Tailoring quantum gases by Floquet engineering
TTailoring quantum gases by Floquet engineering
Christof Weitenberg, ∗ Juliette Simonet
ILP – Institut f¨ur Laserphysik, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, GermanyThe Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: February 16, 2021)
Floquet engineering is the concept of tailoringa system by a periodic drive. It has been verysuccessful in opening new classes of Hamiltoni-ans to the study with ultracold atoms in opticallattices, such as artificial gauge fields, topolog-ical band structures and density-dependent tun-neling. Furthermore, driven systems provide newphysics without static counterpart such as anoma-lous Floquet topological insulators. In this reviewarticle, we provide an overview of the exciting de-velopments in the field and discuss the currentchallenges and perspectives.
Introduction
The philosophy of quantum simulation with ultra-cold atoms is to engineer the Hamiltonian of interest byadding the relevant terms step by step, e.g., tailored po-tentials, additional internal states or controlled interac-tions. A new dimension is opened by time-dependentcontrol of the system, which allows adding terms suchas artificial gauge fields or density-dependent tunneling.Modifying a system by periodic driving is called Floquetengineering and has been proven to be a very power-ful tool [1, 2]. While driving a system at particular fre-quencies has always been a preferred method for probingit, Floquet engineering radically changes the perspectiveand uses the same technique to modify it. When peri-odically shaking an optical lattice, instead of performingspectroscopy of the band structure, one can hybridize thebands to produce bands with new physical properties.Beyond such engineering of static Hamiltonians, Floquetdriving can also give rise to new phases without staticcounterpart. A classical analogy of this Floquet idea isKapitza’s pendulum, which has the upright state as anew stable equilibrium position due to a fast drive of thepivot point.Floquet engineering as a general concept is used inmany areas including solid-state physics and syntheticsystems such as photonic waveguides [3–5] and it stimu-lates intense exchange between these areas. In cold atomresearch, this approach is particularly fruitful due to thehigh control over these systems, which allows implement-ing a multitude of driving schemes, as well as due to ∗ [email protected] the easily accessible time scales. E.g., driving a solid-state crystal at the relevant scales requires laser beamsof extremely high intensity, which can be provided onlyby pulsed lasers and requires suitably fast measurementschemes [6]. In cold atoms, the driving frequency is typ-ically in the Kilohertz regime and the displacement am-plitude can span over many lattice sites, such that themain limitation is the heating inherently associated withFloquet systems. In this review article, we want to givea short overview of the techniques known as Floquet en-gineering with a focus on optical lattices and discuss thenew physics that has been explored with quantum gasexperiments. Effective Hamiltonian and renormalized tunneling
Floquet systems are periodically driven systems de-scribed by a Hamiltonian ˆ H ( t + T ) = ˆ H ( t ) with the driv-ing period T = 2 π/ Ω. The primary interest in Floquetsystems is the fact that despite being time dependent,they can be described by a time-independent effectiveHamiltonian ˆ H eff , if one probes at multiples of the driv-ing period, i.e. the time-evolution operator is given by U ( t + T, t ) = e i ˆ H eff T/ (cid:126) [1, 2]. The dynamics withinone driving period, the so-called micromotion, can of-ten be separated from the long-time dynamics. The ef-fective Hamiltonian can be calculated, e.g., by a high-frequency expansion in powers of 1 / Ω [1, 2]. In lowestorder, which is a suitable approximation in the high-frequency limit, it is given by the average over one drivingperiod ˆ H eff = < ˆ H ( t ) > T .As a simple but insightful example, let us consider ul-tracold atoms in a driven one-dimensional optical lattice,i.e. a periodic potential formed by the interference of twolaser beams. In the tight-binding description, the disper-sion relation is given by the energy E ( k ) = − J cos( kd )with the tunneling element J , the quasimomentum k andthe lattice constant d . Lattice shaking can be easily real-ized by periodically changing the frequency of one laserbeam and it results in a oscillating inertial force of theform F cos(Ω t ), i.e. with zero average force. For anatom prepared in a certain quasimomentum state, latticeshaking leads to a periodic oscillation in quasimomentumspace. In the high frequency limit, the effective energyis given by the average over the varying energy exploredduring one oscillation (Fig. 1a). The effective Hamilto-nian is now described by a tight-binding dispersion E eff with a renormalized tunneling element J eff = J J ( K ), a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b where J is the Bessel function of the first kind of orderzero and K = dF (cid:126) Ω is the driving strength. This renor-malization of J with the Bessel function was mapped outexperimentally by studying the rate of expansion of anatomic cloud, which is directly related to J (Fig. 1a)[7].The phenomenon of complete suppression of tunneling atthe zero crossing of the Bessel function is known as dy-namical localization. The same effect as lattice shakingcan also be produced in a static lattice by applying an os-cillating external force, e.g. from an oscillating magneticfield gradient [8].For sufficiently large driving amplitudes where J ( K ) <
0, one obtains an inverted band structure withnew minima at the edges of the Brillouin zone. A Bose-Einstein condensate (BEC) in a shaken lattice will re-condense at the new minima, which can be directly ob-served in the quasimomentum distribution obtained by atime-of-flight expansion (Fig. 1a inset). This sign inver-sion of the tunneling elements is particularly interestingin a triangular lattice, where it can lead to frustrationand intriguing magnetic phases of the classical XY model[9, 10]. An alternative scheme is to dress the lowest bandwith the first excited band by near-resonantly driving thelattice with the band energy difference [11, 12]. This pro-duces, e.g., a double-well dispersion in the lowest band,which can be mapped to a ferromagnetic spin model[11]. Starting from this basic concept of tunnel renormal-ization, one can realize increasingly complex Hamiltoni-ans by employing more and more sophisticated drivingschemes, as we will see in the following.
