Stroboscopic aliasing in long-range interacting quantum systems
SStroboscopic aliasing in long-range interacting quantum systems
Shane P. Kelly,
1, 2, 3, ∗ Eddy Timmermans, Jamir Marino, and S.-W. Tsai Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Department of Physics and Astronomy, University of California Riverside, Riverside, California 92521, USA Institut f¨ur Physik, Johannes Gutenberg Universit¨at Mainz, D-55099 Mainz, Germany XCP-5, XCP Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
We unveil a mechanism for generating oscillations with arbitrary multiplets of the period of a givenexternal drive, in long-range interacting quantum many-particle spin systems. These oscillationsbreak discrete time translation symmetry as in time crystals, but they are understood via twointertwined stroboscopic effects similar to the aliasing resulting from video taping a single fastrotating helicopter blade. The first effect is similar to a single blade appearing as multiple bladesdue to a frame rate that is in resonance with the frequency of the helicopter blades’ rotation;the second is akin to the optical appearance of the helicopter blades moving in reverse direction.Analogously to other dynamically stabilized states in interacting quantum many-body systems, thisstroboscopic aliasing is robust to detuning and excursions from a chosen set of driving parameters,and it offers a novel route for engineering dynamical n -tuplets in long-range quantum simulators,with potential applications to spin squeezing generation and entangled state preparation. Introduction.
The field of dynamical stabilization hasa long tradition tracing back to the Kapitza pendulumin the mid 60s [1]: a rigid rod can be stabilized in an in-verted position by parametrically driving its suspensionpoint with a tuned oscillation amplitude and at high fre-quency. The working principle of a dynamically stabi-lized upside-down pendulum is the building block for re-alizing periodic motion in atomic physics, plasma physicsand in the theory of dynamical control in cyberneticalphysics. Periodic drives are a versatile tool that can beemployed to stabilize systems in configurations prohib-ited at equilibrium. Applications in the quantum domainrange from cold atoms to trapped ions [2–9]: a drive withlarge amplitude and fast frequency can stabilize an entireband of excitations, turning the dynamics of a collectivemode from a runaway trajectory into a periodic orbit. Inthis work, we propose a flexible route to engineer peri-odic dynamical responses characterized by arbitrary in-teger fractions of the period of the drive, relevant for abroad class of quantum many-body simulators.Periodic dynamics in isolated many-particle systems,can be also found in the absence of an external drive.Examples range from quantum ’scars’ [10–14] to the dy-namical confinement of correlations [15–21] and encom-pass the role of dynamical symmetries [22–26] in evok-ing persistent temporal oscillations. The quest for timetranslation breaking in periodically driven quantum sys-tems [27, 28] has recently morphed into the search forquantum time crystals [29–31]. A discrete time crystal(DTC) occurs when the discrete time translation sym-metry of a periodically driven system is spontaneouslybroken into a smaller symmetry subgroup. One iconicexample [32, 33] of DTC occurs when the spins of a dis-ordered spin chain are flipped at periodic intervals, andtheir local magnetization oscillates with a period twice ∗ Corresponding author: [email protected] n=1 n=2
REC
Appearence n=3
Motion
FIG. 1. [Color Online] The top left figure shows the clas-sical stroboscopic dynamics for an n = 1 resonance with( t , t ) = (2 . , . H trajec-tory with period τ = nt . In the region labeled “Forward”the stroboscopic dynamics appear to move forward along thistrajectory (analogous behaviour holds for the region labeled“Backward”). This apparent reversal of motion is equivalentto the stroboscopic aliasing effect observed when the framerate of a camera is faster than the rotation rate of a heli-copter blade. The top right figure shows example of an n = 2resonance with ( t , t ) = (1 . , . n = 2 subharmonic response. The cartoon depicts an exam-ple of stroboscopic aliasing effects that occurs when the framerate of the camera is n = 3 − | (cid:15) | times the rotation rate of theblade. the one of the spin flips. In this model, the stabilityof the time crystalline behaviour is provided by the ex-tensive set of quasi-local integrals of motion which arecharacteristic of many-body localized phases occurringat strong disorder [34, 35].Since original experiments in trapped atomic ions andin nitrogen-vacancy centers [36, 37], many other mech-anisms for time crystals have been proposed [34, 38–48] a r X i v : . [ c ond - m a t . o t h e r] N ov and observed [49–53]. In all of these systems, the periodicdynamics are split into two parts: the natural dynamicsof a system that possesses a Z n symmetry, and a kickprocess that sequentially switches among the n symme-try sectors. An n -period DTC (or ‘n-tuplets dynamics’)occurs since it takes n of such kick processes to bring thesystem back to its original configuration [54].In this work we show how to engineer dynamics witharbitrary n -tuplets that are not distinguished by the sec-tors of a Z n symmetry. Differently from time crystals,their stability emerges as a cooperative effect betweenthe natural dynamics and the kick process. Subharmonicresponse with any value of n can be generated providedthat the kick period is in resonance with the n th harmonicof a collective mode, and this collective mode remainsstable, though deformed, during the kicked process. Thisresults in stroboscopic dynamics which display n periodoscillations between n emergent dynamical fixed points.By considering the kick akin to the sampling performedby a video camera, we identify this subharmonic responseas similar to a type of stroboscopic aliasing that occurswhen filming a single blade helicopter: when the heli-copter blade is rotating at the n th subharmonic frequencyof the camera’s frame rate, its video will appear to have n stationary blades. Unlike the sampling performed bythe camera, the kick acts on the long-range simulator in-creasing or decreasing the frequency of the system. Thisresults in another stroboscopic aliasing effect in which theapparent n stationary blade appear to slowly move for-ward or backwards depending on if the blade frequencywas increased or decreased (cf. with Fig. 1). We showthat for a general class of kicks, both forward and back-ward aliasing appears and generate a set of n strobo-scopic fixed points that stabilize the subharmonic re-sponse. Stroboscopic aliasing produces also a set of n unstable dynamical fixed points which we argue could beused for generating spin squeezing and entangled states. Stroboscopic Aliasing.
