Cold atoms in cavity-generated dynamical optical potentials
Helmut Ritsch, Peter Domokos, Ferdinand Brennecke, Tilman Esslinger
CCold atoms in cavity-generated dynamical optical potentials
Helmut Ritsch ∗ Institut f¨ur theoretische Physik,Universit¨at Innsbruck,Technikerstr. 25,A-6020 Innsbruck,Austria
Peter Domokos † Wigner Research Centre for Physics,Hungarian Academy of Sciences,H-1525 Budapest P.O. Box 49,Hungary
Ferdinand Brennecke ‡ and Tilman Esslinger § Institute for Quantum Electronics,ETH Z¨urich, CH-8093 Z¨urich,Switzerland (Dated: 28. September 2012)
We review state-of-the-art theory and experiment of the motion of cold and ultracoldatoms coupled to the radiation field within a high-finesse optical resonator in the dis-persive regime of the atom-field interaction with small internal excitation. The opticaldipole force on the atoms together with the back-action of atomic motion onto the lightfield gives rise to a complex nonlinear coupled dynamics. As the resonator constitutesan open driven and damped system, the dynamics is non-conservative and in generalenables cooling and confining the motion of polarizable particles. In addition the emit-ted cavity field allows for real-time monitoring of the particle’s position with minimalperturbation up to sub-wavelength accuracy. For many-body systems, the resonatorfield mediates controllable long-range atom-atom interactions, which set the stage forcollective phenomena. Besides correlated motion of distant particles, one finds criticalbehavior and non-equilibrium phase transitions between states of different atomic or-der in conjunction with superradiant light scattering. Quantum degenerate gases insideoptical resonators can be used to emulate opto-mechanics as well as novel quantumphases like supersolids and spin glasses. Non-equilibrium quantum phase transitionsas predicted by e.g. the Dicke Hamiltonian can be controlled and explored in real-timevia monitoring the cavity field. In combination with optical lattices, the cavity fieldcan be utilized for non-destructive probing Hubbard physics and tailoring long-rangeinteractions for ultracold quantum systems.
CONTENTS
I. INTRODUCTION 2II. SINGLE ATOMS IN A CAVITY 4A. Mechanical effects of light on atoms in a cavity 41. A two-level atom in a cavity 42. Dispersive limit 53. Semiclassical description of atomic motion 6B. Cavity cooling 81. Cavity cooling with blue-detuned probe light 82. Cavity cooling and trapping with far red-detunedlight 93. Temperature limit 114. Cooling in multimode cavities 11 ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] C. Extensions of cavity cooling 121. Cooling trapped atoms and ions 122. Cooling nanoparticles and relation tooptomechanics 123. Cooling molecules 134. Cooling and lasing 145. Monitoring and feedback control 14III. COLD ATOMIC ENSEMBLES IN A CAVITY 15A. Collective coupling to the cavity mode 151. Cavity-mediated atom-atom interaction 152. Collective cooling, scaling laws 173. Back-action, nonlinear dynamics 18B. Non-equilibrium phase transitions and collectiveinstabilities 181. Spatial self-organization into a Bragg-crystal 182. Collective atomic recoil laser 21C. Phase-space and mean-field descriptions for largeparticle numbers 231. Critical point 242. Stability analysis and phase diagram 253. Non-equilibrium steady-state distributions 27 a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t IV. QUANTUM GASES IN OPTICAL CAVITIES 28A. Experimental realizations 28B. Theoretical description 30C. Cavity opto-mechanics with ultracold atomicensembles 321. Experimental realizations 322. Nonlinear dynamics and bistability for low photonnumber 343. Quantum-measurement back-action upon collectiveatomic motion 354. Cavity cooling in the resolved sideband regime 37D. Non-equilibrium phase transitions 371. Self-organization of a Bose-Einstein condensate 382. Open-system realization of the Dicke quantumphase transition 403. Phases in highly degenerate cavities 42E. Extended Hubbard-type models for ultracold atoms incavities 441. Bose-Hubbard model with cavity-mediatedatom-atom interactions 442. Cavity-enhanced light scattering for quantummeasurement and preparation 453. Self-consistent Bose-Hubbard models in cavitymean-field approximation 45V. OUTLOOK 50References 51
I. INTRODUCTION
Laser light is a versatile tool to cool, prepare and ma-nipulate atoms. Laser cooling (Chu, 1998; Cohen Tan-noudji, 1998; Phillips, 1998) and optical pumping (Hap-per, 1972) relies on spontaneous emission, which is par-ticularly important if the laser frequency is tuned closeto the energy of an atomic transition. It is suppressed ifthe laser frequency is tuned far from any internal excitedatomic state. In this limit coherent scattering of photonsdominates and the resulting light force, the dipole force,can be derived from an optical potential proportional tothe laser intensity inducing a Stark-shift. This forms thebasis for trapping and the manipulation of cold atoms(Grimm et al. , 2000), Bose-Einstein condensates (Cor-nell and Wieman, 2002; Ketterle, 2002), quantum gases(Bloch et al. , 2008; Giorgini et al. , 2008) and mesoscopicparticles (Gordon and Ashkin, 1980), where spontaneousemission has to be avoided. In free space the back-actionof the particles onto the trapping laser is negligible. In amicroscopic picture, this means that the probability of aphoton to be scattered by a particle is so small, that thechance for a second scattering event involving the samephoton is negligibly small. Hence, the modifications ofthe field are not felt by the particles and the light formsa conservative optical potential.The situation changes drastically when the light fieldis confined in a high-quality optical resonator. Due tomultiple round trips of intracavity photons not only thedipole force gets strongly enhanced, but also the back-action of the atoms on the light gets significant. Sinceatomic motion and cavity field dynamics influence each other, they have to be treated on equal footing. In mostcases the dipole force then can no longer be derived froma conservative potential (Horak et al. , 1997) and the fielddynamics get nonlinear (Vukics et al. , 2009).To get an intuitive picture, consider for example a mov-ing point-like atom, or an entire atomic cloud, forminga dielectric medium with refractive index inside a cavity.This induces a phase shift on the light field that dependson the position and shape of the medium relative to theresonator mode structure. Correspondingly, the cavityresonance frequency is dynamically shifted with respectto the empty cavity. If this shift is comparable to the cav-ity linewidth, the cavity field intensity, induced by an ex-ternal pump laser, can undergo a resonant enhancementand so can the back-action on the motion of the medium.For several atoms this coupled atom-field dynamics hasthe character of a long-range inter-particle interaction. Italso generates a strong nonlinear field response, even ifthe particles are linearly polarizable, as e.g. atoms in thelow saturation regime. Coupling to further light modesgives rise to interference effects, which are the origin ofcollective instabilities and self-organization phenomena.Photons leaking out of the cavity cause a damping of thiscoupled dynamics. This designable decay channel can beutilized to cool the motion of the medium independentof its specific characteristics.Historically, cavity quantum electrodynamics (QED)was born as a research field devoted to studying the ra-diation properties of atoms when boundaries are present(Berman, 1994; Haroche, 1992; Purcell, 1946). Advancesin cavity technology over more than 30 years allowedto reach, both in the microwave (Raimond et al. , 2001;Walther, 2002) and optical (Kimble, 1998; Mabuchi andDoherty, 2002) frequency domains, the strong couplingregime where the coherent interaction between an atomictransition and a single radiation field mode dominatesover all dissipation processes. As a next step, cold andslow atoms have been integrated successfully within op-tical cavity QED experiments, which led to significantcoupling of the atomic motion to the cavity field. Itgot possible to generate sufficiently strong forces in orderto trap an atom in the field of a single photon (Hood et al. , 2000; Pinkse et al. , 2000). Several experimentsachieved strong coupling even in the dispersive regime ofcavity QED where the detuning between the light fieldand the internal atomic transitions is large. Althoughthe resonant energy exchange between atom and field issuppressed in this regime, the position-dependent cavityfrequency shift exceeds the cavity linewidth. Motion-induced changes of the effective resonator frequency andits back-action on mechanical motion is also the physicalground of cavity optomechanics (Kippenberg and Vahala,2008), which can be considered as an extension of disper-sive cavity QED towards macroscopic objects.In this review we survey the recent advancements ofcavity QED systems in which coherent momentum ex-change between particles and radiation field is the domi-nating effect of the light-matter interaction. The externaldegree of freedom of the material component ranges fromthe center-of-mass motion of a single atom, or a cloud ofcold atoms, to the density distribution of a continuousmedium such as the quantized matter-wave field of anultracold gas.The review illuminates different generic features of thecavity-generated optical dipole force and is structuredin three main sections. Briefly summarizing, in Sect. IIwe discuss the consequences of the retardation betweenatomic motion and the cavity field dynamics. This timedelay leads to an irreversible dynamics that can be thebasis of cooling schemes, as presented for single atomsin a cavity. Sect. III discusses how the field modificationinduced by an atom acts back on the motion of otheratoms moving within the cavity. This cavity-mediatedatom-atom interaction is a source of collective effects inatomic clouds. Finally, in Sect. IV we consider the col-lective dynamics of an ultracold gas induced by its strongcoupling to the cavity field. Owing to the low temper-ature, the dynamics involves a reduced set of motionaldegrees of freedom, and the system becomes a realizationof various paradigmatic models of quantum many-bodyphysics and quantum optics (Lewenstein et al. , 2007).The most elementary situation that we will discussis the dispersive atom-field dynamics of a single atom,or polarizable particle, inside a laser-driven high-finessecavity (Domokos and Ritsch, 2003; Pinkse and Rempe,2002). The cavity field dynamically responds to the po-sition and velocity of the particle, thereby generating atime-dependent dipole force acting back onto the parti-cle motion. It is the finite response time of the cavityfield which gives rise to the velocity dependent compo-nent of the force. It can have the character of a frictionforce shuffling kinetic energy from the particle to thecavity field and dissipating it via the cavity loss chan-nel. (Vuleti´c and Chu, 2000). This allows for coolingand self-trapping of particles in the cavity field (Maunz et al. , 2004; Nussmann et al. , 2005). Sub-recoil cavitycooling of an ultracold atomic cloud has been achievedrecently (Wolke et al. , 2012), which paves the way to-wards reaching quantum degeneracy without relying onevaporative cooling techniques. Cavity cooling allows forslowing of any sufficiently polarizable particle with smallabsorption, without the need of a cycling transition. Thepossibility of extending the applicability of cavity coolingbeyond atoms has been the subject of extensive researchin the past years. The light field leaking out the cav-ity carries information on the trajectory of the particle(Hood et al. , 1998; Maunz et al. , 2003). Continous mon-itoring of the atomic motion, in turn, can be used forfeedback control (Fischer et al. , 2002), which became bytoday the standard tool to capture single atoms inside acavity for quantum manipulation (Kubanek et al. , 2009).For cold atomic ensembles inside a laser-driven cavity (Elsasser et al. , 2004; Kruse et al. , 2003b) the atom-fieldcoupling strength increases and the dynamics becomesmore complex. In many cases the effective couplingstrength between particles and cavity field scales with thesquare root of the particle number (Raizen et al. , 1989;Sauer et al. , 2004; Tavis and Cummings, 1968; Tuchman et al. , 2006). As a consequence of this, the cooling of thecenter-of-mass motion is correspondingly more effective.Additional complexity arises from the relative motion ofthe particles, as the local intensity of the cavity field ex-perienced by one atom depends on the position of allother atoms (Horak and Ritsch, 2001c). This gives riseto an effective long-range (M¨unstermann et al. , 2000),or global atom-atom interaction, described by an overalldispersive shift. The contribution of each particle to thisshift depends on the local field intensity, which is pro-portional to the square of the cavity mode function atthe position of the atom. In the low excitation regimethis can be captured by a collective potential. Further-more, dissipative forces acting on the relative motion ofthe particles have been identified (Chan et al. , 2003) andinteresting correlations between particles can build up(Asb´oth et al. , 2004).The cavity-meditated long-range interactions have adifferent character when the mode of the driving fieldis not identical with the cavity mode. In this case,the atoms can be considered as sources for the intra-cavity field and interference between these sources be-comes crucial. Correspondingly, the effective cavity driv-ing strength depends on the position of all atoms withinthe cavity mode profile and it is the field amplitude ratherthan its intensity, which mediates the long-range interac-tion. For the case of a transversally laser-driven atomicensemble in a linear cavity, the long-range interactioncauses a phase transition to a self-organized phase, inwhich the atoms arrange themselves in a checkerboardpattern, thereby maximizing scattering into the cavitymode (Black et al. , 2003; Domokos and Ritsch, 2002).In a unidirectionally driven ring cavity geometry, col-lective scattering between the two counter-propagatingcavity modes results in a collective instability, referredto as collective atomic recoil lasing (Kruse et al. , 2003a).Various mean-field type theories can be used to describethe non-equilibrium dynamics and asymptotic behaviorof large atomic ensembles, including the derivation ofscaling laws characterizing the above described criticalphenomena (Asb´oth et al. , 2005; Grießer et al. , 2010).Coupling ultracold atomic ensembles or Bose-Einsteincondensates to the radiation field inside a high-finesseresonator, requires a quantized description of the atomicmotion and reduces the number of relevant external de-grees of freedom (Brennecke et al. , 2007; Colombe et al. ,2007; Gupta et al. , 2007; Slama et al. , 2007a). In thecase of a laser-driven cavity, situations can be realizedwhere the cavity field couples dominantly to a single col-lective motional mode of the atomic ensemble, provid-ing a direct analogy to cavity optomechanics (Brennecke et al. , 2008; Murch et al. , 2008; Stamper-Kurn, 2012).Coupling a laser-driven Bose-Einstein condensate to thevacuum field of a cavity, leads to a quantum phase tran-sition between a superfluid and a self-organized phase(Baumann et al. , 2010; Nagy et al. , 2008). This providesan open-system realization of the Dicke Hamiltonian andits quantum phase transition (Dicke, 1954; Dimer et al. ,2007; Hepp and Lieb, 1973; Nagy et al. , 2010). The self-organized state can also be considered as a supersolidresulting from a broken Ising-type symmetry. More com-plex situations occur in highly degenerate multi-modecavities (Gopalakrishnan et al. , 2009, 2011b; Strack andSachdev, 2011).Ultracold gases in optical lattices are one of the mostintriguing systems in which the power of atomic and laserphysics can be exploited to explore generic phenomenaof solid-state physics (Bloch et al. , 2008). The Hubbardmodel describing the dynamics of periodically arrangedbosons or fermions can be de facto realized with ad-justable parameters and variable dimensionality. Whenthe optical lattice potential is created by the field sus-tained by an optical high-finesse cavity, the correspond-ing cavity Hubbard-model predicts exotic new phases ofmatter (Larson et al. , 2008a; Maschler and Ritsch, 2005).In many cases the cavity fields provide for a convenient,built-in real-time observation tool. Analyzing the emit-ted fields allows for dynamical monitoring of quantumphase transitions with minimum and well controlled mea-surement back-action (Mekhov et al. , 2007c).
II. SINGLE ATOMS IN A CAVITY
A central objective of cavity quantum electrodynam-ics (QED) is the perfect control of light–matter inter-action at the single-atom and single-photon level in theregime of strong coupling where atom and cavity fieldform a single entity. A long lifetime of such an ‘atom-photon molecule’ requires slow and very cold atoms toensure long interaction times and precise control of theatomic position. At sufficiently small kinetic energies,however, the light forces induced by even a few intra-cavity photons influence the atomic trajectory. The firstcavity QED experiments with cold atoms have alreadymanifested that the cavity light forces guide or deflectslowly moving atoms. In addition, extra diffusion takesplace in cavity-sustained dipole traps which may removethe atom from the interaction volume. Clear signaturesof such effects have been observed in transmission spec-troscopy experiments (Hood et al. , 1998; Mabuchi et al. ,1996; M¨unstermann et al. , 1999). Time-resolved detec-tion of the transmitted light signal allowed for the re-construction of atomic trajectories (Hood et al. , 2000;Pinkse et al. , 2000). These experiments set the stage toinclude the atomic center-of-mass degrees of freedom and the optical forces in the cavity QED theory. In the fol-lowing decade, the theoretical and experimental effortsresulted in an extension of the interaction time from thetransit-time range of microseconds to the range of min-utes (Figueroa et al. , 2011; Kubanek et al. , 2011).
A. Mechanical effects of light on atoms in a cavity
The theoretical description of the coupled atom-fielddynamics has been presented in detail by Domokos andRitsch (2003). Here, we recapitulate the notations andmethods. Within the vast field of single-atom cavityQED, we restrict ourselves to the atomic motion in acavity, in particular, to the important concept of cavitycooling. We review the recent experiments demonstrat-ing cavity cooling of single atoms. It is a manifestationof the time-delayed action of the electric dipole force onatoms within the cavity. The understanding at a single-atom level complements nicely another facet of cavitycooling which we will encounter in the case of many atomsystems, where it appears in the form of the imaginarypart of the collective excitation spectrum.
1. A two-level atom in a cavity
We consider a single two-level atom with transitionfrequency ω A coupled to a single mode of the electro-magnetic field inside an optical resonator with resonancefrequency ω C . These frequencies will be referenced tothe frequency ω of an external pump laser by definingthe cavity detuning ∆ C = ω − ω C and the atomic detun-ing ∆ A = ω − ω A . The two relevant atomic states are theground state | g (cid:105) and the excited state | e (cid:105) . We introducethe atomic raising and lowering operators, σ † = | e (cid:105) (cid:104) g | and σ = | g (cid:105) (cid:104) e | . The cavity mode variables are the pho-ton creation and annihilation operators, a † and a , re-spectively. In the electric-dipole and the rotating-waveapproximations and in a frame rotating at the angularfrequency ω , the atom-field coupling is described by H JC / ¯ h = − ∆ C a † a − ∆ A ( r ) σ † σ + ig (cid:0) σ † a f ( r ) − f ∗ ( r ) a † σ (cid:1) , (1)which is usually referred to as the Jaynes–CummingsHamiltonian (Jaynes and Cummings, 1963) and, in thequantum optical context, has been reviewed by Shoreand Knight (1993). The emphasis here is on that theposition r of the atom is explicitly taken into account.The spatial dependence of the atomic detuning, ∆ A ( r ) =∆ A − ∆ S ( r ), may account for a differential AC-Stark shift∆ S ( r ) which can be induced by auxiliary, far-detunedoptical trapping fields. The coupling strength in Eq. (1)is spatially modulated according to the intracavity elec-tric field strength which is proportional to the cavitymode function f ( r ). For the effects reviewed in this pa-per, it is sufficient to consider modulations on the op-tical wavelength scale, thus writing f ( r ) = cos( kx ) fora standing-wave mode of a Fabry-P´erot resonator, or f ( r ) = e ± ikx for the running-wave modes sustained bya ring resonator ( k = ω/c is the optical wavenumber).The maximum coupling strength is given by the single-photon Rabi frequency, g = d (cid:112) ¯ hω C / (cid:15) V , where d isthe atomic dipole moment along the cavity mode polar-ization and V = (cid:82) d r | f ( r ) | denotes the effective cavitymode volume (the maximum of | f ( r ) | is set to 1). Therotating wave approximation relies on that the character-istic frequencies of H JC are much smaller than the opticalfrequency, i.e. , ( | ∆ A | , | ∆ C | , g (cid:28) ω ). The atomic center-of-mass (CM) motion is a dynamical component of thesystem, which is described by the Hamiltonian H mech = p m + V cl ( r ) , (2)where m is the mass of the atom and the term V cl canrepresent an arbitrary external trapping potential. Forthe case of a far off-resonance optical dipole trap, thisterm, together with the differential AC-Stark shift ∆ S ( r )in Eq. (1), fully describe the effect of the trapping laser.The characteristic frequency of the CM motion is givenby the kinetic energy of an atom carrying one unit ofphoton momentum, | p | = ¯ hk . We will use throughout thepaper the notion of recoil frequency (Cohen-Tannoudji,1992), with the notation ω R ≡ ¯ hk m . The system can be excited with a coherent laser fieldat frequency ω , which either drives the cavity mode withdriving amplitude η , or directly the atomic internal de-gree of freedom at Rabi frequency Ω, described by H pump / ¯ h = iη (cid:0) a † − a (cid:1) + i Ω h ( r ) (cid:0) σ † − σ (cid:1) . (3)For the case of pumping the atom with a standing-wavelaser field from a transverse direction perpendicular tothe cavity axis, the spatial mode function is given by h ( r ) = cos( kz ). H pump is effectively time-independentsince we work in the frame rotating at the angular fre-quency ω of the monochromatic pump laser.Cavity QED systems in the optical domain are stronglyinfluenced by dissipative coupling to the vacuum modesof the electromagnetic field environment (thermal pho-tons can be neglected at optical frequencies). Corre-spondingly, the dynamics of the system is described by aquantum master equation (Carmichael, 2003)˙ ρ = − i ¯ h (cid:2) H, ρ (cid:3) + L cav ρ + L atom ρ , (4)with H = H JC + H mech + H pump and ρ denoting thedensity operator for the atomic (motional and internal)and cavity degrees of freedom. The dissipative processes are captured by the Liouville operators in Born-Markovapproximation L cav ρ = − κ (cid:0) a † aρ + ρa † a − aρa † (cid:1) , (5a)describing decay of the cavity field at rate κ , and L atom ρ = − γ (cid:16) σ † σρ + ρσ † σ − (cid:90) d u N ( u ) σe − ik A ur ρe ik A ur σ † (cid:17) , (5b)describing spontaneous decay of the excited state | e (cid:105) atrate γ accompanied by the emission of a photon intothe free-space modes of the electromagnetic field environ-ment. This process involves a recoil of k A = ω A /c ≈ k op-posite to the direction u of the emitted photon, which isaveraged over the directional distribution function N ( u )characterizing the given atomic transition.In general, the full quantum dynamics of the systemdefined by Eq. (4) including all degrees of freedom – theCM motion, the internal electronic dynamics and the cav-ity photon field – cannot be solved analytically even fora single atom.
2. Dispersive limit
For a broad class of cavity QED parameters, atomicsaturation effects are negligible and the atoms can beconsidered as linearly polarizable particles. This holdstrue when the internal atomic variables σ , σ † evolve on amuch faster time scale as compared to the other variablesdue to a large atomic detuning ∆ A or a large spontaneousdecay rate γ . In either case, following the usual techniqueof adiabatic elimination, the atomic polarization operator σ can be ‘slaved’ to the cavity mode and atomic position‘master’ variables. In the absence of direct atom driving,i.e. , Ω = 0 in Eq. (3), one obtains σ ≈ gf ( r ) a − i ∆ A + γ . (6)This approximation is valid if the population in the ex-cited atomic state is negligible (low saturation regime).By inserting the slaved variable σ into H JC and into theLiouville operator Eq. (5b), an effective master equationis obtained. Of particular interest is the large detuninglimit in which the CM motion and the cavity mode arecoupled dispersively by H eff = − ¯ h (cid:0) ∆ C − U | f ( r ) | (cid:1) a † a . (7)It captures, on the one hand, the atom-induced disper-sive shift of the cavity mode resonance frequency whichdepends on the momentary position of the atom. On theother hand, the cavity field gives rise to an optical po-tential ∝ | f ( r ) | felt by the atom whose depth dependson the dynamical photon number. Dissipation can betreated analogously and the effective Liouville operatorwas presented by Domokos et al. (2001). The dispersiveand absorptive effects of the atom are expressed in termsof the parameters U = g ∆ A ∆ A + γ = − ω C V χ (cid:48) , (8a)Γ = g γ ∆ A + γ = − ω C V χ (cid:48)(cid:48) , (8b)respectively. These relations reveal the connection be-tween the cavity QED parameters and the complex sus-ceptibility χ = χ (cid:48) − iχ (cid:48)(cid:48) of a linearly polarizable objectwith electric polarization P = ε χ E . With this con-nection at hand, the theory presented in this sectioncan be used to describe a much broader class of parti-cles than only two-level atoms, and most of the findingscan directly be applied to polarizable particles of sub-wavelength size. In Sect. II.C.3, the linear polarizabilitypicture is refined for the case of molecules.Using the dispersive interaction Hamiltonian, Eq. (7),the quantized one-dimensional motion of a single atomstrongly coupled to single-mode cavity field has been nu-merically simulated (Vukics et al. , 2005). The calculationconfirmed the basic assumption of semiclassical theories(see Sec. II.A.3), stating that the coherence length ofthe atomic wavefunction reduces well below the opticalwavelength after a few irreversible scattering events. Thishappens although in the dispersive limit the coupling tothe environment is provided by cavity photon loss ratherthan spontaneous photon scattering into free space. Anefficient numerical code has been developed providing ageneral framework for Monte-Carlo wavefunction simula-tions of systems composed of the ‘quantum optical tool-box’ (Vukics, 2012; Vukics and Ritsch, 2007).If the atom is laser-driven from a transverse direction,i.e. , Ω (cid:54) = 0 in Eq. (3), the adiabatic elimination of theinternal degrees of freedom leads to σ ≈ g f ( r ) a + Ω h ( r ) − i ∆ A + γ . (9)Consequently, additional terms appear in the effec-tive adiabatic Hamiltonian Eq. (7) and the LiouvilleanEq. (5b). In particular, coherent photon scattering be-tween the transverse laser field and the cavity mode givesrise to the effective cavity pump term H pump / ¯ h = η eff h ( r ) (cid:0) f ∗ ( r ) a † + f ( r ) a (cid:1) , (10)with the effective cavity drive amplitude η eff = ∆ A g Ω∆ A + γ . The atomic recoil accompanied by photon scattering isaccounted for by the spatial dependence of this term.
3. Semiclassical description of atomic motion
In many cavity QED experiments, cold atoms are re-leased from a magneto-optical trap into the resonator vol-ume. As the temperature T of the atoms is well above therecoil temperature, k B T (cid:29) ¯ hω R , where k B is the Boltz-mann constant, one can assume that the reduced densitymatrix is almost diagonal both in position and momen-tum representation. This allows to treat the position r and momentum p of the atom as stochastic c-numbervariables. The regime of ultracold atoms k B T < ∼ ¯ hω R will be treated in Sec. IV. a. Langevin-type equation The separation of the quan-tized internal and the classical motional degrees offreedom was developed for the description of laser-cooling (Cohen-Tannoudji, 1992; Dalibard and Cohen-Tannoudji, 1985; Gordon and Ashkin, 1980). This ap-proach has been adopted to the cavity QED scenario byextending the internal degrees of freedom to the com-bined space of the atomic polarization and cavity mode(Hechenblaikner et al. , 1998; Horak et al. , 1997). Byeliminating the internal degrees of freedom, the dynam-ics of the CM variables can be formulated in terms of astochastic differential equation˙ r = p m (11a)˙ p = f + β p m + Ξ , (11b)where f denotes the classical force and β a friction co-efficient in a non-conservative and velocity dependentforce term. In general, β can be a tensor in the three-dimensional space as atomic motion along any directiongives rise to friction in all three spatial directions (Vukics et al. , 2004). When the eigenvalues of the tensor β (orscalar in 1D) are negative, one encounters cavity cooling .The noise term Ξ induces the stochastic behavior. Ithas vanishing mean value and is defined via the diffusionmatrix D according to (cid:104) Ξ ( t ) ◦ Ξ ( t (cid:48) ) (cid:105) = D δ ( t − t (cid:48) ) , (12)where ◦ denotes the dyadic product. The exact noisecorrelation function has a width in the range of the dis-sipative parameters κ and γ of the internal dynamics.Therefore, it can be approximated by a Dirac- δ onlyon the much slower CM motion time scale set by theinverse of the recoil frequency, ω − R (cid:29) min { κ − , γ − } .The method of calculating the c -number parameters f , β , and D of the Langevin-type equation from the masterequation concerning the internal degrees of freedom waspresented by Hechenblaikner et al. (1998) for the one-dimensional, and by Domokos and Ritsch (2003) for thethree-dimensional case. This method accounts for thequantum effects of the internal dynamics; hence the fullapproach is semiclassical.The practical use of this method is strongly limited:the nonlinear quantum master equation for the internaland cavity degrees of freedom has to be solved numeri-cally and for all atomic positions r . Moreover, the Hilbertspace of the photon field has to be truncated at low pho-ton numbers. This approach was adopted by Doherty et al. (2000) and Fischer et al. (2001) to simulate the ex-periments conducted by Hood et al. (2000) and Pinkse et al. (2000).Analytical approximations can be obtained for the lowatomic saturation regime (Murr, 2003), where the atomicpolarization can be replaced by a bosonic operator andhence the internal dynamics is described by linear equa-tions of motion. It is then possible to calculate the fric-tion coefficient β and, corresponding to its sign, the cool-ing versus heating regions can be mapped as a functionof the detunings ∆ A and ∆ C , as shown for example inFig. 1. D C [ i n un it s o f ] g D A [in units of ] g coolingheating −10 −5 0 5 10−20−1001020 ∆ C γ [ i n un it s o f ] ∆ A γ [in units of ]20−20−10010−10 −5 0 5 10 CC HC HH
FIG. 1 (Color online) Cooling and heating regions as a func-tion of atom and cavity detunings. Shown are contour plotsof the friction coefficient β (averaged over an optical wave-length) acting along the cavity axis on a laser-driven atom.Left: Bad-cavity regime, g = γ/ κ = 10 γ ; Right: Goodcavity regime, g = 3 γ , κ = γ , where the dressed-state picturecan be invoked for interpretation. Blue contour lines indicatecooling (C, β < β >
0) regions. Notethat the spatially averaged friction coefficient is shown here;tightly confined atoms localized within a small fraction of thewavelength can follow completely different behavior depend-ing on their position. b. Semiclassical theory in the dispersive limit
In the dis-persive limit of atom-cavity coupling presented in Sect.II.A.2, there is an alternative semiclassical approach(Domokos et al. , 2001). The Wigner quasi-probabilitydistribution function can be defined in the joint phasespace of the atomic CM motion and the cavity field am-plitude. The quantum master equation translates theninto a partial differential equation for the Wigner func-tion. By dropping all terms containing higher thansecond-order derivatives, the resulting Fokker-Planck equation corresponds to the evolution of classical stochas-tic variables associated with the atomic motion and thecavity field. One can consider this approach as the con-struction of a semiclassical model which lies closest to thetrue quantum dynamics. As compared to Eq. (11), herelarge intracavity photon numbers are allowed, in fact, thevalidity of this approach requires photon numbers largerthan 1.Consider the generic example of a single atom movingin one dimension along the axis of an externally drivenlinear cavity with the mode function f ( x ) = cos( kx ),described by the equations (Domokos et al. , 2001)˙ x = pm , (13a)˙ p = − ¯ hU | α | ∂∂x f ( x ) + ξ p , (13b)˙ α = η − i (cid:0) U f ( x ) − ∆ C (cid:1) α − (cid:0) κ + Γ f ( x ) (cid:1) α + ξ α . (13c)Apart from the noise terms ξ p and ξ α , these equations co-incide with the classical description in the initial cavitycooling paper by Horak et al. (1997). The force in (13b)acting on the atom is formally identical to the gradient ofthe optical dipole potential of the cavity mode. The am-plitude α , however, depends not only on the momentaryposition of the atom but has a memory effect because ofthe finite bandwidth κ of its linear response. The actualforce has therefore a velocity-dependent character whichcan give rise to a viscous friction force, that is, cavitycooling. Within this approach the friction cannot be de-termined by a single coefficient β , on the other hand, thefriction effect is correctly described for arbitrary velocity.The noise sources are taken into account in a consis-tent way as imposed by quantum mechanics. This resultsin non-trivial correlations (cid:104) ξ p ξ α (cid:105) (cid:54) = 0. The result for thediffusion matrix as well as the generalization for severalatoms was published by Asb´oth et al. (2005). The generalmodel is used for numerically studying many-body sys-tems, c.f. Sect. III, well above the temperature of quan-tum degeneracy. c. Scattering model The semiclassical Langevin-typeequation (11) can be constructed without mode decom-position of the radiation field. This approach is requiredwhen, instead of a simple Fabry–P´erot-type cavity geom-etry, one considers an atom interacting with the radiationfield of an interferometer which is composed of an ar-bitrary one-dimensional configuration of beam splitters.To deal with this situation a scattering model has beenestablished by Xuereb et al. (2009) and solved for theforce terms acting on a particle as in Eq. (11) by Xuereb et al. (2010). In the scattering model the atoms andbeam splitters are treated on an equal footing as ‘scat-terers’ characterized by a single polarizability parame-ter. Thereby, a unified framework is created to describeopto-mechanical systems in general, which has revealedthe close relationship between cavity cooling of atomsand radiation-pressure cooling of mirrors (Arcizet et al. ,2006; Gigan et al. , 2006; Kleckner and Bouwmeester,2006; Metzger and Karrai, 2004; Schliesser et al. , 2009).