Artificial gauge fields and topological bandstructures
When the driving scheme breaks time-reversal symme-try (TRS), it can generate a complex tunneling element J = | J | e iθ with a Peierls phase θ [13, 14] (Fig. 1b). TRScan be broken, e.g., by the multi-step scheme shown inFig. 1b or by circular lattice shaking of a two-dimensionallattice [10, 15, 16]. In a one-dimensional lattice, thePeierls phase leads to an effective band structure with aminimum shifted to a finite quasimomentum θ/d , whichagain can be directly revealed via condensation of bosonsat this minimum (Fig. 1b). When switching on the driveabruptly, one can also use the finite group veloctiy atquasimomentum zero of a BEC to obtain a net displace-ment with zero net force, know as quantum ratchet [17].Peierls phase are most relevant in two-dimensional lat-tices, where they give rise to artificial gauge fields andtopological bands and therefore allow accessing a com-pletely new class of important Hamiltonians for quantumsimulation [18–22].This can be understood as follows: In quantum me-chanics, a particle with charge e acquires an Aharonov-Bohm phase when encircling a magnetic flux. In thePeierls substitution on a lattice, this phase is attached tothe tunneling elements and the sum of the Peierls phases around a plaquette yields the magnetic flux Φ throughthe plaquette according to 2 π Φ / Φ = (cid:80) Plaquette θ i withthe magnetic flux quantum Φ = h/ e . Using Flo-quet engineering, these Peierls phases are directly im-plemented independently of a real magnetic field. Thismeans that the effects of a magnetic field on a chargedparticle become accessible for neutral particles such ascold atoms. Furthermore, one can easily imprint anyvalue of the Peierls phases and thereby reach a magneticflux quantum per plaquette, which would require a mag-netic field of thousands of Tesla in a solid state crystal.For real magnetic fields, a central concept is the gaugefreedom, which means that different vector potentials de-scribe the same magnetic field. In Floquet engineering,however, one experimentally implements a specific gaugeby directly imprinting certain Peierls phases on certainbonds around the plaquette. For this reason, one usu-ally speaks of artificial gauge fields. This gauge choice ofthe Floquet protocol can in fact make a difference, e.g.in the adiabaticity time scale for ramping up a magneticflux [23]. The same considerations also hold for artificalgauge fields in bulk systems [18].Another way to induce Peierls phases is laser-assisted(or Raman-assisted) tunneling [24, 25]. In this scheme,the tunneling is suppressed by applying a tilt with anenergy shift of ∆ per lattice site and then resonantly re-stored by a Bragg-transition of two laser beams with wavevectors (cid:126)k , (cid:126)k and frequencies ω , ω , where the resonanceis ensured via the two-photon detuning δ ≡ ω − ω for (cid:126) δ = ∆ (Fig. 1c). The effect of the Bragg laserbeams can also be viewed as a secondary moving lattice,which induces a time-periodic amplitude modulation ata lattice site l at position (cid:126)r l with locally varying phase φ l = ( (cid:126)k − (cid:126)k ) · (cid:126)r l . The effective tunneling is describedby a Bessel function of the first kind of order one J ( K )and a Peierls phase on a bond between the sites l and l (cid:48) , which stems from the local phases of the two laserbeams, i.e. the local phase of the amplitude modulationat the respective lattice sites as θ ll (cid:48) = ( φ l + φ (cid:48) l ) /
2. For asmall driving amplitude K , where the Bessel function canbe linearized, the tunneling element is therefore given by Ke iθ ll (cid:48) [2].In a two-dimensional square lattice, one can arrangethe laser-assisted tunneling such that the Peierls phasesyield a net magnetic flux Φ, the value of which depends onthe angle between the laser beams (Fig. 1c) [26–30]. Thismodel is known as the Harper-Hofstadter model, a latticeversion of the quantum Hall effect, which has topologi-cally non-trivial bands with non-zero Chern number (Fig.2a). Experiments have revealed the extended periodic-ity of the magnetic unit cell [28](Fig. 2b) and measuredthe Chern number of the lowest band via the transverseHall drift in an accelerated lattice [29](Fig. 2c). Inter-estingly, this quantized response originally predicted forfermionic band insulators also appears for a homogeneousfilling of the lowest band with thermal bosons. FiniteHofstader ribbons were also realized with one artificialdimension formed by internal spin states of the atoms, a b c Forcing amplitude (K)k (2π/d)-1 0 1 E e ff ( k ) E ( k , t ) E ( k , t ) |J|e i � d F -F T T F o r c i n g a m p l i t u d e ( K ) Quasimomentum k (2π/d) | J e ff / J | xy eff k (2π/d) Ke i � x |J|e -i � Ke i � m,n Φ Figure 1.