We consider a long-range in-teracting Ising model [55–60] in which the interactionstrength is periodically kicked U ( m ) = ( U U ) m . Wedefine U a as a unitary generated by the following hamil-tonian H a = − N (cid:88) k =1 σ xk + Λ a N − α N (cid:88) k,j =1 σ zk σ zj | k − j | α , (1)where N is the number of spin-halfs, (cid:126)σ k , which live ona one dimensional lattice, and the unitaries are evolvedfor different times t and t and for different interactionstrengths Λ and Λ (i.e. U a = e it a H a , with a = 1 , N − α is to ensure theextensivity of the hamiltonian in the thermodynamiclimit [61]. The subharmonic response emerges when t isin resonance with a collective mode of H and t (cid:28) t .Focusing our attention to this limit, we will refer to U as the kick.The emergent subharmonic response is most clearlyexplained in the α = 0 infinite range limit in which the Unstable Fixed
Stable Fixed Points
500 505 510 515 520m1.00.50.00.51.0 J x J y J z |J| 500 505 510 515 520m0.50.00.51.0 J x J y J z |J| FIG. 2. Stroboscopic classical Poincar´e section (top) and ex-act stroboscopic quantum dynamics (bottom) for N = 500,Λ = 10, Λ = 0: with ( t , t ) = (0 . , .
2) (left), and(0 . , .
1) (right). The color (brightness) in the top plots dis-tinguishes the initial state. The top plot depicts the emergentclassical fixed points for n = 6 (left), and n = 3 (right). Inthe bottom plots, we show the n = 6 and n = 3 subharmonicoscillations due to U moving between the different emergentfixed points. | J | is plotted to illustrate that the fixed pointsstabilize the system against quantum dephasing. model reduces to the LMG model [62–64]. In the large N limit, dynamics reduce to the motion of the collectivemagnetization J α = N (cid:80) i σ αi [65]. The phase space ofthis collective mode has conjugate variables given by z (the projection of the spin onto the z axis) and by thephase φ of the spin in the x - y plane. The non-linear clas-sical dynamics of H are integrable and can display a sep-aratrix for strong enough Λ . When t and t are large,the classical dynamics has a chaotic structure in the sameuniversality class as the standard map [66]. When t issmall, most of the integrable trajectories of H remainunchanged except for when the kick frequency is in reso-nance with a harmonic of a trajectory of H ; in this case, t ≈ τ /n , where τ is the period of a trajectory of H .When this condition is met for an integer n >
1, the dy-namics display persistent subharmonic oscillations, anda few instances are shown in Fig. 1 and Fig. 2 (withΛ = 10 and Λ = 0). To understand why these oscil-lations occur and to assess their stability, we will firstwork in the limit Λ = 0, and turn our attention tothe first plot of Fig. 1 where we have shown a set of U ( m ) stroboscopic trajectories near an emergent fixedpoint with a n = 1 resonance. There we have also plot-ted the resonant ( n = 1) trajectory of H in black. Since t = τ ( E ), U completes one period of the trajectory andevolves a spin initialized on this trajectory back to its ini-tial point. Thus, ignoring for the moment 1 /N quantumcorrections [67], we can approximate U ≈ H trajectory with period slightly lessthan t , they appear to move slightly forward along thetrajectory by a time t − τ . Again, we can approximate U ( t ) ≈ U ( t − τ ) when U acts in this region of phasespace. Similarly when t < τ , the state appears to moveslightly backwards by a time τ − t and we can approx-imate U ( t ) ≈ U † ( τ − t ). This inspires us to label thetrajectories with τ < t as ‘forward’ trajectories and thetrajectories with τ > t as ‘backward’ trajectories. Thisapparent forward and backward motion is the same stro-boscopic aliasing effect that occurs when video taping ahelicopter blade with a frame rate similar to the rotationfrequency.We now consider the action of the U kick. For Λ = 0,the kick is a J x rotation, and in the region of phase spaceshown in the first plot of Fig. 1, a J x rotation increases z and keeps φ approximately constant. Therefore, when z > z <
