Quantifying and controlling prethermal nonergodicity in interacting Floquet matter

The use of periodic driving for synthesizing many-body quantum states depends crucially on the existence of a prethermal regime, which exhibits drive-tunable properties while forestalling the effects of heating. This motivates the search for direct experimental probes of the underlying localized nonergodic nature of the wave function in this metastable regime. We report experiments on a many-body Floquet system consisting of atoms in an optical lattice subjected to ultrastrong sign-changing amplitude modulation. Using a double-quench protocol we measure an inverse participation ratio quantifying the degree of prethermal localization as a function of tunable drive parameters and interactions. We obtain a complete prethermal map of the drive-dependent properties of Floquet matter spanning four square decades of parameter space. Following the full time evolution, we observe sequential formation of two prethermal plateaux, interaction-driven ergodicity, and strongly frequency-dependent dynamics of long-time thermalization. The quantitative characterization of the prethermal Floquet matter realized in these experiments, along with the demonstration of control of its properties by variation of drive parameters and interactions, opens a new frontier for probing far-from-equilibrium quantum statistical mechanics and new possibilities for dynamical quantum engineering.

In this work, we experimentally probe a universal quantitative measure for non-ergodic localization of a driven quantum system based on a stroboscopically time-averaged return probability to an initial energylocalized state. This quantity directly corresponds to an inverse participation ratio quantifying localization in the eigenbasis of the approximate Floquet Hamiltonian, and can be experimentally measured using a double-quench protocol starting from an almost fully Bose-condensed undriven ground state. The experiments we report use tunably-interacting 7 Li atoms in an amplitude-modulated optical lattice to create and probe prethermal Floquet matter. Uniquely, our experiment * weld@ucsb.edu enables the use of amplitude modulation extending from 10 to 1000 percent of the static lattice depth and drive frequencies extending from 0.1 to 10 times the lattice band gap. This regime of ultrastrong drive amplitudes is previously unrealized experimentally, and this range of drive frequencies extends through and well beyond typical "high-frequency" regimes of Floquet engineering. Realization of these extreme parameter values allows us to map out a sharp threshold between differently-localized regimes of prethermal ergodicity breaking. Observed characteristics of the prethermal state quantitatively confirm theoretical predictions based on a periodic Gibbs ensemble. Tracking the time evolution of the driven system we observe and quantify not only the formation of a prethermal non-ergodic plateau, but also the long-time departure from it, either by a transition to a second prethermal plateau or, for stronger interactions, by the onset of ergodicity.
The initial condition for all experiments discussed here is a Bose-condensed gas of lithium loaded into the ground band of a static 1D optical lattice of depth V 0 and wavelength λ = 1064 nm at zero quasimomentum. Interatomic interactions are set to the desired value using a magnetic Feshbach resonance. The system is quenched into the Floquet Hamiltonian by applying lattice amplitude modulation at some frequency ω and relative amplitude α, keeping the cycle-averaged lattice depth fixed at V 0 . After some modulation time, the atoms are quenched back to the original optical lattice, bandmapped, and imaged in order to measure the resulting distribution in the eigenbasis of the undriven lattice. This double-quench protocol provides a direct probe of the evolution of system properties under the Floquet Hamiltonian, and of localization in the Floquet state basis. It is not restricted to our particular experimental setup, and therefore provides a general strategy for measuring ergodicity breaking that arXiv:1809.05554v5 [quant-ph] 27 Aug 2019 can be useful also in other experimental contexts. More details on the experimental protocol appear in the appendices.
If interactions are tuned to zero, the Hamiltonian of this driven system is where x is position along the lattice, m is the atomic mass, k L = 2π/λ is the wavenumber of the lattice laser, and V 0 = 10E R is the static lattice depth, with recoil energy E R = 2 k 2 L /2m. It is intuitively useful to note that this is a quantum mechanical version of the Hamiltonian for a rigid pendulum with a vertically modulated pivot point, with the compact phase variable of the pendulum replaced by the position x. The role of classical rigid pendula as archetypes for the study of stability and instability in driven systems provides natural motivation for the quantum mechanical experiments reported here.