Effective Hamiltonian and renormalized tunneling. a , In a driven optical lattice, the inertial force leads to anoscillation of the atoms in momentum space and in high-frequency approximation the effective energy is given by the averageover one driving period. This leads to an effective dispersion relation described by a renormalized tunneling element J eff , whichcan become negative for sufficiently strong driving. Bottom: The renormalization by the Bessel function J as measured by thesuppressed diffusion of a BEC. Adapted from [7]. b , Shaking protocols breaking TRS allow realizing tunable Peierls phases θ of the tunneling elements beyond 0 and π . Increasing the driving amplitude K continuously shifts the minimum of the lowestBloch band and gives rise to superfluids at finite quasi-momentum. Adapted from [13]. c , In an optical lattice tilted, e.g. by amagnetic field gradient B (cid:48) , the tunneling can be reestablished using laser-assisted tunneling imprinting Peierls phases given bythe local phases of the laser beams. Suitable geometries in a square lattice yield a finite flux Φ through the plaquettes givingrise to the Harper-Hofstadter model. which obtain a coupling with spatially dependent Peierlsphases by replacing the laser-assisted tunneling with Ra-man transitions within the same lattice site [31]. Theartificial dimension has sharp edges, which allows for theobservation of the skipping orbits at the edges [32–35].Another important model is the Haldane model on thehoneycomb lattice [36], which also has Peierls phases, butno net magnetic flux (Fig. 2d). It can be realized by el-liptical lattice shaking [15, 16] or similarly in graphenesheets by illumination with circularly polarized Terahertzradiation [37], as well as in helically propagating pho-tonic wave guides [38]. Experiments have mapped out thephase diagram via the closing of the band gap [15](Fig.2e) as well as the Chern number via the quantized re-sponse in circular dichroism spectroscopy between thebands [39, 40](Fig. 2f).Topological systems are an active area of current re-search and many models and phenomena are exploredwith cold atoms using Floquet engineering [18–22] suchas non-Abelian gauge fields and synthetic spin-orbit cou-pling both in theory [41–44] and experiment [45–49].Cold atom research has developed various new detec- tion techniques complementary to those of solid-statephysics, which allow directly revealing fundamental topo-logical concepts such as Berry phase [50], Berry curvature[16], skipping orbits [32, 33], quantized Thouless pump-ing [51, 52], as well as completely new concepts such aslinking numbers [53–57] or Hopf invariants [58], whichappear in quench dynamics. Current efforts aim at un-derstanding the interplay of topology and interactions[59–63] and the challenges in the context of Floquet re-alizations of the latter [64–66]. Floquet schemes in correlated systems
A completely new class of Hamiltonians can be ac-cessed by driving correlated Hubbard systems involvingan on-site interaction energy U [19]. In laser-assistedtunneling, the resonance condition for restoring tunnel-ing in a tilted lattice now becomes dependent on theoccupation of the lattice sites involved, i.e. dependenton whether the initial and final states involve the in-teraction energy U . For example starting from one D i ff e r en t i a l s h i ft x ( t ) / d Φ=0
Bloch oscillation time (ms)
C=1 Г + ∆Г ± Г ± ( H z ) Г ± ( H z ) Г – –180° –90° 0° 90° 180°–1000100 Max( ξ −) Max( ξ +) C= +1 C= –1 C= 0 AB / h ( H z ) Δ xy π π ππ π A BA B AB J Column density (a.u.)
Φ=0 ad Spectroscopy frequency ω /2 π (Hz)Shaking phase φ J´ e i θ C=0 fe cb J´ e i θ E ne r g y o ff s e t sp Figure 2.
Artificial gauge fields and topological band structures. a , Harper-Hofstadter model. Square lattice withPeierls phases on the horizontal bonds imprinted via laser-assisted tunneling realizing a flux Φ = π . The Peierls phases havea larger periodicity than the lattice and produce a larger magnetic unit cell. b , The quasi-momentum distribution of a BECin the lattice without flux (left) and with flux (right) directly reveals this reduced symmetry. The squares denote the originaland the magnetic Brillouin zone and the arrows denote the respective reciprocal lattice vectors. Adapted from [28]. c , Theeffective bands have a non-trivial Chern number, which can be revealed by the transverse Hall drift in response to a latticeacceleration. The figure shows the differential center-of-mass shift x ( t ) of the cloud as a function of the acceleration time along y both with a magnetic flux (here Φ = π/
2, grey circles) and without magnetic flux (Φ = 0, blue circles). The drift is initiallylinear with the slope given by the Chern number. Adapted from [29]. d , Haldane model. Honeycomb lattice with A and Bsublattices and a staggered flux in the sub-plaquettes resulting from a Peierls phase θ on the next-nearest-neighbor tunnelingelement of strength J (cid:48) . e , The topological phase diagram with different Chern numbers C as a function of the sublattice energyoffset ∆ AB and the phase ϕ of the elliptical lattice shaking, which maps onto the Peierls phase θ in the Floquet realization.The topological phase transitions are indicated by the gap closing between the two lowest bands measured by Landau-Zenertransitions. Adapted from [15]. f , Chern number of the Haldane model for two different parameters measured via quantizedcircular dichroism, i.e. via the area under the difference of the depletion rate spectra Γ ± ( ω sp ) obtained by additional circularshaking with ω sp with the two chiralities ± . Adapted from [40]. atom at both sites, a tunnel process in the lattice tiltedby ∆ becomes resonant for a two-photon detuning of (cid:126) δ = ∆ + U (Fig. 3a). This allows restoring the tun-neling processes for different occupations with separatepairs of laser beams and therefore to address them sep-arately and imprint occupation-dependent (or density-dependent) Peierls phases. Such processes have beenproposed as building block for many interesting exoticHamiltonians. The density-dependent Peierls phases can,e.g., be mapped onto a one-dimensional anyon-Hubbardmodel with the Peierls phase becoming the statistical ex-change phase of the effective anyons [67, 68](Fig. 3a).The same processes can be realized by restoring the tun-neling in the tilted lattice via multicolor lattice shaking[69] or amplitude modulation [70, 71]. For an appropri-ate choice of the laser beams, the laser-assisted tunnelingcan additionally flip the spin of the atoms [72]. This wasemployed in an experimental demonstration of density-dependent tunneling in a fermionic Mott insulator [73].Another possibility to obtain density-dependent tunnel-ing is to periodically modulate the interaction strength it- self taking advantage of Feshbach resonances. This strat-egy was employed in an experimental demonstration ofcorrelated tunneling processes in a bosonic Mott insula-tor [74].Density-dependent tunneling processes are essential forthe implementation of dynamical gauge fields and latticegauge theories [75–77], which include a feedback of theneutral matter onto the synthetic gauge fields. Recentexperiments have realized the first steps in this direc-tion by implementing density-dependent tunneling pro-cesses in isolated double wells using sophisticated shakingschemes, e.g. involving two spin-states and spin-selectivetilts [78, 79]. Density-dependent gauge fields have alsobeen realized by combining lattice shaking and modula-tion of the interactions [80]. Such a combination of modu-lations allows engineering a broad class of unconventionalHubbard models with correlated tunneling [81–83].Periodic driving can also be employed to modify thespin interactions in Hubbard models (Fig. 3b). Usingtwo spin states in a regime of a Mott insulator withone atom per site, spin interactions arise as a super-exchange process consisting of two virtual tunneling pro-cesses with tunneling element J , where the intermediatestate with both atoms at the same lattice site is detunedby U . The superexchange coupling is therefore given by J ex ∼ J /U and renormalizing J and U will change J ex accordingly. High-frequency lattice shaking will only re-duce J and therefore J ex . However, lattice shaking withΩ in near resonance with U will renormalize the interac-tion to U eff = U − (cid:126) Ω and therefore reduce, enhance andeven reverse the spin interactions depending on its fre-quency [84]. A recent experiment has measured both theeffect on the superexchange dynamics in isolated doublewells [85](Fig. 3b) as well as the resulting spin correla-tions in a fermionic many-body system on a honeycomblattice changing from antiferromagnetic to ferromagnetic[85, 86].
Physics beyond an equilibrium description
The physics of driven systems is richer than what iscaptured by the effective Hamiltonian. While the slowdynamics of Floquet systems can often be mapped tostatic Hamiltonians, Floquet systems are inherently non-equilibrium in nature with new properties beyond staticconcepts and they possess new phases without a staticcounterpart. These situations are of particular interestin nonequilibrium quantum statistical physics and theycan be assessed with cold atoms.To get some insight into these issues, let us startwith the Floquet theorem for time-periodic Hamiltoni-ans, which states that the eigenstates are Floquet states | Ψ n ( t ) (cid:105) in analogy to Bloch states for spatially periodicpotentials: they can be written as a time-periodic parttimes a phase evolution | Ψ n ( t ) (cid:105) = | u n ( t ) (cid:105) e − i(cid:15) n t/ (cid:126) withthe Floquet modes | u n ( t + T ) (cid:105) = | u n ( t ) (cid:105) . The statesdo not have energies, but quasi-energies (cid:15) n , which aredefined modulo (cid:126) Ω, because the system can always ex-change energy quanta of (cid:126)
Ω with the drive: Energy isnot conserved in a driven system. In the extended zonescheme, this leads to many copies of the spectrum andcorrespondingly to new band gaps (see Fig. 4b). Thesenew band gaps in the quasi energies can have importantconsequences such as new collision processes becomingresonant [87, 88].An illustrative example of the consequences of the Flo-quet band gaps is the anomalous Floquet topological in-sulator, where the Floquet-nature of the phase gives riseto new topological properties by redefining the connec-tion between the bulk topological index (e.g. the Chernnumber) and the existence of chiral edge states. Accord-ing to the bulk-edge correspondence of static systems,chiral edge states appear for non-trivial bulk bands. Butin the Floquet system, one can have anomalous edgestates, which appear in a system with zero Chern num-ber [89–91]. The game changer comes from the addi-tional gap between bands around the Brillouin-zone ofquasi-energies (see Fig. 4a,b). The formal description requires the introduction of winding numbers to charac-terize the gaps and not surprisingly, the calculation ofthese winding numbers cannot be done from an effectivestatic Floquet Hamiltonian, but requires to take into ac-count the full time-dependency of the system [90]. The-ses states have been realized with photonic waveguides[92, 93] and recently with cold atoms [94], which promiseaccess to bulk and edge observables in the same system.This example shows that established principles have tobe reexamined in a non-equilibrium context. In fact, anew classification of topological insulators has been in-troduced for Floquet systems [95, 96].A consequence of the absence of energy conservationin driven systems is that in the long-time limit the sys-tem will heat up to unconstraint temperature [97]. How-ever, there is a prethermal regime at an intermediate timescale, where the system is in an equilibrium-like state andwhich can be employed to study the physics of the effec-tive Floquet Hamiltonian [98–100]. The time scale of thisprethermal regime, which can be up to 10 driving peri-ods, depends critically on drive frequency and interactionstrength and generally increases for high-frequency driv-ing. Therefore experimental studies with cold atoms haveinvestigated heating in various conditions [101, 102] andmapped out the parameter space to identify and charac-terize the prethermal regime, in particular for bosons indriven optical lattices finding exponentially suppressedheating rates [103–105](Fig. 4c). Outlook
In this review article, we have summarized how Flo-quet engineering of quantum gases has evolved into avery active field of research exploring ever more complexsystems. One ubiquitous challenge is the heating in thedriven system, which has to be controlled. Therefore therecent systematic studies of the prethermal regime are animportant benchmark. Current efforts aim at exploringways to circumvent heating, be it by using non-ergodicsystems [106, 107] or by employing destructive interfer-ence of excitation paths via a two-color drive [108]. Con-trolling heating would allow accessing strongly correlatedphases such as fractional quantum Hall states [63] or half-integer Mott insulators [67], as well as extending quan-tum simulation to high-energy physical concepts such asfull-fledged dynamical gauge fields [75–77]. Furthermore,Floquet systems allow accessing totally new systems suchas discrete time crystals, which spontaneously break thediscrete time symmetry of the Floquet system to a re-duced symmetry with to a period, which is a multiple ofthe driving period [109].Floquet techniques have been introduced to ultracoldatoms as a way to tailor the systems and to add newproperties such as artificial gauge fields. After more thanone decade of intense research and cross-fertilization be-tween theory and experiments, we can say that Floquettechniques do much more than that. They have opened a b J e i � J e x / h ( H z ) U | J e x | / h ( H z ) J k , ω k , ω k , ω k , ω J e i � JJ Figure 3.