0, a spin on abackwards trajectory is kicked towards the forward tra-jectories. Thus, in this region of phase space, the inter-play of stroboscopic aliasing and the kick causes the spinto switch back and forth between the forward and back-ward trajectories and creates a new stroboscopic fixedpoint. For small t , these non-trivial tori are separatedby the perturbed LMG tori by two separatrices that meetat n unstable fixed point (See Fig. 2).When the resonance condition occurs for n > U only completes a fraction (1 /n ) of a trajectory. There-fore, we should define the forward and backward trajec-tories based off the classical trajectories of the unitary, U (cid:48) = ( U U ) n − U . In the perturbative limit of small t , the classical periods and trajectories of U (cid:48) will onlybe slightly shifted from the LMG trajectories, and wecan follow similar arguments as above. The dynamicsdefined by U (cid:48) ( m ) = ( U (cid:48) U ) m will then have a similarfixed point structure and trajectories as shown in Fig. 1,but will only capture the dynamics when looking every n steps of U . Looking at every step, we see that U willshift the fixed point and resonant trajectories of U (cid:48) to n different U (cid:48) fixed points in phase space before returningto the original U (cid:48) fixed point. This shows that, at theresonances, there must be n stroboscopic fixed points ofthe U (cid:48) dynamics, and this is confirmed in Fig. 2. Sincethese are fixed points of the U (cid:48) dynamics, the U dynamicsdisplay a period- n oscillation due to U moving the spinbetween the n different fixed points of U (cid:48) . In the anal-ogy to stroboscopic aliasing, this subharmonic responseis similar to a filmed single blade helicopter apparentlyshowing multiple n blades when the frame rate 1 /t is n times the frequency of the helicopter 1 /τ . Stability . Unlike the stroboscopic aliasing that occurswhile filming helicopters, the stroboscopic aliasing sub-harmonic response is actively stabilized by the interplaybetween aliasing and kicking, and it does persist whenthe drive parameters are slightly detuned. First, we dis-cuss the stability of stroboscopic aliasing to the accu-mulation of quantum fluctuations in the course of long-time dynamics. In the bare LMG model H , fluctuations lead to the collapse of periodic oscillations [68], while inthe exact[69] numerical calculations, we find that suchcollapse does not occur for the aliasing subharmonic re-sponse. This can be understood in a semiclassical picturewhere quantum fluctuations are captured by a quantumdiffusion process that spreads the wave function along theconservative classical trajectory [70]. Collapse of periodicoscillations occurs when the diffusion process reaches asteady state with the wave function completely spreadout along the periodic trajectory performed by the clas-sical dynamics.For the stroboscopic aliasing subharmonic response,the steady state contains an oscillation that moves thespin between the n dynamical fixed points. These oscilla-tions remain quantum because the wave function remainslocalized around these fixed points. Qualitatively, this isexpected by regarding quantum corrections as quantumjumps that move the spin off of its classical trajectory.In the large N limit, these jumps are exponentially sup-pressed [70], and so they can only move a spin withinthe well of an emergent fixed point, but not betweenthem. Thus, we expect that quantum corrections can-not spread the state between the different stable emer- m a x f J y ( f ) FIG. 3. In this plot we demonstrate the stability of the n = 2Stroboscopic Aliasing subharmonic oscillations to variation ofhamiltonian parameters and many body perturbations. Cal-culations are done for the hamiltonian (1) in one dimension.The bottom two panels are for α = 0 and are computed usingexact quantum dynamics. They show the order parametermax f J y ( f ) discussed in the text as a function of Λ , t (left)and t and the initial phase φ (right). In these plots, thebrightest yellow corresponds to J y ( f ) = 1, while the darkestblue to J y ( f ) = 0. The top panel is computed for finite α us-ing DTWA. It shows the same order parameter as a functionof α , and its insets show J y ( t ) at the points indicated by thearrows. gent fixed points and that the subharmonic response tobe robust to quantum fluctuations. This is confirmed bythe stability of the subharmonic response after m = 500oscillations, and the dynamics of | J | = (cid:80) α (cid:104) J α (cid:105) , whichshows that spins move along the surface of the Blochsphere (See Fig. 2).Therefore, one should expect the stroboscopic alias-ing subharmonic response to be stable to variations in t a and Λ a as long as they only deform the emergentfixed point structure. To test the extent of this stabil-ity, we focus on the n = 2 case shown in Fig. 1 andwork with an initial state completely polarized along the J y direction. As shown in the same figure, the sub-harmonic response is observed in oscillations of J y be-tween 1 and −