The drive frequency can be expressed dimensionlessly as Ω = ω/ω 0 where ω 0 = 2 √ V 0 E R / is the frequency of harmonic motion in a single lattice site, which approximates the lowest band gap of the static lattice. Our experiments explore four square decades of drive parameter space, with a dimensionless frequency from Ω = 0.1 to Ω = 10 and a dimensionless amplitude from α = 0.1 to α = 10. This wide range represents a challenge both for theory, due to the absence of any reliably small scale in the problem, and for experiment, due to the difficulty of attaining modulation amplitudes greater than 100%. We realize the strongly-driven regime α > 1 by simultaneously modulating two orthogonally-polarized co-axial 1D optical lattices with a relative spatial phase shift of λ/4. Exploration of this ultrastrong driving regime is a key experimental novelty of this work, and it is required for access to the majority of the parameter space we explore in maps of prethermal Floquet matter properties like those shown in Fig. 1.
We characterize and quantify prethermalization in our Floquet system via the fraction f 0 of atoms which occupy the single-particle ground state after the modulation is quenched off. This quantity naturally contains information about heating in the Floquet system since those atoms which are excited out of the initial ground state automatically lead to a reduced f 0 . Beyond this intuitive argument, we find that f 0 in fact represents a powerful quantitative measure for localization and therefore for prethermalization and the absence of heating: specifically, f 0 can be directly related to an inverse participation ratio (IPR) in our experimental context. An IPR, defined as IPR = n | ψ 0 |n | 4 , quantifies how strongly the state |ψ 0 , here representing the initial condition, is localized in the basis |n , which in our case is the eigenbasis of the Floquet time-evolution operator. The participation ratio 1/IPR measures the number of Floquet states |n required to represent the initial state. In a localized prethermal state the IPR takes on a nonzero value whereas in the delocalized ergodic regime the IPR becomes vanishingly small. We find, crucially, that if interactions are neglected the long-time average of f 0 measured stroboscopically at integer multiples of the driving period,f 0 , is exactly equivalent to the IPR; this is demonstrated in detail in appendix D. While the IPR is a standard diagnostic of localization, the experimental measurement of such a quantity is, in general, very challenging for a many-body system. For an interacting system in the prethermal regime described by the PGE, the exact identification of f 0 as an IPR does not hold, since for example the condensate fraction will be depleted slightly already in the ground state by quantum fluctuations, but f 0 remains a useful and experimentally accessible metric for characterizing the properties of the interacting driven system. In particular, a non-zero f 0 still indicates a non-ergodic interacting prethermal state. We note that the IPR-based interpretation off 0 as a quantitative measure for prethermal ergodicity breaking proposed here applies to a wide range of bosonic quantum systems, since it essentially relies only on a small quantum depletion of the pre-quench condensate.
The dependence of Floquet matter attributes on drive properties can be directly calculated in the noninteracting case. We calculatef 0 (α, Ω) from the periodic Gibbs ensemble ρ ∝ exp(− i η ini (t)) characterized by integrals of motion given by the occupationŝ n i (t) of the single-particle Floquet modes |i(t) of the multi-band Hamiltonian. Here, the coefficients are are directly given by singleparticle overlaps with the undriven (single-particle) ground state |0 . The predicted dependencef 0 (α, Ω) is shown in Fig. 1A as a function of dimensionless driving amplitude α and frequency Ω. The theory predicts thatf 0 1 for a large region in (α, Ω) space, but as Ω decreases from large values for any given drive amplitude, there is always some α-dependent drive frequency below which f 0 sharply decreases to a lower but still non-ergodic value between 0 and 1. Intriguingly, this sharp crossover corresponds approximately to the stableunstable crossover of the corresponding classical system: a rigid pendulum with a vertically modulated pivot point. The dashed line in Fig. 1A shows the classical boundary of stability for the downward-pointing pendulum; exploration of a possible quantum analogue of the upwardpointing Kapitza state [32,33] is an interesting potential direction for future work.
The wide tunability afforded by our flexible experimental platform enables measurement of the full prethermal map of Floquet material properties predicted in Fig. 1A, and direct observation of the Floquet delocalization crossover. Figures 1B and 1C show experimentally measured maps of f 0 (α, Ω) after 500 µs for noninteracting and interacting samples respectively. Because experimental quasimomentum resolution is limited by the finite initial spatial size of the condensate and a finite time-of-flight, the experimental maps are based on integrals over the central 40% of the Brillouin zone. Dashed line shows PGE prediction, solid line shows TDSE prediction of the fluctuating evolution of a spatially homogeneous non-interacting system. Each point represents a single run; error bar shows representative estimated fractional error due to shot-to-shot number fluctuation. For these data, the optical lattice is immediately quenched back to the initial 10 ER static lattice at the indicated modulation time and then band-mapped.
