Observation of a prethermal discrete time crystal
Antonis Kyprianidis, Francisco Machado, William Morong, Patrick Becker, Kate S. Collins, Dominic V. Else, Lei Feng, Paul W. Hess, Chetan Nayak, Guido Pagano, Norman Y. Yao, Christopher Monroe
OObservation of a prethermal discrete time crystal
A. Kyprianidis * † , F. Machado * , , W. Morong , P. Becker , K. S. Collins , D. V. Else ,L. Feng , P. W. Hess , C. Nayak , , G. Pagano , N. Y. Yao , , and C. Monroe Joint Quantum Institute, Dept. of Physics and Joint Center for Quantum Informationand Computer Science, University of Maryland, College Park, MD 20742 USA Dept. of Physics, University of California, Berkeley, CA 94720 USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA Dept. of Physics, Middlebury College, Middlebury, VT 05753 USA Microsoft Quantum, Station Q, Santa Barbara, CA 93106 USA Dept. of Physics, University of California, Santa Barbara, CA 93106 USA Dept. of Physics and Astronomy, Rice University, Houston, TX 77005 USA † To whom correspondence should be addressed; E-mail: [email protected].
The conventional framework for defining andunderstanding phases of matter requires ther-modynamic equilibrium. Extensions to non-equilibrium systems have led to surprising in-sights into the nature of many-body thermaliza-tion and the discovery of novel phases of mat-ter, often catalyzed by driving the system peri-odically. The inherent heating from such Floquetdrives can be tempered by including strong disor-der in the system, but this can also mask the gen-erality of non-equilibrium phases. In this work,we utilize a trapped-ion quantum simulator toobserve signatures of a non-equilibrium drivenphase without disorder: the prethermal discretetime crystal (PDTC). Here, many-body heating issuppressed not by disorder-induced many-bodylocalization, but instead via high-frequency driv-ing, leading to an expansive time window wherenon-equilibrium phases can emerge. We observea number of key features that distinguish thePDTC from its many-body-localized disordered * These authors contributed equally to the preparation of thismanuscript. counterpart, such as the drive-frequency con-trol of its lifetime and the dependence of time-crystalline order on the energy density of the ini-tial state. Floquet prethermalization is thus pre-sented as a general strategy for creating, stabiliz-ing and studying intrinsically out-of-equilibriumphases of matter.
The periodic modulation of a system represents aversatile technique for controlling its behavior, en-abling the emergence of phenomena ranging fromparametric synchronization to dynamic stabiliza-tion [1, 2]. Periodic driving has become a sta-ple in fields ranging from nuclear magnetic reso-nance spectroscopy to quantum information process-ing [3, 4, 5, 6]. On a more fundamental level,such a periodic Floquet drive also causes the sys-tem to exhibit a discrete time-translational symme-try. Remarkably, this symmetry can be sponta-neously broken to form time-crystalline order andcan also be utilized to protect novel Floquet topo-logical phases [7, 8, 9, 10, 11, 12].The realization of many-body Floquet phases ofmatter requires overcoming two crucial challenges.1 a r X i v : . [ qu a n t - ph ] F e b irst, the system must not absorb energy from thedriving field. In the presence of a periodic drive,dynamics are not constrained by energy conserva-tion, and Floquet heating causes a generic many-body system to approach infinite temperature, pre-cluding the existence of any non-trivial order [13].Second, genuine late-time dynamics must be clearlydifferentiated from early-time transient behavior: aphase of matter can only be characterized after dy-namical processes lead to steady state behavior.The conventional strategy for addressing the first(heating) challenge is to utilize strong disorder toinduce many-body localization (MBL), where thepresence of an extensive set of conserved local quan-tities prevents Floquet heating [14, 15, 16]. How-ever, requiring many-body localization leads to dif-ficulties, including stringent constraints on both thedimensionality and the range of interactions [17, 18].Moreover, the presence of strong disorder furtherslows down equilibration, making it even more dif-ficult to overcome the second (timescale) challengeand distinguish between early- and late-time dynam-ics.Recently, an alternate, disorder-free frameworkfor addressing both challenges has emerged: Flo-quet prethermalization [20, 21, 22, 23, 24]. For suf-ficiently high Floquet drive frequencies, energy ab-sorption by the many-body system requires multiplecorrelated local rearrangements, strongly suppress-ing the heating rate. The Floquet heating time τ ∗ scales exponentially with the drive frequency and canthus be prolonged beyond experimentally practicaltimescales; of course, directly observing such an ex-ponential is challenging in any experiment becauseof finite decoherence timescales. For time t < τ ∗ ,the system dynamics is then captured by an effectiveprethermal Hamiltonian H eff [20, 21]. This prether-mal Hamiltonian defines an effective energy for theFloquet system and also determines the nature ofthe prethermal state, which is reached at the muchshorter local equilibration time τ pre . Thus, by focus-ing on times between τ pre and τ ∗ , the dynamics areguaranteed to reflect the actual thermodynamic prop-erties of the Floquet phase. This intermediate prethermal regime is not nec-essarily trivial. New symmetries, protected by thediscrete time translation symmetry of the drive, canemerge and lead to intrinsically non-equilibriumphases of matter [25, 26]. One example of sucha phase is the prethermal discrete time crystal(PDTC), in which the many-body system sponta-neously breaks the discrete time translation symme-try of the drive and develops a robust sub-harmonicresponse.A disorder-free PDTC