1. We therefore use the Fourier spectrum, J y ( f ) = M (cid:80) n =1 e − ifn J y ( n ) of the y component of thespin to asses the stability of the stroboscopic aliasing sub-harmonic response. When oscillations are stable for longtimes, the discrete Fourier spectrum, J y ( f ) will be sin-gularly peaked around f = π . Thus, similar to [45], wetake max f J y ( f ) as our order parameter for the n = 2stroboscopic aliasing oscillation phase.A phase diagram of this order parameter in the t andΛ parameter space is shown in Fig 3. The pronouncedstability to variation in Λ reflects the fact that any U that connects the forward and backward trajectories inthis region of phase space is sufficient to stabilize thefixed point there. When t becomes large, the majority ofthe resonant trajectories around the fixed points becomechaotic and the phase is destroyed. Fig 3 also shows thatthe phase is stable to variations in t . This is becausethere is a continuum of periods with τ = 2 t which canbe in resonance with U .Up to now, we have discussed the limit of α = 0 inthe hamiltonian (1). In this case, dynamics are well ap-proximated by the motion of a single large spin, and theevolved states are constrained to a Hilbert space wherethe spins at different sites are indistinguishable by permu-tation symmetry. This Hilbert space has only N statesand does not fully reflect the many body nature of arealistic experiment. Therefore, we study the robust-ness of the subharmonic response at finite α . We usethe Discrete Truncated Wigner Approximation (DTWA)which yields accurate results in long-range interactingmodels [71–77]. DTWA evolves the dynamics accordingto classical equations of motion, but treats exactly quan-tum fluctuations in the initial state by sampling over adiscrete Wigner distribution [70].We again compute max f J y ( f ) and the results areshown in Fig. 3. For N = 100, quantum diffusion oc-curs on observable time scales. As shown in the insetand discussed above for α = 0, this decreases the am-plitude of the subharmonic response but does not resultin a complete decay. For N = 200, our numerics showthat, up to computable time scales, the oscillations arealmost perfect up to α = 0 . α , many body effects relax the oscilla- n=4 n=3 FIG. 4. Stroboscopic aliasing subharmonic response in thepresence of collective spin emission. Depending on initialconditions an n = 4 or an n = 3 subharmonic oscillationcan occur. Dynamics are computed using the same methodsas in [41]. tions before quantum diffusion in the collective Hilbertspace occurs. As we increase N , this critical α grows tolarger values indicating that these many body effects area finite size effect and are suppressed at large N . Whilethese numerics cannot identify the critical value in thethermodynamic limit, they do show that oscillations arestable for finite α , finite N and within observable timescales. Generality and Perspectives . We believe that the stro-boscopic aliasing subharmonic response discussed in thiswork is a general phenomenon provided a few require-ments are satisfied. The collective mode should haveonly one dominant frequency, otherwise the kick can-not be in resonance with a single period. Further-more, the kick must deform the collective mode, althoughnot completely destroy it. The trajectory of the de-formed collective mode should cross the bare trajectoryin two points since this will allow for the dynamics of U (cid:48) = ( U U ) n − U to cross back and forth across theresonant trajectory. Notice that these requirements areeasily satisfied when the classical phase space of the col-lective mode is two dimensional because this guaranteesregular trajectories with only one frequency. Despitesuch required regularity in the collective mode dynam-ics, integrability is not required as demonstrated by therobustness of the subharmonic response to many bodyperturbations at finite α . Furthermore, the dynamicsof the collective mode is not required to be conserva-tive either. We demonstrate this aspect by consideringthe effect of a global spin decay modeled by a Lindbladjump operator proportional to J − , which occurs natu-rally in cavity QED experiments [78–81]. For κ = 0 . J x = − t .Choosing t to be in resonance with the period of thesecollective modes, we are able to find a subharmonic re-sponse and have plotted examples for n = 4 and n = 3in Fig. 4.To conclude, we remark that the stroboscopic aliasingeffects discussed so far should be observable in experi-ments. The hamiltonian (1) is used to describe trappedion experiments [82, 83] in which the transverse field iseasily controlled and can be employed to implement thekicks