B. Cavity cooling
One of the most promising results from the under-standing of the complex cavity QED dynamics involvingatomic motion is the realization of cavity cooling, i.e. ,the dissipation of kinetic energy through the cavity pho-ton loss channel in a controlled manner.Early ideas about using an optical resonator to en-hance the efficiency of laser-cooling relied on the modi-fication of the spectral mode density of the electromag-netic radiation field in the presence of spatial boundaryconditions (Lewenstein and Roso, 1993; Mossberg et al. ,1991). In the most general form, the notion of cavitycooling in the perturbative regime has been expressed byVuleti´c and Chu (2000). If an atom, placed inside anoptical cavity, is laser driven at a frequency below thecavity resonance, ∆ C <
0, scattering favors the emis-sion of photons at frequencies higher than the pump fre-quency due to the increased mode density around thecavity resonance. The energy needed to upshift the pho-ton frequency is provided by the loss in kinetic energyin processes of inelastic scattering. With this very sim-ple picture, a robust three-dimensional cooling effect canbe interpreted (Vuleti´c et al. , 2001). However, the pic-ture holds only true in the regime of weak atom-photoncoupling, see the left panel of Fig. 1. When the reabsorp-tion of a photon starts to become non-negligible, whichhappens in a high-finesse cavity, the cooling mechanismsubstantially changes. This drastic change is illustratedin Fig. 1, where the right panel presents the friction co-efficient for a ratio g/κ in the the single-atom strong-coupling regime. In the case of a standing-wave cavity,the dynamical cavity cooling effect can be interpreted inthe frequency domain by means of a Sisyphus-type argu-ment (Horak et al. , 1997), using the dressed-state pictureof the strong-coupling regime of cavity QED (Haroche,1992), see Fig. 2. For the case of a ring cavity, inter-estingly, the intuitive photon scattering picture can bepursued also in the strong-coupling regime and the fullvelocity dependence of the radiation pressure for arbi-trary coupling constant g can be obtained (Murr, 2006).In the following we will survey two regimes where cav-ity cooling has been demonstrated experimentally, andpresent the corresponding intuitive pictures of the cool-ing effect. Both regimes are in the dispersive limit ofatom-photon interaction keeping the atomic saturationlow. x / l ww AC |+>|−>
FIG. 2 Sisyphus-type cooling mechanism underlying thehyperbolic-shaped cooling region in the right panel of Fig. 1.The atomic motion leads to a modulation of the internaldressed-state energy levels |±(cid:105) which are linear combina-tions of the | g, (cid:105) and | e, (cid:105) states with mixing determinedby the mode function f ( x ) = cos( kx ). This amounts to astate-dependent potential for the CM degree of freedom. For∆ A < C ≈ − κ + g ∆ A , the transition from the groundstate | g, (cid:105) to the lower dressed state |−(cid:105) is resonantly excitedat an antinode. Thus the excitation happens more likely atthe minimum of the potential wells, whereas spontaneous orcavity decay transfers the atom-cavity system back to theground state homogeneously in space. From Domokos andRitsch (2003).
1. Cavity cooling with blue-detuned probe light
Cavity cooling of single atoms has been first demon-strated by Maunz et al. (2004) via the observation ofextended storage times and improved localization of sin-gle Rb atoms in an intracavity dipole trap. The trapfield was red-detuned with respect to the atom, however,the cooling was induced by a weak, blue-detuned probefield. The cooling rate has been estimated to exceed thatachieved in free-space cooling methods by at least a fac-tor of five, for comparable excitation of the atom. Maunz et al. (2004) present a very intuitive interpretation of thecooling effect in terms of the classical notion of the refrac-tive index. Consider a standing-wave optical cavity res-onantly excited by a weak probe laser, ∆ C = 0, which isblue-detuned from the atomic resonance by ∆ A = 2 π × >
0, see Fig. 3. The resulting light shift parameter,Eq. (8a), far exceeds the cavity linewidth, U > κ , sothat even one atom can significantly influence the opticalpath length between the cavity mirrors. Due to Eq. (7),the atom placed at a node of the standing-wave modeprofile does not couple to the cavity field and the intra-cavity intensity is maximum. In contrast, placed at anantinode, the atom shifts the cavity resonance towardshigher frequency, i.e. , out of resonance with the probelaser, resulting in a reduced intracavity intensity. In ahigh-finesse cavity, however, the intensity cannot dropinstantaneously when the atom moves away from a node.The induced blue-shift of the cavity frequency at almostconstant photon number leads to an increase of the en-ergy stored in the field, before the photons are able toleak out of the cavity. This occurs at the expense ofkinetic energy of the atoms. The reverse, acceleratingeffect occurring when the atom moves from an antinodetowards a node is much weaker, because the cavity isinitially out of resonance with the probe laser and conse-quently only a small number of photons are present andundergo a corresponding red-shift. This argument alsoreveals that the delicate correlation between the atomicmotion and the photon number variation, underlying thecooling effect, imposes an upper bound on the atomic ve-locity, kv < κ , which sets the velocity capture range ofcavity cooling. FIG. 3 Experimental scheme used for the observation of cav-ity cooling. Single atoms are captured by an optical dipoletrap formed by far red-detuned light in a longitudinal cavitymode which is different from the one used for cavity cooling.The characteristic parameters of the interaction between theweak probe and the atoms are ( g, κ, γ ) = 2 π × (16 , . , et al. (2004). In the experiment, single atoms injected into the cavityare trapped at the field antinodes of a strong intracavitydipole trap. To be detectable in cavity transmission ofthe weak probe beam, the atoms simultaneously have tobe close to an antinode of the probe field mode. As theprobe field induces cavity cooling, the resulting strongerconfinement can be directly read out of the transmittedsignal, as shown in Fig. 4. Time-resolved detection ofthe cavity transmission allowed to extract a cooling rateof β/m = 21 kHz, which is large compared to the es-timated cooling rate of 4 kHz expected for blue-detunedSisyphus cooling of a two-level atom in free space, or withthe Doppler cooling rate of 1 .
2. Cavity cooling and trapping with far red-detuned light
Far-off-resonance dipole traps are commonly used forlong-time capturing and localization of neutral atoms(Grimm et al. , 2000). The suppression of spontaneousemission results in an almost conservative trapping po-tential. However, with the elimination of spontaneousemission ( | ∆ A | (cid:29) γ ), any free-space cooling mechanismalso disappears. The far-off-resonant trapping scheme FIG. 4 Demonstration of cavity cooling. Time-resolved re-duction of the averaged cavity transmission of a weak resonantprobe beam indicating improved localization of the atom atthe antinodes of a far red-detuned trapping field. The closerthe atom resides at an antinode, the larger the detuning of thecavity resonance with respect to the probe frequency. FromMaunz et al. (2004). has been revisited for a strongly coupled atom-cavity sys-tem where the cavity mode provides a new dissipationchannel.Surprisingly, cavity cooling can remain very efficient inthe limit of large atomic detuning | ∆ A | → ∞ . For opti-mal cooling the driving frequency has to be set slightlybelow the cavity resonance frequency ∆ C ≈ − κ + U .Underlying the cooling mechanism is a polariton reso-nance of the strongly coupled atom-cavity system (corre-sponding to the dressed state |−(cid:105) in the weak excitationlimit where only the lowest excitations manifold of theJaynes-Cummings spectrum matters, see Fig. 2). Even if ω A and ω C are very different, the bare cavity resonanceis slightly modified because the photon excitation mixeswith a small amount of the atomic excitation. In an inho-mogeneous system, the mixing leads to a dependence ofthe polariton resonance on the atomic position (see Fig.2). Although the modulation is tiny in amplitude, theresonance is comparably narrow, having a width in therange of κ for the cavity-type polariton. Thus the sys-tem can be very sensitive to the atomic motion and evenslow atom velocities induce large non-adiabatic modula-tions of the steady-state field amplitude (Domokos et al. ,2004; Murr, 2006).For demonstration, we consider the simplest case of anatom moving along the cavity axis in the field generatedby an external driving laser. It has been shown that thestanding-wave cavity field simultaneously traps and coolsthe atom (Vukics and Domokos, 2005). For a standing-0wave mode, f ( x ) = cos( kx ), the cooling rate is given by β γP e = ω R γ ( kx ) ×× g (∆ C − U cos ( kx ))( κ + Γ cos ( kx )) (cid:16) (∆ C − U cos ( kx )) + ( κ + Γ cos ( kx )) (cid:17) , (14)where P e denotes the mean population in the excitedstate | e (cid:105) . Choosing the cavity detuning as ∆ C ≈ − κ + U leads to the optimum friction coefficient which, spatiallyaveraged, reads β γP e = ω R γ (cid:16) gκ (cid:17) . (15)On the left-hand side the cooling rate is normalized to therate of spontaneous photon scattering. This expressionshows that the friction coefficient at a fixed saturation P e is independent of the atomic detuning, which gives riseto the perspective of cooling molecules or other objectswithout closed cycling transition (see Sec. II.C.3). FIG. 5 Transverse pump scheme of cavity QED. Atoms aretransported into the cavity using an optical conveyor belt.Instead of driving the cavity directly, the atoms are trans-versely laser-driven giving rise to photon scattering into thecavity mode. The standing-wave dipole trap yields a largedifferential AC-Stark shift, i.e. , a modulation of the atomicdetuning ∆ A ( r ), see Eq. (1). In this geometry the cavityvacuum field, the weak driving laser tuned according to theoptimum choice in Eq. (15), and the trap laser together forma very efficient three-dimensional cooling scheme. From Nuss-mann et al. (2005). Further studies revealed that the previous setting ofdetunings and intensities can be extended to more gen-eral geometries, including the motion perpendicular tothe cavity axis, or the external driving of the atom in-stead of the cavity. What is required for cooling is an in-homogeneity in the system on the wavelength scale whichleads to a position-dependent steady-state of the coupledatom-cavity system. This inhomogeneity can arise fromthe cavity mode function, as for the result in Eq. (14),but also from a standing-wave pump field, or from thespatially modulated AC-Stark shift in a strong standing-wave laser field (Murr et al. , 2006b). All these sources contribute to the cooling efficiency. The resulting cool-ing effect has been demonstrated experimentally by Nuss-mann et al. (2005), making use of an orthogonal arrange-ment of a cooling laser, a trapping laser and a cavityvacuum mode (see Fig. 5). This combination gives riseto friction forces along all three spatial directions. Theachieved cooling efficiency led to microkelvin tempera-tures and to an average single-atom trapping time in thehigh-finesse cavity as long as 17 seconds, during whichthe strongly coupled atom could be observed continu-ously, see Fig. 6.
FIG. 6 Demonstration of cavity cooling and long-time trap-ping of a controlled number of atoms inside the cavity. (a)Shown is a single trace of the recorded photon-count rate in-dicating the capture of an atom 75 ms after switching on thepump laser (see Fig. 5). Within 100 µ s, the scattering ratereaches a steady-state value. (b) The recorded photon-countrate allows for determining the atom number and the trap-ping time. (c) The analysis of 50 traces, each of 6 s durationand starting with one atom, yields an average lifetime τ of17 s (upper curve), whereas single atoms that are not exposedto the pump laser only reside for 2 . et al. (2005). In the experiment (Fig. 5), a far-detuned standing-wave dipole trap which is oriented perpendicular to thecavity axis is used to transport atoms into the cavity(Dotsenko et al. , 2005; Kuhr et al. , 2003). The combi-nation of controlled insertion of single atoms into andretrieval out of a high-finesse optical resonator with cav-ity cooling led to a deterministic strategy for assemblinga permanently bound and strongly coupled atom-cavitysystem. Long storage times well above 10 seconds and thecontrolled positioning of single or a given small numberof atoms on the submicrometer scale are simultaneouslyavailable (Khudaverdyan et al. , 2008; Nussmann et al. ,12005).The exploration of cavity cooling was a stimulatingand essential step for the experimental achievement ofstrongly-coupled cavity QED systems combined with thecontrol over the atomic motion. With the implementa-tion of free-space laser cooling and trapping techniquesin cavity experiments sufficiently long atom-cavity inter-action times were demonstrated. These achievements ledto remarkable experimental breakthroughs and applica-tions in single-atom cavity QED, recently. For exam-ple, high-precision measurements demonstrated the ba-sic cavity QED model in the optical domain, i.e. , by re-solving the doublet of the lowest-lying excitations of theatom-cavity system Boca et al. (2004) and Maunz et al. (2005), as well as the quantum anharmonic domain ofthe Jaynes–Cummings spectrum (Kubanek et al. , 2008;Schuster et al. , 2008) where squeezed light can be readilygenerated (Ourjoumtsev et al. , 2011). Furthermore, theachieved trapping times permitted the development ofa deterministic single-photon source (Kuhn et al. , 2002;McKeever et al. , 2004), for having full polarization con-trol (Wilk et al. , 2007b), and to realize the long timesought atom-photon quantum interface (Boozer et al. ,2007; Wilk et al. , 2007a) and single-atom quantum mem-ory (Specht et al. , 2011). Many prosperous directions cangrow out from the realization of the elementary case ofelectromagnetically induced transparency with a singleatom (Figueroa et al. , 2011; Kampschulte et al. , 2010),such as for example all-optical switching with single pho-tons.
3. Temperature limit
The extra friction term induced by the cavity is closelyconnected to the modification of the zero-point field fluc-tuations. Indeed, even for large atom-field detuning thediffusion within a cavity-sustained optical dipole trap canbe an order of magnitude larger than for a free-spacefield (van Enk et al. , 2001; Murr et al. , 2006a; Puppe et al. , 2007b). The heating rate due to fluctuations canbe explicitly calculated in a semiclassical approach (seeSec. II.A.3) which allows to estimate the stationary tem-perature attained by the atoms. Under optimal condi-tions one finds the intuitive result k B T ≈ ¯ hκ , (16)which is independent of the atomic parameters. This re-sult was confirmed by numerical simulations (Domokos et al. , 2001) and fits very well to experimental obser-vations. Interestingly, the result remains largely validin the limit where the temperature reaches the recoillimit k B T ≈ ¯ hω R which is not governed by the semi-classical description anymore. For a particle trapped in a harmonic potential with vibrational frequency ν > κ (resolved-sideband regime) efficient ground state cool-ing was proposed by Zippilli and Morigi (2005). Quan-tum interference effects in the spontaneous emission of atrapped particle in a cavity allow for ground-state coolingeven in the bad-cavity regime where ν < κ (Cirac et al. ,1995). This prediction does not contradict Eq. (16), sinceit was made for a strongly localized trapped atom (Lamb-Dicke regime) at a precisely given position, whereas thetemperature limit above assumes spatial averaging overthe cavity wavelength.According to Eq. (16), there seems to be no lowerbound on the temperature as long as the cavity finessecan be increased. However, with decreasing loss rate,the capture range of the cavity cooling mechanism alsoshrinks. This relation is thoroughly discussed by Murr(2006), based on an explicit expression for the frictionforce obtained for arbitrary velocity. When applying verystrong cavity fields, α (cid:29)
1, in close analogy to the meanfield treatment of optomechanical models (Genes et al. ,2008), it is possible to effectively enhance the weak atom-field interaction appearing at very large detunings to aneffective strong coupling g eff = g α at the expense ofintroducing extra fluctuation terms (Nimmrichter et al. ,2010). This setting can considerably speed up the coolingprocess and enhance the capture range, while still leadingto a similar final temperature as given by Eq. (16).
4. Cooling in multimode cavities
The atom-field dynamics qualitatively changes wheninvoking several cavity modes to participate as dynam-ical degrees of freedom. In simple terms, not only themagnitude but also the spatial shape of the optical poten-tial and the associated light forces become a dynamicalquantity.This can be easily demonstrated at the generic ex-ample of a ring cavity geometry (Gangl and Ritsch,2000). In the regime of dispersive atom-field coupling,the atom not only modifies the resonance frequencies ofthe two counter-propagating cavity modes, thereby tun-ing their field amplitudes, but also gives rise to phaselocking by coherent photon redistribution between thecavity modes. This determines the position of the nodesand antinodes of the emergent standing-wave interferencepattern of the cavity radiation field. For a red-detunedpump field, ∆ A <
0, the particle is drawn to an antinodeof the field which, at the same time, gets dragged alongwith the slowly moving atom (assuming kv < κ, γ ). Dueto the delayed response of the intracavity field, however,the particle is permanently running uphill and thus ex-periences a friction force. The two-mode geometry of aring cavity has been shown to result in faster cooling andlarger velocity capture range as compared to a single-mode standing-wave cavity (Gangl et al. , 2000; Schulze2 et al. , 2010). Moreover, the laser pump configurationused for polarization gradient cooling or velocity-selectivecoherent population trapping can be envisaged within aring cavity, for which case very efficient cavity cooling ispredicted without fundamental lower limit on the tem-perature (Gangl and Ritsch, 2001)The more modes in a cavity are available in the vicin-ity of the pump frequency, the smaller is the trans-verse length scale on which the field shape gets mod-ulated in the presence of an atom. On the one hand,this leads to stronger three-dimensional localization ofatoms around their self-generated intensity maximum(Salzburger et al. , 2002). On the other hand, the cool-ing time reduces in a highly-degenerate confocal cavitymore or less quadratically, whereas the diffusion increasesonly linearly with the effective number of modes involved(Domokos et al. , 2002; Nimmrichter et al. , 2010).The scope of cavity-mediated optical manipulation ofatoms significantly enlarges also in the case of many-atomsystems, which we will review in Sect. IV.D.3.
C. Extensions of cavity cooling
The general principle of cavity cooling is expected tobe applicable in a broad range of other systems with dif-ferent radiation field geometries or other material com-ponents.
1. Cooling trapped atoms and ions
There are several experimental systems in whichtrapped atoms are strongly coupled to a high-finesse cav-ity. Ion trap setups have been combined with high-finesse cavities in the moderate coupling regime (Her-skind et al. , 2009; Keller et al. , 2004). There are all-optical schemes, too, where different longitudinal modesof a standing-wave cavity are used to separate the op-tical trap modes from the cooling ones (Maunz et al. ,2004; Schleier-Smith et al. , 2011). State-insensitive cool-ing and trapping of single atoms employing light fieldat magic wavelengths, which induces an almost identi-cal AC-Stark shift of the two relevant electronic states,has been demonstrated (McKeever et al. , 2003). Further,trapping of atoms in low field regions of a blue-detunedintracavity dipole potential has been investigated exper-imentally (Puppe et al. , 2007a). In a similar intracavitydipole trap, the axial atomic motion was cooled down tothe ground state by way of coherent Raman transitionson the red vibrational sideband, meanwhile the atomicmotion was inferred from the recorded Raman spectrumby Boozer et al. (2006).The cavity cooling mechanism operates also in the caseof tightly confined particles. In the Lamb-Dicke regimefor tightly confined particles, (cid:112) ω R /ν (cid:28) ν de- notes the harmonic trap frequency, explicit expressionsfor the cooling and heating rates of the CM motion of anatom, trapped in an optical resonator and driven by alaser field, have been derived both in the regime of weakand strong atom-cavity coupling. In the former, a variantof sideband cooling appears (Cirac et al. , 1995; Vuleti´c et al. , 2001). Experimentally, the cavity cooling of a sin-gle trapped Sr + ion in the resolved-sideband regimehas been demonstrated and quantitatively characterizedrecently (Leibrandt et al. , 2009). The spectrum of cav-ity transmission, the heating and cooling rates, and thesteady-state cooling limit have been measured in perfectagreement with a rate equation theory. The final tem-perature corresponding to 22.5(3) occupied vibrationalquanta was limited by the moderate coupling betweenthe ion and the cavity.The calculations have been extended to the strong-coupling regime, where higher-order transitions betweeneigenstates of the coupled system have been identifiedand novel non-trivial parameter regimes leading to cool-ing have been revealed (Blake et al. , 2011; Zippilli andMorigi, 2005). In the resolved-sideband regime, ν (cid:29) κ, γ ,the discreteness of the vibrational spectrum, which is thesame for the electronic ground and excited states, givesrise to interference between different transition paths inanalogy to the cooling of trapped multilevel atoms (Mo-rigi et al. , 2000). Ground state cooling is achievable ac-cording to the theoretical predictions (Zippilli and Mo-rigi, 2007).
2. Cooling nanoparticles and relation to optomechanics
The fact that cavity cooling requires only linear po-larizability suggests that it could be directly applicableto large objects, such as nanobeads (Barker and Shnei-der, 2010; Chang et al. , 2009), thin reflective membranes(Genes et al. , 2009), or even small biological objects suchas viruses (Romero-Isart et al. , 2010). Moreover, sincemembranes, being macroscopic objects, can have largestatic polarizability (refractive index), the cooling can bemuch more efficient than for single atoms or molecules.Indeed, there is a strong connection between cavity cool-ing of atoms and dispersive cavity optomechanics (Jayich et al. , 2008; Thompson et al. , 2008), which can easily beseized in the framework of the scattering models (Xuereb et al. , 2009). Cavity cooling of membranes experimen-tally shows great success down to the vibrational quan-tum ground state (Jayich et al. , 2011).As the local field strength is strongly enhanced insidea resonator, optical dipole traps can be operated at verylarge detunings, where only the static polarizability ofthe particle is relevant (Deachapunya et al. , 2008; Nimm-richter et al. , 2010). In such a setting of coupled opticaland mechanical systems, the ring cavity with degeneratepairs of counterpropagating modes, or other configura-3tions where degenerate modes are available, can offer therealization of various effective models.Consider, for example, a symmetrically pumped ringcavity. The field can be written as a superposition of thestrongly pumped and thus highly excited cosine modeand the empty sine mode. The cosine mode fulfills twopurposes: (i) it generates the trapping potential, and (ii)it feeds the sine mode through photon scattering off theparticle (atom, molecule, membrane). The model Hamil-tonian is of the form (Schulze et al. , 2010): H = p m − ¯ h ∆ C (cid:0) a † c a c + a † s a s (cid:1) − ¯ hU ( x )+ i ¯ h (cid:0) ηa † c − η ∗ a c (cid:1) , (17)where U ( x ) is the dispersive interaction potential, and a c ( a s ) denote the field amplitudes of the cosine (sine) mode,respectively. Linearizing the position around the trapminimum, we can recover the standard optomechanicalHamiltonian, H = (cid:20) p m + 12 m hU a † c a c ( kx ) (cid:21) − ¯ h (∆ C − U ) a † c a c − ¯ h ∆ C a † s a s − ¯ hU (cid:48) ( a s + a † s ) x , (18)with quadratic coupling to the cosine trapping mode andlinear coupling to the sine cooling mode. As the particlecouples the two modes there appears an energy splittingwhich allows to extract via inelastic scattering kinetic en-ergy from the vibrational motion in the optical trap (El-sasser et al. , 2003). As for standard cavity cooling thefinal temperature in the classical regime is again limitedby the cavity linewidth k B T ≈ ¯ hκ . However, in a verygood cavity, when the pump field is sufficiently strong,one can reach the resolved-sideband regime, where thetrap frequency ν exceeds the cavity linewidth, and thefinal temperature would correspond to less than a sin-gle excitation k B T < ¯ hν . In this ground-state coolinglimit one has to resort to a quantum description of mo-tion and the optical fields. Interestingly, the sine modeautomatically acts as a built-in monitoring system whichcontinuously observes the vibrational quantum state ofthe particle in the cosine mode. Hence close to T = 0one can observe quantum jumps of the particle via thesine mode photon counts (Schulze et al. , 2010).
3. Cooling molecules a. Cooling the translational motion of molecules
Molecularstructure fundamentally alters and complicates the pic-ture conceived for laser-cooling two-level atoms. Uponexcitation from the pump field, the molecule can relaxby either Rayleigh scattering back to the ground state | g (cid:105) at the rate γ Ry or Raman scattering to metastablestates at the rate γ Rn . There is a multitude of metastable molecular states (spin-orbit, rotational, and vibrational)available via inelastic Raman scattering. The generallylow free-space branching ratio γ Ry /γ Rn results in popu-lation shelving after only a few photon scattering events,thereby prematurely quenching the cooling process. Be-cause of the prohibitive expense of building multiple re-pumping laser systems, optical cooling of molecules viafree-space dissipative scattering of photons is thought notto be practicable.Since cavity-assisted laser-cooling relies on the cavitydissipation channel, it has been suggested as a potentialmethod to mitigate Raman loss. Spontaneous photonscattering, in principle, can be entirely suppressed by us-ing large detuning. However, as discussed in Sec. II.B.2,in order to keep the cooling efficiency constant, one needsto preserve a given level of excitation in the atom ormolecule. Therefore, merely the large detuning does notsolve the branching ratio problem of molecules (Lev et al. ,2008). To overcome this severe problem, the use of anoptical cavity with cooperativity parameter much largerthan unity is mandatory, in accordance with Eq. (15). Inthis case the enhanced coherent Rayleigh scattering intoa decaying cavity mode can ensure a vanishingly smallprobability of the molecule to Raman scatter during thecooling time. For CN diatomic molecules, the cavity cool-ing process has been calculated numerically (Lu et al. ,2007). b. Cooling the rotation and vibration of molecules Whiletheoretical models and experiments have mostly concen-trated on the center-of-mass motion of structureless po-larizable particles or two-level atoms, the complex rovi-brational structure of molecules is one of the central ob-stacles preventing efficient laser-cooling of molecules. Inmany common beam sources the initial temperature canbe designed to be low enough to freeze most vibrationsand only leave few rotational quanta (Rangwala et al. ,2003). Nevertheless, the interaction with the coolinglaser light will in general start to redistribute the popula-tion within the rovibrational manifolds strongly alteringthe optical properties of the molecules and hamperingfurther cooling. Only a few exceptions of this rule havebeen discovered and investigated lately (Shuman et al. ,2010). Cavity cooling, however, can in principle be de-signed to counteract this heating process and even fur-ther cool the rovibrational energy of molecule. As anenourmous spread of transition frequencies is required tofacilitate this, it proves advantageous to simultaneouslyapply a multitude of different longitudinal cavity modes(Kowalewski et al. , 2007; Morigi et al. , 2007). Simula-tions show that the rovibrational cooling can be com-bined with motional cooling, e.g., in a trap (Kowalewski et al. , 2011), to get a cold molecular gas in all degrees offreedom. At this point a practical implementation wouldrequire precooling by other methods, such as the opto-4electrical scheme proposed by Zeppenfeld et al. (2009), toachieve sufficient interaction times and densities withinthe cavity mode volume.