Floquet schemes in correlated systems. a , Top: Laser-assisted tunneling in presence of an on-site interaction U allows selectively restoring tunneling processes depending on the site occupations giving rise to density-dependent Peierlsphases. Bottom: Resulting tunneling processes, which carry a Peierls phase θ if the target site is already occupied. The twoconsecutive processes leading to an exchange of two particles therefore yield the phase θ , which can be interpreted as an anyonicexchange phase, mapping the bosonic atoms to anyonic particles. b , Floquet engineering of the spin interactions in doublewells of a fermionic Hubbard system. Top: Lattice shaking in the high frequency limit ( (cid:126) Ω >> U ) leads to the expectedrenormalization of the superexchange coupling J ex ∼ J /U due to the renormalization of J . Bottom: For a driving in nearresonance with U , the behavior is radically different: red-detuned driving ( (cid:126) Ω < U , red diamonds) enhances the superexchangecoupling J ex for increasing driving amplitude K , while blue-detuned driving ( (cid:126) Ω > U , blue and white diamonds) turns J ex negative for larger driving amplitudes. Adapted from [85]. a new avenue to exotic states of matter and to excitingnon-equilibrium physics and they will continue to playan important role. Acknowledgements
This work was supported by the European ResearchCouncil (ERC) under the European Union’s Horizon 2020 research and innovation programme under grantagreement No. 802701 and by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) viaResearch Unit FOR 2414 ’Artificial gauge fields and in-teracting topological phases in ultracold atoms’ underproject number 277974659 and via the SFB 925 ’Lightinduced dynamics and control of correlated quantum sys-tems’ under project number 170620586. [1] M. Bukov, L. D’Alessio, and A. Polkovnikov, Advancesin Physics , 139 (2015).[2] A. Eckardt, Rev. Mod. Phys. , 011004 (2017).[3] T. Ozawa, H. M. Price, A. Amo, N. Goldman,M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Si-mon, O. Zilberberg, and I. Carusotto, Reviews of Mod-ern Physics , 015006 (2019), arXiv:1802.04173.[4] T. Oka and S. Kitamura, Annual Review of CondensedMatter Physics , 387 (2019), arXiv:1804.03212.[5] M. S. Rudner and N. H. Lindner, Nature ReviewsPhysics , 229 (2020), arXiv:1909.02008.[6] J. W. McIver, B. Schulte, F. U. Stein, T. Matsuyama,G. Jotzu, G. Meier, and A. Cavalleri, Nature Physics , 38 (2020), arXiv:1811.03522. [7] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini,O. Morsch, and E. Arimondo, Phys. Rev. Lett. ,220403 (2007).[8] G. Jotzu, M. Messer, F. G¨org, D. Greif, R. Desbuquois,and T. Esslinger, Phys. Rev. Lett. , 073002 (2015),arXiv:arXiv:1504.05573v1.[9] J. Struck, C. Olschlager, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger,and K. Sengstock, Science , 996 (2011).[10] J. Struck, M. Weinberg, C. ¨Olschl¨ager, P. Wind-passinger, J. Simonet, K. Sengstock, R. H¨oppner,P. Hauke, A. Eckardt, M. Lewenstein, and L. Mathey,Nature Physics , 738 (2013), arXiv:arXiv:1304.5520v1. E ne r g y Q ua s i ene r g y arXiv:1912.09443 Number of drive cycles N cyc D en s i t y o f ho l e s ρ h Increasing frequency
C=0Quasimomentum k ( π /d )C=0C=-1C=1 π T π T a bc II Quasimomentum k ( π /d ) II Figure 4.