exhibits a number of keydifferences compared to the MBL discrete time crys-tal, despite the similarity of their sub-harmonic re-sponse [27, 28]. Stabilized by MBL, time-crystallineorder is independent of the initial state and persiststo arbitrarily late times, but is believed to only oc-cur in low dimensions with sufficiently short-rangeinteractions [17, 18]. By contrast, the PDTC life-time is limited by τ ∗ and depends on the energydensity of the initial state; this energy density deter-mines the prethermal state to which the system equi-librates for times t > τ pre . Crucially, if the prether-mal state spontaneously breaks an emergent symme-try of H eff , then the many-body system will also ex-hibit robust time-crystalline order, corresponding toan oscillation between the different symmetry sec-tors [25, 26]. On the other hand, if the prethermalstate is symmetry-unbroken, any signatures of time-crystalline order will decay by τ pre and the system isin a trivial Floquet phase.The energy density dependence of the PDTCphase can also be understood as the necessity for H eff to host a symmetry-breaking phase. This againcontrasts with the MBL discrete time crystal, be-cause symmetry breaking is more easily realizedin higher dimensions . Indeed, in one dimension,Landau-Peierls arguments rule out the existence ofa PDTC with short-range interactions [29, 30], andlong-range interactions are a necessary ingredient tostabilize a prethermal time crystal [26].We exploit the controlled long-range spin-spin in-teractions of an ion trap quantum simulator to ob-serve signatures of a one-dimensional prethermaldiscrete time crystal. Our main results are three-2igure 1: Experimental setup and protocol . A. Schematic of a linear array of 25 trapped atomic ion spins, fixedin space, as the physical platform for the experiments [19]. Top: single-site addressing enables the preparation ofarbitrary product states of the spins. Middle: two global Raman beams off-resonantly couple to motional modes andgenerate long-range Ising interactions. Bottom: state-dependent fluorescence provides single-site readout, enablingthe measurement of both magnetization and energy density. B. For intermediate times between τ pre and τ ∗ , the systemapproaches an equilibrium state of the prethermal Hamiltonian H eff . After τ pre , the magnetization in the trivial Floquetphase remains constant. Meanwhile, in the PDTC phase, the magnetization oscillates each period leading to a robustsub-harmonic response. For both phases, at times t (cid:29) τ ∗ , Floquet heating eventually brings the many-body systemto a featureless infinite temperature ensemble. C. Left: Schematic of the stroboscopic magnetization dynamics in thetrivial [red] and PDTC [blue] phase (full/dashed curves represent even/odd driving periods). In the trivial phase, anytransient time-crystalline-order decays by the prethermal equilibration time τ pre , while in the PDTC phase, the orderremains robust until τ ∗ , the frequency-controlled heating timescale. Right: Starting from a product state with zeroentropy, the dynamics under H eff bring the system to an equilibrium state at time τ pre . The PDTC behavior is robustif the initial state thermalizes to a prethermal equilibrium state, which spontaneously breaks an emergent symmetry of H eff . In our system, this occurs if the energy density of the initial state is above a critical value ε crit (i.e. to the rightof the dashed line), wherein the system equilibrates to a ferromagnetic state. Regardless of the initial state, for t > τ ∗ ,Floquet heating eventually brings the system to the maximum entropy state at zero energy density. τ pre . Second, we measure the time dynam-ics of the energy density as a function of the driv-ing frequency. By preparing states at both posi-tive and effectively negative temperatures (Fig. 1c),we observe either the gain or loss of energy as thesystem heats to infinite temperature (correspondingto zero energy density). Importantly, we find thatthe heating timescale, τ ∗ , increases with the driv-ing frequency (Fig. 2). Finally, to probe the natureof prethermal time-crystalline order, we study theFloquet dynamics of initial states, spanning acrossthe entire energy spectrum, that equilibrate to eithera symmetry-breaking or a symmetry-unbroken en-semble. The former exhibits robust period-doublingbehavior up until the frequency-controlled heatingtimescale, τ ∗ (Fig. 3B). In comparison, for the lat-ter, all signatures of period doubling disappear by thefrequency-independent timescale τ pre (Fig. 3A). Byinvestigating the lifetime of the time-crystalline or-der as a function of the energy density of the initialstate, we identify the phase boundary for the PDTC.This boundary is consistent with independent quan-tum Monte Carlo calculations for the location of thephase transition in H eff .Our system consists of a one-dimensional chainof 25 Yb + ions. Each ion encodes an effectivespin-1/2 degree of freedom in its hyperfine levels | F = 0 , m F = 0 (cid:105) and | F = 1 , m F = 0 (cid:105) (Fig.1a). Long-range Ising interactions are generated viaa pair of Raman laser beams [31, 32]. Arbitrary ef-fective magnetic fields can be applied either locallyor globally and single-site readout can be performedsimultaneously across the full chain [19], enablingthe direct measurement of the Floquet dynamics ofboth the magnetization and the energy density.The Floquet drive alternates between two types ofHamiltonian dynamics (Fig. 1b): (i) a global π -pulse around the ˆ y axis and (ii) evolution for time T undera disorder-free, long-range, mixed-field Ising model.This is described by the two evolution operators, U = exp (cid:34) i π N (cid:88) i σ yi (cid:35) U = exp iT N (cid:88) i
20 40 60
Drive Frequency ω/J H ea t i n g t i m e τ ∗ J Figure 2:
Characterizing the prethermal regime. A, B.