The non-interacting measurement of Fig. 1B shows good agreement with the PGE-based theoretical prediction of Fig. 1A, thus experimentally confirming the conjectured applicability of the PGE as a model for strongly-driven systems. For these measurements, theory-experiment discrepancies can arise from experimental imperfections or from the fact that the theory plot represents the predicted long-time average occupation over the stroboscopic dynamics as computed from the PGE whereas the experimental data is a snapshot of the occupation at a fixed point in time. The interacting data in Fig. 1C display a qualitatively similar but not quantitatively identical behavior to those in Fig. 1B; the crucial effects of interactions are explored further in the time-evolution measurements discussed below. Both measurements clearly show the predicted amplitude-dependent Floquet delocalization crossover.
Measuring higher-band observables in addition to f 0 allows a fuller comparison between experimental and theoretical descriptions of the prethermal state. Over the timescales shown in Fig. 2 the dynamics of the noninteracting driven system mainly redistribute population among the lowest few even Bloch bands, as expected from parity conservation at k = 0. Fig. 2A shows the results of a PGE-based calculation of fractional projected occupations f 2ν , ν ∈ (1, . . . 6) of the first six even excited bands, as a function of drive parameters α and Ω. The theory predicts a rich dependence of Floquet material properties on drive parameters, with a distinct map for each projected band population. Our experiment can quantitatively test these predictions by directly imaging such maps. Fig. 2B shows the experimentally measured fractional population of the first six even-parity excited bands at each (α, Ω) point, using the same doublequench protocol used to produce Fig. 1. The observed close match between theory and experiment lends further support to the PGE-based theoretical description of the prethermal state. We do not observe significant occupation above the twelfth band, in agreement with the PGE model. These detailed experimental maps of the properties of prethermal Floquet matter throughout a wide range of drive parameter space reveal the intricate structure of the amplitude-dependent Floquet delocalization crossover in the strong-driving regime and constitute the first main result of this report.
Moving beyond such fixed-time maps, it is possible to experimentally explore the full time evolution of f 0 and higher-band observables at any point in drive parameter space. This allows direct measurement of the temporal emergence of the prethermal state, as well as investigation of its long-term fate. As shown in Fig. 2C, the measured time evolution of f 0 for various drive parameters shows remarkably rapid attainment of an average value in close agreement with the PGE theoretical prediction, on timescales near a single drive period [19]. For a drive with Ω = 0.3 and α = 0.5, f 0 remains close to 1, while for higher drive amplitudes and frequencies it fluctuates around a lower average value which agrees well with the PGE prediction. Since the inverse of the IPR provides a measure for the number of eigenstates of the time-evolution operator the system can access, it might appear surprising that even at large driving strengths the predicted and measured f 0 remains so high throughout the dynamics; as discussed in appendix E this can be understood as a consequence of the eventual dominance of higher-band kinetic energy splittings over any fixed coupling matrix element.
Return probabilities such as f 0 yield on general grounds fluctuations of the same order as the mean for unitary dynamics; thus, the fact that we not only measure the expected temporal mean but also observe large temporal fluctuations can be interpreted as a signature of the unitary quantum character of the real-time dynamics of the experiment. In principle, such fluctuations also contain information on the spectrum of the Floquet Hamiltonian. To enable qualitative comparison to the expected form of these fluctuations, the solid lines in Fig. 2C show the result of time-dependent Schrödinger equation integration for a non-interacting perfectly spatially homogeneous sample. While we do not expect the spatially-inhomogeneous experiment to perfectly reproduce this very simple theory, we do observe good qualitative agreement for the mean value and typical fluctuation amplitude, supporting the picture of these fluctuations as a signature of unitary dynamics.
The rapid but not instantaneous nature of the dephasing responsible for the emergence of the prethermal plateau can be revealed by measuring a different observable: the interference patterns after the system is quenched back to a static lattice which is snapped off immediately rather than band-mapped. Fig. 3 shows that over the course of a few dozen drive cycles, the occupations of the first two interference peaks approach their quasi-steady-state values. We interpret this as a signature of the complex drive-dependent dephasing pro- cesses which create the prethermal state, and note that the timescale of the measured relaxation lies between the static lattice tunneling time and the inverse of typical static interband energy scales.