Physics beyond an equilibrium description. a , b Illustration of the anomalous Floquet topological insulator.Energy spectrum of a two-dimensional lattice in a strip geometry as a function of the quasimomentum k (cid:107) parallel to the strip( a ). In the case of the driven system ( b ), the energy is replaced by a quasienergy and the spectrum contains many copies ofthe bands as well as new gaps at ± π/T . The bulk-edge correspondence makes a connection between the topological index ofthe bulk bands (here the Chern number C ) and the existence of chiral edge states at the edges of the system (red arrow in theinset), which lie in the bulk band gaps (red lines): the Chern number is given by the difference of the outgoing and ingoingedge states. In a static system (top), edge states can only occur, when the bulk bands have non-zero Chern number. In aFloquet-system with its additional gaps, one can have two chiral edge states despite the zero Chern numbers of the bulk bands.These edge states are therefore called anomalous. c , Floquet thermalization in a driven interacting Bose-Hubbard system.Measurement of the density of holes as a function of the driving cycles N cyc measured after ramping the system adiabaticallyinto the atomic limit of a Mott insulator. Heating processes revealed by the formation of holes set in after 10 − N cyc .Strikingly, the heating rate decreases for increasing shaking frequency Ω, which is an indication of the exponential slow downof heating characteristic of Floquet prethermalization. Adapted from [105].[11] C. V. Parker, L. C. Ha, and C. Chin, Nature Phys. ,769 (2013).[12] C. J. Fujiwara, K. Singh, Z. A. Geiger, R. Senaratne,S. V. Rajagopal, M. Lipatov, and D. M. Weld, PhysicalReview Letters , 10402 (2019), arXiv:1806.07858.[13] J. Struck, C. ¨Olschl¨ager, M. Weinberg, P. Hauke, J. Si-monet, A. Eckardt, M. Lewenstein, K. Sengstock, andP. Windpassinger, Phys. Rev. Lett. , 225304 (2012).[14] K. Jim´enez-Garc´ıa, L. J. Leblanc, R. A. Williams, M. C.Beeler, A. R. Perry, and I. B. Spielman, Physical Re-view Letters , 225303 (2012), arXiv:1201.6630.[15] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat,T. Uehlinger, D. Greif, and T. Esslinger, Nature ,237 (2014).[16] N. Fl¨aschner, B. S. Rem, M. Tarnowski, D. Vogel, D.-S.L¨uhmann, K. Sengstock, and C. Weitenberg, Science , 1091 (2016).[17] T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, in Science , Vol. 326 (2009) pp.1241–1244. [18] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. ¨Ohberg,Rev. Mod. Phys. , 1523 (2011).[19] M. Lewenstein, A. Sanpera, and V. Ahufinger, Ultracoldatoms in optical lattice: simulating quantum many-bodysystems (Oxford Univ. Press, 2012).[20] N. Goldman and J. Dalibard, Phys. Rev. X , 031027(2014).[21] D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan,and S. L. Zhu, Advances in Physics , 253 (2018),arXiv:1810.09228.[22] N. R. Cooper, J. Dalibard, and I. B. Spielman, Reviewsof Modern Physics , 015005 (2019), arXiv:1803.00249.[23] B. Wang, X.-Y. Dong, F. N. ¨Unal, and A. Eckardt,(2020), arXiv:2009.00560.[24] D. Jaksch and P. Zoller, New J. Phys. , 56.1 (2003),arXiv:0304038 [quant-ph].[25] F. Gerbier and J. Dalibard, New Journal of Physics ,033007 (2010), arXiv:0910.4606.[26] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur-ton, and W. Ketterle, Phys. Rev. Lett. , 185302(2013). [27] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro,B. Paredes, and I. Bloch, Phys. Rev. Lett. , 185301(2013).[28] C. J. Kennedy, W. C. Burton, W. C. Chung, andW. Ketterle, Nature Phys. , 859 (2015).[29] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala,J. T. Barreiro, S. Nascimb`ene, N. R. Cooper, I. Bloch,and N. Goldman, Nature Physics , 162 (2015),arXiv:1407.4205.[30] M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, T. Menke,Dan Borgnia, P. M. Preiss, F. Grusdt, A. M. Kauf-man, and M. Greiner, Nature , 519 (2017),arXiv:1612.05631.[31] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B.Spielman, G. Juzeli?nas, and M. Lewenstein, Phys.Rev. Lett. , 043001 (2014).[32] M. Mancini, G. Pagano, G. Cappellini, L. Livi,M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio,M. Dalmonte, and L. Fallani, Science , 510 (2015).[33] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, andI. B. Spielman, Science , 1514 (2015).[34] F. A. An, E. J. Meier, and B. Gadway, Science Ad-vances , 1 (2017), arXiv:1609.09467.