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arXiv:2106.02219 · Physical Review A

Discrete time crystals in Bose-Einstein condensates and the symmetry-breaking edge in a simple two-mode theory

Jia Wang, Krzysztof Sacha, Peter Hannaford, Bryan J. Dalton

Published Nov 29, 2021• 8 citations
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Abstract

Discrete time crystals (DTCs) refer to a novel many-body steady state that spontaneously breaks the discrete time-translational symmetry in a periodically driven quantum system. Here, we study DTCs in a Bose-Einstein condensate bouncing resonantly on an oscillating mirror, using a two-mode model derived from a standard quantum field theory. We investigate the validity of this model and apply it to study the long-time behavior of our system. A wide variety of initial states based on two Wannier modes are considered. We find that in previous studies the investigated phenomena in the evolution time window ($⪅2000$ driving periods) are actually ``short-time'' transient behavior though DTC formation signaled by the subharmonic responses is still shown if the interboson interaction is strong enough. After a much longer (about 20 times) evolution time, initial states with no ``long-range'' correlations relax to a steady state, where time-symmetry breaking can be unambiguously defined. Quantum revivals also eventually occur. This long-time behavior can be understood via the many-body Floquet quasieigenenergy spectrum of the two-mode model. A symmetry-breaking edge for DTC formation appears in the spectrum for strong enough interaction, where all quasieigenstates below the edge are symmetry breaking while those above the edge are symmetric. The late-time steady state's time-translational symmetry depends solely on whether the initial energy is above or below the symmetry-breaking edge. A phase diagram showing regions of symmetry-broken and symmetric phases for differing initial energies and interaction strengths is presented. We find that, according to this two-mode model, the discrete time crystal survives for times out to at least $250\phantom{\rule{0.16em}{0ex}}000$ driving periods.