The dynamics of the energy density for a low-energy N´eelstate (left) and a high-energy polarized state (right). Both states exhibit Floquet heating toward infinite temperature,albeit from opposite sides of the many-body spectrum. In addition, in both cases, faster drive frequencies ω suppressthe heating rate. Statistical error bars are of similar size as the point markers. C. The heating time τ ∗ for theN´eel (red) and polarized (blue) states increases with frequency. τ ∗ is extracted by fitting the dynamics of the energydensity to ∼ e − t/τ ∗ ; exponential fits are shown as solid curves in (A) and (B). At high frequencies this behaviorsaturates owing to the presence of external noise. Error bars for the heating time correspond to fit errors. D. Theprethermal equilibration time, τ pre , can be characterized by observing the local ˆ x -magnetization dynamics for evenFloquet periods. Top: The middle two spins (purple), initially prepared along a different axis, rapidly align with theirneighbors (orange) at time τ pre J ≈ . The homogenization of the magnetization signals local equilibration to aequilibrium state of H eff . Bottom: The magnetization of each ion as a function of time. M ( t ) Even PeriodsOdd Periods ω/J A B C τ pre J
12 15 38 55 ω/J τ pre J
22 28 38 67
Time tJ S p i n N u m b e r Time tJ Energy Density › H eff fi / ( NJ ) D eca y T i m e J τ PDTCTrivial ω/J = 38 ω/J = 55 ω/J = 67 τ PDTC τ ∗ Figure 3:
Characterizing the PDTC phase. A, B.
Upper plots: The magnetization dynamics, M ( t ) , for the N´eelstate (left) and the polarized state (right). For the N´eel state, M ( t ) quickly decays to zero at time τ pre (dashed verticalline), independent of the drive frequency. For the polarized state, the sub-harmonic response (2 T -periodicity) persistswell-beyond τ pre and its lifetime is extended upon increasing the drive frequency. We characterize the lifetime ofthe prethermal time-crystalline order by fitting the magnetization dynamics to an exponential and extracting a decaytime, τ PDTC [32]. Statistical error bars are of similar size as the point markers. Lower plots: The ˆ x -magnetizationdynamics for each ion in the chain at ω/J = 38 . C. Heating ( τ ∗ ) and magnetization decay ( τ PDTC ) times for fourdifferent initial states at varying energy densities [32]. For low energy densities, τ PDTC (orange) are short, independentof frequency, and significantly shorter than τ ∗ (magenta), highlighting the trivial Floquet phase. For high energies, τ PDTC is similar to τ ∗ , highlighting the long-lived, frequency-controlled nature of the PDTC behavior. The location ofthe observed crossover in energy density is in agreement with an independent quantum Monte Carlo calculation (redand blue shaded regions) [32]. Error bars for the decay time correspond to fit errors, while error bars for the energydensity correspond to statistical errors. (cid:104) H eff (cid:105) / ( N J ) ≈ , in agreement with inde-pendent numerical calculations via quantum MonteCarlo [32].In this work we report the experimental observa-tion of robust prethermal time-crystalline behaviorthat persists beyond any early-time transient dynam-ics. By varying the energy density of the initial state,we study the crossover between the trivial Floquetphase and the PDTC phase. Our results highlight thepotential of periodic driving, in general, and prether-malization, in particular, as a framework for realizingand studying out-of-equilibrium phenomena. Evenin the presence of noise, we find that the prether-mal dynamics remain stable, suggesting the possi-bility that an external bath at sufficiently low tem-perature can stabilize the prethermal dynamics forinfinitely long times [25]. This stands in contrast tolocalization-based approaches for stabilizing Floquetphases, in which the presence of an external bathtends to destabilize the dynamics. Our work opensthe door to a number of intriguing future directions:(i) exploring generalizations of Floquet prethermal-ization to a quasi-periodic drive[33], (ii) stabilizingFloquet topological phases [34, 35], and (iii) lever-aging non-equilibrium many-body dynamics for en-hanced metrology [36]. References [1] S. H. Strogatz,