of Λ i . Furthermore, the emergent unstable fixedpoints could also be used to create squeezing or moregeneral entangled states in a way similar to the bareunstable fixed points of H . Similar to Refs. [65, 84–86] such fixed points have two stable directions and twounstable directions. A quantum state initialized on theunstable fixed point, compresses in the two stable direc-tions and expands in the two unstable direction creating,on short times, a squeezed state. At longer times, thestate is stretched further apart and no longer resembles asqueezed state, yet it might show non-gaussian entangle-ment with properties controlled by the shape of the sepa-ratrix [85]. Since separatrices in the stroboscopic aliasingdiscussed here, have different topologies, they can openopportunities to generate new classes of entangled statesin trapped ions simulators or in ultracold atoms experi-ments [65, 86], potentially with novel metrological uses. Finally, studying the critical properties of the transitionaway from the stroboscopic aliasing response, and ana-lyzing its interplay with quantum fluctuations [87, 88]remains an interesting future direction of research. ACKNOWLEDGMENTS
Acknowledgments:
S. P. K. would like to acknowl-edge stimulating discussions with Levent Subasi andDavid Campbell. S. P. K. acknowledges financial sup-port from the UC Office of the President through theUC Laboratory Fees Research Program, Award NumberLGF-17- 476883. S. P. K. and J. M. acknowledge sup-port by the Dynamics and Topology Centre funded bythe State of Rhineland Palatinate. S. W. T. acknowledgesupport by National Science Foundation (NSF) RAISETAQS (award no. 1839153). The research of E. T. inthe work presented in this manuscript was supported bythe Laboratory Directed Research and Development pro-gram of Los Alamos National Laboratory under projectnumber 20180045DR. [1] Pyotr Leonidovich Kapitza, “Dynamical stability of apendulum when its point of suspension vibrates, and pen-dulum with a vibrating suspension,” Collected papers ofPL Kapitza , 714–737 (1965).[2] Hiroki Saito and Masahito Ueda, “Dynamically stabi-lized bright solitons in a two-dimensional bose-einsteincondensate,” Physical review letters , 040403 (2003).[3] Fatkhulla Kh Abdullaev, Jean Guy Caputo, Robert AKraenkel, and Boris A Malomed, “Controlling collapsein bose-einstein condensates by temporal modulation ofthe scattering length,” Physical Review A , 013605(2003).[4] Wenxian Zhang, Bo Sun, MS Chapman, and L You,“Localization of spin mixing dynamics in a spin-1 bose-einstein condensate,” Physical Review A , 033602(2010).[5] TM Hoang, CS Gerving, BJ Land, M Anquez,CD Hamley, and MS Chapman, “Dynamic stabiliza-tion of a quantum many-body system,” arXiv preprintarXiv:1209.4363.[6] F Kh Abdullaev and Roberto Andr´e Kraenkel, “Macro-scopic quantum tunneling and resonances in coupledbose–einstein condensates with oscillating atomic scat-tering length,” Physics Letters A , 395–401 (2000).[7] Erez Boukobza, Michael G Moore, Doron Cohen, andAmichay Vardi, “Nonlinear phase dynamics in a drivenbosonic josephson junction,” Physical review letters ,240402 (2010).[8] Roberta Citro, Emanuele G Dalla Torre, Luca D?Alessio,Anatoli Polkovnikov, Mehrtash Babadi, Takashi Oka,and Eugene Demler, “Dynamical stability of a many-body kapitza pendulum,” Annals of Physics , 694–710 (2015).[9] Alessio Lerose, Jamir Marino, Andrea Gambassi, and Alessandro Silva, “Prethermal quantum many-bodykapitza phases of periodically driven spin systems,” Phys-ical Review B , 104306 (2019).[10] Soonwon Choi, Christopher J. Turner, Hannes Pich-ler, Wen Wei Ho, Alexios A. Michailidis, Zlatko Papi´c,Maksym Serbyn, Mikhail D. Lukin, and Dmitry A.Abanin, “Emergent su(2) dynamics and perfect quantummany-body scars,” Phys. Rev. Lett. , 220603 (2019).[11] Michael Schecter and Thomas Iadecola, “Weak ergodic-ity breaking and quantum many-body scars in spin-1 xy magnets,” Phys. Rev. Lett. , 147201 (2019).[12] Christopher J Turner, Alexios A Michailidis, Dmitry AAbanin, Maksym Serbyn, and Zlatko Papi´c, “Weak er-godicity breaking from quantum many-body scars,” Na-ture Physics , 745–749 (2018).[13] Hannes Bernien, Sylvain Schwartz, Alexander Keesling,Harry Levine, Ahmed Omran, Hannes Pichler, Soon-won Choi, Alexander S Zibrov, Manuel Endres, MarkusGreiner, et al. , “Probing many-body dynamics on a 51-atom quantum simulator,” Nature , 579–584 (2017).[14] Sanjay Moudgalya, Stephan Rachel, B. Andrei Bernevig,and Nicolas Regnault, “Exact excited states of noninte-grable models,” Phys. Rev. B , 235155 (2018).