4. Cooling and lasing
Collective coherent emission of a laser-driven atomicensemble into the field of an optical cavity accompa-nied by a very fast and efficient cooling of atomic mo-tion was observed in an experiment conducted by Chan et al. (2003). Although the effect has not yet been fullyunderstood, it is attributed to Raman gain within a Zee-man manifold. The combination of cavity cooling withintracavity gain is an intriguing prospect. It was initiallysuggested by Vuletic (2001) to transform a bad cavity ef-fectively into a good cavity with fast cooling towards aneven lower temperature. While the principle idea provesto be correct, a more realistic and detailed modeling,which accounts for fluctuations to consistently treat thegain, gives a higher limit of the achievable temperature(Salzburger and Ritsch, 2006). This observation was alsoconfirmed in the optomechanical regime of cavity cooling,where intracavity gain leads to faster cooling but a higherfinal temperature (Genes et al. , 2009).In a standard setup, the intracavity gain could be gen-erated by an additional inverted medium placed withinthe cavity. This would lead to a technically challeng-ing setup, if one aims to operate in the strong-couplingregime. Interestingly, it turns out that in a conceptuallymuch simpler configuration, the gain can also be providedby the same atomic medium which is aimed to be cooledin the setup. Of course, such a scheme requires a suit-able pumping mechanism which transfers atoms from thelower to the upper level of the cooling transition, with-out introducing too much extra noise. In the ultimatelimit one can envisage a single atom, which is exter-nally pumped within a high-finesse cavity. Stimulatedemission into the cavity mode provides gain to createa trapping potential for the atom. For a blue-detunedcavity this gain simultaneously extracts motional energyfrom the particle and thus provides cooling (Salzburgerand Ritsch, 2004). Fortunately, an inverted atom is ahigh-field seeker in the blue-detuned light field, so thatit will be trapped close to optimal gain. Hence thissetup provides for lasing, trapping and cooling of a sin-gle atom within a resonator forming the most minimal-istic implementation of a laser (Salzburger et al. , 2005).The system can be generalized to several particles, whichstrongly reduces the requirements on the pump mecha-nism (Salzburger and Ritsch, 2006). In the limit of ul-tracold gases in an optical lattice, stimulated optical gainoccurs concurrent with Bose enhanced coherent popula-tion of the lowest energy band. While for a pulsed setupthis constitutes in principle a very fast and efficient cool-ing method, a CW setup could provide a possible route towards the realization of a CW atom laser (Salzburgerand Ritsch, 2007, 2008).
5. Monitoring and feedback control
Starting from the early days of cavity QED, a stronglycoupled atom-cavity system was considered as a number-resolving neutral particle detector (Mabuchi et al. , 1996),a concept which is still being developed and implementedin miniaturized devices (Teper et al. , 2006). Going onestep further, the high-finesse resonator acts as a micro-scope with which the trajectory of individual atoms canbe reconstructed from the recorded cavity transmissionwith high spatial ( < µ m) and temporal ( < µ s) resolution(Hood et al. , 2000). The method can be considerablyimproved by the use of multimode cavities. The particledoes not only modify the phase and intensity of the intra-cavity field, but redistributes light between the differentspatial modes. The output field imaged on a CCD cam-era therefore allows to monitor directly and in real timethe motion of the particle (Horak et al. , 2002; Maunz et al. , 2003). Note that even for incomplete position in-formation at any given time, the most likely trajectoryof single atoms can be reconstructed with the help ofinversion algorithms based on the coupled equations ofmotion.Once the position and motion of the particle areknown, it is straightforward to apply feedback on themotion of a single atom by adjusting the pump lasers tosteer the particle motion within the cavity and increaseits trapping time (Fischer et al. , 2002). The cavity fieldboth provides particle detection and mediates the feed-back force. This method has been successfully refinedby several groups and resulted in an increase of single-particle trapping times by several orders of magnitude(Kubanek et al. , 2009, 2011). By applying controlledand delayed feedback forces on the particle, its kineticenergy can be reduced as well. This kind of feedbackcooling resembles stochastic cooling techniques applied inhigh-energy physics. Strongly enhanced cooling has beenpredicted when the feedback scheme, consisting of time-dependent switching of the trapping field as a functionof the intracavity intensity, is operated in the dispersivebistability regime (Vilensky et al. , 2007). This methodshould also give new prospects to optomechanical setups.5
III. COLD ATOMIC ENSEMBLES IN A CAVITY
New research directions opened in cavity QED whencold and ultracold atomic ensembles were successfullyprepared within high-finesse optical resonators. In themany-body configuration, the common coupling of atomsto the cavity field creates a wealth of new possibilitiesto implement tailored atom-atom interactions over largedistances, an ingredient which usually is absent in free-space cold atom experiments.The atom-atom coupling is mediated by the cavity ra-diation field between the AC electric dipole moments.However, its nature is inherently different from the free-space dipole-dipole interaction. In a cavity, the inter-action strength does not decay with the interatomic dis-tance and depends only on the local coupling of the atomsto the cavity field. Fundamentally, the interaction is notbinary: the ensemble of atoms collectively acts onto thestate of the radiation field which then reacts back on theindividual atoms. This scenario is generally referred to as global coupling . The range of the interaction is given bythe size of the cavity mode, which can be macroscopic. Incases where single-atom strong coupling is not achieved,the collective energy exchange still can be dominated bycoherent interaction.After discussing the nature of the long range atom-atom interaction mediated by a cavity field in variousgeometries, we consider first the many-body influenceon the cavity cooling scheme. Then we address themost spectacular collective effects realized by cold atomswithin linear and ring cavities. Critical phenomena, in-stability thresholds, and scaling laws will be discussedby means of various mean field theories in the end of thissection.
A. Collective coupling to the cavity mode
Resonant coherent coupling between an ensemble of N two-level atoms and a single standing-wave cavity modeis described by the many-body generalization of Eq. (1) H/ ¯ h = − ∆ C a † a − (cid:88) j ∆ A ( r j ) σ † j σ j + (cid:88) j igf ( r j )( σ † j a − a † σ j ) (19)where j = 1 . . . N labels the atoms, and the mode func-tion f ( r ), for simplicity, is real. The atomic ensemble canbe represented by a single collective dipole with effectivecoupling strength only if (i) the atomic motion can be av-eraged out (ii) only the cavity mode is laser-driven, and(iii) the atoms are in the low saturation regime. In thiscase the atoms collectively couple to the cavity mode withan effective strength of g eff = g (cid:113)(cid:80) j f ( r j ) (the sum-mation index runs from 1 to N ). Correspondingly, an N -fold enhancement appears for the many-atom systemin terms of the single-atom cooperativity, C = g / (2 κγ ),which measures the ratio of light scattering into cavitymode versus surrounding vacuum modes (Tuchman et al. ,2006). For example, the strong distortions of the single-atom normal mode splitting in the cavity transmissionspectrum induced by a thermal beam of atoms crossingthe cavity could be interpreted by such a collective modepicture (Raizen et al. , 1989). Employing an optical con-veyor belt, an adjustable number, N = 1 . . . et al. , 2004).In general, however, one has to consider the many-body system composed of a large number of internal andmotional degrees of freedom. We will exhibit this in thefollowing at the simplest nontrivial case of two atoms inthe same mode.
1. Cavity-mediated atom-atom interaction
Let us discuss the character of the cavity-mediatedatom-atom interaction in two different pump geometries,namely, pumping the cavity field either directly or indi-rectly via light scattering off the laser-driven atoms. a. Cavity pumping
Consider N atoms moving in the fieldof a laser-driven optical cavity. The detuning between thedriving laser and the dispersively shifted cavity resonancedepends on the position of all atoms, which in turn expe-rience the optical dipole force of the intracavity field. Forsmall atomic velocities and in the low saturation limit, anadiabatic potential can be deduced (Fischer et al. , 2001) V ( r , . . . r N ) = ¯ h ∆ A | η | ∆ A κ + ∆ C γ atan γκ − ∆ A ∆ C + g ∆ A κ + ∆ C γ , (20)which is analogous to the Born-Oppenheimer approxima-tion used for describing the motion of nuclei in moleculesin the averaged electronic potential. The potential V de-pends on the atomic positions solely via the collectivecoupling strength g eff , and thus is valid for any num-ber of atoms. This is not surprising as the adiabaticforce is calculated by freezing the atomic motion. Theresulting cavity-mediated long-range atom-atom inter-action gives rise to an asymmetric deformation of thenormal-mode splitting as was observed experimentally byM¨unstermann et al. (2000).In the case of two atoms with positions x and x theinteraction potential landscape V ( x , x ) along the cavityaxis is shown in Fig. 7 for two different parameter set-tings. The upper graph corresponds to the experimentalparameters used in the Garching group (M¨unstermann6 [ (cid:104) ] 0 0.2 0.4 0.6 0.8 1x [ (cid:104) ] 0 1 2 3 4V [h (cid:97) /2 (cid:47) ] [ (cid:104) ] 0 0.2 0.4 0.6 0.8 1x [ (cid:104) ] 0 0.1 0.2 0.3 0.4 0.5 0.6V [h (cid:97) /2 (cid:47) ] FIG. 7 The adiabatic cavity potential V as a function of theatomic positions x and x . The cavity is quasi-resonantlyexcited by a pump laser with detuning ∆ C = − κ + U . Thedetuning from the atomic resonance is set to ∆ A = − γ to ensure the suppression of spontaneous photon scattering.In the upper graph, typical experimental cavity parameters( κ = γ/ g = 5 γ ) (M¨unstermann et al. , 2000) have beenused, whereas in the lower graph g was increased fourfold. Inthe first case, the potential is well approximated by a sum oftwo single-particle potentials. In the second case, either bothatoms are trapped or free. From Asb´oth et al. (2004). et al. , 2000). Although the single-atom light shift is com-parable with the cavity linewidth | U | ≈ κ , the effec-tive interaction between the atoms is relatively weak andthe potential resembles the familiar “egg-carton” surfaceproportional to sin ( kx ) + sin ( kx ). For ‘artificially’enlarged atom-field coupling g = 20 γ , shown in Fig. 7(lower graph), the atom-atom interaction strongly affectsthe potential landscape felt by the second atom, depend-ing on the position of the first atom and vice versa. Forthis parameter setting the single-atom light shift is suf-ficiently large, U (cid:29) κ , that removing one atom fromthe cavity antinode makes the potential experienced bythe other atom vanish. Note that the trap is deeper forthe smaller coupling of the upper graph. Interestingly,as was shown by Asb´oth et al. (2004), the motion of thetwo atoms gets correlated even for the parameter settingof the upper graph as a consequence of additional non-conservative forces (see Sec. III.A.2). b. Atom pumping The situation drastically changes ifthe atoms are laser-driven from a direction perpendic-ular to the cavity axis. Intracavity photons are then cre-ated by Rayleigh scattering of laser photons into the cav-ity mode. Due to light interference, the scattered intra-cavity field exhibits a very sensitive dependence on theinteratomic distance. For two atoms separated by oddinteger multiples of the half-wavelength, the correspond-ing scattering amplitudes into the mode have the samemagnitude but opposite sign, resulting in destructive in-terference and a vanishing cavity field amplitude. On theother hand, for atoms separated by even integer multiplesof the half-wavelength, the field components scattered offthe two atoms interfere constructively. Compared to thefield intensity created by a single scatterer, the latter caseyields a fourfold enhancement of the intensity, referred toas superradiance (DeVoe and Brewer, 1996; Dicke, 1954).In the case of pumping directly the atoms the forcealong the cavity axis acting on the individual atoms dueto light scattering cannot be expressed as a gradient ofa collective potential, at variance to Eq. (20). One canadmit this by checking that ∇ i F j (cid:54) = ∇ j F i , where ∇ i isthe gradient with respect to the coordinate r i , and F j is the force acting on atom j . If there was a potential V such that F j = −∇ j V , the two sides should be equalas they are the second derivatives of the potential andthe order of taking the derivatives is irrelevant accordingto Young’s theorem. The fact that the force can not bederived from a potential is not so surprising, in hindsight,as we are dealing with an open system with continuousenergy exchange with the environment and an unlimitedenergy resource in the form of the pump laser. Actually,the existence of a potential Eq. (20) for the cavity-drivinggeometry is the exceptional case.Approximately, in the limit of U , Γ →
0, more pre-cisely N U (cid:28) ( κ, | ∆ C | ), the motion of the atoms is gov-erned by the collective potential V ( r , . . . r N ) = ¯ h η ∆ C ∆ C + κ N (cid:88) j =1 cos( kx j ) cos( kz j ) , (21)where cosine mode functions were assumed for the cav-ity and the pump laser field. The interference effect ismanifest: when scanning the atom-atom distance over awavelength, the contrast of the interference in the cavityfield intensity is unity regardless the atom-cavity cou-pling constant g . This is not the case for cavity pump-ing, where, in the limit of small coupling constant g , theatoms cause only a small modulation of the cavity in-tensity. Therefore, the atom pumping geometry lendsitself to observe spectacular many-body effects even inthe weak-coupling regime.The superradiant light scattering into the cavity is thebasis of various collective dynamical effects, which havebeen more profoundly studied theoretically. Although we7 -3 -4 Π Π Π i /x λ /x λ aa (a) (b) photon number atomic populations FIG. 8 Collective scattering of two atoms into the cavitymode including internal atomic excitation. (a) Mean numberof photons scattered into the cavity mode at pump–cavity res-onance, ∆ C = 0, with coupling strength g cos( kx ). The atomsare placed at x = 0 and x , respectively. (b) Excited statepopulations Π and Π of the two atoms (dashed and solidlines). The parameters are κ = 0 . γ, η eff = γ, g = 10 γ, ∆ A =100 γ . From Zippilli et al. (2004b). mostly neglect atomic saturation effects in this review, itis important to reveal modifications of the interferenceeffect in the collective scattering when a small but finiteatomic saturation is taken into account. Since the satura-tion also depends on the relative distance of the particles,new types of nonlinear behavior take place. For exam-ple, as shown in Fig. 8, the destructive interference for aseparation of half-wavelength between the atoms is notperfect any more and the photon scattering generates anonclassical cavity field with zero amplitude but finitephoton number (Vidal et al. , 2007; Zippilli et al. , 2004a).
2. Collective cooling, scaling laws
As was discussed in Sec. II.B.1, the cavity cooling forceon single atoms stems from a delicate correlation betweenthe atomic motion and the retarded dynamics of the cav-ity field. In a many-atom system it is at first unclearwhat happens to these correlations in the presence ofother moving atoms. Furthermore, the cavity-mediatedcrosstalk between atoms has a component sensitive to theatomic velocities (Domokos and Ritsch, 2003), i.e., atom1 moving at velocity v induces a linear friction force onatom 2, which might yield correlations in velocity space.To answer this question, one can straightforwardly gen-eralize the semiclassical model, presented in Sec. II.A.3,for many atoms. In general, however, this leads to ananalytically intractable problem. The dynamics of themany-atom system cannot be reduced to that of an ef-fective mode, as was the case for the adiabatic poten-tial Eq. (20) for atoms at rest. The two-atom case hasbeen discussed in detail by Asb´oth et al. (2004), whofound, using the parameter regime of Fig. 7a, a buildupof strong correlations in the motion of two atoms due tothe velocity-dependent cavity forces.The scaling of the cavity cooling efficiency with the number of particles has been studied by means of numer-ical simulations for N = 1 . . .
100 in the limit of a weaklydriven single-mode field, where the optical dipole poten-tial negligibly perturbs the free motion of atoms alongthe cavity axis (Horak and Ritsch, 2001c). If the param-eter U is chosen sufficiently small so that the collectivelight shift is still below the cavity linewidth, N U < κ ,the rate of kinetic energy dissipation is independent ofthe number of atoms. This suggests that the individualatoms in the cloud are cooled independently from eachother, although they are all coupled to the same cavitymode. This holds only in the weak-coupling limit whichis not practical for cooling since the cooling time is long.When the collective coupling to the cavity mode is sig-nificant with respect to the linewidth κ , the scaling be-havior of cooling with the number of atoms has beenstudied by keeping N U and η/ √ N constant while vary-ing the atom number N . The former ensures an identi-cal maximum collective light shift induced by the atoms,the latter amounts to a nearly constant optical potentialdepth (proportional to U η /κ ). With this rescalingof the parameters, the effect of individual atoms on thecavity field diminishes as the number of atoms increases.The final temperature was found invariant, however, thecooling time increases linearly with the atom number N .As long as the driving η is weak enough to result in ashallow optical potential depth, in which the atoms movealmost freely, all the motional degrees of freedom alongthe cavity axis are cooled.In the limit of tightly confined atoms, both theoreti-cal calculations (Asb´oth et al. , 2004; Nagy et al. , 2006a)and experiments (Schleier-Smith et al. , 2011) proved thatonly the center-of-mass motion is damped by the cavity-induced friction force (Gangl and Ritsch, 1999, 2000). Anefficient sideband cooling scheme has been proposed byElsasser et al. (2003) for particles confined in the opticallattice potential generated by two counter-propagatingdegenerate modes of a ring cavity. The scheme relies onthe collective atom-field coupling which lifts the degen-eracy and creates two standing-wave modes phase-lockedby the back-scattering of light. The lower-lying mode sus-tains the optical lattice with an intensity adjusted suchthat the upper-lying mode becomes resonant with thevibrational anti-Stokes Raman transition. The sidebandcooling allows to reach the vibrational ground state.A collective enhancement of friction on the center-of-mass motion has been demonstrated experimentally inthe transverse pump configuration, as an accompanyingeffect of the self-organization into a Bragg-scattering lat-tice (see Sec. III.B.1). Peak decelerations of − m / s have been observed, and the damping effect has beendemonstrated with light-atom detunings up to ∆ A / π = − A / π ≈ − − and temperatures as low as7 µ K) have been observed in another set of experiments8
For short T off , the relative phase is predominantly ! ! ! . As T off increases, relative phases of ! ! ! " appear, and the difference f " f " decays with a timeconstant ! $ % $ s . Optical-path-length drifts inthe interferometer then cause a slow decay in f and f " ,and an increase in f " = to the 10% random-phase level.Having confirmed a mechanism for the emission, weinvestigate the forces on falling atoms interacting withthe pump light after the MOT is extinguished. The col-lective emission into the cavity leads to strong damping ofthe c.m. motion of the atoms, an effect qualitativelysimilar to that predicted by Gangl et al. for atoms in amultimode ring cavity [11]. The TOF measurements in-dicate that, following emission, a large fraction of theatoms is substantially decelerated (inset of Fig. 4). Foratoms with an initial velocity v !
15 cm = s at the time ofillumination, decelerations of about
300 m = s are ob-served, with typically a third of the atoms being slowed.The TOF trace shown indicates a final velocity of = s ,while those with the largest delays indicate that the cloudis stopped before the emission ends.For driven atoms falling at velocity v > , the mea-sured intracavity power is modulated at a frequency giveninitially by twice the atomic Doppler effect ! d ! kv (Fig. 4). This modulation can be explained as the beat notebetween fields emitted upward and downward, which areDoppler shifted by & ! d . Alternatively, it can be ex-plained by the spatial variation with period % = of theatom-cavity coupling, whereby falling atoms cannot emitat a node of the intracavity standing wave. The changingmodulation frequency then indicates the atomic accelera-tion, as displayed in Fig. 4. These measured accelerationsare consistent with the TOF measurements. The peakaccelerations, up to " = s , occur shortly after theonset of the collective emission. Heating of the delayedatoms is consistent with recoil heating at the photonscattering rate into the cavity. While most of our dataare taken in the range " : ’ & a = " ’ " :
57 GHz , we observe damping for pump-atom detunings up to & a = " ( " : , limited only by the pump intensityneeded to reach threshold. For blue pump-atom detuningsup to & a = " ( : there is collective emission butno c.m. damping.Since the pump-atom detuning & a is much larger thanthe atoms’ excited-state hyperfine splitting [Fig. 1(a)], thetransition F g ! ! F e can be treated as an open, two-level system with decay to the F g ! ground state. If therepumper counteracting the decay is turned off during theapplication of the pump beam, the ensuing TOF traces aresimilar to that displayed in the inset of Fig. 4, with asimilar fraction of 40% of the sample being slowed. Foran initial c.m. velocity of v !
15 cm = s the Bragg scat-tering lasts up to $ s , slowing the sample to a final µ s) D e c e l e r a t i on ( m / s ) t (ms) T O F E m i tt ed P o w e r ( µ W ) FIG. 4. The emitted power (thin line) during the illuminationof a falling atomic cloud. The beat frequency is used tocalculate the deceleration (thick line). The inset shows thetime-of-flight signal, in arbitrary units, of atomic clouds with-out (gray line) and with (black line) a $ s exposure to thepump beam. Here the pump I=I s ! , & a = " !" :
580 GHz , and ! c = " ! "
10 MHz . The atom number is N ! : ) . E m i ss i on π φ (a)(b) FIG. 2. Simultaneous time traces of the intracavityintensity (a) (arbitrary units) and field phase (b). The param-eters are N ! : ) , & a = " ! " :
59 GHz , ! c = " !"
20 MHz , and
I=I s ! . off ( µ s) F r a c t i on o f P u l s e P a i r s λλ xz ∆φ = π /2 ∆φ = 0 ∆φ = π FIG. 3. The fraction of pulse pairs with relative phase shift ! ! is plotted versus pulse separation time. The solid circles,open circles, and solid squares correspond to ! ! ! $ " = , " $ " = , and " = $ " = , respectively. The parameters arethe same as for Fig. 2, except here N ! : ) . The insetshows the two possible lattice configurations producing relativephase shift ! ! ! " in the emitted light. P H Y S I C A L R E V I E W L E T T E R S week ending14 NOVEMBER 2003 V OLUME
91, N
UMBER
FIG. 9 Observation of collective friction force on the center–of-mass motion. The cavity output power (thin line, leftscale) is shown during illumination of a freely falling atomiccloud (initial velocity 15 cm/s) with a transverse laser beam.The initial increase signifies the self-ordering into a Bragg-scattering lattice (see Sec. III.B.1). Deceleration of the center-of-mass motion is recorded via the beat signal recorded over300 µ s. The modulation stems from the spatial variation of theatom-cavity coupling with period λ/
2, whereby atoms can-not scatter at a node of the intracavity standing wave. Thechanging modulation frequency indicates the atomic deceler-ation (thick line, right scale). The inset shows the densityprofile (a.u.) of the atomic cloud after free expansion with-out (gray line) and with (black line) a 400 µ s exposure to thepump beam. A fraction of about one third of the atoms isdelayed significantly in accordance with the measured decel-eration. Here the pump I/I s = 420 ( I s = 1 . is theD line saturation intensity of Cs), ∆ A / π = − .
58 GHz, and∆ C / π = −
10 MHz. The atom number is N = 2 . × .From Black et al. (2003). (Chan et al. , 2003). Both the large friction and the lowtemperature cannot be explained in terms of the interac-tion between single atoms and the cavity field. In this de-tuning regime a theoretical description is more involvedbecause the entire hyperfine manifold has to be takeninto account and the interaction can lead to Raman las-ing between different magnetic sublevels.
3. Back-action, nonlinear dynamics
In general, the interplay between the mechanical ef-fect of light on the atomic motion and light scatteringinside the cavity off the spatial atomic density distribu-tion can lead to highly non-linear dynamics. Having alarge atomic ensemble organized in a lattice structure,for example, the collective Bragg scattering much moreefficiently redistributes the light between modes thanRayleigh scattering from the individual atoms. The en-hancement factor, being on the order of the number ofatoms, can give rise to significant sensitivity of light scat-tering to small variations of the spatial distribution. Asan example, the back scattering between the counter-propagating modes (denoted by + and − ) of a ring cavitywas found to depend strongly on the bunching parameter of the atomic distribution around the trapping sites of anoptical lattice. In experiments performed by the Ham-burg group (Elsasser et al. , 2004; Nagorny et al. , 2003),the amplitude α + in one of the modes was actively sta-bilized by a feedback loop, ˙ α + = 0, and thus the othermode obeyed the nonlinear equation of motion˙ α − = iN U B α − α + − κα − − iN U α + B ∗ + η − . (22)Here, η − denotes the driving amplitude of thismode and B = (cid:104) e − ikz (cid:105) the bunching parame-ter (see also Sec. III.B.2). For a thermal cloudthe atomic bunching follows approximately B ∝ α ∗− / | α − | exp (cid:110) − const / (cid:112) | α − | (cid:111) . The resulting nonlineardynamics was exemplified by a new kind of optical bista-bility in the dispersive atom-field coupling regime, whichis outside the range of optical bistability effects relyingon the nonlinearity of the internal atom-field coupling(Lugiato, 1984). Subsequent experiments (Klinner et al. ,2006) also revealed the mechanical effect of light on theatomic distribution in the dispersive regime through thenormal-mode splitting. B. Non-equilibrium phase transitions and collectiveinstabilities
The nonlinear collective dynamics of thermal atoms ina high-finesse resonator can give rise to non-equilibriumphase transitions and collective instabilities. In the fol-lowing we present two, experimentally evidenced exam-ples which have been theoretically studied both in thethermodynamic limit and by means of microscopic mod-els.