Nonlinear dynamics and chaos withstudent solutions manual: With applications to physics, biology, chemistry, and engineering (CRCpress, 2018).[2] L. D. Landau, E. M. Lifshitz,
Mechanics, ThirdEdition: Volume 1 (Course of Theoretical Physics) (Butterworth-Heinemann, 1976), third edn.[3] P. Mansfield,
Journal of Physics C: Solid StatePhysics , 1444–1452 (1971).[4] L. M. K. Vandersypen, I. L. Chuang, Reviews ofModern Physics , 1037–1069 (2005).[5] H. Zhou, et al. , Physical Review X , 031003(2020).[6] J. Choi, et al. , Physical Review X , 031002(2020).[7] T. Oka, S. Kitamura, Annual Review of CondensedMatter Physics , 387–408 (2019).[8] A. C. Potter, T. Morimoto, A. Vishwanath, PhysicalReview X , 041001 (2016).[9] F. Nathan, D. Abanin, E. Berg, N. H. Lindner, M. S.Rudner, Physical Review B , 195133 (2019).[10] D. V. Else, B. Bauer, C. Nayak, Physical ReviewLetters , 090402 (2016).[11] V. Khemani, A. Lazarides, R. Moessner, S. Sondhi,
Physical Review Letters , 250401 (2016).[12] N. Yao, A. Potter, I.-D. Potirniche, A. Vishwanath,
Physical Review Letters , 030401 (2017).[13] L. D’Alessio, M. Rigol,
Physical Review X ,041048 (2014).[14] R. Nandkishore, D. A. Huse, Annual Review of Con-densed Matter Physics , 15–38 (2015).[15] P. Ponte, Z. Papi´c, F. Huveneers, D. A. Abanin, Physical Review Letters , 140401 (2015).[16] D. A. Abanin, E. Altman, I. Bloch, M. Serbyn,
Re-views of Modern Physics , 021001 (2019).[17] N. Yao, et al. , Physical Review Letters , 243002(2014).[18] W. De Roeck, F. Huveneers,
Physical Review B ,155129 (2017).
19] C. Monroe, et al. , arXiv:1912.07845 (2019).[20] T. Kuwahara, T. Mori, K. Saito, Annals of Physics , 96–124 (2016).[21] D. A. Abanin, W. De Roeck, W. W. Ho, F. Huve-neers,
Physical Review B , 014112 (2017).[22] F. Machado, G. D. Kahanamoku-Meyer, D. V. Else,C. Nayak, N. Y. Yao, Physical Review Research ,033202 (2019).[23] A. Rubio-Abadal, et al. , Physical Review X ,021044 (2020).[24] P. Peng, C. Yin, X. Huang, C. Ramanathan, P. Cap-pellaro, Nature Physics p. 1–4 (2021).[25] D. V. Else, B. Bauer, C. Nayak,
Physical Review X , 011026 (2017).[26] F. Machado, D. V. Else, G. D. Kahanamoku-Meyer,C. Nayak, N. Y. Yao, Physical Review X , 011043(2020).[27] D. V. Else, C. Monroe, C. Nayak, N. Y. Yao, AnnualReview of Condensed Matter Physics (2020).[28] V. Khemani, R. Moessner, S. L. Sondhi, arXiv:1910.10745 (2019).[29] R. Peierls, Mathematical Proceedings of the Cam-bridge Philosophical Society , 477–481 (1936).[30] L. Landau, Zh. Eksp. Teor. Fiz. , 19 (1937).[31] K. Mølmer, A. Sørensen, Physical Review Letters (1999).[32] See Supplementary material.[33] D. V. Else, W. W. Ho, P. T. Dumitrescu, PhysicalReview X , 021032 (2020).[34] I.-D. Potirniche, A. C. Potter, M. Schleier-Smith,A. Vishwanath, N. Y. Yao, Physical Review Letters , 123601 (2017).[35] D. V. Else, P. Fendley, J. Kemp, C. Nayak,
PhysicalReview X , 041062 (2017).[36] S. Choi, N. Y. Yao, M. D. Lukin, arXiv:1801.00042 (2017).[37] K. Kim, et al. , Physical Review Letters (2009). [38] D. James,
Applied Physics B: Lasers and Optics (1998).[39] S. Olmschenk, et al. , Physical Review A (2007).[40] K. R. Brown, A. W. Harrow, I. L. Chuang, PhysicalReview A (2004).[41] A. C. Lee, et al. , Physical Review A (2016).[42] D. Wineland, et al. , Journal of Research of the Na-tional Institute of Standards and Technology (1998).
Acknowledgments
We acknowledge fruitful discussions with C. Lau-mann, W. L. Tan, A. Vishwanath, D. Weld, andJ. Zhang.
Funding:
This work is supported bythe DARPA Driven and Non-equilibrium QuantumSystems (DRINQS) Program D18AC00033, NSFPractical Fully-Connected Quantum Computer Pro-gram PHY-1818914, the DOE Basic Energy Sci-ences: Materials and Chemical Sciences for Quan-tum Information Science program DE-SC0019449,the DOE High Energy Physics: Quantum Infor-mation Science Enabled Discovery Programs DE-0001893, the AFOSR MURI on Dissipation En-gineering in Open Quantum Systems FA9550-19-1-0399, the David and Lucile Packard foundation,the W. M. Keck foundation, and the EPiQS Ini-tiative of the Gordon and Betty Moore FoundationGBMF4303.
Author contributions:
A.K., W.M.,P.B., K.S.C., L.F., P.W.H., G.P., and C.M. designedand performed experimental research, F.M., D.V.E.,C.N., and N.Y.Y. analyzed the data theoretically, andall authors wrote the paper.