[15] Alessio Lerose, Federica M Surace, Paolo P Mazza,Gabriele Perfetto, Mario Collura, and Andrea Gambassi,“Quasilocalized dynamics from confinement of quantumexcitations,” Physical Review B , 041118 (2020).[16] Neil J Robinson, Andrew JA James, and Robert MKonik, “Signatures of rare states and thermalization in atheory with confinement,” Physical Review B , 195108(2019).[17] Titas Chanda, Jakub Zakrzewski, Maciej Lewenstein,and Luca Tagliacozzo, “Confinement and lack of thermal-ization after quenches in the bosonic schwinger model,” Phys. Rev. Lett. , 180602 (2020).[18] Marton Kormos, Mario Collura, Gabor Tak´acs, andPasquale Calabrese, “Real-time confinement followinga quantum quench to a non-integrable model,” NaturePhysics , 246–249 (2017).[19] Fangli Liu, Rex Lundgren, Paraj Titum, Guido Pagano,Jiehang Zhang, Christopher Monroe, and Alexey V. Gor-shkov, “Confined quasiparticle dynamics in long-rangeinteracting quantum spin chains,” Phys. Rev. Lett. ,150601 (2019).[20] Paolo Pietro Mazza, Gabriele Perfetto, Alessio Lerose,Mario Collura, and Andrea Gambassi, “Suppression oftransport in nondisordered quantum spin chains due toconfined excitations,” Phys. Rev. B , 180302 (2019).[21] Riccardo Javier Valencia Tortora, Pasquale Calabrese,and Mario Collura, “Relaxation of the order-parameterstatistics and dynamical confinement,” arXiv preprintarXiv:2005.01679 (2020).[22] Marko Medenjak, Berislav Buˇca, and Dieter Jaksch,“Isolated heisenberg magnet as a quantum time crystal,”Phys. Rev. B , 041117 (2020).[23] Koki Chinzei and Tatsuhiko N. Ikeda, “Time crystals pro-tected by floquet dynamical symmetry in hubbard mod-els,” Phys. Rev. Lett. , 060601 (2020).[24] Daniel K. Mark and Olexei I. Motrunich, “ η -pairingstates as true scars in an extended hubbard model,” Phys.Rev. B , 075132 (2020).[25] Sanjay Moudgalya, Nicolas Regnault, and B. AndreiBernevig, “ η -pairing in hubbard models: From spectrumgenerating algebras to quantum many-body scars,” Phys.Rev. B , 085140 (2020).[26] Berislav Buca, Archak Purkayastha, Giacomo Guarnieri,Mark T Mitchison, Dieter Jaksch, and John Goold,“Quantum many-body attractor with strictly local dy-namical symmetries,” arXiv preprint arXiv:2008.11166(2020).[27] Frank Wilczek, “Quantum Time Crystals,” Phys. Rev.Lett. , 160401 (2012).[28] Haruki Watanabe and Masaki Oshikawa, “Absence ofQuantum Time Crystals,” Phys. Rev. Lett. , 251603(2015).[29] Vedika Khemani, Roderich Moessner, and S. L. Sondhi,“A Brief History of Time Crystals,” arXiv:1910.10745[cond-mat, physics:hep-th] (2019), arXiv:1910.10745[cond-mat, physics:hep-th].[30] Dominic V. Else, Christopher Monroe, Chetan Nayak,and Norman Y. Yao, “Discrete Time Crystals,”arXiv:1905.13232 [cond-mat] (2019), arXiv:1905.13232[cond-mat].[31] Krzysztof Sacha and Jakub Zakrzewski, “Time crystals:A review,” Rep. Prog. Phys. , 016401 (2017).[32] Vedika Khemani, Achilleas Lazarides, Roderich Moess-ner, and S. L. Sondhi, “Phase Structure of Driven Quan-tum Systems,” Phys. Rev. Lett. , 250401 (2016).[33] Dominic V. Else, Bela Bauer, and Chetan Nayak,“Floquet Time Crystals,” Phys. Rev. Lett. , 090402(2016).[34] Dmitry Abanin, Wojciech De Roeck, and Fran¸cois Huve-neers, “Exponentially slow heating in periodically drivenmany-body systems,” Phys. Rev. Lett. , 256803(2015), arXiv:1507.01474.[35] Rahul Nandkishore and David A. Huse, “Many-BodyLocalization and Thermalization in Quantum Statisti-cal Mechanics,” Annual Review of Condensed Matter Physics , 15–38 (2015).[36] J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee,J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter,A. Vishwanath, N. Y. Yao, and C. Monroe, “Obser-vation of a discrete time crystal,” Nature , 217–220(2017).[37] Soonwon Choi, Joonhee Choi, Renate Landig, GeorgKucsko, Hengyun Zhou, Junichi Isoya, Fedor Jelezko,Shinobu Onoda, Hitoshi Sumiya, Vedika Khemani, Curtvon Keyserlingk, Norman Y. Yao, Eugene Demler,and Mikhail D. Lukin, “Observation of discrete time-crystalline order in a disordered dipolar many-body sys-tem,” Nature , 221–225 (2017).[38] Tongcang Li, Zhe-Xuan Gong, Zhang-Qi Yin, HT Quan,Xiaobo Yin, Peng Zhang, L-M Duan, and Xiang Zhang,“Space-time crystals of trapped ions,” Physical reviewletters , 163001 (2012).[39] Angelo Russomanno, Fernando Iemini, Marcello Dal-monte, and Rosario Fazio, “Floquet time crystal in theLipkin-Meshkov-Glick model,” Phys. Rev. B , 214307(2017).[40] Zongping Gong, Ryusuke Hamazaki, and MasahitoUeda, “Discrete Time-Crystalline Order in Cavity andCircuit QED Systems,” Phys. Rev. Lett. , 040404(2018).