DOI
10.1103/physreva.104.053327

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Discrete time crystals (DTCs) refer to a novel many-body steady state that spontaneously breaks the discrete time-translational symmetry in a periodically driven quantum system. Here, we study DTCs in a Bose-Einstein condensate bouncing resonantly on an oscillating mirror, using a two-mode model derived from a standard quantum field theory. We investigate the validity of this model and apply it to study the long-time behavior of our system. A wide variety of initial states based on two Wannier modes are considered. We find that in previous studies the investigated phenomena in the evolution time window (⪅2000⪅2000⪅2000 driving periods) are actually short-time'' transient behavior though DTC formation signaled by the subharmonic responses is still shown if the interboson interaction is strong enough. After a much longer (about 20 times) evolution time, initial states with no long-range'' correlations relax to a steady state, where time-symmetry breaking can be unambiguously defined. Quantum revivals also eventually occur. This long-time behavior can be understood via the many-body Floquet quasieigenenergy spectrum of the two-mode model. A symmetry-breaking edge for DTC formation appears in the spectrum for strong enough interaction, where all quasieigenstates below the edge are symmetry breaking while those above the edge are symmetric. The late-time steady state's time-translational symmetry depends solely on whether the initial energy is above or below the symmetry-breaking edge. A phase diagram showing regions of symmetry-broken and symmetric phases for differing initial energies and interaction strengths is presented. We find that, according to this two-mode model, the discrete time crystal survives for times out to at least 250000250\phantom{\rule{0.16em}{0ex}}000250000 driving periods.
Abstract We present a fully comprehensive multi-mode quantum treatment based on the truncated Wigner approximation (TWA) to study many-body effects and effects of quantum fluctuations on the formation of a discrete time crystal (DTC) in a Bose–Einstein condensate (BEC) bouncing resonantly on a periodically driven atom mirror. Zero-range contact interactions between the bosonic atoms are assumed. Our theoretical approach avoids the restrictions both of mean-field theory, where all bosons are assumed to remain in a single mode, and of time-dependent Bogoliubov theory, which assumes boson depletion from the condensate mode is small. We show that the mean-field and time-dependent Bogoliubov approaches can be derived as approximations to the TWA treatment. Differing initial conditions, such as a finite temperature BEC, can also be treated. For realistic initial conditions corresponding to a harmonic trap condensate mode function, our TWA calculations performed for period-doubling agree broadly with recent mean-field calculations for times out to at least 2000 mirror oscillations, except at interaction strengths very close to the threshold value for DTC formation where the position probability density differs significantly from that determined from mean-field theory. For typical attractive interaction strengths above the threshold value for DTC formation and for the chosen trap and driving parameters, the TWA calculations indicate a quantum depletion due to quantum many-body fluctuations of less than about two atoms out of a total of 600 atoms at times corresponding to 2000 mirror oscillations, in agreement with time-dependent Bogoliubov theory calculations. On the other hand, for interaction strengths very close to the threshold value for DTC formation, the TWA calculations predict a large quantum depletion—as high as about 260 atoms out of 600. We also show that the mean energy per particle of the DTC does not increase significantly for times out to at least 2000 mirror oscillations and typically oscillates around an average value close to its initial value; so TWA theory predicts the absence of thermalisation. Finally, we find that the dynamical behaviour of our system is largely independent of whether the boson–boson interaction is attractive or repulsive, and that it is possible to create a stable DTC based on repulsive interactions.
Abstract We study discrete time crystal (DTC) formation in a system driven periodically by an oscillating atomic mirror, consisting of two distinct ultracold atomic clouds in the presence of a gravitational field. The intra-species interactions are weak and attractive, while the inter-species interactions are infinitely strong and repulsive. The clouds are arranged in a one-dimensional stack, where the bottom cloud bounces on an oscillating atomic mirror, which effectively acts as a driving force for the upper cloud due to the infinite inter-species repulsion. Using a Jastrow-like variational ansatz for the many-body wavefunction, we show numerically that sufficiently strong attractive intra-species interactions drive each subsystem to spontaneously break discrete time translation symmetry, resulting in the formation of a cascading DTC evolving with a period different than the driving period. Since the bottom cloud serves as the effective periodic driving for the upper cloud, this leads to a cascade of spontaneous symmetry breaking. With increasing intra-species interactions, we first observe a pronounced effect of spontaneous time translation symmetry breaking in the upper cloud, followed by a similar effect in the lower atomic cloud.
Periodically driven quantum systems host a range of non-equilibrium phenomena which are unrealizable at equilibrium. Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, a putative quantum phase of matter. We present the observation of discrete time crystalline order in a driven system of paramagnetic PPP -donor impurities in isotopically enriched 28Si^{28}Si28Si cooled below 101010 K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of DTC order. We propose a theoretical model based on the paradigmatic central spin model which is in good agreement with experimental observations, and investigate the role of dissipation in the stabilisation of the DTC. Both experiment and theory indicate that the order in this system is primarily a dissipative effect, and which persists in the presence of spin-spin interactions. We present a theoretical phase diagram as a function of interactions and dissipation for the central spin model which is consistent with the experiments. This opens up questions about the interplay of coherent interaction and dissipation for time-translation symmetry breaking in many-body Floquet systems.
We report progress towards realizing discrete time crystals using a BEC of potassium-39 atoms bouncing on an oscillating mirror. In the absence of interactions, each bouncing atom performs classical motion in which there are resonant islands in phase space located around periodic orbits whose periods match an integral multiple s of the driving period T of the mirror. If all the atoms are prepared in a single wave-packet, in the absence of interactions they tunnel to neighbouring wave-packets on a timescale determined by the driving amplitude and period of the oscillating mirror. In the presence of a sufficiently strong attractive interaction, the Floquet eigenstates of the periodic system become Schrödinger cat-like states, so that measurement of the position of just a single atom is sufficient to break the timetranslation symmetry and cause the system to collapse into one of the s wave-packets bouncing with period sT. Such a system allows dramatic breaking of discrete time translation symmetry where the symmetry-broken state evolves with a period typically s=40 times longer than the driving period and does not decay. This system provides also a suitable platform for demonstrating a wide range of condensed matter phenomena in the time domain.
Discrete time crystals (DTCs) are non-equilibrium phases of matter that break the discrete time-translation symmetry and is characterized by a robust subharmonic response in periodically driven quantum systems. Here, we explore the DTC in a disorder-free, periodically kicked XXZ spin chain, which is stabilized by the Floquet strong Hilbert space fragmentation. We numerically show the period-doubling response of the conventional DTC order, and uncover a multiple-period response with beating dynamics due to the coherent interplay of multiple πππ-pairs in the Floquet spectrum of small-size systems. The lifetime of the DTC order exhibits independence of the driving frequency and a power-law dependence on the ZZ interaction strength. It also grows exponentially with the system size, as a hallmark of the strong fragmentation inherent to the Floquet model. We analytically reveal the approximate conservation of the magnetization and domain-wall number in the Floquet operator for the emergent strong fragmentation, which is consistent with numerical results of the dimensionality ratio of symmetry subspaces. The rigidity and phase regime of the DTC order are identified through finite-size scaling of the Floquet-spectrum-averaged mutual information, as well as via dynamical probes. Our work establishes the Floquet Hilbert space fragmentation as a disorder-free mechanism for sustaining nontrivial temporal orders in out-of-equilibrium quantum many-body systems.
Time crystals are quantum many-body systems which are able to self-organize their motion in a periodic way in time. Discrete time crystals have been experimentally demonstrated in spin systems. However, the first idea of spontaneous breaking of discrete time translation symmetry, in ultracold atoms bouncing on an oscillating mirror, still awaits experimental demonstration. Here, we perform a detailed analysis of the experimental conditions needed for the realization of such a discrete time crystal. Importantly, the considered system allows for the realization of dramatic breaking of discrete time translation symmetry where a symmetry broken state evolves with a period tens of times longer than the driving period. Moreover, atoms bouncing on an oscillating mirror constitute a suitable system for the realization of dynamical quantum phase transitions in discrete time crystals and for the demonstration of various nontrivial condensed-matter phenomena in the time domain. We show that Anderson localization effects, which are typically associated with spatial disorder and exponential localization of eigenstates of a particle in configuration space, can be observed in the time domain when ultracold atoms are bouncing on a randomly moving mirror.