Competing interests:
C.M. is the co-founder and Chief Scientist at IonQ,Inc.
Data availability:
All data needed to evalu-ate the conclusions in the paper are present in thepaper. Additional data related to this paper may berequested from the corresponding author.9 upplementary materials
The trapped-ion quantum simulator
The quantum simulator used in this work is based ona chain of Yb + ions trapped in a 3-layer Paul trapat room temperature [37]. The ions are confined inall three directions by a combination of static and os-cillating electric fields. The interplay of the repulsiveCoulomb force and the trapping potential arrange theions in a linear configuration. The transverse center-of-mass (COM) motional mode frequency along the ˆ x axis is f COM = 4 . MHz and the axial COMfrequency is f z = 0 . MHz. The axial trappingstrength is set such that 25 ions settle in a linearconfiguration with inter-ion spacings varying from ∼ µ m (chain center) to ∼ . µ m (chain edges)[38]. Spin and motional state preparation
Between experiments, the ions are Doppler cooledby a 369.5 nm laser red-detuned from the S / to P / transition by 10 MHz, one-half of the tran-sition linewidth. This laser projects onto all threeprinciple axes of the trap, ensuring that the ions arecooled along all directions. To begin an experiment,the ions are initialized in the low-energy hyperfinequbit state |↓(cid:105) ≡ S / | F = 0 , m F = 0 (cid:105) by an inco-herent optical pumping process [39]. Optical pump-ing requires approximately µ s and initializes allions to |↓(cid:105) with at least 99 % fidelity. At this pointthe individual spin states of the ions are well-known,while the shared motional state is a thermal distri-bution with ¯ n ≤ average motional quanta in thetransverse ˆ x axis modes. Resolved sideband cool-ing on multiple motional modes brings the ions neartheir motional ground state ( ¯ n ≤ . average mo-tional quanta).With the ions cooled and their spin states initial-ized, we prepare the spins in product states alongthe ˆ x axis of the Bloch sphere with a combina-tion of global rotations and individual σ z rotations.Global rotations are driven with a pair of Ramanlaser beams, intersecting at a ◦ angle. These lasers produce a beatnote that drives oscillations betweenthe qubit states with Rabi frequency Ω when tuned onresonance with the ions’ S-manifold hyperfine split-ting. The phase of this beatnote determines the Blochsphere axis about which the spins are rotated. The BB1 pulse sequence
Each Raman beam has a Gaussian intensity profilewith waists of µ m by µ m at the ion plane. A25-ion chain has a length of about µ m. Each ionsamples a slightly different intensity from the Ramanlasers, resulting in different rates of rotation acrossthe chain. To minimize rotation errors caused by thisinhomogeneity, we employ BB1 dynamical decou-pling sequences[40] to ensure that all spins along thechain are rotated by the same amount.A traditional ˆ y rotation unitary has the form ˆ U yθ = e − iθσ yi / , where θ is the desired angle of rotationabout the ˆ y axis. The angle θ ≡ Ω i t , where Ω i is theRabi frequency experienced by spin i , is sensitive tothe spacially-inhomogeneous intensity profiles of theRaman lasers. We instead apply the following BB1unitary, consisting of 4 sub-rotations: ˆ U yθ, BB1 = e − i π σ φi e − iπσ φi e − i π σ φi e − i θ σ yi . (5)The phase φ depends on the desired rotation angle θ : φ = arccos (cid:18) θ π (cid:19) . (6)While a π -rotation using this sequence takes fivetimes longer than a traditional rotation pulse, it re-duces rotation errors significantly and prevents de-phasing across the chain. This allows us to ap-ply hundreds of π -pulses with negligible loss ofcontrast—a requirement for the time-crystal exper-imental sequences presented in this manuscript. Arbitrary product state preparation
An individual addressing beam focused to a waist of nm generates rotations on each spin with rel-atively low crosstalk. A high-bandwidth acousto-optical deflector (AOD) steers the beam, and the10OD’s rf drive frequency maps the beam to a lo-cation along the ion chain. This beam applies afourth-order AC Stark shift to the hyperfine qubitsplitting[41], creating an effective σ zi rotation on asingle spin i . This rotation is mapped to a rotationabout any axis using the appropriate global analy-sis π/ rotations, allowing for preparation of productstates with arbitrary spin flips such as the antiferro-magnetic N´eel state. Qubit readout
At the end of an evolution, we measure the mag-netization of each spin using state-dependent flu-orescence. A 369.5 nm laser resonant with the S / | F = 1 (cid:105) ↔ P / | F = 0 (cid:105) transition (linewidth γ/ π ≈ . MHz) causes each ion to scatter pho-tons if the qubit is projected to the | ↑(cid:105) state. Ionsprojected to the | ↓(cid:105) qubit state scatter a negligiblenumber of photons because the laser is detuned fromresonance by the S / hyperfine splitting. By apply-ing global π/ -rotations, we rotate the x and y basesinto the z basis. This allows us to measure all in-dividual magnetizations and many-body correlatorsalong any single axis. In the experiments reported inthis work, we repeat the experimental sequence andthe measurement for 50-600 times to reduce quan-tum projection noise.For each measurement, a finite-conjugate NA =0 . objective lens system (total magnification of × ) collects scattered 369.5 nm photons and im-ages them onto an Andor iXon Ultra 897 EMCCDcamera. Before taking data, high-contrast calibra-tion images of the ion chain, illuminated by Dopplercooling light, are used to identify a region of inter-est (ROI) on the camera sensor for each ion. Duringdata collection, fluorescence is integrated for µ s,after which a pre-calibrated binary threshold is ap-plied to discriminate the qubit state of each ion withapproximately 98 % accuracy per ion.The dominant error sources for the qubit readout,ordered by decreasing significance, are: mixing ofqubit states caused by off-resonant coupling duringthe µ s camera exposure window, crosstalk be- tween ion ROIs due to small inter-ion spacings nearthe center of the chain, electronic camera noise, andlaser power fluctuations. No state preparation andmeasurement (SPAM) correction has been applied todata presented in this work. Simulating the transverse field Ising Hamil-tonian
We generate the effective spin-spin interactionHamiltonian by applying spin-dependent dipoleforces with the pair of 355 nm Raman beams men-tioned earlier. These beams produce a beatnote withwavevector ∆ (cid:126)k aligned along a principle axis of thetrap. The frequencies of these beams are controlledwith acousto-optical modulators (AOMs) to generatea pair of beatnote frequencies detuned by − µ (redbeatnote) and + µ (blue beatnote) from the resonantqubit transition frequency. For µ − f COM (cid:29) η Ω ( η is the Lamb-Dicke parameter[42]) and η (cid:28) ,the experiment operates in the far-detuned Mølmer-Sørensen (MS) regime[31, 37]. Here, excited mo-tional states are adiabatically eliminated and thelaser-ion interaction takes the form of a spin-spin, ef-fective long-range interacting Ising Hamiltonian H = N (cid:88) i 10 15 200.00.51.01.52.02.5 A B Figure S1: The interaction matrix J ij . A. The position of each column represents a pair of spins { i, j } , while itscolor-coded height the strength of their coupling. B. To illustrate how the interaction strength scales with distancebetween i and j , we average the elements of the diagonals of J (circular markers), e.g. the | i − j | = 2 corresponds to (cid:104){ J , J , J , J , . . . }(cid:105) . The line markers illustrate all the individual couplings for that value of | i − j | . In the reported experiments, the average nearest-neighbor interaction J / (2 π ) = 0 . ± . kHz.We add transverse fields to Eq. 7 in two ways.To create an effective transverse field B z along the ˆ z direction, we apply a global offset of B z to thetwo Raman beatnote frequencies, imposing a rotat-ing frame shift between the qubit and the beatnotesto generate an effective field with strength B z . Athird Raman beatnote, resonant with the qubit tran-sition and applied simultaneously with the Mølmer-Sørensen red and blue beatnotes, creates additionaltransverse fields along the ˆ x or ˆ y directions depend-ing on the beatnote phase. Altogether, the long-range, transverse-field Ising Hamiltonian takes theform H = N (cid:88) i All Raman laser operations and individual-addressing operations are implemented via am-plitude modulation of the rf that drives variousAOMs and AODs. An arbitrary-wavefrom generator(AWG) outputs this amplitude-modulated rf signal to switch lasers and beatnotes on and off accordingto the experimental sequence. If the control rfis modulated with square pulses, the sharp edges(limited by the rise/fall time of the AOM/D) causesignificant spectral broadening of the signal in theFourier domain. This effect is more pronounced forshorter pulses, such as the Raman pulses used togenerate the unitary U . This spectral broadeningcan be on the order of MHz, and causes undesirabledriving of qubit motional and spin transitions.We suppress spectral broadening by applyingTukey window pulse shaping, where the first and last10 µ s of the pulse are multiplied respectively by arising and falling sinusoidal envelope. We accountfor the resulting reduction in the magnitude of eachterm of the Hamiltonian by scaling it down by an ap-propriate factor, which varies with the total durationof that pulse. Longer pulses need to be scaled downless, since the ramp time is fixed. Error sources The dynamics observed in this work are the combi-nation of ideal Hamiltonian evolution as in Eq. 1 andother terms of smaller magnitude that we refer to as“error sources”. The combined effect of the latter,when measuring the chain magnetization, manifests12s decoherence.The most significant error source is fluctuating ACStark shifts of the hyperfine qubit frequency. Thisfluctuation is mostly caused by power instability ofthe 355 nm laser light at the ions’ location. Eventhough there is a power PI locking scheme in ef-fect for the 355 nm light, the sampling point for thelock is at a more upstream location that the ions. Asthe beams propagate downstream from that point, ac-tive elements, acoustic noise, and air turbulence in-troduce extra power noise. At the ions location, thelight’s red and blue beatnotes ideally produce exactlyopposite AC Stark shifts of the qubit levels and can-cel each other, in practice they are not always per-fectly balanced. In this case, common-mode powerfluctuations will make the sum of their Stark shiftsfluctuate. This manifests as an effective fluctuatingmagnetic field term B ( AC ) ( t ) (cid:80) i σ zi , common for allspins i and