[41] F. Iemini, A. Russomanno, J. Keeling, M. Schir`o, M. Dal-monte, and R. Fazio, “Boundary time crystals,” Phys.Rev. Lett. , 035301 (2018), arXiv:1708.05014.[42] Diego Barberena, Robert J. Lewis-Swan, James K.Thompson, and Ana Maria Rey, “Driven-dissipativequantum dynamics in ultra-long-lived dipoles in an opti-cal cavity,” Phys. Rev. A , 053411 (2019).[43] Bihui Zhu, Jamir Marino, Norman Y. Yao, Mikhail D.Lukin, and Eugene A. Demler, “Dicke time crystals indriven-dissipative quantum many-body systems,” New J.Phys. , 073028 (2019).[44] Federica Maria Surace, Angelo Russomanno, MarcelloDalmonte, Alessandro Silva, Rosario Fazio, and Fer-nando Iemini, “Floquet time crystals in clock models,”Phys. Rev. B , 104303 (2019), arXiv:1811.12426.[45] Andrea Pizzi, Johannes Knolle, and Andreas Nun-nenkamp, “Period- $ n $ discrete time crystals and qua-sicrystals with ultracold bosons,” Phys. Rev. Lett. ,150601 (2019), arXiv:1907.04703.[46] David J. Luitz, Roderich Moessner, S. L. Sondhi, andVedika Khemani, “Prethermalization without Tempera-ture,” Phys. Rev. X , 021046 (2020).[47] Dominic V. Else, Bela Bauer, and Chetan Nayak,“Prethermal Phases of Matter Protected by Time-Translation Symmetry,” Phys. Rev. X , 011026 (2017).[48] Kristopher Tucker, Bihui Zhu, Robert J. Lewis-Swan,Jamir Marino, Felix Jimenez, Juan G. Restrepo, andAna Maria Rey, “Shattered Time: Can a DissipativeTime Crystal Survive Many-Body Correlations?” New J.Phys. , 123003 (2018), arXiv:1805.03343.[49] Soham Pal, Naveen Nishad, T. S. Mahesh, and G. J.Sreejith, “Temporal Order in Periodically Driven Spinsin Star-Shaped Clusters,” Phys. Rev. Lett. , 180602(2018).[50] Jared Rovny, Robert L. Blum, and Sean E. Barrett, “31PNMR study of discrete time-crystalline signatures in anordered crystal of ammonium dihydrogen phosphate,”Phys. Rev. B , 184301 (2018).[51] Jared Rovny, Robert L. Blum, and Sean E. Barrett, “Observation of Discrete-Time-Crystal Signatures in anOrdered Dipolar Many-Body System,” Phys. Rev. Lett. , 180603 (2018).[52] Antonio Rubio-Abadal, Matteo Ippoliti, Simon Hollerith,David Wei, Jun Rui, S. L. Sondhi, Vedika Khemani,Christian Gross, and Immanuel Bloch, “Floquet Prether-malization in a Bose-Hubbard System,” Phys. Rev. X ,021044 (2020).[53] Samuli Autti, Petri J Heikkinen, Jere T M¨akinen, Grig-ori E Volovik, Vladislav V Zavjalov, and Vladimir BEltsov, “Ac josephson effect between two superfluid timecrystals,” arXiv preprint arXiv:2003.06313 (2020).[54] Vedika Khemani, C. W. von Keyserlingk, and S. L.Sondhi, “Defining time crystals via representation the-ory,” Phys. Rev. B , 115127 (2017).[55] Bruno Sciolla and Giulio Biroli, “Quantum quenches, dy-namical transitions, and off-equilibrium quantum criti-cality,” Physical Review B , 201110 (2013).[56] Arnab Das, K Sengupta, Diptiman Sen, and Bikas KChakrabarti, “Infinite-range ising ferromagnet in a time-dependent transverse magnetic field: Quench and ac dy-namics near the quantum critical point,” Physical ReviewB , 144423 (2006).[57] Amit Dutta, Gabriel Aeppli, Bikas K Chakrabarti, UmaDivakaran, Thomas F Rosenbaum, and Diptiman Sen, Quantum phase transitions in transverse field spin mod-els: from statistical physics to quantum information (Cambridge University Press, 2015).[58] F Tonielli, R Fazio, S Diehl, and J Marino, “Orthog-onality catastrophe in dissipative quantum many-bodysystems,” Physical Review Letters , 040604 (2019).[59] Alessio Lerose, Bojan ˇZunkoviˇc, Jamir Marino, AndreaGambassi, and Alessandro Silva, “Impact of nonequilib-rium fluctuations on prethermal dynamical phase tran-sitions in long-range interacting spin chains,” PhysicalReview B , 045128 (2019).[60] Alessio Lerose, Jamir Marino, Bojan ˇZunkoviˇc, AndreaGambassi, and Alessandro Silva, “Chaotic dynamicalferromagnetic phase induced by nonequilibrium quantumfluctuations,” Physical review letters , 130603 (2018).[61] M. Kac, G. E. Uhlenbeck, and P. C. Hemmer, “On thevan der Waals Theory of the Vapor-Liquid Equilibrium.I. Discussion of a One-Dimensional Model,” Journal ofMathematical Physics , 216–228 (1963).[62] A. J. Glick, H. J. Lipkin, and N. Meshkov, “Validity ofmany-body approximation methods for a solvable model:(III). Diagram summations,” Nuclear Physics , 211–224 (1965).[63] H. J. Lipkin, N. Meshkov, and A. J. Glick, “Validity ofmany-body approximation methods for a solvable model:(I). Exact solutions and perturbation theory,” NuclearPhysics , 188–198 (1965).[64] N. Meshkov, A. J. Glick, and H. J. Lipkin, “Validity ofmany-body approximation methods for a solvable model:(II). Linearization procedures,” Nuclear Physics , 199–210 (1965).