1. Spatial self-organization into a Bragg-crystal
A thermal cloud of cold atoms interacting with a sin-gle mode of a high-finesse Fabry-P´erot cavity undergoesa phase transition upon tuning the power P of a far-detuned laser beam (wavelength λ ) which illuminates theatoms from a direction perpendicular to the cavity axis(Asb´oth et al. , 2005; Domokos and Ritsch, 2002). Belowa threshold power P cr , the thermal fluctuations stabi-lize the homogeneous density distribution of the atomiccloud, and light which is scattered off the atoms into thecavity destructively interferes, rendering the mean cavityfield amplitude to be zero. Above threshold, P > P cr , theatoms self-organize into a λ -periodic crystalline checker-board order which is bound by the interference betweenthe pump field and the macroscopic cavity field, resultingfrom Bragg scattering into the cavity mode.This self-organization effect can be described in termsof a semiclassical model similar to Eq. (13), generalized tomany atoms. A set of variables p j and r j is introduced,9the index j = 1 , . . . , N labeling the atoms. For simplic-ity, the atomic motion is considered in two dimensionsspanned by the cavity axis and the pump laser direction,with coordinates x and z , respectively. The equation ofmotion for the coherent cavity field amplitude α is givenby ˙ α = i (cid:104) ∆ C − U (cid:88) j cos ( kx j ) (cid:105) α − (cid:104) κ +Γ (cid:88) j cos ( kx j ) (cid:105) α − iη eff (cid:88) j cos( kx j ) cos( kz j ) + ξ α , (23a)where the effective pumping strength of the cavity modeis denoted by η eff = Ω g ∆ A ∆ A + γ , see Eq. (10). Due to the in-terference term (cid:80) j cos( kx j ) cos( kz j ) light scattering intothe cavity vanishes for a homogeneous atomic density dis-tribution. It can be small even if all the atoms are max-imally coupled but the signs of the summands alternate.The light forces exerted on the individual atoms alongthe cavity and pump direction are given by˙ p xj = − ¯ hU | α | ∂∂x j cos ( kx j ) − ¯ hη eff ( α + α ∗ ) ∂∂x j cos( kz j ) cos( kx j ) + ξ xj , (23b)˙ p zj = − ¯ hU (Ω /g ) ∂∂z j cos ( kz j ) − ¯ hη eff ( α + α ∗ ) ∂∂z j cos( kx j ) cos( kz j ) + ξ zj , (23c)These equations include Langevin noise terms ξ α , ξ xj ,and ξ zj , defined by the non-vanishing second-order cor-relations, (cid:104) ξ ∗ α ξ α (cid:105) = κ + N (cid:88) j =1 Γ cos ( kx j ) , (24a) (cid:104) ξ n ξ α (cid:105) = i ¯ h Γ ∂ n E ( r j ) cos( kx j ) , (24b) (cid:104) ξ n ξ m (cid:105) = 2¯ h k Γ |E ( r j ) | u n δ nm + ¯ h Γ (cid:104) ∂ n E ∗ ( r j ) ∂ m E ( r j ) + ∂ n E ( r j ) ∂ m E ∗ ( r j ) (cid:105) , (24c)with indices n, m = x j , z j . The noise terms associatedwith different atoms are not correlated. The complexdimensionless electric field is given by E ( r ) = cos( kx ) α +cos( kz )Ω /g .In Fig. 10 a numerical simulation of the trajectoriesof 40 atoms during the first 50 µ s of illumination areshown. The initial configuration is given by an ensembleof thermal atoms with random positions from a uniform,and velocities from a thermal distribution. The cavitymode initially is in the vacuum state ( α = 0). With theright choice of parameters the emergence of a periodicpattern in the spatial density distribution of the atoms FIG. 10 (Color online) Self-organization of laser-driven atomsin a cavity. Numerically simulated two-dimensional trajecto-ries during the first 50 µ sec of transverse illumination. Acheckerboard pattern of trapped atoms emerges, in which theoccupied trapping positions are separated by even multiplesof λ/ γ = 20 /µ sec,( g, κ ) = (2 . , . γ , atomic detuning ∆ A = − γ , cavity de-tuning ∆ C = − κ + NU , and the pumping strength Ω = 50 γ .From Asb´oth et al. (2005). is observed, accompanied by the buildup of a coherentcavity field amplitude (Fig. 11). In the emerging config-uration, assuming a red-detuned pump laser, the trappedatoms are oscillating about intensity maxima of the in-terfering pump-cavity field. Along the cavity and pumpdirection these are separated by even multiples of the op-tical wavelength. Since only the black or white fields ofthe underlying checkerboard lattice pattern are occupied,constructive interference leads to efficient Bragg scatter-ing of pump photons into the cavity.As will be shown in more detail later, the process ofself-organization relies on the right choice of the detun-ing δ C = ∆ C − N U / C . For thecase δ C <
0, the potential term cos( kx j ) cos( kz j ) inEqs. (23b,c) attracts atoms towards the “majority” sitesand repels them from the “minority” sites, providing pos-itive feedback. Initiated by density fluctuations, one ofthe two possible Bragg lattices is then formed in a run-away process. For the case δ C >
0, the scattered cavityfield creates potential maxima (minima) at the positionsof the majority (minority) sites, counteracting the ampli-fication of density fluctuations and preventing a dynam-0
0 1 2 3 4 5time [ms] N = 40 N = 160 p h o t o n n u m b e r , N =
100 150 50 0 2400 1600 800 0 p h o t o n n u m b e r , N = z p z / _ h time [ms] FIG. 11 (Color online) Cavity cooling in the self-organizedphase. The time evolution of the photon number in the cav-ity (left) and the phase space density of the atoms (right) ona long time scale, for N = 40 and N = 160 atoms (differ-ent vertical scalings are used for illustrating the superradi-ance | α | ∝ N ). In the right panel, the green curve fluc-tuating around a constant value corresponds to a uniformlydistributed N = 40 atoms driven below the self-organizationthreshold. The parameters are the same as in Fig. 10. FromAsb´oth et al. (2005). ical instability. Furthermore, in this regime the delayedcavity response causes cavity heating of the atomic mo-tion which obscures an equilibrium situation in the lackof other dissipative process.For ∆ C − N U <
0, the initial fast buildup of a coher-ent cavity field continues over a longer timescale. Thekinetic energy of the oscillating and the untrapped atomsdissipates owing to the cavity cooling mechanism, whichleads to an increase of the number of trapped atoms anda stronger localization in the potential wells. This furtherimproves coherent scattering into the cavity, as indicatedby the slow increase in the cavity field intensity shownin Fig. 11. Comparing the time evolution of the intra-cavity photon number for the self-organization processof 40 and 160 atoms (rescaled in Fig. 11 by a factor of16) demonstrates the superradiance effect, i.e. the fieldintensity scales cooperatively as the square of the par-ticle number. The right panel of the figure shows thatthe cooling rate, described by the decrease of the phasespace density of the atoms, is also similar for N = 40 and N = 160 and that the self-organization leads to smaller phase space densities than the homogeneous distributionbelow threshold.Self-organization of laser-cooled atoms has been ob-served in experiments of the MIT group with N ≈ Cs atoms prepared at a temperature of 6 µ K in a nearlyconfocal Fabry-P´erot cavity (Black et al. , 2003). Abovea threshold intensity of the transverse pump beam, col-lective emission of light into the cavity was observed ata rate which exceeds the free-space single-atom Rayleighscattering rate by a factor of up to 10 . This experimentclearly demonstrated the process of spontaneous symme-try breaking by measuring π -jumps in the phase of theemitted cavity field relative to the transverse pump field,corresponding to self-organization into the black or whitelattice sites of a checkerboard pattern (Fig. 12). Retar-dation between the cavity field and the atomic motion re-sulted in a collective friction force on the center-of-massdegree of freedom. A deceleration of up to 1000 m / s has been achieved with atom-cavity detunings as largeas ∆ A = − π × .
58 GHz.For finite atom number N and finite measurementtime, an interesting hysteresis effect accompanies self-organization, as shown in Fig. 13. The thermodynamiclimit N → ∞ is approached by simulations of Eq. (23)with the atomic density N/V ∝ N g and the cavity lossrate kept constant. The percentage of defect atoms af-ter 4 ms of simulation time as a function of the pump-ing laser strength clearly shows the transition. However,the transition point is dependent on N and on whether For short T off , the relative phase is predominantly ! ! ! . As T off increases, relative phases of ! ! ! " appear, and the difference f " f " decays with a timeconstant ! $ % $ s . Optical-path-length drifts inthe interferometer then cause a slow decay in f and f " ,and an increase in f " = to the 10% random-phase level.Having confirmed a mechanism for the emission, weinvestigate the forces on falling atoms interacting withthe pump light after the MOT is extinguished. The col-lective emission into the cavity leads to strong damping ofthe c.m. motion of the atoms, an effect qualitativelysimilar to that predicted by Gangl et al. for atoms in amultimode ring cavity [11]. The TOF measurements in-dicate that, following emission, a large fraction of theatoms is substantially decelerated (inset of Fig. 4). Foratoms with an initial velocity v !
15 cm = s at the time ofillumination, decelerations of about
300 m = s are ob-served, with typically a third of the atoms being slowed.The TOF trace shown indicates a final velocity of = s ,while those with the largest delays indicate that the cloudis stopped before the emission ends.For driven atoms falling at velocity v > , the mea-sured intracavity power is modulated at a frequency giveninitially by twice the atomic Doppler effect ! d ! kv (Fig. 4). This modulation can be explained as the beat notebetween fields emitted upward and downward, which areDoppler shifted by & ! d . Alternatively, it can be ex-plained by the spatial variation with period % = of theatom-cavity coupling, whereby falling atoms cannot emitat a node of the intracavity standing wave. The changingmodulation frequency then indicates the atomic accelera-tion, as displayed in Fig. 4. These measured accelerationsare consistent with the TOF measurements. The peakaccelerations, up to " = s , occur shortly after theonset of the collective emission. Heating of the delayedatoms is consistent with recoil heating at the photonscattering rate into the cavity. While most of our dataare taken in the range " : ’ & a = " ’ " :
57 GHz , we observe damping for pump-atom detunings up to & a = " ( " : , limited only by the pump intensityneeded to reach threshold. For blue pump-atom detuningsup to & a = " ( : there is collective emission butno c.m. damping.Since the pump-atom detuning & a is much larger thanthe atoms’ excited-state hyperfine splitting [Fig. 1(a)], thetransition F g ! ! F e can be treated as an open, two-level system with decay to the F g ! ground state. If therepumper counteracting the decay is turned off during theapplication of the pump beam, the ensuing TOF traces aresimilar to that displayed in the inset of Fig. 4, with asimilar fraction of 40% of the sample being slowed. Foran initial c.m. velocity of v !
15 cm = s the Bragg scat-tering lasts up to $ s , slowing the sample to a final µ s) D e c e l e r a t i on ( m / s ) t (ms) T O F E m i tt ed P o w e r ( µ W ) FIG. 4. The emitted power (thin line) during the illuminationof a falling atomic cloud. The beat frequency is used tocalculate the deceleration (thick line). The inset shows thetime-of-flight signal, in arbitrary units, of atomic clouds with-out (gray line) and with (black line) a $ s exposure to thepump beam. Here the pump I=I s ! , & a = " !" :
580 GHz , and ! c = " ! "
10 MHz . The atom number is N ! : ) . E m i ss i on π φ (a)(b) FIG. 2. Simultaneous time traces of the intracavityintensity (a) (arbitrary units) and field phase (b). The param-eters are N ! : ) , & a = " ! " :
59 GHz , ! c = " !"
20 MHz , and
I=I s ! . off ( µ s) F r a c t i on o f P u l s e P a i r s λλ xz ∆φ = π /2 ∆φ = 0 ∆φ = π FIG. 3. The fraction of pulse pairs with relative phase shift ! ! is plotted versus pulse separation time. The solid circles,open circles, and solid squares correspond to ! ! ! $ " = , " $ " = , and " = $ " = , respectively. The parameters arethe same as for Fig. 2, except here N ! : ) . The insetshows the two possible lattice configurations producing relativephase shift ! ! ! " in the emitted light. P H Y S I C A L R E V I E W L E T T E R S week ending14 NOVEMBER 2003 V OLUME
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FIG. 12 Observation of spontaneous symmetry breaking inthe self-organization phase transition. Simultaneous timetraces of the (a) intracavity intensity (in arbitrary units)and (b) the relative pump-cavity phase. Drops in the in-tensity correspond to time intervals during which the beamsof the magneto-optical trap (MOT) are switched on forcingthe atomic density distribution to randomize. After switch-ing off the MOT beams, the atoms self-organize again intoone of the two possible checkerboard patterns, as indicatedby the relative phase signal. Experimental parameters were N = 8 . × , ∆ A = − π × .
59 GHz, ∆ C = − π ×
20 MHz,and
I/I sat = 440. From Black et al. (2003). FIG. 13 (Color online) Hysteresis effect for finite measure-ment time. Ratio of atoms in the “defect” positions againstpumping strength η , 4ms after the loading of the trap witha uniform (“up”) or organized (“down”) gas of atoms. Thedifferent curves show the approach towards the thermody-namic limit. The parameters are κ = γ/ , ∆ A = − γ , Ng = 200 γ ∆ C = − κ − Ng / | ∆ A | , k B T = ¯ hκ . FromAsb´oth et al. (2005). the initial positions were uniformly distributed (“up”)or at “odd” points of maximal coupling (“down”). Thebreadth of the hysteresis increases with the atom num-ber, but decreases with the measurement time. Thisbehavior can be explained by taking into account sta-tistical fluctuations arising from the finite atom number N . Assuming that the self-organization from a uniformdistribution (“up” curves) is triggered when the fluctu-ating energy difference between the even and odd sitesmomentarily exceeds the mean kinetic energy, a scaling N g of the threshold with the pump intensity was foundin accordance with the numerical results of the plot. Thedisappearance of the lattice pattern for decreasing pumppower (“down” curves) when the system is started fromthe ordered phase occurs at the half of the mean-fieldthreshold (see Eq. (33) in the next subsection), indepen-dently of the atom number N .Self-organization of laser-driven atoms occurs also ina transversally driven ring-resonator geometry support-ing two running-wave modes (Nagy et al. , 2006b). Incontrast to the linear single-mode cavity case, here thetransition from the homogeneous to the organized densitydistribution involves spontaneous breaking of a continu-ous (rather than a discrete) translational symmetry (seealso IV.D.3).
2. Collective atomic recoil laser
Collective atomic recoil lasing (CARL) is the promi-nent many-body instability effect in a ring cavity, origi-nally predicted by Bonifacio et al. (1994). An ensemble ofcold atoms couples to two counter-propagating modes ofa unidirectionally pumped high-finesse ring cavity. Lightscattering off the atomic ensemble between these cavity modes leads to a collective instability corresponding toan exponential gain for the back-propagating field modeamplitude in conjunction with an atomic bunching atthe antinodes of a self-organized optical lattice. In thepresence of dissipation of the atomic kinetic energy, asteady-state operation of CARL can be achieved with aself-determined atomic drift velocity and back-reflectedlight frequency.The CARL scheme involves an interplay between theinfluence of the atomic motion on the Rayleigh scatter-ing of light and, reversely, the mechanical effect of lightupon the atomic motion. The former effect has beenpreviously seen, for example, as the so-called recoil in-duced resonance (RIR) in the transmission spectrum of aprobe beam making a small angle with a one-dimensionallin ⊥ lin optical molasses (Courtois et al. , 1994), and alsoin a optical dipole trap formed by counter-propagatingmodes of a ring cavity (Kruse et al. , 2003b). This nar-row, dispersion-like resonance around the pump field fre-quency originates from a two-photon Raman transitionbetween different momentum states of the atoms. Differ-ent populations of the corresponding momentum stateslead to gain or attenuation of a probe beam. For athermal velocity distribution, a probe frequency tunedslightly below the pump frequency gives rise to gain,whereas for negative detuning there is an attenuation ofthe probe. The probe transmission spectrum measure-ment provides information about the temperature, andeven more about the velocity distribution (Brzozowska et al. , 2006).It has been predicted that, based on the RIR gain ef-fect, a weak probe field injected in the direction oppo-site to the strong pump field would exponentially am-plify due to the self-bunching of a part of the atomsinto a lattice which Bragg reflects the strong pump beammore efficiently than single atoms (Bonifacio et al. , 1994).However, one needs long enough interaction time so thatback-action of the light scattering on the velocity dis-tribution can have a significant effect (Berman, 1999).This can be accomplished, for example, by confiningthe light modes into a cavity. Lasing mediated by thecollective atomic recoil between the counter-propagatingmodes of a unidirectionally ring cavity has been observed(Kruse et al. , 2003a). Back-action on the atomic mo-tion has been demonstrated by detecting the displace-ment of the atoms accelerated by the momentum trans-fer process. The self-consistent solution is an accelerat-ing Bragg-lattice of atoms co-moving with the standingwave formed by the pump and the back-reflected com-ponent having a Doppler-shifted frequency ∆ ω = 2 kv with respect to the pump. The phase dynamics of thecounter-propagating modes can be monitored as a beatsignal between the outcoupled beams, which reveals theacceleration by an increasingly red-detuned probe as afunction of time (Fig. 14).The runaway process can be counteracted by introduc-2 T ! : " , see Fig. 1). The outcoupled light power isrelated to the intracavity power by P $ out %& ! TP $ cav %& ! T ! h! ! j " & j . The phase dynamics of the two counterpro-pagating cavity modes is monitored as the beat signalbetween the two outcoupled beams. Any frequency dif-ference between pump and probe, " ! ’ ! ( ! , i.e.,propagation of the standing wave nodes inside the ringcavity, is translated into an amplitude variation of theobserved interference signal P beat ! T ! h! ! j " ( ( " j .In order to observe a CARL signal, we load atoms fromthe MOT into the symmetrically pumped ring cavitystanding wave ( ! : ). The detuning from thenearest atomic transition ( D line at 794.8 nm) is 1 THz.After 30 ms trapping time one beam ( $ ) is switched offvia a mechanical shutter. With no atoms the interferencesignal P beat then drops to T ! h! ! j " ( j within a time of % s , limited by the finite shutter closing time. Anobserved residual low-frequency fluctuation of P beat isassigned to scattering at the imperfect mirror surfaces.In contrast, if atoms are loaded into the ring cavity stand-ing wave, oscillations appear on the interference signalshortly after the switch-off, as shown in Fig. 2(a). Theiramplitude is rapidly damped, however they are still dis-cernible after more than 1.5 ms, which exceeds the cavitydecay time & ! $ ’ % by more than a factor of 600. Weverified that the oscillations do not occur for low cavityfinesse, which is realized by rotating the polarization ofthe incoupled lasers from s to p polarization [4], therebyreducing the finesse from 80 000 to 2500. Figure 2(c)shows the frequency of the interference signal as a func-tion of time. From the observations we can deduce (i) thatthe probe mode " is fed with light in the presence ofatoms, thus leading to a standing wave superposed to thepump mode " ( . (ii) Having verified that the oscillationsare solely due to a relative phase shift of the cavity modes(no oscillations are detected in either mode), we knowthat the detuning between probe and pump increases intime, and therefore the standing wave is displaced andaccelerated by the presence of atoms. (iii) With time thecontrast of the standing wave reduces to zero, i.e., after a fast initial decay, the probe fades out on a time scale of1.5 ms. (iv) The atoms are dragged by the moving stand-ing wave. The frequency shift of the backscattered lightfield by up to 1 MHz corresponds to an atomic velocity of
40 cm = s . At interaction times of a few ms, the atomicmotion should lead to a detectable displacement. To verifythis we have taken time-of-flight absorption images of theatomic cloud at various times after one-sided switch-off,shown in Figs. 2(d) and 2(e). We observe that, even thougha major fraction of the cloud stays at the waist of the laserbeam, its center of mass is shifted along the propagationdirection of the standing wave. In the case of low cavityfinesse, the displacement is almost zero.The observations can be understood in terms of lasingmediated by collective atomic recoil, despite the differ-ences between our system and the original CARL pro-posal [5]. The original idea is based on injecting ahomogeneously distributed monokinetic atomic beamcounterpropagating to the pump and a seed for the probe,which predetermines the probe frequency. In this system,the signature for CARL action is a blue-detuned amplifiedprobe pulse followed by a highly irregular evolution ofthe system. In contrast, in our system the probe frequencyand the atomic momentum distribution evolve in a self-consistent manner. Initially the atoms are highly bunched FIG. 2. ( a ) Recorded time evolution of the observed beatsignal between the two cavity modes with N ! and P $ cav %& ! . At time t ! the pumping of the probe " hasbeen interrupted. ( b ) Numerical simulation according to Eq. (1)with the temperature adjusted to % K . (c) The symbols ( " )trace the evolution of the beat frequency after switch-off. Thedotted line is based on a numerical simulation. The solid line isobtained from Eq. (3) with the assumption that the fraction ofatoms participating in the coherent dynamics is = to ac-count for imperfect bunching. (d) Absorption images of a cloudof " atoms recorded for high cavity finesse at 0 msand (e) 6 ms after switching off the probe beam pumping.All images are taken after a 1 ms free expansion time. (f) Thisimage is obtained by subtracting from image (e) an absorptionimage taken with low cavity finesse 6 ms after switch-off. Theintracavity power has been adjusted to the same value as in thehigh finesse case.FIG. 1. Scheme of the experimental setup. A Ti-sapphire laseris locked to one of the two counterpropagating modes ( " ( ) of aring cavity. The beam " $ in % can be switched off by means of amechanical shutter (S). The atomic cloud is located in the free-space waist of the cavity mode. We observe the evolution of theinterference signal between the two light fields leaking throughone of the cavity mirrors and the spatial evolution of the atomsvia absorption imaging. P H Y S I C A L R E V I E W L E T T E R S week ending31 OCTOBER 2003 V OLUME
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FIG. 14 Observation of lasing mediated by collective atomicrecoil. (a) Recorded time evolution of the observed beat sig-nal between the α + and α − cavity modes of a ring cavity.Initially both modes are pumped to form an optical lattice of N = 10 atoms. The initial drop is due to the decay of theunpumped mode after switching off the α − pump at t = 0.Well beyond the ring-down time of about 10 µ s, the persist-ing oscillations demonstrate the coherent backscattering ofthe pumped mode. (b) Numerical simulation with the tem-perature adjusted to 200 µ K. (c) The symbols ( × ) trace theevolution of the beat frequency after switch-off (dotted lineis from numerical simulation). The increase of the beat fre-quency corresponds to the acceleration of the Bragg lattice ofback reflecting atoms, which lasts until the Doppler-shiftedfrequency drops out of cavity resonance. Absorption imagesof a cloud of 6 × atoms recorded at (d) 0 ms and (e) 6 msafter switching off the probe beam pumping. All images aretaken after a 1 ms free expansion time. (f) This image is ob-tained by subtracting from image (e) an absorption imagetaken 6 ms after switch-off with low cavity finesse for whichno collective recoil is expected. The intracavity power hasbeen adjusted to the same value as in the high finesse case.From Kruse et al. (2003a). ing some external friction force on the atom motion. Thedissipation gives rise to a steady-state solution which in-volves a constant drift of the entire atomic cloud at aspeed v . Accordingly, we transform the atomic positionvariables such as z j = ˜ z j + vt , and the coherent field am-plitude of the running-wave field mode propagating op-posite to the pumped field mode as α − = ˜ α − e ikvt . Thedrift velocity v is to be determined in a self-consistentmanner. The semiclassical equations describing atomicmotion are given by˙ p j = − βm p j + ¯ hU ik (cid:0) α ∗ + ˜ α − e − ik ˜ z j − ˜ α ∗− α + e ik ˜ z j , (cid:1) (25a)where β represents the linear friction arising, e.g., fromcollisions with a buffer gas (Bonifacio, 1996; Perrin et al. ,2001) or laser-cooling in an optical molasses (Kruse et al. , 2003a). The cavity field amplitudes evolve as˙ α + = ( iδ C − κ ) α + − iN U B ˜ α − + η , (25b)˙˜ α − = [ i ( δ C − kv ) − κ ] ˜ α − − iN U B ∗ α + , (25c)where δ C = ∆ C − N U is the effective detuning of thepump frequency from the atoms-shifted cavity resonance.The atomic positions enter through the bunching param-eter B = 1 N N (cid:88) j =1 e − ik ˜ z j ≡ be − iϕ . (25d)A closed set of equations can be formed in which theatomic cloud is characterized by the three real param-eters v , b , and φ . A trivial solution of these equationscorresponds to the case where the atoms are uniformlydistributed in space ( b = 0, v = 0) and the counter-propagating field mode amplitude vanishes ( α − = 0). Anon-trivial steady-state solution can be obtained numer-ically from the coupled algebraic equations (time deriva-tives set to zero). This solution can be approximatedanalytically by assuming perfect bunching, b = 1 and ϕ = 0. The only remaining free parameter is then givenby the steady-state drift velocity v which obeys the alge-braic equation 2 kv = 8 m ω R β N U κ | η | | D | , (26)where D = ( iδ C − κ )[ i ( δ C − kv ) − κ ] + N U b . Inorder to gain insight into the solution, the following sim-plifications can be made: (i) neglect in D the last U term originating from (second-order) scattering of the α − mode back into the pumped α + one, and (ii) con-sider resonance δ C = 0. Then one can identify the typ-ical solutions of two different regimes. In the limit ofa large Doppler shift, kv (cid:29) κ , the drift velocity andthe back reflected power scale with the atom number as v ∝ N / and | α − | ∝ N / , respectively. This is re-ferred to as the CARL limit in the literature. In the op-posite limit of a small Doppler shift, kv (cid:28) κ , the velocityobeys v ∝ N and the intensity exhibits superradiant be-havior, | α − | ∝ N , corresponding in this geometry toBragg retro-reflection. The relation between these twokinds of superradiant instabilities in the collective inter-action of light with an atomic gas has been establishedexperimentally by Slama et al. (2007a,b). While superra-diant Rayleigh scattering from atomic clouds is normallyobserved only at very low temperatures, i.e., well below1 µ K, (Inouye et al. , 1999), the presence of the ring cav-ity enhances cooperativity and allows for superradiancewith thermal clouds as hot as several 10 µ K.In the experiments of the T¨ubingen group (Cube et al. ,2004; Kruse et al. , 2003a), an optical molasses has beenused to impose a motional damping force on the atoms.In such a viscous CARL system, there is a steady-state3 reaches a peak density of about ! cm " and atemperature of a few ! K .A genuine problem of the CARL is the following: In theabsence of damping for the external degrees of freedomthe CARL process continuously accelerates the atomiccenter of mass [2,9]. Even though the acceleration de-creases because the Doppler-shifted CARL frequencyeventually drops out of the cavity resonance, it neverreaches a stationary value. In fact, being focused onstudies of transient phenomena, the original CARL model[1] does not consider relaxation of the translational de-grees of freedom. On the other hand, standard methods ofoptically cooling atoms are based on controlled dissipa-tion, e.g., optical molasses. Close to resonance the motionof atoms in an optical molasses is well described by afriction force. In our experiment, we harness this dissi-pation mechanism and subject the dipole-trapped atomiccloud to an optical molasses.We use the laser beams of themagneto-optical trap and tune them 50 MHz below thecooling transition ( D , F ! F ). In this situation,the beat signal oscillations quickly reach a stable equilib-rium frequency between ! != "
100 kHz and170 kHz, which corresponds to an atomic velocity ofseven to
13 cm = s .In order to observe a threshold behavior in experiment,we adiabatically ramp up and down the intensity of thepump laser. The CARL radiation is monitored by record-ing the time evolution of the beat frequency of the coun-terpropagating modes. The curves shown in Fig. 1(a)represent frequency spectra obtained by Fourier trans-forming the beat signal restricted to successive time-intervals. The peaks’’ locations then denote the instanta-neous frequency shift of the probe beam, and their heightsreflect the standing wave’s contrast. The pronounced de-pendence of the CARL frequency on the pump intensityrevealed by the series of Fourier spectra is emphasized inFig. 1(b). The intensity of the CARL radiation shown inFig. 1(c) decreases with the pump intensity, but moreimportant is the fact that the curve exhibits a minimumpump intensity required to initiate CARL lasing. Just likein common lasers, only if the energy fed to the systemexceeds the losses, the laser emits coherent radiation.The experimental observations may be discussed invarious ways. The model of Ref. [2] to describe the impactof optical molasses on CARL consisted of simply addinga friction force, proportional to a coefficient fr , to theequations governing the atomic dynamics. This proce-dure certainly represents a coarse simplification. For ex-ample, it predicts that the atoms quickly bunch under theinfluence of the molasses and are cooled until the tem-perature of the cloud is T , and it denies the presenceof any threshold. In reality, the molasses temperature islimited by diffusion in momentum space, i.e., heating. Toaccount for this heating, one may supplement the dy-namic equations for the trajectories of individual atoms [Ref. [2], Eq. (1)] with a stochastically fluctuatingLangevin force, $ n $ t % with h $ n $ t % i and h $ n $ t % $ m $ % % i fr D & mn & $ t " % % , where the diffusion coefficient D ’ = fr is proportional to the atoms’ equilibrium tem-perature, which is related to the Maxwell-Gaussian ve-locity spread by ’ & k !!!!!!!!!!!!!!! k B T=m p : " ( n "iU ) ’ $ ) " e " i ( n " ) (" e i ( n % " fr _ ( n ’ $ n : (2)Here we defined ( n & kx n as the position of the n th atomalong the optical axis normalized to the optical wave-length and assumed the pump laser stabilized on reso-nance with the cavity ) ’ is set real and constant. N is thetotal atom number and " & hk =m is twice the recoilfrequency shift. The coupling strength U (or single-photon light shift) is related to the one-photon Rabi-frequency g and to the laser detuning from resonanceby U & g = ! a . The functional dependence of the quan-tity ) " on the order (or bunching) parameter b & j b j e i & N " P m e i ( m is determined by the differentialequation _ ) " *) " " iNU ) ’ b: (3)An alternative to simulating trajectories of individualatoms is to calculate the dynamics of distribution func-tions. Particularly adequate to the problem of diffusion inmomentum space induced by optical molasses is aFokker-Planck approach. Here the thermalization of the ∆ω /2 π (kHz) t ( m s ) (a) 0 5 10050100150 ∆ ω / π ( k H z ) (b)0 5 10050100150 P − ( µ W ) (c)0 5 1000.51 P + (W) | b | (d) FIG. 1. (a) Sectionwise Fourier-transform of the interferencesignal P beat with the pump laser power being ramped up anddown. The dotted line is proportional to the pump laser power.(b) Dependence of the CARL frequency on the intracavitypump power. (c) Dependence of the probe field intensity onthe pump power. The CARL laser threshold is around P ’ intracavity power. The fitted curves are based on a Fokker-Planck theory outlined in [8]. (d) Calculated bunchingparameter. The parameters are fr * , N , ! a " % : , and T ! K . V OLUME
93, N
UMBER week ending20 AUGUST 2004
FIG. 15 Pump power threshold for the collective atomic recoillasing. (a) Sectionwise Fourier-transform of the interferencesignal P beat with the pump laser power being ramped downand up (the dotted line is proportional to the pump laserpower). At t = 0 the system is in the ordered CARL phase,by gradually decreasing the power the drift velocity decreases(the peaks in the Fourier spectra shift downward) until theback reflection ceases at a threshold pump power. Rampingup the pump power from about t = 5 ms, the appearance of apeak is delayed and occurs at about the same threshold valueof the pump power. (b) Dependence of the CARL frequencyon the intracavity pump power. (c) Dependence of the probefield intensity on the pump power. The CARL laser thresholdis around P + = 4 W intracavity power. The fitted curves arebased on a Fokker-Planck theory outlined in Sec. III.C.1. (d)Calculated bunching parameter. The parameters are β = 4 κ , N = 10 atoms, ∆ A = − π × . T = 200 µ K. FromCube et al. (2004). operation with a self-consistent drift velocity, accordingto Eq. (26), at which the friction compensates the accel-eration due to back scattering of photons. The Fourierspectrum of the beat signal between the pumped modeand the counter-propagating one in steady-state is shownin Fig. 15a for various pump strengths. The drift veloc-ity can be deduced from the Doppler-shift of the back-reflected field as shown in Fig. 15b, the correspondingrange of velocities is from 7 to 13 cm / s. The beat fre-quency as well as the drift frequency vary as a functionof the pump power. However, the dramatic feature ofFig. 15 is the clear appearance of a threshold: the self-bunching and backscattering starts only above a well-defined threshold pump power. This measurement pro-vides thus an experimental evidence of a phase transitionto a state of synchronized atomic motion (Cube et al. ,2004). Underlying the critical behavior is the diffusionaccompanying friction: the spontaneous photon scatter-ing of the molasses laser beams leads to a random heatingforce which stabilizes the homogeneous distribution andthereby can prevent the formation of a Bragg-lattice forweak pump power (Robb et al. , 2004). Above thresh- old, the dissipation and the fluctuations together lead toa position distribution which exhibits a finite bunchingparameter. These effects are discussed in the frameworkof various mean-field theories for the atomic position dis-tribution (see Sect. III.C).We note that the CARL system and its phase transi-tion has been studied also in the case where the atomictransition frequency is close-to-resonance with the pumplaser field (Perrin et al. , 2001). Then the atomic polar-ization plays a dynamical role and the transition does notrequire spatial bunching but the emergence of coherentpolarization grating (Perrin et al. , 2002).The viscous CARL transition exhibits analogy withthat of the generic Kuramoto model which describes theself-synchronization of coupled oscillators with differentfrequencies (Kuramoto, 1975; Strogatz, 2000). To re-veal the analogy, one can transform the CARL equa-tions using the following assumptions: (i) the motion isoverdamped ( ˙ p j = 0); (ii) the pumped field amplitude α + is a constant of time ( α + ≈ η/κ ); (iii) the counter-propagating mode amplitude is stationary, oscillating atfrequency ω , i.e. , effectively ˙˜ α − = − iω ˜ α − with thefrequency ω being determined by the constant drift ve-locity; and (iv) κ (cid:28) ω . With these assumptions and byusing the notation θ j = 2 k ˜ z j , Eqs. (25) simplify to˙ θ j = 2 km β ξ + Kb sin ( ϕ − θ j ) , (27)which is formally equivalent with the Kuramoto model.The Langevin-type random noise ξ , associated with thefriction term − βp in Eq. (25a), introduces the randomfrequencies present in the Kuramoto-type systems. Thecoupling strength is K = 2 ω R N U | α + | / ( ω β ). Themean-field character is obvious: each oscillator couplesonly to the mean-field quantities b and ϕ . The phase θ j is pulled toward the mean-field phase ϕ , which increasesthe order parameter b . The coupling is proportional to b ,which sets a positive feedback loop. With the increasingcoherence b , even more oscillators can be recruited tothe synchronized pack (those being within the bandwidth Kb ), further increasing b . Such a runaway process startsonly above a critical coupling K . C. Phase-space and mean-field descriptions for largeparticle numbers
Due to the nonlinearity of the coupled atom-field dy-namics, exact analytic results for atomic ensembles cou-pled to optical cavities are rather sparse and the com-putational demand often hinders simulations at realisticparticle numbers (Deachapunya et al. , 2008; Salzburgerand Ritsch, 2009). In the following, we will present mean-field methods which allow to predict instability thresh-olds and to model the effective dynamics in the thermo-dynamic limit. Starting with the assumption of a quasi-4thermal distribution, the critical behavior can be revealedand the threshold for criticality can be approximated. Wewill continue by adopting phase-space methods that canaccount for general position and velocity distributions. AVlasov-type equation allows for a more accurate estima-tion of the threshold, and also for performing stabilityanalysis and setting up a phase diagram. This analy-sis leads to such intriguing predictions that the cavity-mediated interaction combined with the cavity coolingeffect can be the basis of a generally applicable sym-pathetic cooling scheme. Finally, a Fokker-Planck-typeequation can be constructed in order to determine thesteady-state of the non-equilibrium systems in the ther-modynamic limit. The methods show similarities withplasma physics where equally complex, coupled dynam-ics of particles and fields occurs (Montgomery, 1971).