is present in every stage of the experi-mental sequence. A fortunate side effect of the π -rotations of the drive is that they echo out part of thisnoise. However, the spectral portion of B ( AC ) ( t ) thatis faster than ω/ is not echoed out and differs be-tween different repetitions, manifesting as decoher-ence in the final averaged signal. In numerics pre-sented in the next section, we model this noise basedon experimental evidence and reasonable simplifica-tions, and present numerical simulations that includeit. Imperfect qubit state readout also impacts the finalfidelity of the simulation. During the finite readoutwindow of µ s, there is a small probability that a |↓(cid:105) state will be off-resonantly pumped, and read outas a | ↑(cid:105) state, and vice versa. For the experimentspresented in this work, the average readout error was . for each ion.Another error source comes from a term combin-ing the spin and the motional part of the qubit wave-function that acts in parallel with the effective Isinginteraction in Eq. (7). This term represents entangle-ment between these two parts; when we measure thequbit spin, we effectively trace out the entangled mo-tional state, resulting in a probabilistic mixed state.The probability for such an erroneous spin flip to oc- cur is proportional to N (cid:88) m =1 (cid:18) η m b im Ω δ m (cid:19) (11)Therefore, by increasing this detuning, we minimizethe undesired spin-motion entanglement, but we arealso decreasing the strength of our spin-spin inter-action term (see Supplement, Simulating the trans-verse field Ising Hamiltonian). We set the balancebetween these effects by keeping the sum in Eq. (11)less than . for two spins, which for the 25 spins re-sults in approximately . flip probability per spin.This effect is somewhat amplified by the finite dura-tion of the Hamiltonian quenches in the second termof the Floquet drive, whose spectral decompositionhas nonzero components in the motional frequencies.We considerably mitigate this effect by applying theTukey window shaping to the relevant pulses (seeSupplement, Tukey window pulse shaping), whichreduces these undesired spectral terms. Supporting numerical evidence Dynamics in the absence and presence ofnoise In this section, we present a numerical investiga-tion of the dynamics simulated by the experimentalplatform, highlighting the physics of prethermaliza-tion and the PDTC. We first focus on the noiselesscase, where we observe the expected exponential fre-quency dependence of τ ∗ on the drive frequency ω ,and then turn to studying the effect of noise on theobserved dynamics.We consider an N = 19 spin chain with inter-actions given by the experimentally determined in-teraction matrix for 25 spins, truncated to the mid-dle spins. We perform the exact time evolutionof the full quantum system using Krylov subspacemethods, simulating the entire experimental protocol(which includes the Tukey window pulse shaping).For all states considered (Fig. S2), we observe thesame effect of the frequency on the heating timescale13 .60.40.20.0 › H e ff fi / ( J N ) A ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ω/J = 67 ω/J = 38 ω/J = 28 ω/J = 22 ↓ ↓ ↓ ↓ ↓ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ Time tJ M ( t ) E 0 10 20 Time tJ Time tJ Time tJ Figure S2: PDTC dynamics in the absence of noise A-D. Dynamics of the energy density (cid:104) H eff (cid:105) /N J for differentinitial states with increasing energy density. The decay of the energy density to the late time infinite temperature valueis exponentially sensitive to the frequency of the drive. E-H. Dynamics of the magnetization M ( t ) , for different initialstates. Full[dashed] line corresponds to even[odd] periods. For low energy density, the magnetization is approximatelyfrequency-independent, highlighting the trivial nature of the dynamics. For high energy density, the magnetizationdecay follows the decay of the energy density and exhibits a robust period doubling behavior—the two hallmarks ofthe PDTC. τ ∗ —the larger the frequency, the slower the en-ergy approach to its infinite temperature value. Bycontrast, the dynamics of the magnetization can bestarkly different. For the states at low enough energydensity, where H eff does not exhibit a spontaneoussymmetry-breaking phase, the dynamics of M ( t ) aremostly frequency-independent, and much faster thanthe heating. For states near the top of the spectrum,where a spontaneous symmetry-breaking phase ex-ists, the dynamics of the magnetization exhibit ro-bust period doubling whose decay matches that ofthe energy density—this is PDTC behavior. We sum-marize the timescales observed in Fig. S3, where thefrequency dependence of τ ∗ occurs across all initialstates, while the frequency dependence of τ TC onlyoccurs at the top of the spectrum.We now turn to simulating the effect of noisein the observed dynamics. As mentioned in Errorsources, the most significant noise arises from laserpower fluctuations. We parameterize these fluctua-tions at the ions’ location with the random variable (cid:15) ( t ) , characterized by a flat spectrum and standard deviation σ over a given duration (we have found thatresults are independent of the upper frequency cutoffas long as it is much larger than the drive frequency). We model this effect in the dynamics by adding atime dependence on the different parameters, B y , B z