[35] T. Chalopin, T. Satoor, A. Evrard, V. Makhalov, J. Dal-ibard, R. Lopes, and S. Nascimbene, Nature Physics(2020), 10.1038/s41567-020-0942-5.[36] F. D. Haldane, Physical Review Letters , 2015 (1988).[37] T. Oka and H. Aoki, Phys. Rev. B , 081406(R) (2009).[38] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer,D. Podolsky, F. Dreisow, S. Nolte, M. Segev, andA. Szameit, Nature , 196 (2013).[39] D. T. Tran, A. Dauphin, A. G. Grushin, P. Zoller, andN. Goldman, Science Advances (2017), 10.1126/sci-adv.1701207, arXiv:1704.01990.[40] L. Asteria, D. T. Tran, T. Ozawa, M. Tarnowski,B. S. Rem, N. Fl¨aschner, K. Sengstock, N. Goldman,and C. Weitenberg, Nature Physics , 449 (2019),arXiv:1805.11077.[41] J. Ruseckas, G. Juzeliunas, P. ¨Ohberg, andM. Fleischhauer, Phys. Rev. Lett. , 010404 (2005),arXiv:0503187 [cond-mat].[42] P. Hauke, O. Tieleman, A. Celi, C. ¨Olschl¨ager, J. Si-monet, J. Struck, M. Weinberg, P. Windpassinger,K. Sengstock, M. Lewenstein, and A. Eckardt, Phys.Rev. Lett. , 145301 (2012).[43] M. Burrello, I. C. Fulga, E. Alba, L. Lepori, andA. Trombettoni, Phys. Rev. A , 053619 (2013).[44] V. Galitski and I. B. Spielman, Nature , 49 (2013).[45] Z. Wu, L. Zhang, W. Sun, X. T. Xu, B. Z. Wang, S. C.Ji, Y. Deng, S. Chen, X. J. Liu, and J. W. Pan, Science , 83 (2016).[46] J. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas,F. C¸ . Top, A. O. Jamison, and W. Ketterle, Nature , 91 (2017), arXiv:1610.08194.[47] L. Huang, P. Peng, D. Li, Z. Meng, L. Chen, C. Qu,P. Wang, C. Zhang, and J. Zhang, Physical Review A , 013615 (2018).[48] S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, andI. B. Spielman, Science , 1429 (2018).[49] B. Song, C. He, S. Niu, L. Zhang, Z. Ren, X. J.Liu, and G. B. Jo, Nature Physics , 911 (2019),arXiv:1808.07428. [50] L. Duca, T. Li, M. Reitter, I. Bloch, M. Schleier-Smith, and U. Schneider, Science , 288 (2015),arXiv:1407.5635.[51] S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa,L. Wang, M. Troyer, and Y. Takahashi, Nature Physics , 296 (2016), arXiv:1507.02223.[52] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidels-burger, and I. Bloch, Nature Physics , 350 (2016),arXiv:1507.02225.[53] N. Fl¨aschner, D. Vogel, M. Tarnowski, B. S. Rem, D. S.L¨uhmann, M. Heyl, J. C. Budich, L. Mathey, K. Sen-gstock, and C. Weitenberg, Nature Physics , 265(2018).[54] C. Wang, P. Zhang, X. Chen, J. Yu, and H. Zhai, Phys.Rev. Lett. , 185701 (2017).[55] W. Sun, C. R. Yi, B. Z. Wang, W. W. Zhang, B. C.Sanders, X. T. Xu, Z. Y. Wang, J. Schmiedmayer,Y. Deng, X. J. Liu, S. Chen, and J. W. Pan, Physi-cal Review Letters , 1 (2018), arXiv:1804.08226.[56] M. Tarnowski, F. N. ¨Unal, N. Fl¨aschner, B. S. Rem,A. Eckardt, K. Sengstock, and C. Weitenberg, NatureCommunications , 1728 (2019).[57] M. McGinley and N. R. Cooper, Physical Review Re-search , 33204 (2019), arXiv:1908.06875.[58] F. N. ¨Unal, A. Eckardt, and R.-J. Slager, Physical Re-view Research , 022003(R) (2019), arXiv:1904.03202.[59] N. Regnault and B. Andrei Bernevig, Physical ReviewX , 021014 (2011), arXiv:1105.4867.[60] A. G. Grushin, ´A. G´omez-Le´on, and T. Neupert, Phys.Rev. Lett. , 156801 (2014).[61] T. I. Vanhala, T. Siro, L. Liang, M. Troyer, A. Harju,and P. T¨orm¨a, Phys. Rev. Lett. , 225305 (2016),arXiv:1512.08804.[62] L. Stenzel, A. L. Hayward, C. Hubig, U. Schollw¨ock,and F. Heidrich-Meisner, Physical Review A , 53614(2019), arXiv:1903.06108.[63] S. Rachel, Reports on Progress in Physics , aad6a6(2018), arXiv:1804.10656.[64] E. Anisimovas, G. ˇZlabys, B. M. Anderson, G. Juzeliu-nas, and A. Eckardt, Phys. Rev. B , 245135 (2015).[65] T. Qin and W. Hofstetter, Physical Review B ,075134 (2017).[66] K. Plekhanov, G. Roux, and K. Le Hur, Physical Re-view B , 045102 (2017), arXiv:1608.00025.[67] T. Keilmann, S. Lanzmich, I. McCulloch, andM. Roncaglia, Nature Communications , 361 (2011),arXiv:1009.2036.[68] S. Greschner and L. Santos, Physical Review Letters , 053002 (2015), arXiv:1501.07462.[69] C. Str¨ater, S. C. Srivastava, and A. Eckardt, PhysicalReview Letters , 205303 (2016), arXiv:1602.08384.[70] R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon,and M. Greiner, Phys. Rev. Lett. , 095301 (2011).[71] L. Cardarelli, S. Greschner, and L. Santos, PhysicalReview A , 023615 (2016), arXiv:1604.08829.[72] A. Bermudez and D. Porras, New J. Phys. , 103021(2015).[73] W. Xu, W. Morong, H.