1. Critical point
The simplest mean-field model is based on the assump-tion that the atomic motion is overdamped and the distri-bution function of the atomic positions ρ ( x, t ) is a ther-mal distribution. The critical point of the CARL in-stability (Sec. III.B.2) and that of the self-organization(Sec. III.B.1) can be calculated with this approach. a. Dynamical equations The motional damping is char-acterized by a linear friction coefficient β (half of thekinetic energy damping rate is β/m with m being theatomic mass) and a temperature T . Then the mean atomdensity distribution obeys the Smoluchowski-equation ∂ρ ( x, t ) ∂t = − β ∂∂x (cid:20) F ( x ) ρ ( x, t ) − k B T ∂ρ ( x, t ) ∂x (cid:21) . (28)In the ring cavity geometry of CARL (Kruse et al. ,2003a), for example, the force F ( x ) is given by the lastterm on the right-hand side of Eq. (25a). It contains thefield mode amplitudes which couple back to the atomicdensity distribution via the bunching parameter B ,˙˜ α − = − κ ˜ α − − iU α + B , (29a)with B = (cid:90) λ dx ρ ( x, t ) e ikx . (29b)For simplicity, the center-of-mass velocity and the de-tuning δ C were set to zero. Linear perturbation calculusleads to the instability threshold of the homogeneous so-lution (Robb et al. , 2004). Since the spatial couplingfunctions are sinusoidal, only a few Fourier componentsare involved in the initial dynamics. In particular, inorder to determine the instability of the homogeneousdistribution, only the single mode function e − ikx of thecounter-propagating cavity mode needs to be taken intoaccount. In the so-called CARL limit (see Sec. III.B.2), one obtains the following threshold condition for the cav-ity pump amplitude: η ≥ (cid:18) k B T ¯ h (cid:19) / (cid:114) mω R β κ / N U . (29c) b. Canonical distribution A further simplification can bemade in the mean-field approach if κ is the largest rate inthe dynamics. Adiabatic elimination of the cavity fielddynamics then results in a self-consistent optical poten-tial V ( x ) in which the spatial density ρ ( x ) of the atomsis determined by a canonical distribution, ρ ( x ) = 1 Z exp( − V ( x ) / ( k B T )) , (30)with the partition function Z = (cid:82) exp( − V ( x ) / ( k B T )) dx ensuring normalization of ρ ( x ) to unity. The tempera-ture could be identified with the one which is achieved incavity cooling, k B T ≈ ¯ hκ , but in general it can be set byother means, e.g., by laser-cooling in an external opticalmolasses. The nonlinearity enters the equations throughthe dependence of the potential V ( x ) on the atomic den-sity ρ ( x ) itself, i.e. , V = V ( x, ρ ( x )).In principle, the optical force acting on the atoms inthe cavity does not derive from a potential when theback-action of the atomic motion on the radiation fieldamplitude is significant. As was noted by Asb´oth andDomokos (2007), dynamical equations based on forces,such as Eq. (28) and Eq. (29a) have to be used. How-ever, in the spirit of the mean-field approach, the effectof an individual atom on the field amplitude is negligi-ble with respect to the summed effect of all the others.In the limit of many atoms with small single-atom cou-pling, the motion of a single atom is very well describedby an effective potential determined by the many-bodyensemble.For the example of self-organization in a standing-wavecavity, see Sec. III.B.1, the light potential along the cav-ity axis is given by V ( x ) = U cos ( kx ) + U cos( kx ) , (31)which is composed of the sum of a λ/ λ -periodic onearising from the interference between cavity and pumpfields. The depths of these potentials are U = N (cid:104) cos( kx ) (cid:105) ¯ hI U (32a) U = 2 N (cid:104) cos( kx ) (cid:105) ¯ hI (∆ C − N U (cid:104) cos ( kx ) (cid:105) ) , (32b)where I is a dimensionless single-atom scattering pa-rameter, I ∝ η . Equation (30) has to be solved ina self-consistent manner by iteration. In Asb´oth et al. (2005), the threshold has been analytically determinedto be η , c = k B T ¯ h ( κ + δ C ) N | δ C | . (33)5
2. Stability analysis and phase diagram
In the following we will account explicitly for the ef-fect of the velocity distribution on the dynamics andon the instability threshold. A mean-field model basedon the Vlasov-equation for the phase space distribution f ( x, v, t ) has been derived from a microscopic theory forthe infinite system size by Grießer et al. (2010). For one-dimensional motion along the cavity axis, the dynamicalequation reads ∂f∂t + v ∂f∂x − ∂ x φ ( x, α ) ∂f∂v = 0 , (34)where φ ( x, α ) is the potential corresponding to a mo-mentary field amplitude α . For the generic example of alaser-driven cold atomic cloud in a single-mode standing-wave resonator with mode function cos( kx ), the potentialis φ ( x, α ) = 2¯ hm (cid:18) U | α | cos(2 kx ) + η eff Re ( α ) cos( kx ) (cid:19) . (35)Similarly to the adiabatic potential, Eq. (31), there isa λ/ λ -periodic term, this latter originates frominterference between the transverse pump laser and theintracavity field. However, in this more general approachthe cavity field amplitude is kept dynamical obeying theself-consistent equation˙ α = ( − κ + iδ C ) α + η − iα N U (cid:90) ∞−∞ dv (cid:90) λ cos(2 kx ) f ( x, v, t ) dx − iN η eff (cid:90) ∞−∞ dv (cid:90) λ cos( kx ) f ( x, v, t ) dx , (36)where δ C = ∆ C − N U /
2. This approach based on theVlasov-equation is well suited to study the mean-fielddynamics at short times in order to test the stability ofstationary states. Over longer time scales, statistical fluc-tuations have to be taken into account in the frameworkof a Fokker–Planck equation for the velocity distribution,which is presented in Sec. III.C.3. a. Nonlinear response of a cold atomic cloud in a drivenFabry-P´erot cavity
For cavity pumping only ( η eff =0 , η (cid:54) = 0), the self-consistent steady-state solution ex-hibits a strong nonlinear optical response. Underlyingthe nonlinearity, the particle distribution and thus theeffective refractive index of the cloud depends on the cav-ity pump intensity. Above a sufficient pump strength,multiple stationary solutions appear, reminiscent of op-tical bistability.One can perform a systematic stabilityanalysis of these solutions by studying the dynamics ofsmall fluctuations of the field and the particle distribu-tion (Grießer and Ritsch, 2011). As shown in Fig. 16, the FIG. 16 (Color online) Normalized solutions for the steady-state photon number (red) I = | α | versus effective cavitydetuning δ = δ C − NU for a thermal gas in a driven standing-wave cavity. The driving strength is η = 13 κ (a) and η = 18 κ (b). Those parts of the response curve that lie inside theinstability region (shaded area) are depicted in dashed andcorrespond to linearly unstable steady states. The intervalsdesignated A correspond to bistability, the interval designatedB supports no stable steady state at all. The parameters are N = 10 , U = 0 . κ , η = 18 κ , κ = 2000 ω R and k B T = ¯ hκ .From Grießer and Ritsch (2011). stability analysis reveals regions of bistability as well asparameter ranges where no stable solutions exist.In the parameter regions of instability, the numericalsolution of the dynamical Vlasov equation reveals a limitcycle behavior with subsequent appearance of higher fre-quencies than the fundamental cycle (Grießer and Ritsch,2011). In the quantum regime essentially the same be-havior is retrieved with the recoil frequency determiningthe oscillation frequencies ν ≈ ω R (Ritter et al. , 2009)(see Sec. IV.C). b. Self-organization of a laser-driven cloud of atoms Fora purely transverse pump geometry ( η = 0 , η eff (cid:54) = 0),one can systematically recalculate the critical pump am-plitude η eff , c that marks the transition from the stableregime to the unstable one, where small fluctuations areamplified and grow exponentially.The Vlasov equation Eq. (34) together with the equa-tion for the coherent cavity field amplitude α possessesan infinite number of stationary solutions with a spa-tially homogeneous density distribution and zero cavityfield but different velocity distribution, which, however,are not necessarily stable against fluctuations. Indeed,any symmetric velocity distribution g ( v/v T ) = Lv T f ( v )for δ C < N | η eff | k B T vp (cid:90) ∞−∞ g (cid:48) ( ξ ) − ξ dξ < δ C + κ ¯ h | δ C | , (37)where vp denotes the Cauchy principal value. Here wehave defined the thermal velocity v = 2 k B T /m ; L de-notes the cavity length. For a Gaussian distribution theintegral evaluates to one, and the condition is equivalentto Eq. (33).6 δ C = − κδ C = κ FIG. 17 (Color online) Laser illuminated cold gas in a ring-cavity: time evolution of the intra-cavity field intensity (left),and the instantaneous spatial n ( x ) (upper right) and veloc-ity F ( v ) (lower right) distributions of the particles along thecavity axis at times t = 0 and t e = 28 /κ for δ C = − κ (upperrow) and δ C = κ (bottom row). For δ C <
0, after a transientexponential growth, the field intensity saturates accompaniedby the trapping of atoms in the self-organized pattern. Bycontrast, for δ C > N = 10 , U = − κ/N , η eff = 0 . κ , kv T = κ , and v R = v T /
5. From Grießer et al. (2010).
Figure 17 shows the results of a numerical simulationof Eqs. (34), (35) and (36), initialized with a perturbedgaussian distribution f ( x, v,
0) = 1 λ √ πv T e − v /v T (1 − (cid:15) cos( kx )) , (38)with (cid:15) (cid:28)
1, for the case of a transversely pumped ringcavity, where the light can be scattered into a superposi-tion of two resonant cavity modes. Apart from possess-ing continuous translational symmetry, the dynamics isqualitatively very similar to the single mode case (Grießer et al. , 2010) shown in Fig. 11 of the previous section. Oneclearly recognizes the striking difference in the dynamicalbehavior for positive and negative values of δ C . While wehave an instability in both cases, self-organization is onlyfound for δ C < c. Sympathetic self-ordering and cooling Self-organizationand collective coherent light scattering into a high-finessecavity in principle allows for trapping and cooling of anykind of polarizable particles. In practice, however, therequired phase-space densities and laser intensities to ini-tiate the ordering process are hard to achieve for atomic (d) p/ ( m v ) nu m b e r o f p a r t i c l e s p/ ( m v ) nu m b e r o f p a r t i c l e s kv t θ , x/λ s p a t i a l d i s t r i bu t i o n FIG. 18 (Color online) Simultaneous self-organization of twospecies. The system is started from a perturbed uniform stateabove the instability threshold, Eq. (39), in such a way thatspecies one (two) itself would be pumped six times above (farbelow) the critical point. Figure (a) shows the position dis-tributions in the final state, (c) and (d) the momentum dis-tributions initially (dashed lines) and after self-organization(solid line). Solid lines depict the results of stochastic tra-jectory simulations for ensembles of particles as in Eq.(23b),while open circles show the predictions of the correspondingVlasov model. Figure (b) shows the time evolution of the twoorder parameters θ approaching theoretical steady-statevalues. Parameters are N = 10 , N = 500, m = 10 m , k B T = 10 ¯ hκ , k B T = 2 . × ¯ hκ , η = 2 . κ , η = 27 . κ and ω R = 10 − κ . From Grießer et al. (2011). species, molecules or nanoparticles which cannot be ef-ficiently optically precooled (Deachapunya et al. , 2008;Lev et al. , 2008). As an alternative approach, the self-organization threshold can be achieved by inserting dif-ferent species simultaneously into an optical resonator.The Vlasov-type model can be generalized to a dilutegas of various kinds of N s polarizable point particles ofmass m s illuminated by a single transverse standing-wavelaser field. For the threshold value, a condition analogousto Eq. (37) can be obtained (Grießer et al. , 2011). Thehomogeneous distribution is unstable if and only if S (cid:88) s =1 ¯ h N s η s k B T s (cid:18) vp (cid:90) ∞−∞ g (cid:48) s ( u ) − u du (cid:19) > κ + δ C | δ C | , (39)where k B T s = m s v s /
2. Note that the right-hand sideof Eq. (39) depends only on cavity parameters, and allterms in the sum on the left-hand side are positive andproportional to the pump intensity. This guarantees thatinserting any additional species into the cavity always in-creases the total light scattering rate and thus lowers theminimum power needed to start the self-organization pro-cess, regardless of temperature and polarizability or den-sity of the additional particles. Moreover, the different7 photon numberheavy aloneheavylight alonelight ω R,1 t/N ! | α | " k B T k i n / ¯ hκ photon numberheavy alone, optimal η heavylight ω R,1 t/N ! | α | " k B T k i n / ¯ hκ FIG. 19 (Color online) Sympathetic cavity cooling. Timeevolution of the kinetic temperatures of a heavy and a lightspecies. The blue dashed line represents the heavy particlealone and the blue solid line the enhanced cooling in thepresence of a lighter species below self-organization thresh-old. Parameters are m = 200 m , N = 200, N = 200, √ N η = 134 ω R , √ N η = 134 ω R , κ = 200 ω R and δ C = − κ .From Grießer et al. (2011). species can be located at different regions within the cav-ity. Assisted self-organization of a species which, alone,would be pumped below threshold is shown in Fig. 18.Below the self-organization threshold cooling occursthereby equalizing the stationary momentum distribu-tions for all species. Fig. 19 exhibits the enhanced decayof the kinetic energy of the heavy particles in presenceof cavity field and a cold species. Although the distri-butions get independent in stationary equilibrium, thecooling process itself involves energy exchange betweendifferent species. Thus if any of the species is cold or canbe cooled by different means, the other components aresympathetically cooled in parallel.
3. Non-equilibrium steady-state distributions
Over longer time scales, diffusion has to be accountedfor in terms of a nonlinear Fokker–Planck equation for thestatistically averaged velocity distribution. This allowsfor the calculations of cooling time scales and the uniquesteady-state distribution. a. Transverse pump configuration below threshold
Belowthe instability threshold, Eq. (37), the mean spatial dis-tribution f is homogeneous, i.e., independent of x . Thestatistical fluctuations of the potential and the actualatomic distribution gets important. For the spatiallyaveraged distribution, a lengthy calculation (Niedenzu et al. , 2011) leads to a nonlinear Fokker–Planck equationfor the velocity distribution F ( v, t ) = f ( x, v, t ), ∂∂t F + ∂∂v (cid:16) A [ F ] F (cid:17) = ∂∂v (cid:18) B [ F ] ∂∂v F (cid:19) , (40) with coefficients A [ F ] = 2¯ hkδ C κη m kv | D ( ikv ) | (41a) B [ F ] = ¯ h k η κ m κ + δ C + k v | D ( ikv ) | . (41b)These functionals depend on (cid:104) F (cid:105) via the dispersion rela-tion D ( s ) = ( s + κ ) + δ C −− i ¯ hkδ C N Lη m (cid:90) ∞−∞ dv (cid:18) F (cid:48) ( v ) s + ikv − F (cid:48) ( v ) s − ikv (cid:19) , (42)which encodes all cavity-mediated long-range particle in-teractions. Far below threshold the dispersion relationreduces to D ( ikv ) (cid:39) ( ikv + κ ) + δ C , which correspondsto the case of independent particles.Steady-state solutions of Eq. (40) exist only for neg-ative detuning δ C <
0, where light scattering is accom-panied by kinetic energy extraction from atomic motion.Below threshold one obtains non-thermal q -Gaussian ve-locity distribution functions (de Souza and Tsallis, 1997): F ( v ) ∝ (cid:18) − (1 − q ) mv k B T (cid:19) − q , (43)with q = 1 + ω R / | δ C | and the effective temperature k B T = ¯ h κ + δ C | δ C | ≥ ¯ hκ . (44)The minimum temperature is reached for δ C = − κ . Themagnitude of the detuning | δ C | /ω R determines the shapeof the distribution. For | δ C | = ω R , it is a Lorentziandistribution, whereas for | δ C | /ω R → ∞ , i.e. q →
1, itconverges to a Gaussian distribution with kinetic tem-perature k B T kin = m (cid:10) v (cid:11) .Inserting the steady-state q -Gaussian (43) distributioninto the threshold condition (37) gives a self-consistentstability criterion. As a result, the homogeneous distri-bution is stable only if √ N η eff ≤ κ (cid:114) − q , (45)where the equality is reached for δ C = − κ , i.e., for op-timum cavity cooling given by Eq. (14). The stabilitycriterion can be rewritten in the intuitive form N | U | V p ≤ κ , (46)where V p = Ω / ∆ A is the optical potential depth createdby the pump laser, and N U is the total dispersive shiftof the cavity resonance. Note that even if the initialtemperature is too high for the homogeneous distributionto be unstable, cavity cooling induced self-organizationis possible.8 selforganisedhomogeneous strongly organised, thermalq-Gaussian Gaussianweaklyorganisedweakly organised C e ff FIG. 20 (Color online) Schematic view of the phase di-agram in the weak-coupling limit ( N | U | (cid:28) κ ) for κ =100 ω R . Equilibrium solutions exist only for δ C < − ω R / | δ C | = ω R . For large negative values of the detun-ing δ C , strongly organized equilibrium solutions exist alreadyfor pump strengths slightly above the critical value. Adaptedfrom Niedenzu et al. (2011) b. Transverse pump configuration above threshold Abovethe self-organization threshold, the inhomogeneous spa-tial distribution can still be derived from a Fokker–Planckequation similar to Eq. (40) by using action-angle vari-ables (Chavanis, 2007; Luciani and Pellat, 1987). In thelimit of deep trapping and a harmonic approximation forthe potential, the steady state is a thermal distributionwith a temperature depending both on the effective trapfrequency ω and on the cavity linewidth κ , k B T = ¯ h κ + δ C + 4 ω | δ C | δ C = − ω ≈ ¯ hω . (47)The effective trap frequency ω can be approximated by ω (cid:39) √ N η eff ω R (cid:32) η eff η eff , c + (cid:115) η η , c − (cid:33) , (48)which is valid in the regime | δ C | (cid:29) ω R with η eff , c givenby the equality of Eq. (45). As the temperature dependsexplicitly on the pump strength, a stronger pump laserbeam results in more confined particles with increasedkinetic energy. The system has the interesting propertythat the more particles we add, the deeper the opticalpotential gets, which shows analogy to self-gravitatingsystems (Posch and Thirring, 2005).Finally, putting all this together one obtains the self-consistent phase diagram for self-organization which ac-counts for cavity cooling, see Fig. 20. IV. QUANTUM GASES IN OPTICAL CAVITIES
Quantum gases are considered as ideal model sys-tems to study quantum many-body phenomena underwell-controlled experimental conditions. The possibilitieswhich arise from loading ultracold atomic ensembles ofdifferent particle statistics into various optical potentiallandscapes and to tune the strength of the contact atom-atom interaction, make these system well suited for quan-tum information and simulation research (Bloch et al. ,2008). The merger of the field of ultracold gases with thatof cavity QED provides a set of additional possibilities.Cavity-mediated atom-atom interactions can be tailoredby choosing different resonator and pump geometries andgive rise to novel quantum phases. Closely related, theatomic back-action upon the cavity-generated lattice po-tentials can be significant, which paves the way to studyphonon or soft-condensed matter physics with ultracoldgases (Lewenstein et al. , 2007). Further, coherent scat-tering into the cavity field can be used for non-destructiveand real-time probing of different many-body phases.The coupling between a Bose-Einstein condensate andan optical cavity is conceptually fundamental since a sin-gle mode of a matter wave field interacts with a singlemode of the light field: as all atoms occupy the samemotional quantum state they couple identically to theoptical cavity field. This situation can substantially re-duce the number of degrees of freedom necessary to de-scribe the system. Therefore the experimental situationcan often be almost perfectly described by fundamen-tal Hamiltonians of matter-light interaction. These in-clude the Tavis-Cummings or Dicke model, as well as thegeneric model for cavity optomechanics.