and J ij : B y ( t ) = B static y × [1 + (cid:15) ( t )] (12) B z ( t ) = B static z + (cid:15) ( t ) × π × (13) J ij ( t ) = J static ij × [1 + 2 (cid:15) ( t )] (14)The dependence of each term on (cid:15) ( t ) is determinedby the way that the laser power relates to that term,eg. the B y ( t ) field depends linearly on the 2-photonRabi frequency Ω , which is proportional to laserpower. In Fig. S4, we highlight the effect of noise intwo states at opposite sides of the spectrum and alsoconsidered in the main text: the polarized state andthe N´eel state. The effect on either is qualitativelysimilar: noise reduces the frequency control of theFloquet heating leading to a plateau in the achievable τ ∗ , in agreement with the experimental observation14 Energy Density › H eff fi / ( NJ ) T i m e s ca l e τ J τ ∗ τ PDTC ω/J = 22 ω/J = 28 ω/J = 38 Figure S3: PDTC timescales in absence of noise . Summary of the decay time scales of energy density ( τ ∗ ) andmagnetization ( τ PDTC ). In the region where H eff exhibits a spontaneous symmetry broken phase (blue shading), τ PDTC follows the frequency dependence of τ ∗ . By contrast, outside this region, the τ PDTC is much smaller than τ ∗ and is mostly frequency-independent. (we highlight this feature in Fig. S5). It is impor-tant to emphasize two points. First, while for largefrequency the noise dominates the heating towardsthe infinite temperature state, the frequency still pro-vides a control of the Floquet heating highlightingthe importance of of the drive. Second, the dynam-ics of the magnetization is distinct in the trivial andthe PDTC regimes. In the former, the magnetizationquickly decays before the heating time scale while inthe latter, the magnetization decay follows the energydensity decay. Computing the crossover between trivial andPDTC behavior Since the PDTC behavior is dependent on a sponta-neous symmetry-broken phase in H eff , we can calcu-late the boundary between trivial and PDTC behaviorby mapping the location of the transition (which forfinite system size will emerge as a crossover). In par-ticular, owing to the antiferromagnetic interactions J ij > , we are interested in the properties of thetop of the spectrum, where the system orders ferro-magnetically. To this end, we perform a quantumMonte Carlo simulation of the Hamiltonian − H eff in an N = 25 spin chain (Eq. 3), which maps thecalculation to the more common problem of findingthe low-temperature phase boundary of the para-to-ferromagnetic phase. In Fig. S6 we present the re-sults in terms of the original energy density. By thesystem extending into the imaginary time dimension,we can perform the calculation at finite temperature /β and extract the crossover temperature and thenmap it to the relevant crossover energy density. Inparticular, we identify the location of the crossoverby the location of the peak in the heat capacity ofthe system which we fit to a Lorentzian peak, whilethe width is taken to be the quarter-width half-maxof the peak. The crossover location is then given by: ε/J = 2 . ± . .15 ncreasing NoiseA BC D Figure S4: Impact of noise on the PDTC and trivial Floquet dynamics Heating dynamics of the polarized state ( A-B ) and the N´eel state ( C-D ) for different strength of the noise σ . The presence of noise hastens heating to the infinitetemperature state, setting an upper bound on the time scales. Nevertheless, the trivial dynamics can be distinguishedfrom the PDTC behavior. In the former, the magnetization decay remains frequency independent and much fasterthan the energy decay. In the latter, the magnetization decay has a similar frequency dependence to the energy densitydecay. .40.20.0 › H e ff fi / ( N J ) A Néel State Time tJ M ( t ) ω/J =51 ω/J =38 ω/J =28 ω/J =22 ω/J =12 ω/J =10 B Polarized State ω/J =51 ω/J =38 ω/J =28 ω/J =22 ω/J =12 ω/J =10 Time tJ Frequency ω/J T i m e s ca l e s τ C τ ∗ τ PDTC ω/J =51 ω/J =38 ω/J =28 ω/J =22 ω/J =12 ω/J =10 10 20 30 40 50 Frequency ω/J D Figure S5: Frequency dependence of the heating τ ∗ and time-crystalline τ PDTC timescales for fixed strength ofthe noise. A-B. Energy (top) and mangetization (bottom) dynamics for the N´eel and the polarized initial state, in thepresence of a fluctuating B z field with moderate strengths σ = 0 . . C-D. Upon increasing the frequency of the drive,we observe an increase in the heating time scale τ ∗ for both initial states, up until very high-frequencies where heatingbecomes dominated by the fluctuating noise. Crucially, the dynamics of the time-crystalline order is very different.For the N´eel state, the time-crystalline order parameter decays quickly and is frequency independent—the system isin a trivial Floquet phase. For the polarized state, the increased heating time is mirrored by a increase of lifetime ofthe time-crystalline order parameter—the system is in the PDTC Floquet phase. Temperature (1 /βJ ) C V = N ∂ h − H e ff i ∂ T A B 0 5 10 15 Temperature (1 /βJ ) h H e ff i / ( N J ) Figure S6: Calculation of the para-to-ferromagnetic crossover region in H eff . Quantum Monte Carlo calculationfor the N = 25 spin chain considered in the experiment. A. We can locate the location and width of the crossover bycharacterizing the peak in the heat capacity C V . B. Armed with the crossover temperature, we can directly map it intothe crossover energy density which yields: H eff / ( N J ) = 2 . ± . ..