To probe the effects of varying interparticle interactions, we measure the evolution of strongly-driven Floquet matter over much longer timescales than those shown in Figs. 2 and 3. The long-time evolution of interacting driven systems is both especially relevant for the realization of useful many-body Floquet engineering, and especially challenging to address theoretically. Fig. 4A compares the initial evolution of samples with different Feshbach-tuned interaction strengths under the same drive parameters (Ω = 2.6, α = 3). For all three values of the interaction strength, the early-time dynamics are in agreement with the PGE description. As the system continues to evolve to large times over thousands of drive cycles, the PGE plateau decays. Strikingly, for a wide range of parameters we observe, as shown in Fig. 4B, that the system enters a second plateau in which f 0 is non-zero but smaller than in the first plateau. Noninteracting and weakly-interacting samples remain in this second plateau for at least twenty thousand drive cycles at Ω = 2.6. We hypothesize that this second plateau can be understood as a consequence of a slow spreading in both position and quasimomentum: one relevant effect among others is that increasing quasimomentum extent will break the selection rules that prohibit odd-band occupation at k = 0. To account in part for this spreading, all measurements of f 0 in Figs. 4 and 5 integrate over the entirety of the first Brillouin zone. The strongestinteracting samples, subjected to the same drive, behave in a fundamentally different way: they do not exhibit this second plateau but instead are observed to attain ergod-icity, heating up indefinitely with no detectable atoms in the ground band after modulation. The second main result of this report, after the maps of the IPR and drivedependent material properties, comprises these measurements of the detailed time evolution of Floquet matter: at short time scales, we observe a rapid emergence of a PGE prethermal state exhibiting large fluctuations, and at long time scales, we observe a second prethermal plateau and interaction-dependent ergodicity.
These results raise a crucial question: is the rapid onset of ergodicity for the strongest-interacting system inevitable? To quantitatively explore the possibility of delaying the onset of ergodicity in driven interacting systems, we measured the long-time evolution of the participation ratio 1/f 0 for the strongest-interacting samples at α = 3 and increasing values of Ω. The results are shown in Fig. 5. At Ω = 2, the observed long-time evolution of 1/f 0 is consistent with the √ t dependence naively expected from Joule heating, and no significant plateau is observed. As the drive frequency is increased holding α constant, we observe the emergence of a quasistatic prethermal plateau lasting thousands of drive cycles, at a value of f 0 consistent with the PGE prediction. This plateau too eventually decays. The lifetime of the high-frequency plateau is approximately ten times the timescale associated with the initial Thomas-Fermi interaction energies. We note that this high-frequency stabilization of the prethermal plateau cannot be simply explained by the exponential suppression of Floquet heating, predicted for single-band and spin models [14][15][16][17], though it may be related; neither is the prethermal state characterized by simple high-frequency Floquet-Magnustype expansions.
Intriguingly, for the highest-frequency drives, the long- time departure from this plateau is significantly slower than for the lower drive frequencies, and clearly rules out the √ t dependence expected for Joule heating. In this regime 1/f 0 obeys an approximate t 1/4 time dependence after departure from the prethermal plateau. This unusual behavior, while still poorly understood, is consistent with the sub-Joule heating predicted by recent related theoretical analyses [21,22]. Future work measuring the dynamics of heating away from the prethermal plateau in other regimes of drive strength, interaction strength, and modulation type should further elucidate the generality of these predictions and experimental observations. The recovery of the prethermal plateau by increasing the drive frequency in the presence of interactions, and the observation of anomalously slow heating dynamics away from the recovered plateau, together constitute the third and final main result of this report.
In conclusion, we have used a flexible platform for studying strongly driven interacting quantum systems to acquire a complete prethermal map of the drivedependent properties of tunable Floquet matter, revealing a Floquet delocalization crossover. The quantitative agreement between measured data and theoretical calculations provides an experimental confirmation that the prethermal state is describable by a periodic Gibbs ensemble. The measurement of an IPR via a double quench protocol introduced here represents a powerful quantitative tool for characterizing strongly-driven quantum systems in a variety of experimental contexts. Measuring the evolution of driven ensembles at both short and long times, we have observed two prethermal plateaux and a long-time transition to ergodicity at a rate and onset time which depend critically on drive frequency and interaction strength.