A. Experimental realizations
Experimentally, there have been different approachesto realize and study Bose-Einstein condensates or bosonicatomic ensembles close to quantum degeneracy in opti-cal high-finesse cavities (Brennecke et al. , 2007; Colombe et al. , 2007; Gupta et al. , 2007; Purdy et al. , 2010; Slama et al. , 2007a). So far, all groups used Rb atoms. In theT¨ubingen group (Slama et al. , 2007a) a Bose-Einsteincondensate was loaded for the first time into a ring cav-ity with large mode volume using magnetic trapping andtransport. This experiment extended prior work on thecollective atomic recoil laser (see Sect. III.B.2) with laser-cooled atoms (Cube et al. , 2004; Kruse et al. , 2003a) intothe ultracold regime.Loading ultracold quantum gases or Bose-Einstein con-densates into ultra-high finesse optical cavities of small mode volume, which operate in the single-atom strongcoupling regime of cavity QED, has been achieved byapplying different concepts. The Berkeley group (Gupta et al. , 2007) prepared an ultracold gas of up to 10 atoms9 FIG. 21 (Color online) Different experimental schemes forpreparing ultracold atoms and Bose-Einstein condensates in-side high-finesse optical Fabry-P´erot resonators. (a) Ultracoldatoms are prepared in a magnetic trap, formed using electro-magnets coaxial with the vertically oriented high-finesse cav-ity (length = 194 µ m) and delivered along the x -axis towardsthe cavity center. Once overlapping with the cavity mode,the atoms are loaded into a deep intracavity lattice poten-tial provided by a far-detuned cavity pump field. Adaptedfrom Murch et al. (2008). (b) Ultracold atoms, preparedin a magnetic trap placed above the optical resonator, areloaded into a vertically oriented optical lattice potential andtransported into the cavity by controlled frequency chirp-ing the counter-propagating laser beams. Once in the cav-ity (length = 176 µ m), the atoms are loaded into a crossed-beam harmonic dipole trap where Bose-Einstein condensa-tion is achieved. From Brennecke et al. (2007). (c) A Bose-Einstein condensate is prepared in an atom-chip-based mag-netic trap and positioned afterwards with subwavelength pre-cision in the mode of a fibre-based Farby-P´erot cavity whichhas a length of 39 µ m. Figure courtesy of J. Reichel. (d)A magnetic QUIC trap is used to prepare and to transfer aBose-Einstein condensate into the field of a vertically orientedFabry-P´erot resonator with a length of 5 cm. Figure courtesyof A. Hemmerich. within an ultra-high finesse Fabry-P´erot resonator byloading it into a vertically oriented, deep intracavity op-tical lattice potential, see Fig. 21a. A cavity with similarparameters was used in the approach of the Z¨urich group(Brennecke et al. , 2007). Here, Bose-Einstein conden-sates of typically 2 × atoms were transported intothe mode volume of the optical cavity using an opticalelevator formed by two counter-propagating laser beamswith controlled frequency difference, see Fig. 21b. TheParis group (Colombe et al. , 2007) used an atomic chip toproduce Bose-Einstein condensates of up to 3000 atomsand control its position on a sub-wavelength scales withina novel type of fibre-based Fabry-P´erot cavity with highmirror curvature and reduced mode volume, see Fig. 21c.Also the Berkeley group (Purdy et al. , 2010) achievedsub-wavelength positioning of Bose-Einstein condensatesof a few thousands of atoms inside a conventional small-volume high-finesse optical cavity using an atomic chip.A novel BEC-cavity system operating in an interestingand so far unexplored parameter regime was presented re-cently by the Hamburg group (Wolke et al. , 2012). Here,Bose-Einstein condensates of typical 2 × atoms areprepared and transferred magnetically into the field of a5 cm long near-concentric Fabry-P´erot resonator result-ing in a large single-atom cooperative C (cid:29) ω R , see Fig. 21d.The lowest electronic excitation spectrum of degen-erate and non-degenerate atomic samples strongly cou-pled to the cavity field has been studied by Brennecke et al. (2007) and Colombe et al. (2007). The presenceof N atoms which collectively couple to the cavity fieldresults in an enhanced collective coupling which scalesas √ N . A correspondingly large vacuum Rabi-splittingwas measured in the experiments (Brennecke et al. , 2007;Colombe et al. , 2007). The energy spectrum of a coupledBEC-cavity system together with the square-root depen-dence of the energy splitting on the atom number areshown in Fig. 22.The electronic excitation spectrum is sensitive to theeffective number of atoms coupled maximally to the cav-ity mode, i.e. it depends on the density distribution ofthe atoms integrated over the cavity mode profile. How-ever, the above described measurements of the electronicexcitation spectrum have an energy resolution given bythe excited state and cavity lifetimes, which is too largeto probe the low-energy excitations of the external degreeof freedom of a Bose-Einstein condensate. Probing quan-tum statistics and quantum correlations in atomic many-body states using the dispersive coupling to far-detunedlaser and cavity fields is discussed in Sect. IV.E.2.0 −8 −6 −4 −2 Cavity detuning ∆ c /2 π (GHz) P r obe de t un i ng ∆ p / π ( G H z ) F’ F=1 F’ F=2 σ + σ− (a)(b) | gn i nu t ed ebo r P ∆ P | / π ) z H G ( Atom number N ( ) σ + σ− FIG. 22 (Color online) Collective vacuum Rabi-splitting of acoupled BEC-cavity system. The displayed data was obtainedby cavity transmission spectroscopy using a weak probe laserbeam (Brennecke et al. , 2007). The detuning of the probebeam with respect to the frequency ω A of the | F = 1 (cid:105) →| F (cid:48) = 1 (cid:105) transition of the D line of Rb is denoted by ∆ p .Two orthogonal circular polarizations of the transmitted lightwere recorded and are displayed as red circles ( σ + ) and blacktriangles ( σ − ). (a) Position of the probed resonances as afunction of the detuning ∆ c between the empty cavity reso-nance and the atomic transition frequency ω A for 2 . × atoms. Bare atomic resonances are shown as dotted lines,whereas the empty cavity resonance of the TEM is plot-ted as a dashed-dotted line. The solid lines are the resultof a theoretical model including the influence of higher-ordercavity modes. (b) Shift of the lower resonance of the coupledBEC-cavity system from the bare atomic resonance as a func-tion of atom number for ∆ c = 0. The solid lines are fits ofthe square-root dependence on the atom number N . Adaptedfrom Brennecke et al. (2007). B. Theoretical description
This section provides the theoretical basis for a quan-tum many-body description of a coupled and laser-drivenBEC-cavity system at zero temperature. For simplicity,we consider two different pump laser fields with equalfrequency ω which propagate along and transverse to theaxis of a Fabry-P´erot resonator. A similar many-body de-scription for the case of a BEC in a driven ring-cavity hasbeen presented e.g. in Moore et al. (1999). The Hamilto-nian is composed of an atomic, a cavity and an atom-fieldinteraction part H = H A + H C + H AC . (49)In the following, we will assume a sufficiently large de-tuning of the cavity frequency ω C and the pump laserfrequency ω from the atomic transition frequency ω A , sothat the atom-field interaction is of purely dispersive na- ture, see Sec. II.A.2. In this case all excited states canbe adiabatically eliminated and the atom resides most ofthe time in its electronic ground state. Correspondingly,the motional degree of freedom is captured by a scalarmatter-wave field operator Ψ( r ).The atomic many-body Hamiltonian is given by H A = (cid:90) d r Ψ † ( r ) (cid:104) H (1) + u † ( r )Ψ( r ) (cid:105) Ψ( r ) , (50)where u = 4 π ¯ h a s /m denotes the strength of theshort-range s-wave collisions with scattering length a s (Pitaevskii and Stringari, 2003). Here, the single-atomHamiltonian H (1) = p m + V cl ( r ) , (51)includes an external trapping potential V cl ( r ) which alsoincorporates the potential caused by the transverse pumplaser field.The dynamics of a single, coherently laser-driven cav-ity mode with mode function cos( kx ) and resonance fre-quency ω C is described in a frame rotating at the pumplaser frequency ω by the Hamiltonian H C = − ¯ h ∆ C a † a + i ¯ hη ( a † − a ) . (52)As before, the detuning between the pump laser fre-quency and the cavity resonance frequency is denoted by∆ C = ω − ω C . The generalization to multimode cavitieswill be discussed in Sec. IV.D.3.The dispersive interaction between the pump and cav-ity radiation fields and the atoms reads (in the framerotating at ω ) H AC = (cid:90) d r Ψ † ( r ) (cid:104) ¯ hU cos ( kx ) a † a + ¯ hη eff cos( kx ) cos( kz ) (cid:0) a † + a (cid:1) (cid:105) Ψ( r ) . (53)The first term arises from the absorption and stimulatedemission of cavity photons, with U = g ∆ A denoting themaximum atomic light-shift for a single intracavity pho-ton. As before, g denotes the maximum atom-cavitycoupling strength and ∆ A = ω − ω A the pump-atomdetuning. The second term corresponds to the coher-ent redistribution of photons between the standing-wavetransverse pump laser (with mode function cos( kz )) andthe cavity field. The maximum scattering rate for a sin-gle atom is given by the two-photon (vacuum) Rabi fre-quency η eff = Ω g ∆ A , where Ω is the Rabi frequency of thetransverse pump laser. Both interaction terms in Eq. (53)can be viewed as a four-wave mixing of light and matterwave fields (Rolston and Phillips, 2002).The system is subject to dissipation due to photonleakage through the cavity mirrors. The correspondingirreversible evolution can be modeled by the Liouville1terms in the master equation, Eq. (5a), or, equivalently,by a Heisenberg-Langevin equation (Gardiner and Zoller,2004) for the cavity field operator a , ddt a = − i [ a , H ] − κa + ξ , (54)with cavity field decay rate κ . The Gaussian noise oper-ator ξ maintains the commutation relation for the photonoperators in the presence of cavity decay. In the opticaldomain, the temperature of the bath of electromagneticfield modes can be set to zero. Correspondingly, ξ haszero mean value and the only non-vanishing correlationfunction reads (cid:104) ξ ( t ) ξ † ( t (cid:48) ) (cid:105) = 2 κδ ( t − t (cid:48) ) , (55)according to the fluctuation-dissipation theorem. Fur-ther possible dissipation channels can, for example, actdirectly on the atomic cloud.As the cavity field mediates a global coupling amongall atoms, a mean-field approach is well suited to solvethe above set of equations (Horak et al. , 2000; Horak andRitsch, 2001a; Nagy et al. , 2008). The mean-field descrip-tion assumes the presence of a macroscopically populatedmatter wave field ϕ ( r ) = (cid:104) Ψ( r ) (cid:105) ( condensate wavefunc-tion ) and a coherent cavity field with amplitude α = (cid:104) a (cid:105) which can be separated from the quantum fluctuationsaccording to a → α + δa , (56a)Ψ( r ) → (cid:112) N c ϕ ( r ) + δ Ψ( r ) . (56b)Here, N c denotes the number of condensate atoms with ϕ ( r ) being normalized to 1. Quantum fluctuations areassumed to be small and their mean values vanish bydefinition, i.e. (cid:104) δa (cid:105) = 0 and (cid:104) δ Ψ( r ) (cid:105) = 0. The dynamicalequations of motion resulting from Eqs. (56) contain ahierarchy of terms according to the different powers ofthe fluctuation operators. To zeroth order in the fluctu-ations, one obtains a Gross–Pitaevskii-type equation forthe condensate wavefunction ϕ ( r , t ) coupled to an ordi-nary differential equation for α ( t ): i ¯ h ∂∂t ϕ ( r , t ) = (cid:104) − ¯ h ∇ m + V ( r ) + N c u | ϕ ( r , t ) | + ¯ hU | α ( t ) | cos ( kx )+ 2 ¯ hη eff Re { α ( t ) } cos( kx ) cos( kz ) (cid:105) ϕ ( r , t ) . (57a) It is important to note that the original problem is intrinsicallytime-dependent because of the external laser driving, althoughthis time dependence has been formally eliminated by going intoa rotating frame. Nevertheless, the coupling to the reservoiroccurs at the high, optical frequency range and thus the simpleform of the loss description can be used irrespective of the low-frequency dynamics imposed by the effective Hamiltonian. i ∂∂t α ( t ) = [ − ∆ C + N c U (cid:104) cos ( kx ) (cid:105) − iκ ] α ( t ) + iη + N c η eff (cid:104) cos( kx ) cos( kz ) (cid:105) , (57b)where we used the notation (cid:104) f ( r ) (cid:105) = (cid:82) d rf ( r ) | ϕ ( r , t ) | .The Gross–Pitaevskii equation contains potential-liketerms which depend on the amplitude α and intensity | α | of the cavity field, and express the mechanical effectof the cavity light upon the atoms. The dynamics of thecavity field involves spatial averages over the condensatedensity distribution. Because of cavity decay, the timeevolution leads to a self-consistent stationary solution forthe mean fields, which usually is obtained only numeri-cally (Nagy et al. , 2008).For a given condensate wave function and coherentcavity field amplitude in steady state, the quantum fluc-tuations to leading order form a linear system, whichprovides the energy spectrum of excitations. With thenotation R = [ δa, δa † , δ Ψ( r ) , δ Ψ † ( r )], the time evolutionof the fluctuation operators takes the compact form ∂∂t R = MR + Ξ , (58)where M is the linear stability matrix of the mean fieldsolution (Nagy et al. , 2008), and the term Ξ = [ ξ, ξ † , , M is non-normal, i.e. it does notcommute with its hermitian adjoint. Therefore it hasdifferent left and right eigenvectors, denoted by l ( k ) and r ( k ) , that form a bi-orthogonal system with scalar prod-uct ( l ( k ) , r ( l ) ) = δ k,l . The decoupled quasi-normal exci-tation modes defined by ρ k = ( l ( k ) , R ) are mixed excita-tions of the photon and the matter wave fields.The spectrum of excitations was analyzed first from acavity cooling point of view in the cavity pumping geom-etry ( η (cid:54) = 0, η eff = 0). The imaginary part of the spec-trum revealed that excitations of the ultracold atomic gascan be damped through the cavity loss channel (Gardiner et al. , 2001; Horak and Ritsch, 2001a), provided the decayrate κ is on the order of the recoil frequency ω R . The ex-citation spectrum was used in further studies to describecritical phenomena, such as the dispersive optical bista-bility in the cavity pumping geometry (Szirmai et al. ,2010) and the self-organization phase transition (K´onya et al. , 2011; Nagy et al. , 2008) in the atom pumping ge-ometry ( η eff (cid:54) = 0, η = 0), see Sec. IV.C and Sec. IV.D.In the stable regime, second-order correlation func-tions can be derived from Eq. (58). Importantly, therecan be a non-vanishing population of the atomic excitedmodes, (cid:104) δ Ψ † ( r ) δ Ψ( r ) (cid:105) (cid:54) = 0, even at zero temperature.This quantum depletion of the condensate is indepen-dent of collisional interactions, which are known to causea finite population of the Bogoliubov modes (Pitaevskiiand Stringari, 2003). At variance, here the condensatedepletion arises from the cavity-mediated atom-atom in-teractions as well as the dissipative process associated2with cavity decay. The quantum noise accompanying thephoton loss process couples into the atomic system andexcites atoms out of the condensate mode. Formally, itstems from the term containing the photon creation oper-ator a † in the equations of motion of Ψ( r ). This noise am-plification mechanism is analogous to the Petermann ex-cess noise factor in lasers with unstable cavities (Grang-ier and Poizat, 1998). It was shown by Szirmai et al. (2009) that a depletion on the order of (cid:112) ∆ C + κ /ω R can be expected rather independently of the atom–fieldcoupling, even for U → η eff = 0. It is a signa-ture of the global coupling that the quantum depletionis independent of the total atom number N . In mostof the experiments with linear cavities, the ratio κ/ω R is large ( ∼ ). Since cavity decay can also be inter-preted as a continuous weak measurement of the cavityphoton number, the depletion can be attributed to quan-tum back-action upon the atomic many-body state, asdiscussed in Sec. IV.C.3. It is also interesting to notethat the second-order correlation functions reveal an en-tanglement between the matter-wave and the cavity fieldmodes (Szirmai et al. , 2010). C. Cavity opto-mechanics with ultracold atomic ensembles
In this section we focus on the dispersive interactionbetween the collective motion of a quantum gas anda single-mode Fabry-P´erot cavity, which is coherentlydriven with amplitude η by a laser field at frequency ω .In this case the dispersive matter-light interaction, (53),is given by H AC = (cid:90) d r Ψ † ( r ) (cid:104) ¯ hU cos ( kx ) a † a (cid:105) Ψ( r ) . (59)On the one hand, the atomic medium experiences a pe-riodic potential, whose depth is proportional to the in-tracavity photon number a † a . The potential depth for asingle cavity photon is U = g / ∆ A and can be tuned inthe experiment via the detuning ∆ A between the cavitypump frequency ω and the atomic transition frequency.On the other hand, the atom-light interaction causes adispersive shift of the empty cavity frequency, which isdetermined by the spatial overlap between the atomicdensity Ψ † ( r )Ψ( r ) and the cavity intensity mode func-tion cos ( kx ). A change in the atomic density distribu-tion caused by the intracavity dipole force can thereforedynamically act back on the intracavity field intensity byshifting the cavity resonance with respect to the drivingfield.In general, the interplay of these two effects results ina highly nonlinear evolution of the coupled atoms-cavitysystem. For certain limiting situations, however, the sys-tem can effectively be described in the framework of cav-ity optomechanics (Kippenberg and Vahala, 2008), which studies the radiation-pressure interaction between a har-monically suspended mechanical element and the fieldinside an electromagnetic resonator. In a frame rotatingat ω this is described by the generic cavity optomechanicsHamiltonian H OM = ¯ hω m c † c − ¯ h ( δ C − GX ) a † a + i ¯ hη ( a † − a ) (60)where c † and c denote creation and annihilation opera-tors of the mechanical oscillator at frequency ω m . Themechanical element couples via its position quadrature X = ( c + c † ) / √ G to the in-tracavity photon number a † a . The detuning betweenthe driving laser and the cavity resonance frequency forzero displacement X is denoted by δ C . Configurationsin which the harmonic oscillator couples quadratically in X to the cavity field have recently been realized (Purdy et al. , 2010; Thompson et al. , 2008). This offers the pos-sibility to detect phonon Fock states of the mechanicalelement and to prepare squeezed states of the mechanicaloscillator or the optical output field.
1. Experimental realizations
Particular experimental situations allow to realize theoptomechanics Hamiltonian Eq. (60) with an atomic en-semble dispersively coupled to the field inside an opti-cal cavity. This relies on the fact, that the cavity fieldaffects and senses predominantly a single collective mo-tional mode, which matches the spatial cavity mode pro-file and plays the role of the harmonically suspended me-chanical element. Two different approaches for realizingcavity optomechanics with ultracold atoms have so farbeen realized experimentally. a. Collective center-of-mass motion in the Lamb-Dicke regime
In experiments performed by the Berkeley group (Gupta et al. , 2007; Murch et al. , 2008), ultracold atoms areloaded into the lowest band of a far-detuned intracavitylattice potential, forming a stack of hundreds of tightlyconfined atom clouds (see Fig. 23). Each atom cloudis harmonically suspended with oscillation frequency ω m and extends along the cavity axis by only a fraction ofthe optical wavelength, thus realizing the Lamb-Dickeregime. A cavity mode, whose periodicity differs fromthat of the trapping lattice potential, couples stronglyto a single collective center-of-mass mode of the atomicstack. All remaining collective modes decouple from thecavity field and can be considered as a heat bath to whichthe distinguished collective mode is only weakly coupledvia e.g. collisional atom-atom interactions. The systemrealizes the linear optomechanics Hamiltonian Eq. (60),with the optomechanical coupling strength being givenby G = √ N eff kU X ho . Here, k is the cavity wave vec-tor, X ho = (cid:112) ¯ h/ mω m denotes the harmonic oscillator3 FIG. 23 (Color online) Scheme for cavity optomechanics withultracold atoms confined in the Lamb-Dicke regime. A high-finesse cavity supports two longitudinal modes: one withwavelength of about 780 nm that is near the D line of Rb,and another with wavelength of about 850 nm. The latter pro-duces a one-dimensional optical lattice potential, with trapminima indicated in orange, in which ultracold Rb atomsare confined within the lowest vibrational band. The atomicclouds induce, depending on their trapping position z i , disper-sive frequency shifts on the 780 nm cavity resonance, as shownin the bottom line. In turn, the cavity field exerts a positiondependent force f , as indicated by the red arrows. In theLamb-Dicke regime, the collective atoms-cavity interactionreduces to the generic optomechanics Hamiltonian whereina single collective mode of harmonic motion linearly couplesto the cavity field. From Botter et al. (2009). length with atomic mass m , and N eff ≈ N/ N .The quadratic coupling regime of optomechanics withultracold atoms was realized in an atom-chip-based setup(Purdy et al. , 2010) which allows for subwavelength po-sitioning of a tightly confined ultracold ensemble. Bypreparing as low as two atomic clouds, tightly confinedat adjacent lattice sites of a far-detuned intracavity lat-tice potential, and controlling their center-of-mass po-sition along the cavity axis both linear and quadraticoptomechanical coupling can be realized, providing anatoms-based realization of the ”membran-in-the-middle”approach (Thompson et al. , 2008). b. Collective density oscillations in a Bose-Einstein conden-sate A different route to realize linear cavity optome-chanics with an ultracold atomic ensemble was experi-mentally explored in the Z¨urich group (Brennecke et al. ,2008). A Bose-Einstein condensate of typically 10 atomsis prepared in an external harmonic trapping potential,extending over several periods of the cavity standing-wave mode structure (see Fig. 24). In contrast to theLamb-Dicke regime considered before, here a momen-tum picture is more appropriate. Initially, all condensateatoms are prepared – relative to the recoil momentum¯ hk – in the zero-momentum state | p = 0 (cid:105) . The dis-persive interaction with the cavity field diffracts atomsinto the symmetric superposition of momentum states | ± hk (cid:105) along the cavity axis. Matter-wave interferencebetween the macroscopically occupied zero-momentum component and the recoiling component results in aspatial modulation of the condensate density with pe-riodicity λ/
2, which oscillates in time at the frequency4 ω R = 2¯ hk /m . As long as diffraction into higher-ordermomentum modes can be neglected, the dynamics of thecoupled system is again captured by the simple optome-chanics Hamiltonian Eq. (60). Here, collective excita-tions of the recoiling momentum mode play the role ofphonon excitations of a mechanical mode with oscillationfrequency ω m = 4 ω R . The coupling rate, G = √ N U / et al. , 2011, 2010; De Chiara et al. , 2011;Zhang et al. , 2009). collectivedensityoscillation (a) (b) p/ h k 204 h ω R -2 FIG. 24 (Color online) Cavity-optomechanics with a weaklyconfined Bose-Einstein condensate dispersively coupled to thefield of an optical high-finesse resonator. (a) A collectivedensity excitation of the condensate (blue) with periodicity λ/ π/k acts as a mechanical oscillator with oscillationfrequency 4 ω R . Optomechanical coupling to the cavity fieldis provided by the dependence of the optical path length onthe atomic density distribution within the spatially periodiccavity mode structure. (b) Condensate atoms initially pre-pared close to zero momentum, p = 0, are scattered off theintracavity lattice potential into the symmetric superpositionof states with momentum p = ± hk . Matter-wave interfer-ence with the macroscopic zero-momentum component resultsin a harmonic density oscillation evolving at frequency 4 ω R .Adapted from Brennecke et al. (2008). Further realizations of cavity optomechanics withatomic ensembles have been proposed theoretically bydispersively coupling a quantum-degenerate Fermi gas(Kanamoto and Meystre, 2010) or the internal spin-degrees of freedom of a quantum gas (Brahms and Kurn,2010; Jing et al. , 2011) to the field of an optical cavity.The latter system has been shown to exhibit a formalanalogy with a torsional oscillator coupled quadraticallyto the cavity mode. It provides an ideal nondestructivetool for the control of quantum spin dynamics, and wasproposed to resolve the quantum regime of an antiferro-magnetic spin-1 condensate.4The realization of an optomechanical system using ul-tracold atoms offers direct access to the quantum regimeof cavity optomechanics. In contrast to solid-state real-izations of optomechanics, evaporative cooling techniquesavailable for atomic gases allow for a natural and verypure preparation of the mechanical oscillator mode in itsquantum ground state. Correspondingly, these systemspave the way to directly study quantum effects of theoptomechanical interaction (Brahms et al. , 2012; Brooks et al. , 2012; Murch et al. , 2008).The easy tunability of system parameters like e.g. themechanical oscillator frequency ω m (via the external con-fining potential), the optomechanical coupling strength G (via the atom number or the pump-atom detuning) orthe initial temperature of the mechanical oscillator allowsto explore the transition between different regimes of op-tomechanics. Most important, the coupling strengths G achievable with atomic systems open access to the ’granu-lar’ regime of optomechanics (Ludwig et al. , 2008; Murch et al. , 2008), where single excitations in either of the twosubsystems have a non-negligible effect upon the dynam-ics of the other. This can be measured by the granularity(or quantum) parameter which is defined as ζ = G/κ .For ζ = 1, already a single excitation of the mechanicalmode shifts the cavity resonance by half its linewidth,and already a single photon entering the cavity impartsone excitation quantum in the mechanical oscillator. Infuture research, this might allow the generation and de-tection of quantum correlations between the mechanicaland light degrees of freedom. Further research possibili-ties based on atoms-based realizations of optomechanics,are given by possible implementations of precision sen-sors of forces, or devices to manipulate light fields on aquantum level.
2. Nonlinear dynamics and bistability for low photon number
The optomechanical interaction, Eq. (60), being intrin-sically nonlinear gives rise to dispersive optical bistabilityand nonlinear dynamics of the coupled system. Opticalbistability (Lugiato, 1984), a well studied phenomenon innonlinear optics, refers to the co-existence of two stablesteady-state solutions when e.g. driving an optical cavityfilled with a medium whose refractive index depends onthe light intensity. In typical nonlinear Kerr media andsolid-state realizations of optomechanics, the occurrenceof bistability typically requires large intracavity power inorder to significantly alter the system’s optical proper-ties. The large coupling strength achieved in the atomic-ensemble realizations of optomechanics induces opticalbistability at a mean-intracavity photon level below one(Gupta et al. , 2007; Ritter et al. , 2009). This achieve-ment is desirable for applications ranging from opticalcommunication to quantum computation (Cirac et al. ,1997; Imamoglu et al. , 1997). The occurrence of bistability in optomechanical sys-tems can be understood from the corresponding semiclas-sical equations of motion for the oscillator displacement X and the coherent intracavity field amplitude α derivedfrom Hamiltonian Eq. (60)¨ X + ω m X = − ω m G | α | (61)˙ α = ( i ( δ C − GX ) − κ ) α + η . In the bad cavity regime, κ (cid:29) ω R , the atoms move on atimescale which is large compared to the lifetime (2 κ ) − of intracavity photons. Correspondingly, the cavity fieldadiabatically follows the atomic dynamics according to | α | = η κ + ( δ C − GX ) . (62)Retardation effects resulting in dynamical back-actioncooling or heating of the mechanical element are ne-glected in this approximation. Inserting this expres-sion into the equation of motion for X , Eq. (61), yields¨ X = − ddX V OM ( X ). The optomechanical potential givenby V OM ( X ) = 12 ¯ hω m X − ¯ hη κ arctan (cid:18) ∆( X ) κ (cid:19) , (63)captures the combination of the harmonic confinementand the cavity dipole forces. Here, ∆( X ) = δ C − GX denotes the detuning between the driving laser and theatoms-shifted cavity resonance. −10 −5 0 5024 δ C / κ | α | / ( η c r / κ ) V O M ( a . u . ) (e) XX (d)(c)(a) (b) a dcb V O M ( a . u . ) −10−20 0 10 20 −10−20 0 10 20−10−20 0 10 20−10−20 0 10 20 FIG. 25 (Color online) Optomechanical potential and bista-bility. (a-d) Optomechanical potential landscape V OM ( X ) fordifferent pump-cavity detunings δ C , indicated by the dashedlines in (e). The shaded regions show the resonance profileof the cavity. (e) Mean intracavity photon number | α | insteady state. Open and close circles correspond to the situa-tion shown in (b). Parameters are G = 0 . κ and η = √ η cr . The optomechanical potential provides an intuitivepicture to understand the steady-state as well as thedynamical behavior of the coupled system, see Fig. 25.Above a critical cavity pump strength η cr , determinedby η = √ ω m κ G , the optomechanical potential ex-hibits (within a certain detuning range) two local min-ima, which correspond to different intracavity light inten-sities as shown in the bistable resonance curve, Fig. 25e.5Depending on the direction into which δ C is adiabati-cally tuned, the system remains in either of the two localminimum of V OM , following the upper or lower bistableresonance branch. When reaching the critical detuning,where one of the local minima turns into a saddle point,the system starts to perform transient oscillations in theremaining potential minimum, which translate into a pe-riodically modulated cavity light intensity. Due to damp-ing of the collective atomic motion, the system finally re-laxes to the steady-state in the remaining potential min-imum.Optical bistability induced by collective atomic motionwas observed at low intracavity photon number both inthe Berkeley group (Gupta et al. , 2007) and the Z¨urichgroup (Ritter et al. , 2009). The lower and upper bistabil-ity branches were observed in single experimental runs byslowly sweeping the frequency of the driving laser twiceacross resonance, first with increasing detuning and thenwith decreasing detuning, see Fig. 26. Upon increasingthe probe strength, the cavity transmission profile be-comes more and more asymmetric and exhibits hystere-sis. FIG. 26 (Color online) Dispersive optical bistability with col-lective atomic motion. (a) Observed cavity line shapes (red)for increasing cavity input power at low intracavity photonnumber ¯ n and model line shapes (black), based on the Voigt-profile of the bare cavity line (inset). ∆ pc denotes the detun-ing between the probe laser frequency and the empty cavityfrequency. (b) Lower (blue) and upper (red) branches of op-tical bistability observed in single sweeps across resonance.From Gupta et al. (2007). Dispersive optical bistability with collective atomicmotion was also studied in the regime of quadratic op-tomechanical coupling (Purdy et al. , 2010). Here, insteadof displacing the center-of-mass motion of the mechani-cal element, the intracavity dipole forces increase or de-crease the rms width of the compressible atomic ensem-ble, depending on whether the atoms are confined at amaximum or a minimum of the intracavity probe latticepotential. The corresponding change of the dispersivecavity shift leads again to bistable resonance curves asobserved in the experiment (Purdy et al. , 2010).Dynamical optomechanical effects arise in small-
Time (ms) P ho t on c oun t r a t e ( M H z ) (a) −10 −5 5 100 Time (ms) P ho t on c oun t r a t e ( M H z ) (b) FIG. 27 Nonlinear dynamics of a driven BEC-cavity system.(a) Sweeping the driving laser frequency across the bistableresonance curve (indicated in dashed, scaled by a factor of4) excites large-amplitude density oscillations in the conden-sate. (b) The magnified cavity transmission signal indicateshow the density oscillations tune the cavity frequency peri-odically in an out of resonance with the driving laser. Themean intracavity photon number on resonance was 7.3 corre-sponding to a photon count rate of 5.8 MHz. Adapted fromBrennecke et al. (2008). amplitude oscillations of the system around steady state.As a result of the optomechanical interaction the fre-quency of such oscillations is shifted with respect to thebare oscillation frequency ω m , in the literature often re-ferred to as the ”optical spring effect”. In the case oflinear optomechanical coupling this can be inferred froma quadratic expansion around the steady state minima ofthe optomechanical potential V OM , Fig. 25. Experimen-tally, the optomechanical frequency shift for the collectiveatomic motion was observed and quantified in agreementwith theory both in the linear and the quadratic couplingregime (Purdy et al. , 2010). Highly nonlinear oscillationsin the optomechanical potential with relatively large am-plitude have been excited and observed in cavity trans-mission either by a sudden displacement of the optome-chanical potential (Gupta et al. , 2007) or by crossing theinstability point of the bistable curves (Brennecke et al. ,2008), see Fig. 27.
3. Quantum-measurement back-action upon collective atomicmotion
The accuracy of any position measurement of a me-chanical element is limited by quantum mechanics. Re-ferred to as the standard quantum limit , this has been ex-tensively studied in connection with the development of6gravitational-wave detectors (Caves, 1980). In a genericoptomechanical setup, which allows for high-precisionmeasurements of the position of a mechanical element,the standard quantum limit arises from the balance be-tween two noise terms: (i) detection shot noise, givenby the random arrivals of photons on the detector and(ii) radiation-pressure induced displacement noise causedby the quantum fluctuations of the intracavity photonnumber. Whereas detection noise can be decreased byincreasing the light power, this comes at the expenseof increased radiation pressure force fluctuations. Theoptimal sensitivity is achieved if these two noise contri-bution are balanced. A direct experimental observationof the intracavity photon number fluctuations, requireslarge optomechanical coupling strengths between the in-tracavity field and the mechanical element in combina-tion with the suppression of thermal or technical noisesources which perturb the mechanical motion.The utilization of collective atomic motion of an ul-tracold gas strongly coupled to the field inside a Fabry-P´erot resonator, allowed for the first observation ofmeasurement-induced back-action upon a macroscopicmechanical element formed of 10 atoms, caused by intra-cavity quantum force fluctuations (Murch et al. , 2008).In the non-granular regime ζ = G/κ (cid:28)
1, the spectraldensity of intracavity photon number fluctuations (Mar-quardt et al. , 2007; Nagy et al. , 2009) agrees with thatin an empty driven cavity, and reads S nn ( ω ) = 2¯ nκκ + (∆( X ) + ω ) . (64)Here, ¯ n = | α | denotes the steady-state mean-intracavityphoton number given in Eq. (62). These photon num-ber fluctuations are transmitted into the momentum ofthe mechanical element via the optomechanical interac-tion, giving rise to a diffusion-like increase of the phononnumber ddt (cid:104) c † c (cid:105) = κ ζ S nn ( − ω m ) , (65)as can be derived e.g. from an effective master equationfor the mechanical oscillator (Nagy et al. , 2009).Murch et al. (2008) measured the corresponding heat-ing rate of the atomic ensemble in a bolometric way byquantifying the evaporative atom loss, see Fig. 28. Afterpreparing the mechanical oscillator close to its groundstate, the cavity transmission of a weak probe beam atfixed frequency is monitored on a single-photon count-ing module. Continuous background atom loss tunesthe atoms-shifted cavity frequency in resonance with thedriving laser. The atom loss rate is deduced from thecomparison between the recorded transmission curve andthe empty-cavity resonance curve. The correspondingsingle-atom heating rate is found to exceed the free-spacespontaneous heating rate, which was deduced by measur-ing the atom loss rate far from the cavity resonance, by FIG. 28 Observing measurement-induced back-action uponthe collective motion of an ultracold atomic ensemble. (a)Mean-intracavity photon number ¯ n (points), monitored as thesystem is brought across the cavity resonance due to evapora-tive atom loss. The expected photon number including (solidline) and excluding (dashed line) measurement back-actionis shown. (b) Total atom number N as a function of timeas inferred from data shown in (a), using the empty cavityline shape and the linear relation between atom number anddispersive cavity shift (inset). From Murch et al. (2008). a factor of 40, in agreement with the theoretical expec-tation. As cavity-mediated coherent amplification anddamping of the mechanical oscillator is negligible in theexperiment, the observation of back-action heating canbe interpreted as a direct measurement of photon num-ber fluctuations in a coherently driven cavity.Another direct signature of quantum back-action oflight upon collective atomic motion was obtained by mon-itoring the Stokes and anti-Stokes sidebands of the cavitytransmission subsequent to the preparation of the collec-tive motional degree of freedom close to its ground state(Brahms et al. , 2012), see Fig. 29. The observed side-band asymmetry provides a direct measurement of thequantized collective motion and serves as a record of theenergy exchanged between motion and the light in agree-ment with a continuous back-action limited quantum po-sition measurement.The disturbance of collective atomic motion via theintracavity quantum force fluctuations acts back againonto the intracavity light field. In particular, the result-ing motional-induced modulation of the cavity field caninterfere with the coherent or vacuum cavity input fieldgiving rise to nonlinear optical parametric amplificationand – for negligible technical or thermal fluctuations –to ponderomotive squeezing (Fabre et al. , 1994; Manciniand Tombesi, 1994). Only recently, these effects whereobserved for the first time in the Berkeley group utilizingthe optomechanical coupling between collective atomicmotion and an optical cavity field (Brooks et al. , 2012).7 FIG. 29 Optical detection of quantization of collective atomicmotion in the cavity output spectrum. Shown are mea-sured Stokes sidebands (left panels) and anti-Stokes side-bands (right panels) for increasing intracavity photon num-ber (bottom to top) together with the theoretical prediction(solid lines). The observed Stokes asymmetry provides acalibration-free measure for the mean occupation number ofthe mechanical oscillator, which was deduce for the lowestgraph to be 0 .