[65] A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, “Many-particle entanglement in two-component Bose-Einsteincondensates,” Physical Review A , 013607 (2003).[66] Boris V Chirikov, “A universal instability of many-dimensional oscillator systems,” Physics Reports ,263–379 (1979).[67] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy,“Coherent oscillations between two weakly coupled Bose- Einstein condensates: Josephson effects, pi oscillations,and macroscopic quantum self-trapping,” Physical Re-view A , 620–633 (1999).[68] Alessio Lerose, Bojan ˇZunkoviˇc, Jamir Marino, AndreaGambassi, and Alessandro Silva, “Impact of nonequilib-rium fluctuations on prethermal dynamical phase transi-tions in long-range interacting spin chains,” Phys. Rev.B , 045128 (2019).[69] Phillip Weinberg and Marin Bukov, “Quspin: a pythonpackage for dynamics and exact diagonalisation of quan-tum many body systems part i: spin chains,” (2017).[70] Anatoli Polkovnikov, “Phase space representation ofquantum dynamics,” Annals of Physics , 1790–1852(2010).[71] J. Schachenmayer, A. Pikovski, and A. M. Rey, “Many-Body Quantum Spin Dynamics with Monte Carlo Tra-jectories on a Discrete Phase Space,” Phys. Rev. X ,011022 (2015).[72] A. Pi˜neiro Orioli, A. Safavi-Naini, M. L. Wall, and A. M.Rey, “Nonequilibrium dynamics of spin-boson modelsfrom phase space methods,” Phys. Rev. A , 033607(2017), arXiv:1705.06203.[73] Shainen M Davidson, Dries Sels, and AnatoliPolkovnikov, “Semiclassical approach to dynamics of in-teracting fermions,” Annals of Physics , 128–141(2017).[74] OL Acevedo, A Safavi-Naini, J Schachenmayer, ML Wall,R Nandkishore, and AM Rey, “Exploring many-bodylocalization and thermalization using semiclassical meth-ods,” Physical Review A , 033604 (2017).[75] Bhuvanesh Sundar, Kenneth C Wang, and Kaden RAHazzard, “Analysis of continuous and discrete wigner ap-proximations for spin dynamics,” Physical Review A ,043627 (2019).[76] Silvia Pappalardi, Anatoli Polkovnikov, and Alessan-dro Silva, “Quantum echo dynamics in the sherrington-kirkpatrick model,” arXiv preprint arXiv:1910.04769(2019).[77] Reyhaneh Khasseh, Angelo Russomanno, MarkusSchmitt, Markus Heyl, and Rosario Fazio, “Discretetruncated wigner approach to dynamical phase transi-tions in ising models after a quantum quench,” arXivpreprint arXiv:2004.09812 (2020).[78] Emily J. Davis, Gregory Bentsen, Lukas Homeier, TracyLi, and Monika H. Schleier-Smith, “Photon-mediatedspin-exchange dynamics of spin-1 atoms,” Phys. Rev.Lett. , 010405 (2019).[79] Juan A Muniz, Diego Barberena, Robert J Lewis-Swan,Dylan J Young, Julia RK Cline, Ana Maria Rey, andJames K Thompson, “Exploring dynamical phase transi-tions with cold atoms in an optical cavity,” Nature ,602–607 (2020).[80] J. Marino and A. M. Rey, “Cavity-qed simulator of slowand fast scrambling,” Phys. Rev. A , 051803 (2019).[81] Gregory Bentsen, Ionut-Dragos Potirniche, Vir B.Bulchandani, Thomas Scaffidi, Xiangyu Cao, Xiao-LiangQi, Monika Schleier-Smith, and Ehud Altman, “Inte-grable and chaotic dynamics of spins coupled to an opti-cal cavity,” Phys. Rev. X , 041011 (2019).[82] Joseph W. Britton, Brian C. Sawyer, Adam C. Keith,C.-C. Joseph Wang, James K. Freericks, Hermann Uys,Michael J. Biercuk, and John J. Bollinger, “Engineeredtwo-dimensional Ising interactions in a trapped-ion quan-tum simulator with hundreds of spins,” Nature , 489–
492 (2012).[83] Jiehang Zhang, Guido Pagano, Paul W Hess, AntonisKyprianidis, Patrick Becker, Harvey Kaplan, Alexey VGorshkov, Z-X Gong, and Christopher Monroe, “Obser-vation of a many-body dynamical phase transition witha 53-qubit quantum simulator,” Nature , 601–604(2017).[84] Khan W. Mahmud, Heidi Perry, and William P. Rein-hardt, “Quantum phase-space picture of Bose-Einsteincondensates in a double well,” Physical Review A ,023615 (2005).[85] Helmut Strobel, Wolfgang Muessel, Daniel Linnemann,Tilman Zibold, David B. Hume, Luca Pezze’, AugustoSmerzi, and Markus K. Oberthaler, “Fisher information and entanglement of non-Gaussian spin states,” Science , 424–427 (2014).[86] Shane P. Kelly, Eddy Timmermans, and S.-W. Tsai,“Detecting macroscopic indefiniteness of cat states inbosonic interferometers,” Phys. Rev. A , 032117(2019).[87] Alessio Lerose, Jamir Marino, Bojan ˇZunkoviˇc, AndreaGambassi, and Alessandro Silva, “Chaotic dynamicalferromagnetic phase induced by nonequilibrium quantumfluctuations,” Phys. Rev. Lett. , 130603 (2018).[88] Alberto Sartori, Jamir Marino, Sandro Stringari, andAlessio Recati, “Spin-dipole oscillation and relaxationof coherently coupled bose–einstein condensates,” NewJournal of Physics17