2m is the lattice recoil energy. In the static lattice, the tunneling rate between lattice sites is 483 Hz and the lattice site trap frequency is ω 0 = 159 kHz. The transverse confinement is provided by the Gaussian lattice beams, resulting in a transverse trapping frequency of 449 Hz× 1 + α/2. We observe no significant excitation of transverse oscillator modes in the experiments reported here. Any additional forces along the lattice direction arising from magnetic field curvature or lattice beam intensity gradients are nulled out using magnetic shim coils to increase the period of Bloch oscillations to time scales significantly longer than our longest experiments [34]. The interacting experiments are performed at Feshbach-induced s-wave scattering lengths of 2 nm and 30 nm, resulting in Thomas-Fermi interaction energies of 1.3 kHz and 3.8 kHz, respectively, for α = 0. Because the Thomas-Fermi interaction energies grow weakly with the modulation depth due to increased transverse confinement, we characterize interactions by the scattering length, which does not depend on the drive parameters.

Appendix B: Optical lattice for sign-changing modulation
To enable realization of the α > 1 regime of ultrastrong lattice modulation, a combined lattice is formed by overlapping two 1D optical lattices with a relative spatial phase shift of half a period. We use up to 7 W of 1064 nm light per beam and an 88 µm beam waist. The two lattices are separated in frequency by 160 MHz and have orthogonal linear polarizations. The beams are retroreflected by the same mirror to form two independent lattices. The relative phase is controlled by means of a waveplate stack, arranged so that the one of the lattices receives a λ/4 phase shift as it is retroreflected. This causes the two lattices to cancel each other when both beams have the same power, resulting in a featureless optical dipole trap; intentionally imbalancing the power results in a lattice of controllable sign. The depth of the combined lattice and the relative spatial phase between the two lattices are calibrated using matter-wave diffraction. After ramp-up of the combined lattice to an initial depth of 10 E R , the system is quenched into the Floquet Hamiltonian by applying lattice amplitude modulation with some Ω and α. The amplitude of the combined lattice is modulated at up to 2 MHz by simultaneously varying the power of RF signals sent to two acousto-optical modulators from an AD9854 DDS board. Crucially for the results we present, this double lattice modulation allows us to create a combined optical lattice that can change sign, where maxima (minima) become minima (maxima) during a drive cycle. motion given by the number operatorsn j (t) of the singleparticle Floquet modes |j(t) = |j(t + T ) . The expectation values of these operators determine the PGE. To quantify the degree of localization (non-ergodicity), we initially prepare the system in the undriven ground-state |ψ 0 , so that c n = n(0)|ψ 0 , and consider the expectation value of the projectorÔ = |ψ 0 ψ 0 | at stroboscopic times t ν = νT with integer ν, which is equal to the squared overlap | ψ 0 |ψ(t ν ) | 2 with the evolved state |ψ(t ν ) . According to the diagonal ensemble and employing |n(t ν ) = |n(0) , the long-time average (indicated by an overbar) over the stroboscopic dynamics of this quantity gives | ψ 0 |ψ(t ν ) | 2 = n | ψ 0 |n(0) | 4 ≡ IPR. (D1) This quantity is directly identifiable as the inverse participation ratio that quantifies the localization of the ground state |ψ 0 in the basis of the Floquet states. Its inverse, 1/IPR, measures the number of Floquet states required to represent |ψ 0 . For the non-interacting gas, we find that the desired overlap is given by the fraction of atoms populating the single-particle ground state (i.e. the quasimomentum k = 0 mode in the lowest Bloch band), | ψ 0 |ψ(t ν ) | 2 = f 0 (t ν ), which we have measured for example in Fig. 1B. Thus, having relaxed to a quasi-steady state, we have so that the measured observable directly quantifies nonergodicity.
In an interacting many-body system, the diagonal ensemble is still formally characterized by an exponentially large number of probabilities |c n | 2 . However, it is believed that the quasi-steady state is characterized by a periodic Gibbs ensemble (PGE), ρ P GE (t) = Z −1 exp[− j λ jÎj (t)], withÎ j (t) =Î j (t + T ) denoting the integrals of motion of the system [19]. It is important to note that while a generic interacting Floquet system should at sufficiently long times approach a fully ergodic high-temperature state ρ ergodic ∝ 1 [20,37], even the interacting system can approach a prethermal state that is accurately described by the PGE on intermediate, and potentially exponentially long, time scales. This motivates the use of f 0 as a prethermal diagnostic of Floquet localization even in the interacting regime.