49. The mechanical oscillation frequency was ω m = 2 π ×
110 kHz. From Brahms et al. (2012).
4. Cavity cooling in the resolved sideband regime
For the small-volume cavities which were employed inthe experiments performed by the Berkeley and Zurichgroup, the cavity decay rate κ exceeds the mechanicaloscillation frequency ω m of the collective atomic degreeof freedom by more than one order of magnitude. In this non-resolved sideband regime of cavity optomechanicscooling of the mechanical oscillator into its ground stateutilizing cavity dissipation is not possible . Rather, theminimal steady-state occupation number when drivingthe cavity field with a laser field which is red-detuned by (cid:112) ω m + κ from the cavity resonance is given by κ ω m (cid:29) et al. , 2007).Ground-state cooling becomes possible only in the re-solved sideband regime where ω m (cid:29) κ (Kippenberg andVahala, 2008). Here, the cavity is able to resolve theStokes resp. anti-Stokes sidebands which correspond toadding resp. removing motional quanta from the mechan-ical degree of freedom. The large asymmetry between Indeed, in those experiments the preparation of the mechanicaldegree of freedom close to its ground state was achieved directlyby evaporative cooling. these processes which is achieved by driving the cavitynear the anti-Stokes sideband results in a steady-statephonon occupation number of ( κ ω m ) (cid:28) et al. , 2007). Optomechanical cooling in the resolvedsideband regime is equivalent to optical Raman sidebandcooling of tightly confined atoms or ions.Cavity cooling in an optomechanical-type BEC-cavitysystem which ranges in the good cavity regime, κ <ω m = 4 ω R , was demonstrated recently by Wolke et al. (2012). By driving the cavity field selectively close tothe Stokes or anti-Stokes sidebands atoms where trans-ferred via cavity-stimulating backward scattering fromthe macroscopically populated zero-momentum state intoa superposition of momentum states | ± hk (cid:105) and back,see Fig. 30. This experiment paves the way towardsthe achievement of quantum degeneracy starting from athermal gas without relying on evaporative cooling tech-niques. FIG. 30 Observation of sub-recoil cavity cooling with a RbBEC in a narrow-bandwidth Fabry-P´erot resonator. Shownare atomic momentum distributions after driving the cavityfield with a far-detuned laser field at 803 nm. First, a 400 µ slong pulse, blue-detuned from the cavity resonance, transfersatoms into the momentum states | ± hk (cid:105) (left), subsequentlya 200 µ s long red-detuned pulse transfers the atoms back intothe zero-momentum state. Binary atomic collisions result ina substantial depletion of the ± hk -momentum state popu-lations visible as a diffusive halo. From Wolke et al. (2012). D. Non-equilibrium phase transitions
Self-organization of a laser-driven atomic ensemble in-side an optical resonator, as was considered for ther-mal atoms in Section III.B.1, was extended both theo-retically and experimentally into the ultracold regime,where atomic motion becomes quantized. Correspond-ingly, the transition point to the self-organized phase isnot determined anymore by thermal density fluctuations,rather by the competition between kinetic energy costand potential energy gain associated with a spatial mod-ulation of the atomic matter-wave in the cavity-inducedlattice potential. In case of a weakly interacting Bose-Einstein condensate, the reduced number of momentum8states accompanying in the dynamics allows for a sim-plified description in terms of a collective spin degree offreedom, providing a direct link between self-organizationand an open-system realization of the Dicke quantumphase transition.
1. Self-organization of a Bose-Einstein condensate
Self-organization of a dilute Bose-Einstein condensate(BEC), which is located in a single-mode optical cav-ity and illuminated transversally to the cavity axis by afar-detuned laser field, was studied theoretically by Nagy et al. (2008). In terms of a mean-field description, thesteady-state of the system was obtained from the equa-tions of motion for the coherent cavity field amplitude α and the atomic mean-field ϕ ( r ), see Eq. (57b) and (57a),setting the on-axis pump strength η to zero. For simplic-ity only atomic motion along the cavity axis was takeninto account. The numerical solution for the steady-stateorder parameter Θ = (cid:104) ϕ | cos( kx ) | ϕ (cid:105) , obtained by numeri-cally propagating the equations of motion into imaginarytime, is shown in Fig. 31. Above a critical two-photonRabi frequency η eff , the order parameter takes a non-zero value indicating self-organization of the atoms in a λ -periodic density pattern. A stability analysis of thenon-organized steady state, Θ = 0, yields the following x/λ p o t e n t i a l | ϕ ( x ) | √ Nη eff [in units of ω R ] o r d e r p a r a m e t e r Θ FIG. 31 Self-organization of a driven Bose-Einstein con-densate in a standing-wave cavity. Plotted is the steady-state order parameter Θ as a function of the effective cavitypump strength η eff , as obtained from a numerical solution ofthe mean-field equations. Parameters are NU = − ω R ,∆ C = − ω R , κ = 200 ω R and µ = 10¯ hω R in the homoge-neous phase. According to Eq. (66) the homogeneous phase isstabilized in this parameter regime dominantly by collisionalinteraction energy. The inset shows the condensate wave func-tions (solid lines) for √ Nη eff = 100 ω R (broader, brown solid)and 300 ω R (narrower, blue solid) and the corresponding opti-cal dipole potentials (dashed lines). Adapted from Nagy et al. (2008). √ Nη eff [in units of ω R ] e x c i t a t i o n f r e q u e n c i e s [ ω R ] FIG. 32 Collective excitation spectrum of the transversallydriven condensate-cavity system. Shown are the eigenfre-quencies of the lowest six collective atomic-like and the firstcavity-like (divided by 5) excited states as a function ofthe transverse pump amplitude. For vanishing pump am-plitude the Bogoliubov spectrum Ω n = (cid:112) n ω R ( n ω R + 2 µ )for a condensate in a 1D box potential of size λ is retained.Self-organization is indicated by the softening of the lowestlying collective mode towards the critical pump amplitude √ Nη eff , c ≈ . ω R . Parameters are the same as in Fig. 31.From Nagy et al. (2008). analytic expression for the critical point √ N η eff , c = (cid:115) ( δ C + κ )( ω R + 2 µ / ¯ h ) − δ C (66)where µ denotes the chemical potential of the homoge-neous condensate and δ C the detuning of the pump laserfrom the dispersively shifted cavity resonance. In con-trast to the thermal case Eq. (33), the critical transversepump power scales in the zero-temperature limit with therecoil frequency (and the chemical potential), which re-flects the fact that the homogeneous phase is stabilizedby the kinetic energy (and atom-atom collisions).A deeper understanding of the process of self-organization is gained from the collective excitation spec-trum on top of the steady-state mean-field solution. Thiswas calculated in K´onya et al. (2011) and Nagy et al. (2008) using a Bogoliubov-type approach based on theseparation ansatz Eq. (56). The eigenvalues of the lin-earized equations for condensate and cavity fluctuations,Eq. (58), yield the energy spectrum of excitations (polari-tons) shown in Fig. 32. For the considered case where thepump-cavity detuning δ C is large compared to the recoilfrequency ω R , the excitations separate into two classes,according to whether they are dominantly cavity-like oratom-like. The occurrence of self-organization is recog-nized in a characteristic softening of the atom-like excita-tion mode which matches the spatial interference patternbetween cavity and transverse pump mode (Fig. 32, redsolid line).9 FIG. 33 Observation of self-organization with a Bose-Einsteincondensate. Simultaneous time traces of the mean-intracavityphoton number (middle panel) and the relative pump-cavityphase ∆ φ (lower panel) while ramping the transverse pumppower twice across the critical point at ≈ .
35 mW. Theabsorption images (upper panel) show the atomic momen-tum distribution for the indicates times. The line of sightis perpendicular to the pump-cavity plane. Parameters are∆ C = − π ×
20 MHz, κ = 2 π × . N = 10 .Adapted from Baumann et al. (2010, 2011). Self-organization of a Bose-Einstein condensate wasobserved by the Zurich group (Baumann et al. , 2010).A BEC of about 10 atoms, harmonically confined insidea high-finesse optical Fabry-P´erot resonator, was illumi-nated by a far red-detuned standing-wave laser beam.By gradually increasing the power of the transverse laserbeam, the transition to the self-organized phase was ob-served in a sharp raise of the intracavity light intensity ac-companied by the build-up of macroscopic populations inthe momentum states ( p x , p z ) = ( ± ¯ hk, ± ¯ hk ), see Fig. 33.Above the critical pump power, the relative phase ∆ φ between pump field and cavity field is observed to stayconstant, which demonstrates that the system reached asteady state. By controlling the transverse pump power,the system can be transferred repeatedly from the normalin the self-organized phase and back (Fig. 33).The process of symmetry breaking at the transitionpoint was studied in Baumann et al. (2011). In re-peated realizations of the self-organized phase, two pos-sible values of the relative phase ∆ φ with a differenceof π were observed, according to self-organization intoeither the even ( u ( x, z ) >
1) or odd ( u ( x, z ) < u ( x, z ) = cos( kx ) cos( kz ) (Fig. 34). The finite spatialextent of the atomic cloud results in a tiny imbalancebetween the even–odd populations in the non-organizedphase. This effectively acts as a symmetry breaking field, φ ∆ FIG. 34 Observation of symmetry breaking at the self-organization transition with a BEC. Shown is in red the rela-tive pump-cavity phase φ monitored on a heterodyne detectorwhile repeatedly entering the self-organized phase by tuningthe transverse pump power P (dashed). The system orga-nizes into one out of two possible checkerboard patterns cor-responding to the two observed phase values differing by π .From Baumann et al. (2010). which favors the realization of one particular organizedpattern, as was observed in the experiment. The influ-ence of the symmetry breaking field could be overcomeby increasing the speed at which the transition is crossed,in accordance with a simple model description based onthe adiabaticity condition (Baumann et al. , 2011).In the limit where the cavity field adiabatically fol-lows the atomic motion, the process of self-organizationcan also be understood as a result of the cavity-mediatedatom-atom interactions, see III.A.1. On a microscopiclevel, these are induced by the virtual exchange of cavityphotons between different laser-driven atoms, accompa-nied by the creation of atom pairs recoiling with momen-tum ¯ hk along the pump and cavity direction. The result-ing λ -periodic density correlations in the atomic cloudenergetically compete with the cost in kinetic energy,which gives rise to a characteristic roton-type softeningin the dispersion relation of the condensate at momenta( ± ¯ hk, ± ¯ hk ), see also Fig. 32. Once the softened excita-tion energy reaches the ground state energy, the systemself-organizes by macroscopically occupying those mo-mentum states. Such mode softening was observed by theZurich group (Mottl et al. , 2012) using a variant of Braggspectroscopy (Stenger et al. , 1999) where the cavity fieldwas probed with a weak laser pulse whose frequency wasdetuned by a variable amount from the transverse pumplaser field. The observed excitation spectrum as a func-tion of sign and modulus of the cavity-mediated atom-atom interaction strength V is depicted in Fig. 35. Thevanishing of the excitation gap at the transition point to-wards the organized phase is accompanied by a divergingsusceptibility of the system to λ -periodic density pertur-bations (Mottl et al. , 2012). As was theoretically con-sidered by ¨Oztop et al. (2012), the softened excitationspectrum can also be probed parametrically via ampli-tude modulation of the transverse pump laser.Conceptually, the self-organized BEC can be regardedas a supersolid (Gopalakrishnan et al. , 2009; Leggett,1970), similar to those proposed for two-component sys-0 ( P cr ) E x c i t a t i o n e n e r g y E s ( h × k H z ) atom dataphoton dataV < > FIG. 35 (Color online) Observation of mode softening in-duced by cavity-mediated atom-atom interactions in a Bose-Einstein condensate. Shown is the motional atomic excitationenergy at momenta ( ± ¯ hk, ± ¯ hk ) along the cavity and pump di-rection as a function of the transverse laser power P , whichsets the modulus | V | of the cavity-mediated atom-atom inter-action. The sign of V is determined by the sign of δ C . Fornegative interaction strength V , the system organizes at thecritical pump power P cr , while for positive interaction an in-creased excitation energy is observed in accordance with theabsence of a phase transition. From Mottl et al. (2012). tems (B¨uchler and Blatter, 2003). Non-trivial diagonallong-range order is induced by the cavity-mediated long-range interactions, which restricts the periodic densitymodulation to two possible checkerboard patterns, incontrast to traditional optical lattice experiments withlaser fields propagating in free space. Simultaneously,the organized phase exhibits off-diagonal long-range or-der which is not destroyed while crossing the phase tran-sition. Only when deeply entering the organized phase,tunneling between different sites of the optical checker-board potential gets suppressed and phase coherence islost (Vidal et al. , 2010).
2. Open-system realization of the Dicke quantum phasetransition
Self-organization of a laser-driven BEC in an opti-cal resonator can be considered as an open-system re-alization of the Dicke quantum phase transition, wherethe quantized atomic motion acts as a macroscopic spinwhich strongly couples to the cavity field. The Dickemodel goes back to the pioneering work of R. W. Dicke(Dicke, 1954) and describes the collective interaction be-tween matter and the electromagnetic field. Consider N two-level systems with transition frequency ω , forminga collective spin variable J , which couple identically to asingle resonator mode at frequency ω . This system canbe described in terms of the Dicke Hamiltonian (also re-ferred to as the Tavis-Cummings model (Tavis and Cum- mings, 1968)) H Dicke / ¯ h = ωa † a + ω J z + λ √ N ( J + + J − )( a + a † ) (67)with the collective coupling strength denoted by λ ∝√ N . The ladder operators J ± = J x ± iJ y describe cre-ation and annihilation of collective atomic excitations.According to Dicke (1954), a collectively excitedmedium, which carries correlations among the differentatomic dipoles, decays within a much shorter time intoits ground state than a single atom. This phenomenon,termed superradiance (or superfluorescence), originatesfrom spontaneous phase-locking of the different radia-tors resulting in a short radiation burst whose intensityis proportional to the number of atoms squared. Super-radiant emission of laser-excited media has been studiedextensively in the past (Gross and Haroche, 1982).In contrast to this transient phenomenon, the DickeHamiltonian Eq. (67) was shown in 1973 to exhibit also aground-state version of superradiance (Carmichael et al. ,1973; Hepp and Lieb, 1973; Lambert et al. , 2004; Wangand Hioe, 1973). When the collective coupling strength λ reaches the critical value λ cr = √ ωω /
2, the Dickemodel undergoes a quantum phase transition from a nor-mal into a superradiant phase, which is characterized by amacroscopic cavity field amplitude (cid:104) a (cid:105) and a macroscopicpolarization (cid:104) J − (cid:105) of the atomic medium. Apart fromits fragility upon the inclusion of the A term originat-ing from the minimal coupling Hamiltonian (Rza˙zewski et al. , 1975), the experimental realization of the super-radiant Dicke phase transition with direct dipole transi-tions was obscured in the past due to practical limitationsin the available dipole coupling strengths.The proposal by Dimer et al. (2007) circumvents theseissues by considering a pair of stable atomic ground stateswhich are coupled via two different Raman transitions in-volving a single ring cavity mode and external laser fields.This scheme realizes the Dicke model through an effec-tive Hamiltonian in an open-system dynamics, includingexternal driving and cavity loss, where the critical cou-pling strength can be reached for realistic experimentalparameters.The transversally driven BEC-cavity system is for-mally equivalent to this proposal upon replacing the elec-tronic atomic states by a pair of motional atomic states,as was shown in Baumann et al. (2010) and Nagy et al. (2010). The two motional states are given by the flatcondensate mode | p x , p z (cid:105) = | , (cid:105) and the coherent su-perposition of the four momentum states | ± ¯ hk, ± ¯ hk (cid:105) ,where x and z denote the cavity and pump direction, re-spectively. Coherent light scattering between the trans-verse pump beam and the cavity mode couples these twomomentum states via two distinguishable Raman chan-nels, resulting in a dipole-type interaction between cav-ity mode and the corresponding collective spin degreeof freedom, see Eq. (67). The parameters ( ω , ω, λ ) of1 FIG. 36 (Color online) Dicke model phase diagram. (a) Dis-played is in color the recorded mean intracavity photon num-ber ¯ n as a function of the transverse pump power P and thepump-cavity detuning ∆ C . A sharp phase boundary is ob-served in agreement with a mean-field description (dashedline). The dispersively shifted cavity resonance for the non-organized BEC is indicated by a horizontal arrow. (b,c) Timetraces of ¯ n while gradually increasing the pump power to1 . et al. (2010). the corresponding realization of the Dicke Hamiltonianare given by the energy difference 2 ω R between the twomomentum modes (neglecting atom-atom collisions), theeffective detuning − δ C between the pump laser frequencyand the dispersively shifted cavity mode frequency, andthe collective two-photon Rabi frequency √ N η eff / et al. , 2010), δ C exceeds the recoilfrequency by three orders of magnitude, thus realizingthe dispersive regime of the Dicke model. Higher-ordermomentum modes do not contribute in the phase transi-tion dynamics itself and are populated only when deeplyentering the self-organized phase (K´onya et al. , 2011).From the analogy to the Dicke model the followingexpression for the critical coupling strength is obtainedupon including cavity decay (Dimer et al. , 2007) λ cr = (cid:114) ( κ + ω ) ω ω . (68)In the absence of atom-atom collisions, this conditionagrees with the result obtained from the stability analy-sis of the mean-field equations, Eq. (66). Experimentally(Baumann et al. , 2010), the phase boundary was mappedout as a function of pump-cavity detuning ∆ C in agree-ment with the theoretical prediction, see Fig. 36.It is instructive to contrast the Dicke quantum phasetransition, realized with a BEC in a single-mode cavity,with the occurrence of free-space superradiant Rayleighscattering off an elongated BEC which is driven by off-resonant laser light (Inouye et al. , 1999). In this ex-periment, a superradiant light pulse was emitted along the axial direction of the atomic cloud accompanied bythe creation of recoiling matter-wave components, oncethe pump intensity exceeded a critical value. This dy-namical effect is equivalent to Dicke superradiance of acollectively excited medium (Dicke, 1954), where matter-wave amplification phase-locks the spontaneous emissionevents into the continuum of optical field modes. Theminimal pump intensity required for superradiance to oc-cur is determined by the balance between loss and gainprocesses. In contrast, light scattering off the BEC into asingle cavity mode is a reversible process and the criticalpump strength dominantly results from the finite pump-cavity detuning.In the self-organized phase the detuning δ C betweenpump laser and the dispersively shifted cavity resonancebecomes a dynamic quantity. This effect is not takeninto account by the description in terms of the Dickemodel Eq. (67) which is a valid approximation as alongas the maximum dispersive cavity shift U N is smallcompared to the pump-cavity detuning | ∆ C | . In thecase where U N exceeds | ∆ C | the system can exhibitdynamically frustrated behavior characterized by a pe-riodic sign change of the effective pump detuning fromthe dispersively shifted cavity resonance, as was observedin Baumann et al. (2010), see Fig. 36c. Theoreticallythe influence of the additional nonlinear dispersive term ∼ U J z a † a appearing in the Dicke model Eq. (67) uponthe dynamics of the coupled BEC-cavity system wasinvestigated in great detail by Bhaseen et al. (2012);Keeling et al. (2010b); and Liu et al. (2011). Employ-ing a semiclassical description, Bhaseen et al. (2012) re-veal a rich phase diagram including distinct superradiantfixed points, bistable and multistable coexistence phaseand regimes of persistent oscillations, and explore thetimescales for reaching these asymptotic states. It is em-phasized that the behavior of the open system is con-trolled by the stable attractors, which do not necessarilycoincide with the points of minimal free energy. As such,there is a crucial distinction between the κ → κ = 0. Similar conclusion can be drawn in the quantumcase, as discussed in the following.The coupling of the cavity field to the electromag-netic field environment, causing cavity decay, amountsto a weak measurement of the coupled BEC-cavity sys-tem. The corresponding quantum back-action results ina diffusion-like depletion of the ground state of the DickeHamiltonian, even at zero temperature. The underly-ing physics is similar to that described in section IV.C.3,with the important difference that the system gets in-creasingly susceptible to quantum back-action when ap-proaching the critical point.The rate at which the ground state of the Dicke Hamil-tonian initially gets depleted due to cavity decay wascalculated in Nagy et al. (2010) based on the Langevinequation approach Eq. (58). In the dispersive regime2 (a) y/y c | α | h δ a † δ a i y/y c | β | h δ b † δ b i FIG. 37 (Color online) Criticality in the closed and open-system Dicke phase transition. The mean values (thick, rightaxes) and the incoherent excitation numbers (thin, left axes)of the photon (a) and atomic (b) fields are plotted as a func-tion of relative coupling strength y/y cr = λ/λ cr . The incoher-ent excitation numbers in steady-state for the open system(thin solid) diverge at the critical point with exponent − − /
2. From Nagy et al. (2011). | δ C | (cid:29) ω R , the ground state depletion happens mostlyin the atomic space and the corresponding diffusion ratecan be approximated below threshold by the expres-sion ω R κ/ | δ C | ( λ/λ cr ) . Per atom, this corresponds for | δ C | (cid:29) κ to a heating rate of κ η δ C . Note, the formalequivalence of this result with the spontaneous heatingrate in a far-detuned dipole trap. Importantly, the useof a large detuning | δ C | removes the time limitation im-posed by measurement-induced back-action.Measurement-induced back-action drives the BEC-cavity system into a steady state which is a dynamicalequilibrium between diffusion and damping. It is veryinteresting that this limiting state is not the same asthe equilibrium state of the system, i.e., for κ = 0 theground state at T = 0. Namely, the order of the twolimiting procedures, t → ∞ and κ → t → ∞ limit) occupation of the cavity field and the excited momentum state werecalculated in Nagy et al. (2011). Flux and second-ordertime correlations of the cavity output signal were investi-gated theoretically in ¨Oztop et al. (2012). The mean-fieldobtained in the thermodynamic limit is a smooth func-tion of κ and the steady-state solution tends to that ofthe ground state of Eq. (67) for κ →
0. By contrast,the comparison of the quantum fluctuations present inthe ground state of the Dicke model and in the steady-state of the damped-driven system exhibit a significantdifference. In the ground-state of Eq. (67) the second-order correlation functions diverge towards the criticalpoint with the exponent − /
2, indicating a mean-field-type transition, whereas in the non-equilibrium case thequantum fluctuations exhibit a divergence with exponent − et al. , 2004) be-tween cavity and atomic subsystem is regularized at thecritical point by the quantum noise associated with cavitydecay. The non-vanishing entanglement, however, showsthat the quantum character of the Dicke quantum phasetransition (self-organization at zero temperature) is notfully destroyed in case of an open-system dynamics, andit cannot be exactly mapped to a thermal noise-drivenphase transition.
3. Phases in highly degenerate cavities
Self-organization of polarizable particles into periodicstructures induced and stabilized by the intracavity lightfield resembles the process of crystallization. In a cavitywith only a single standing-wave mode tuned into reso-nance with the pump field, only the amplitude of the cav-ity field is a dynamical quantity. The formation of a pe-riodic crystal from a spatially homogeneous distributionbreaks the discrete symmetry corresponding to the evenand odd antinodes of the standing-wave mode profile.Already in the two-mode setting of a ring cavity sustain-ing two degenerate counter-propagating modes, the self-organization is accompanied by spontaneous breaking ofa continuous translational symmetry (Nagy et al. , 2006b;Niedenzu et al. , 2010), which induces rigidity against lat-tice deformations. In the case of highly degenerate multi-mode cavities the field has much more freedom to adjustlocally to the particle distribution.The general structure of the resulting complex phasediagram was studied in Gopalakrishnan et al. (2009) andRitsch (2009). In general such setups allow to realizeconceptually novel systems and to explore and discoverproperties of crystalline and liquid-crystalline ordering,including intrinsic effects as dislocations, the growth andarrangement of crystal grain boundaries (see Fig. 38),and the nature of the phonon spectrum. Multimode cavi-ties also offer a natural connection to models developed inthe field of neural networks as the Hopfield model or sim-3ilar spin models with infinite range statistical couplings.First ideas about this relation have been raised very re-cently in Gopalakrishnan et al. (2011a,b); and Strackand Sachdev (2011). Extensions to fermionic atoms inmultimode cavities have been considered in M¨uller et al. (2012).
FIG. 38 (Color online) Self-ordered states in a concentricmultimode cavity forming two-dimensional patterns. The di-agram shows a regime near threshold, with domains locallypopulating distinct TEM xy cavity modes in the equatorialplane. Domains can be punctuated by dislocations (shownin the left half of the figure), but might also show texturalvariation in space (right half of the figure). The black linesrepresent nodes of the cavity field, which separate ’even’ and’odd’ antinodes. As the atoms are Bose-condensed, the atomicpopulation per site is not fixed. From Gopalakrishnan et al. (2009). Gopalakrishnan and coworkers generalized andadapted a field-theoretical framework, also successfullyused in solid-state physics, to describe many-bodysystems coupled to a multitude of degenerate modes of ahigh-finesse cavity (Gopalakrishnan et al. , 2010; Keeling et al. , 2010a). For a quasi-two-dimensional cloud ofatoms confined in the equatorial plane of a concentricoptical cavity, the transition from the homogeneousdistribution into a spatially modulated one is of theBrazovskii type (Brazovskii, 1975), which describes thephase transitions from isotropic to striped structuresin liquid crystals. The description is based on aneffective equilibrium theory which is valid when theeffective cavity loss rate κη / ∆ C is smaller than therecoil frequency ω − R , see IV.D.2. Here, the dispersivecavity shift was assumed to be much smaller than thepump-cavity detuning ∆ C . Unlike the Landau theoryof crystallization, here the free energy of the systemdoes not have a cubic term that breaks the symmetryat the phase transition. The transition persists at zerotemperature, hence it realizes a quantum phase transi-tion of an unusual university class. The non-equilibriumextension of this theory, which includes the effect ofphoton leakage out of the cavity as a perturbation,leads to the conclusion that the photon loss correspondsto an effective temperature and quantum correlationsare washed out by decoherence on timescales longerthan the cavity decay time. Note that the Bogoliubov-type mean-field model description of the open-systemDicke-model in a single mode cavity also predicts the depletion of the ground state (Nagy et al. , 2010) dueto measurement-induced back-action (Murch et al. ,2008), or, in other words, due to the diffusion inducedby fluctuations accompanying cavity photon loss (Nagy et al. , 2009). However, unlike the Brazovskii transition,the Dicke-model system is driven into a steady-state,significantly different from the ground state, which hasa critical point (Nagy et al. , 2011). K.E. / ζ K.E. / ζ a a uniformsuperfluid supersolid/supersmecticnormalsolid S SS SF single mode casemultimode case
FIG. 39 Schematic zero-temperature phase diagram for aBEC in a concentric cavity. The control parameters are theatomic scattering length a and the inverse effective atom-cavity coupling ζ − with ζ = η / ∆ C . For weak, repulsive in-teractions and increasing atom-cavity coupling, the superfluidfirst undergoes self-organization via the Brazovskii transition,thus forming a supersolid. If the transverse laser intensity isincreased further, the supersolid undergoes a transition intoa normal solid (i.e., a Mott insulator). For strong, repulsiveinteractions, the uniform BEC can lose phase coherence con-currently with a first-order self-organization transition. Thissituation is to be contrasted with that for the case of a single-mode cavity (inset), in which there should always be a su-persolid (SS) region separating the uniform fluid (SF) andnormal solid (S) regions. First- and second-order transitionsare marked by 1 and 2, respectively. From Gopalakrishnan et al. (2010). Similarly to the case of the single-mode experimentperformed in the Zurich group (Baumann et al. , 2010),the emergent crystalline state in a transversally drivenmultimode cavity can be considered as a supersolid phasewhere crystalline order and off-diagonal long-range order(long-range phase coherence) coexist. The phase dia-gram, schematically shown in Fig. 39 for a multimodecavity is strikingly different from the single-mode casein that a region with direct uniform superfluid-to-normalphase transition occurs, whereas in the single-mode cav-ity there is always a supersolid state between the uniformand the normal solid phases (Vidal et al. , 2010). It is alsoobserved that, for a strongly layered three-dimensionalstructure, the inter-layer frustration precludes global or-dering and the system breaks up into inhomogeneous do-mains.4
E. Extended Hubbard-type models for ultracold atoms incavities
The theoretical description of the quantum many-bodydynamics of ultracold atoms confined in optical latticesand strongly interacting with a quantized cavity fieldcan be based on a sophisticated extension of the Bose-Hubbard (BH) model (Fisher et al. , 1989). In static op-tical lattices, the BH model properly accounts for thequantum statistical properties of bosonic atoms at thelattice sites, as well as the inter-particle quantum corre-lations (Bloch et al. , 2008). The basic assumption, validin the limit of very low temperature, is that the dynam-ics can be restricted to the lowest (or lowest few) Blochbands of the periodic optical potential. Correspondingly,the many-body wavefunction can be expressed in termsof Wannier functions localized at individual lattice sites.However, if the optical lattice potential is sustainedby the mode of a high-finesse cavity, thus becoming adynamical degree of freedom, it gets a highly nonlin-ear problem to determine the Wannier functions them-selves and thereby the ground state of the many-bodysystem. For a laser-driven cavity, e.g., the atoms disper-sively shift the cavity resonance, thus affecting the intra-cavity field amplitude which itself determines the latticedepth. Hence, the optical lattice potential and the stateof the atoms have to be evaluated in a self-consistentway (Larson et al. , 2008a; Maschler et al. , 2008; Maschlerand Ritsch, 2005; Vidal et al. , 2010), as will be presentedbelow. We focus on the most studied particular caseof spinless bosons, and mention only that the cases offermions and spin particles are expected to lead to in-teresting novel effects (Larson et al. , 2008c; Sun et al. ,2011).
