 2 πν lmn ∆ c cavity detuning, relative frequency between the pump field and the cav-ity resonance: ∆ c = ω p − ω c ˜∆ c complex cavity detuning including cavity dissipation: ˜∆ c ≡ ∆ c + iκδ c dispersively-shift cavity detuning: δ c ≡ ∆ c − (cid:82) U ( r )ˆ n ( r ) d r ˜ δ c dispersively-shift complex cavity detuning: ˜ δ c ≡ ∆ c − (cid:82) U ( r )ˆ n ( r ) d r + iκ = δ c + iκ = ˜∆ c − (cid:82) U ( r )ˆ n ( r ) d r λ c wave length of the cavity field k c wave number of the cavity field: k c = 2 π/λ c (cid:15) c polarization vector of the cavity field E mn ( r ) Hermite-Gaussian wavefunction of TEM mn mode E maximum electric field strength of a single photon in the TEM modewith volume V : E = (cid:112) (cid:126) ω c / (cid:15) V ϕ Gouy mn ( x ) Gouy phase shift associated with TEM mn mode: ϕ Gouy mn ( x ) = (1 + m + n ) arctan( λ c x/πw ) ϕ offset mn Phase offset due to boundary conditions of the cavity: ϕ offset mn = ( m + n ) arctan( λ c l res / πw )ˆ a ν photonic annihilation operator for mode ν with mode indices ν = ( lmn ) α ν coherent field amplitude for mode ν with mode indices ν = ( lmn ): α ν = (cid:104) ˆ a ν (cid:105)G maximum single-photon vacuum Rabi frequency: G = d ge · (cid:15) c E / (cid:126) G ( r ) position-dependent vacuum Rabi frequency: G ( r ) = G E ( r ) / E (cid:126) U maximum cavity-lattice depth per photon (or maximum intra-cavitylight frequency shift per atom): (cid:126) U = (cid:126) G / ∆ a (cid:126) U ( r ) position-dependent cavity lattice per photon: (cid:126) U ( r ) = (cid:126) G ( r ) / ∆ a κ decay rate of the cavity field through cavity mirrors∆ ν full width at half maximum of cavity resonance peak: ∆ ν = 2 κ/ π F Finesse of the cavity: F = πν FSR /κC cooperativity parameter: C = G /κγ eferences [1] A. Micheli, G.K. Brennen, and P. Zoller, A toolbox for lattice-spin models with polarmolecules , Nature Physics 2 (2006), pp. 341–347, Available at .[2] W. Zwerger,
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