-y. Hui, V. W. Scarola, andB. Demarco, Phys. Rev. A , 023623 (2018).[74] F. Meinert, M. J. Mark, K. Lauber, A. J. Daley, andH. C. N¨agerl, Physical Review Letters , 205301(2016), arXiv:1602.02657. [75] L. Tagliacozzo, A. Celi, A. Zamora, and M. Lewenstein,Annals of Physics , 160 (2013), arXiv:1205.0496.[76] U. J. Wiese, Annalen der Physik , 777 (2013),arXiv:1305.1602.[77] E. Zohar, J. I. Cirac, and B. Reznik, Re-ports on Progress in Physics , 014401 (2015),arXiv:1503.02312.[78] F. G¨org, K. Sandholzer, J. Minguzzi, R. Desbuquois,M. Messer, and T. Esslinger, Nature Physics , 1161(2019), arXiv:1812.05895.[79] C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero,E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger,arXiv:1901.07103 (2019), arXiv:1901.07103.[80] L. W. Clark, B. M. Anderson, L. Feng, A. Gaj, K. Levin,and C. Chin, Phys. Rev. Lett. , 030402 (2018),arXiv:1801.10077.[81] S. Greschner, L. Santos, and D. Poletti, Physical Re-view Letters , 183002 (2014), arXiv:1407.6196.[82] H. Zhao, J. Knolle, and F. Mintert, Physical Review A , 053610 (2019), arXiv:1908.05494.[83] T. Wang, S. Hu, S. Eggert, M. Fleischhauer, A. Pelster,and X.-F. Zhang, Physical Review Research , 013275(2020), arXiv:1807.00015.[84] J. R. Coulthard, S. R. Clark, S. Al-Assam, A. Cavalleri,and D. Jaksch, Physical Review B , 085104 (2017),arXiv:1608.03964.[85] F. G¨org, M. Messer, K. Sandholzer, G. Jotzu, R. Des-buquois, and T. Esslinger, Nature , 481 (2018),arXiv:1708.06751.[86] N. Sun, P. Zhang, and H. Zhai, Physical Review A ,043629 (2019), arXiv:1808.03966.[87] T. Bilitewski and N. R. Cooper, Physical Review A ,033601 (2015), arXiv:1410.5364.[88] M. Reitter, J. N¨ager, K. Wintersperger, C. Str¨ater,I. Bloch, A. Eckardt, and U. Schneider, Phys. Rev.Lett. , 200402 (2017).[89] T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys.Rev. B , 235114 (2010).[90] M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin,Phys. Rev. X , 031005 (2013).[91] A. Quelle, C. Weitenberg, K. Sengstock, and C. M.Smith, New Journal of Physics , 113010 (2017).[92] S. Mukherjee, A. Spracklen, M. Valiente, E. Andersson,P. ¨Ohberg, N. Goldman, and R. R. Thomson, NatureCommunications , 13918 (2017), arXiv:1604.05612.[93] L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Sza-meit, Nature Communications , 13756 (2017). [94] K. Wintersperger, C. Braun, F. N. ¨Unal, A. Eckardt,M. D. Liberto, N. Goldman, I. Bloch, and M. Aidels-burger, Nature Physics (2020), 10.1038/s41567-020-0949-y.[95] A. C. Potter, T. Morimoto, and A. Vishwanath, Phys-ical Review X , 041001 (2016), arXiv:1602.05194.[96] R. Roy and F. Harper, Physical Review B , 155118(2017), arXiv:1603.06944.[97] L. D’Alessio and M. Rigol, Physical Review X , 041048(2014).[98] M. Bukov, M. Heyl, D. A. Huse, and A. Polkovnikov,Phys. Rev. B , 155132 (2016), arXiv:1512.02119.[99] T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda,Journal of Physics B: Atomic, Molecular and OpticalPhysics , 1 (2018), arXiv:1712.08790.[100] K. Seetharam, P. Titum, M. Kolodrubetz, and G. Re-fael, Physical Review B , 1 (2018), arXiv:1710.09843.[101] T. Boulier, J. Maslek, M. Bukov, C. Bracamontes,E. Magnan, S. Lellouch, E. Demler, N. Goldman, andJ. V. Porto, Physical Review X , 011047 (2019).[102] K. Wintersperger, M. Bukov, J. N¨ager, S. Lellouch,E. Demler, U. Schneider, I. Bloch, N. Goldman, andM. Aidelsburger, Physical Review X , 011030 (2020),arXiv:1808.07462.[103] M. Messer, K. Sandholzer, F. G¨org, J. Minguzzi, R. Des-buquois, and T. Esslinger, Physical Review Letters , 233603 (2018), arXiv:1808.00506.[104] K. Singh, C. J. Fujiwara, Z. A. Geiger, E. Q. Sim-mons, M. Lipatov, A. Cao, P. Dotti, S. V. Rajagopal,R. Senaratne, T. Shimasaki, M. Heyl, A. Eckardt, andD. M. Weld, Physical Review X , 041021 (2019),arXiv:1809.05554.[105] A. Rubio-Abadal, M. Ippoliti, S. Hollerith, D. Wei,J. Rui, S. L. Sondhi, V. Khemani, C. Gross, andI. Bloch, Physical Review X , 021044 (2020),arXiv:2001.08226.[106] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Re-views of Modern Physics (2019), 10.1103/RevMod-Phys.91.021001, arXiv:1804.11065.[107] I. D. Potirniche, A. C. Potter, M. Schleier-Smith,A. Vishwanath, and N. Y. Yao, Physical Review Letters , 1 (2017).[108] K. Viebahn, J. Minguzzi, K. Sandholzer, A.-S. Walter,F. G¨org, and T. Esslinger, arXiv:2003.05937 (2020),arXiv:2003.05937.[109] K. Sacha and J. Zakrzewski, Reports on Progress inPhysics81