1. Bose-Hubbard model with cavity-mediated atom-atominteractions
Consider an ensemble of N bosonic particles subject toan optical lattice potential which is generated by the fieldof an optical resonator and possibly by an additional,far off-resonant standing-wave laser field. The latter isrepresented by the external potential term V cl ( r ) in thesingle-atom Hamiltonian Eq. (51). Restricting the mo-tional dynamics to the lowest energy band (lowest vibra-tional state), we expand the atomic field operatorΨ( r ) = M (cid:88) i =1 b i w ( r − r i ) , (69)in the Wannier basis of atomic states localized at sites i = 1 . . . M , where b i denotes the associated annihilationoperator. Upon inserting this expansion into Eqs. (49), one obtains H = (cid:88) m (cid:0) − ¯ h ∆ C,m a † m a m + i ¯ hη m ( a † m − a m ) (cid:1) + M (cid:88) i,j =1 (cid:0) E i,j + V cl J cl i,j (cid:1) b † i b j + ¯ h ∆ A (cid:88) l,m g l g m a † l a m M (cid:88) i,j =1 J lmi,j b † i b j + U M (cid:88) i =1 b † i b i ( b † i b i − , (70)where several cavity modes with mode functions f m ( r )and corresponding photon annihilation operators a m areconsidered. The coefficients E i,j and J cl ij are defined asin the standard BH Hamiltonian E i,j = (cid:90) d r w ( r − r i ) (cid:18) − ¯ h ∇ m (cid:19) w ( r − r j ) , (71a) J cl i,j = (cid:90) d r w ( r − r i ) f cl ( r ) w ( r − r j ) , (71b)where we separated the characteristic magnitude V cl ofthe classical trapping potential (difference between max-imum and minimum) from its spatial form f cl ( r ). Thelast term of Eq. (70) describes the on-site interactionwith U = (4 πa s ¯ h /m ) (cid:82) d r | w ( r ) | . Of primary interestare the extra couplings generated by the cavity modeswith matrix elements J lmi,j = (cid:90) d r w ( r − r i ) f ∗ l ( r ) f m ( r ) w ( r − r j ) . (71c)From the cavity field point of view, the diagonal elements, l = m , correspond to the atomic state-dependent disper-sive shifts of the cavity mode frequency ω C,m , whereasthe off-diagonal elements, l (cid:54) = m , describe photon scatter-ing between different cavity modes. The Wannier func-tions w ( r − r i ) appearing in these integrals, in principledepend on the dynamic potential terms generated by thecavity field. This renders the problem highly non-trivial.In the most general case, the Wannier functions haveto be calculated for each photon number state to define acorresponding manifold of parameters in the BH model,Eq. (70). In other words, the couplings J lmi,j are replacedby operators which can be easily expressed in a Fockbasis. Such a brute force approach is necessary if theeffect of the cavity field on the trapping potential is sig-nificantly different for adjacent Fock states (Horak andRitsch, 2001b). Typically, numerical simulations have tobe performed to study, e.g., microscopic processes under-lying many-body effects that are understood in the mean-field limit (Maschler et al. , 2007; Niedenzu et al. , 2010;Vukics et al. , 2007). Obviously, this approach is limitedto small system sizes of a few particles moving in a few5 mode a probe η probe a θ θ FIG. 40 (Color online) Scheme for quantum non-demolitionmeasurement of atomic many-body states in an optical lattice. N atoms are trapped in a one-dimensional lattice potential(green, M lattice sites) which partially overlaps with a cavitymode a (blue) and a transverse probe mode a (red). Thenumber of illuminated lattice sites is denoted by K . Depend-ing on their many-body state the atoms act as a quantumrefractive index whose statistical distribution with respect tothe probe and cavity modes can be mapped out via transmis-sion or diffraction spectroscopy as a function of the probe-cavity detuning or the angles Θ and Θ . cavity modes. Most of the works, however, used approx-imations to treat the cavity-generated optical potentialin which the localized Wannier functions are defined in aself-consistent manner.
2. Cavity-enhanced light scattering for quantum measurementand preparation
Before addressing the problem of dynamical cavity-induced potentials within the framework of the BHmodel, we note that a lot of applications have beendeveloped based on the coupling of quantized cavityfield modes to trapped, ultracold atomic systems in asimple scattering regime, as was exhaustively reviewedby Mekhov and Ritsch (2012b). In the scattering sce-nario, the external lattice potential V cl ( r ) is taken strongenough to define solely the localized Wannier functions,and their modification due to the cavity light forces isnegligible. The quantized cavity field modes are a pertur-bative probe which can yield a mapping between quan-tum properties of atomic many-body states and light ob-servables. This system gives, for example, a means todetermine the quantum state of ultracold atoms by lightscattering (Miyake et al. , 2011).A typical quantum measurement scheme involving asingle cavity mode is depicted in Fig. 40. It was shownthat various quantum states of ultracold bosons trappedin the lowest band of an optical lattice and having equalmean densities can be distinguished (Chen et al. , 2007;Mekhov et al. , 2007a). As a characteristic example, thevery different transmission spectra of the Mott insula-tor (MI) and the superfluid (SF) states are exhibited inFig. 41. In contrast to standard techniques this measure- (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 Δ p / U (b) FIG. 41 (Color online) Cavity transmission spectra showingthe atom number distribution of an ultracold gas in the in-tracavity part of an optical lattice, see Fig. 40. Shown aretransmission profiles as a function of the probe-cavity detun-ing ∆ p of a MI (red) and a SF (blue) state for (a) a goodcavity with κ = 0 . U and (b) a bad cavity with κ = U ,where U = g / ∆ A . In (b) the satellites are not resolved butthe spectra for SF and MI states are still different. Parame-ters are N = M = 30 and K = 15. (c) Spectra for a SF statewith N = M = 70 and different number of illuminated latticesites K . From (Mekhov et al. , 2007c). ment is non-destructive, limited only by quantum mea-surement back-action. Depending on the chosen geome-try, light scattering is sensitive to the global and localatom number fluctuations (Bhattacherjee et al. , 2010;Chen and Meystre, 2009; Mekhov et al. , 2007c), or tolong-range correlations between two or more lattice sites(e.g. four-point correlations) (Bux et al. , 2011; Mekhov et al. , 2007b).The back-action of repeated quantum non-demolitionmeasurements drives the atomic many-body state to-wards specific states (Mekhov and Ritsch, 2009). Thisis in full analogy to microwave cavity QED experiments(Brune et al. , 1990; Guerlin et al. , 2007) where, comple-mentarily, the field state is driven into nonclassical Fockstates by measuring the state of a train of atoms crossingthe cavity.
3. Self-consistent Bose-Hubbard models in cavity mean-fieldapproximation
Genuine cavity-induced dynamical effects appear whenthe excited cavity modes significantly modify the trap-ping with respect to the external potential V cl ( r ). Evenwithout noticeably reshaping the Wannier-functions atthe trapping sites, the perturbative light probe can mod-ify the tunneling rates as was shown for free-space Braggscattering by Rist et al. (2010). In a cavity, local changesof the atomic distribution influence the whole cavity-sustained optical lattice potential. Thereby a new type oflong-range interaction between the particles appears and6gives rise to resonant nonlocal co-tunneling or momen-tum space pairing (Mekhov et al. , 2007b), effects whichgo far beyond the standard Bose-Hubbard model.When the Wannier functions themselves are dynam-ically influenced by the cavity, a self-consistent mean-field approach, similar to the one in Sec. IV.B, hasbeen broadly adopted to describe the nonlinear dynam-ics of trapped, ultracold atoms in a cavity (Chen et al. ,2009; Larson et al. , 2008b; Larson and Lewenstein, 2009;Maschler et al. , 2008; Maschler and Ritsch, 2005; Nimm-richter et al. , 2010; Vidal et al. , 2010). Splitting thecavity field amplitude a = (cid:104) a (cid:105) + δa into its mean valueand fluctuations, the main assumption is that only thehighly excited mean field can modify the trapping poten-tial, whereas the fluctuations amount to a perturbativeprobe. Since the cavity mean-field amplitude (cid:104) a (cid:105) dependson the momentary atomic quantum state, so does thedepth and shape of the on-site potential. Therefore, theWannier functions in Eq. (69) and hence the coefficientsEq. (71) in the Hubbard-type Hamiltonian Eq. (70), haveto be determined self-consistently in conjunction with theproper cavity mean field (cid:104) a (cid:105) . It is noteworthy that, often,the self-consistent calculation does not lead to a uniquesolution. a. Phases in dynamical optical lattices Consider a linearcavity with only a single mode being driven and over-lapping with a static optical lattice potential V cl ( r ). As-suming only nearest-neighbor hopping to be relevant wekeep adjacent (cid:104) i, j (cid:105) pairs from the double sum over in-dices i and j in Eq. (70). The corresponding many-bodyHamiltonian is given by H = E ˆ N + E ˆ B + (cid:0) ¯ hU a † a + V cl (cid:1) (cid:16) J ˆ N + J ˆ B (cid:17) − ¯ h ∆ C a † a − i ¯ hη (cid:0) a − a † (cid:1) + U C. (72)The relevant atomic degrees of freedom are the total atomnumber ˆ N and the collective nearest-neighbor coherenceˆ B defined asˆ N = M (cid:88) i =1 b † i b i , ˆ B = M (cid:88) i =1 b † i b i +1 + b † i +1 b i , (73)respectively, and the operator ˆ C = (cid:80) i b † i b i (cid:16) b † i b i − (cid:17) forthe two-body on-site interaction. The coefficients E , E , J and J derive from Eq. (71) contracted to a singlecavity mode and assuming uniform coupling along thelattice.To exhibit the underlying physics one may neglect thephoton number dependence of the Wannier functions andadiabatically eliminate the cavity field via the Heisenbergequation of motion˙ a = (cid:110) i (cid:104) ∆ C − U (cid:16) J ˆ N + J ˆ B (cid:17)(cid:105) − κ (cid:111) a + η . (74) FIG. 42 Phase diagram with overlapping Mott insulatorstates. Boundaries of different Mott lobes (shaded regions)as a function of the rescaled chemical potential ˜ µ and theinverse of the pump strength η (in units of κ ) in the 1Dcavity lattice potential of K = 50 sites. Parameters are (a)(∆ C , U ) = (2 κ, − κ ) and (b) (∆ C , U ) = (0 , κ ). The Mottlobes are labeled by the number of atoms per site n . Thedashed lines show the boundaries of zones which are hidden.Outside the shaded parameter regions, the state of the systemis superfluid in most cases. From (Larson et al. , 2008a). This is a good approximation as long as the cavity fielddecays fast compared to the timescale of atomic motion.Since tunneling in deep lattice potentials is slow com-pared to the recoil frequency, this applies widely in ex-perimental setups.An effective atomic Bose-Hubbard model, formallyidentical to the usual one but with coefficients J and J depending on the many-body state, has been consideredsystematically in the thermodynamic limit . By evaluat-ing the stability of the Mott insulator states, the phasediagram has been constructed. As shown in Fig. 42, themodel predicts the existence of competing Mott insula-tor states (Larson et al. , 2008a,b). The overlapping Mottlobes indicate the possibility of bistability in this laser-driven, nonlinear system, see IV.C.2. The state of thesystem can be controlled by fine tuning the pump pa-rameters near the shifted cavity resonance. For certainparameters a state with two atoms per site can lead toa much higher photon number and thus deeper opticalpotential, so that its energy falls below that of the statewith unity filling.7In order to gain insight into the nature of atom-atomcoupling via the cavity field, a simple effective Hamilto-nian can be constructed. The adiabatic field amplitudecan be expanded to second order in the small tunnelingmatrix element Ja ≈ ηκ − iδ C (cid:20) − i U Jκ − iδ C ˆ B − ( U J ) ( κ − iδ C ) ˆ B (cid:21) , (75)where the effective detuning δ C = ∆ C − U J N was in-troduced, and the atom number was set to N . Insertingthis solution back into the Hamiltonian Eq. (72) and theLiouville operator Eq. (5a), which accounts for cavitydamping, leads to an effective adiabatic model. It com-prises the nonlinear Hamiltonian H ad = ( E + JV cl ) ˆ B + U C + ¯ hU Jη κ + δ C (cid:18) ˆ B + U Jδ C κ + δ C κ − δ C κ + δ C ˆ B (cid:19) , (76a)and Liouville operator L ad (cid:37) = κU J η (cid:0) κ + δ C (cid:1) (cid:16) B(cid:37) ˆ B − ˆ B (cid:37) − (cid:37) ˆ B (cid:17) , (76b)which describes decoherence in the basis of the eigen-states of the operator ˆ B . Note that the adiabatic elimi-nation procedure described above is not rigorous mathe-matically, since we adiabatically approximated the solu-tion of a nonlinear dynamical equation in Eq. (75), whichappears also as an ordering ambiguity of the involved op-erators (Larson et al. , 2008b; Maschler et al. , 2008).For a small, numerically tractable system, the lowestenergy eigenstate of H ad can be calculated. As a keyexample, the SF-to-MI quantum phase transition in anoptical lattice sustained entirely by a quantized standing-wave cavity field was analyzed (Maschler et al. , 2008).The dynamical response of the photon number to theatomic motion is able to strongly modify atomic num-ber fluctuations and hence to drive the phase transition.Depending on the cavity parameters (e.g. the detuningbetween cavity and external pump laser), the photon fluc-tuations can either suppress or enhance atomic fluctua-tions and hopping, therefore pushing the system towardsor outwards the MI or SF states. Accordingly, as depictedin Fig. 43, the position of the SF-to-MI phase transitionin a cavity optical lattice potential can be shifted (keep-ing the mean potential depth constant) towards eithersmaller or larger values of the collisional atom-atom inter-action strengths, depending on whether the pump-cavitydetuning is chosen positive (a) or negative (b).Figure 44 demonstrates the importance of the photonnumber fluctuations in a quantum potential by testingthe stability of the Mott phase in a potential of fixed av-erage depth but different mean photon number. Whilein an almost classical field (highly excited coherent state FIG. 43 Mott insulator (MI) to superfluid (SF) phase tran-sition in a cavity optical lattice. The probabilities p MI and p SF to find the atoms in the states | Ψ MI (cid:105) and | Ψ SF (cid:105) as afunction of the dimensionless 1D on-site interaction strength g D / ( dE R ) ( d is the lattice constant, E R = ¯ hω R is the recoilenergy) are compared for two cases: first, for an optical latticesustained by the quantum field of a cavity mode ( V cl = 0),and second, for a purely classical optical lattice ( η = 0). Wechoose η such that in each of the two examples (a and b)both potentials have equal depth for zero on-site interaction g . The quantum (QM) and classical (class) cases are de-picted with solid and dashed lines, respectively. Parametersare ( U , κ, η ) = ( − , / √ , √ . ω R . The detuning betweenthe probe and the dispersively shifted cavity frequencies af-fects the position of the phase transition. In (a) resp. (b)this detuning is positive, ∆ C − U N = κ , resp. negative∆ C − U N = − κ and the transition point is shifted towardssmaller resp. higher interaction strengths in comparison tothat in a classical lattice. From Maschler et al. (2008). with many photons) the Mott phase is stable, photonnumber fluctuations (uncertainty) inherent in a weak co-herent state of few photons enhance tunneling and decayof the perfect order.This is explicitly shown in Fig. 44 depicting the decayof an initially prepared perfectly ordered atomic state.We choose different mean intracavity photon numbersand keep the average depth of the potential constant byreadjusting the coupling strength U . In the classicallimit (very large photon number and small atom-cavitycoupling), the system remains in the initially preparedMI state. For smaller photon numbers ¯ n ∼
20, the ini-8
FIG. 44 (Color online) Effect of photon number granularityupon the Mott insulator (MI) state for two atoms in a cavity-sustained optical lattice. The time evolution of the occupationprobability p MI of the MI state is shown for various meanintracavity photon numbers n . The atom-cavity coupling g is adjusted such that the average potential depth (8 E R ) isidentical for all curves. Whereas for a static optical latticepotential of equal depth the system remains in the initiallyprepared MI state (solid line), photon number fluctuations forlow n deplete the MI state. From Maschler et al. (2008). tial MI state only slowly degrades in time. However,when the photon number fluctuations become compara-ble to the mean, i.e. for mean photon numbers as smallas ¯ n ∼
1, the system quickly escapes from the MI statevia fluctuation induced tunneling. Note that in order tokeep the average optical potential constant, lower pho-ton numbers are connected to a larger potential per pho-ton, so that the potential fluctuations are additionallyenhanced at low photon numbers. The classical limit isalso approached in the bad cavity limit, κ (cid:29) ω R , wherenumber fluctuations occur so fast, that particles only seethe average and do not have the time to tunnel duringan intensity fluctuation.The quantum properties of the cavity light become pre-dominant if already single intracavity photons create anoptical potential of considerable depth, capable of trap-ping numerous atoms. As quantum mechanics allows forthe existence of superpositions of photon number states,one may obtain superpositions of several potentials withdifferent depths (Horak and Ritsch, 2001b).The complete phase diagram of ultracold atoms in two-band BH models coupled to a cavity light field has beencalculated by Silver et al. (2010) by means of a vari-ational approach and the analogy to the Dicke-modelsuperradiant phase transition has been pointed out, seeSec. IV.D.2. b. Self-organization within the Hubbard-model approach
Apossible influence of quantum statistical properties on the rl α rαJJl (a)(b)
FIG. 45 Self-organization as a quantum seesaw effect. (a)Atoms which are trapped in a free-space 1D lattice poten-tial with two adjacent sites (left and right) tunnel with rate J between the corresponding Wannier states | l (cid:105) and | r (cid:105) . (b)Coupling the atoms in addition to the field of a cavity whoseaxis is perpendicular to the optical lattice induces light scat-tering between the optical lattice laser and the cavity fieldwith opposite phases from the two sites. The modified po-tential resulting from the interference of the lattice field andthe cavity field discriminates the two sites and causes positivefeedback and atomic ordering into one of them. The processstarts by spontaneous symmetry breaking, and depends onthe quantum statistics of the initially prepared many-bodystate. spatial self-organization process, described in Sec. III.B.1and IV.D, can be studied within the framework of anextended BH model. For simplicity, the geometry ismodified in comparison with the generic case of self-organization, as depicted schematically in Fig. 45. Atomsare confined in a static optical lattice potential whichis oriented perpendicular to the cavity axis. As before,large atom-laser detuning and negligible atomic satura-tion are assumed. The laser fields providing the opticallattice potential are considered to be tuned close to res-onance with a cavity mode, therefore inducing coherentcavity driving via Rayleigh scattering off the atoms. Thesingle-atom Hamiltonian corresponding to this geometryreads (Maschler et al. , 2007) H = p m + V cl cos ( kx ) − ¯ h (∆ C − U ) a † a + (cid:112) ¯ hV cl U cos( kx ) (cid:0) a + a † (cid:1) , (77)where V cl denotes the depth of the static lattice poten-tial, ∆ C the detuning between the lattice laser and thecavity resonance, and U the light shift of the cavity res-onance frequency per atom. In a simple and intuitivepicture the dynamic cavity field plays the role of a “see-saw” potential. Interference between the cos( kx ) poten-tial, generated through photon scattering, and the staticcos ( kx ) potential determines the overall potential felt bythe atoms. The spatial symmetry of the system allowsfor the emergence of two possible ordered configurations,with all atoms residing at either odd (cos( kx ) = 1) or9 FIG. 46 Entanglement-assisted self-organization in a quan-tum optical lattice potential, see Fig. 45. Shown are atom-light entanglement (solid lines), mean cavity photon number(dotted-dashed lines), and the two-site atom-atom correla-tion function (dotted lines) for two atoms in two adjacentlattice wells, denoted by left ( l ) and right ( r ). Lines with ex-tra circles show the case of exactly one atom in each well atstart (MI state), while the other lines show the evolution foran initially symmetric superposition state for each atom (SFstate). The parameters are U = − κ, ∆ C = − κ, J = κ/ J = 1 . κ . From Maschler et al. (2007). even (cos( kx ) = −
1) lattice sites.The many-body Bose-Hubbard Hamiltonian, Eq. (70),adapted to this scheme reads H = (cid:88) i,j J i,j b † i b j − ¯ h (cid:32) ∆ C − U (cid:88) i b † i b i (cid:33) a † a + (cid:0) a + a † (cid:1) (cid:88) i,j ¯ h ˜ J i,j b † i b j . (78)Here, the standard matrix elements for the kinetic andpotential energy p / m + V cl cos ( kx ) between sites i and j are denoted by J i,j , whereas ˜ J i,j gives the ma-trix elements of the interference term (cid:112) U V cl / ¯ h cos( kx ).On-site interactions are neglected at this point and, forconsistency, we require weak coupling per atom, i.e.¯ h | U | (cid:28) | V cl | .Essential dynamical properties of this system beyondthe mean-field approximation become evident already fortwo atoms. Monitoring the microscopic physics of self-organization as shown in Fig. 46, the process resemblesthe decay of a homogeneously filled lattice with one par-ticle per site on average to the self-ordered state, whereboth particles occupy even or odd sites. Following thedecay of the probability for the two atoms sitting in dif-ferent wells ( (cid:104) n l n r (cid:105) ), we first note that the formation ofthe self-organized state is accompanied by a fast growthof atom-field entanglement. Most importantly, however,one finds a striking dependence of the self-organization dynamics on the initial quantum fluctuations. A SF statewith both atoms prepared in the symmetric superpositionof the two wells, 1 / b † l + b † r ) | (cid:105) , self-organizes muchfaster than a perfectly ordered MI state, b † l b † r | (cid:105) , withexactly one atom per well. In the latter case, the cavityfield remains in the vacuum state until a tunneling eventinduces atomic coherence between the left and right lat-tice site, triggering the decay of the MI state towards theself-organized state.Under some approximations this model can be extrap-olated to the thermodynamic limit where quantum phasetransitions similar to the one predicted in the mean-fieldapproach can be studied. Among various other propertiesthis leads to the coexistence of diagonal long-range orderand long-range coherence (Vidal et al. , 2010), indicat-ing new phases to appear in the gaps between Mott-likestates with different integer filling factors. c. Ring cavity When several independent cavity modesare dynamically interacting with the atoms, not only thedepth but also the shape and the spatial periodicity ofthe potential can change. In the generic case of a ringcavity the depth and the longitudinal position of the lat-tice is dynamical. While, already in standard optical lat-tices the validity of the lowest-band assumption is oftendoubtful and corrections are necessary, this approxima-tion loses its meaning in a ring cavity. Expansions basedon a single set of Wannier functions cannot be consis-tently formulated since the lowest-band Wannier func-tions for a given position contain contributions from alarge number of higher bands for a slightly shifted posi-tion. Hence small lattice shifts immediately involve manyhigher-order bands.The simplest example of two cavity modes sustained bya ring cavity reveals, that a naive crude truncation of theBose-Hubbard model with respect to the pumped cosinemode at the lowest band decouples the atoms from theassociated sine mode. This immediately eliminates cen-tral dynamical effects of the system as overall momentumconservation and nonlocal correlated hopping (Niedenzu et al. , 2010).As an example, Fig. 47 shows that a quantum jumpin the lattice photon number is accompanied by a sud-den change of the tunneling rate between adjacent sites.After the second jump in the trajectory shown in theFigure, the system returns to the original mean values,however, both the position and photon number quanti-ties exhibit much larger noise, which demonstrates theeffect of heating stemming from the photon number fluc-tuations.0 (cid:2) a † a (cid:3)(cid:2) kx (cid:3) ω R t FIG. 47 (Color online) Correlated photon jumps and tunnel-ing of an atom in a symmetrically driven ring cavity. The sam-ple trajectory, showing the position expectation value (cid:104) kx (cid:105) (left axis) and the mean photon number (cid:104) a † a (cid:105) of the un-driven sine mode (right axis), presents two quantum jumpsoccurring at ω R t ≈
70 and ω R t ≈ U = − ω R , α c = √ C = U − κ and κ = 500 ω R . FromNiedenzu et al. (2010). V. OUTLOOK
After almost two decades of active research in opticalcavity QED with cold and ultracold atoms, initially dom-inated by theoretical investigations, the field presentlyexhibits fast growth with several experimental groupsdemonstrating spectacular effects. Single atoms are rou-tinely cavity-cooled and trapped over seconds within op-tical high-finesse resonators, providing a well-controlledquantum system for quantum information science. Ul-tracold quantum gases prepared in magnetic or opti-cal traps are now reliably coupled to high-quality cavitymodes. Even in the far dispersive regime, these systemsare governed by strong back-action effects of the collec-tive atomic motion on the cavity field degrees of freedom.Cavity decay offers a unique channel to monitor the com-plex coupled atom-light dynamics non-destructively andin real time. Many central atomic variables can be ac-cessed by quantum non-demolition measurements whichminimize quantum back-action.With the aim of trapping and cooling ensembles, nano-and even microscopic particles or arrays of thin mem-branes, the research field of cavity QED, reviewed inthis paper, overlaps and unifies more and more with therapidly growing field of optomechanics (Stamper-Kurn,2012). Trapping and cooling arrays of membranes incavities is only one very striking example (Xuereb et al. ,2012). Practical applications as ultra-sensitive detectors of mass, acceleration or magnetic fields or even tests ofgeneral relativity seem within range of current technol-ogy.Initiated by theoretical studies and early experimentswith cold atomic ensembles, a non-equilibrium quantumphase transition between a superfluid and a supersolidphase has been investigated experimentally. Recently,theoretical investigations opened new directions and pos-sibilities towards controlled preparation and investiga-tions of the physics of spin glasses, more complex su-persolid and superglass phases (Gopalakrishnan et al. ,2011b; Strack and Sachdev, 2011). Further prominentsolid-state Hamiltonians involving phonons or polaronscould be studied with unprecedented control and obser-vation possibilities (Mekhov and Ritsch, 2012a).A recent breakthrough experiment demonstrating sub-recoil cavity cooling towards quantum degeneracy (Wolke et al. , 2012) opens the prospect of replacing evaporativecooling techniques by cavity cooling and direct prepa-ration of exotic quantum states from a thermal gas.This also paves the way towards implementing a contin-uous atom-laser as a new tool in ultracold atom physics(Salzburger and Ritsch, 2007).Still, important challenges as cooling and trapping ofmolecular samples or large suspended objects have notbeen experimentally demonstrated. The prospects ofmulti-species implementations in multimode cavity en-vironments still have to be fully evaluated. Apart fromthis point, experiments seem ahead of theoretical and nu-merical simulation possibilities, where theory has to beimproved and better suited models need to be developed.In a long term vision, cavity-sustained light fieldsallow to couple hybrid systems of very different physicalnature like superconducting qubits, cold quantum gasesand micromechanical oscillators without destroyingquantum coherence of the systems, brought aboutby any classical coupling of such systems. In thisway cavity-based setups with ultracold gases coulddevelop into an important building block for quantuminformation processing or other quantum-based futuretechnologies (Henschel et al. , 2010) or a route to an evenbetter atomic lattice clock (Nicholson et al. , 2012).
Acknowledgments
We thank D. Nagy, G. Szirmai, G. K´onya, A. Vukics,J. Asb´oth, I. Mekhov, C. Genes, K. Baumann, R. Mottl,T. Donner and R. Landig for stimulating discussions.H. R. acknowledges support from the Austrian ScienceFund FWF (grant F 4013-N16). P. D. acknowledges thefinancial support from the Hungarian National Office forResearch and Technology under the contract ERC HU 09OPTOMECH, and from the Hungarian Academy of Sci-ences (Lend¨ulet Program, LP2011-016). F. B. and T. E.1acknowledge financial funding from SQMS (ERC ad-vanced grant), NAME-QUAM (EU, FET open), NCCR-QSIT and ESF (POLATOM).
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