We investigate an unconventional symmetry in time-periodically driven systems, the Floquet dynamical symmetry (FDS). Unlike the usual symmetries, the FDS gives symmetry sectors that are equidistant in the Floquet spectrum and protects quantum coherence between them from dissipation and dephasing, leading to two kinds of time crystals: the discrete time crystal and discrete time quasicrystal that have different periodicity in time. We show that these time crystals appear in the Bose- and Fermi-Hubbard models under ac fields and their periodicity can be tuned only by adjusting the strength of the field. These time crystals arise only from the FDS and thus appear in both dissipative and isolated systems and in the presence of disorder as long as the FDS is respected. We discuss their experimental realizations in cold atom experiments and generalization to the SU(N)-symmetric Hubbard models.
We investigate an unconventional symmetry in time-periodically driven systems, the Floquet dynamical symmetry (FDS). Unlike the usual symmetries, the FDS gives symmetry sectors that are equidistant in the Floquet spectrum and protects quantum coherence between them from dissipation and dephasing, leading to two kinds of time crystals: the discrete time crystal and discrete time quasicrystal that have different periodicity in time. We show that these time crystals appear in the Bose- and Fermi-Hubbard models under ac fields and their periodicity can be tuned only by adjusting the strength of the field. These time crystals arise only from the FDS and thus appear in both dissipative and isolated systems and in the presence of disorder as long as the FDS is respected. We discuss their experimental realizations in cold atom experiments and generalization to the SU(N)-symmetric Hubbard models.
A discrete time crystal (DTC) is an essentially nonequilibrium phase that breaks discrete time-translation symmetry in periodically driven (Floquet) systems.In ideal isolated quantum systems without dissipation, many kinds of DTCs have been theoretically proposed and experimentally verified in various artificial quantum systems, such as trapped ions, NV centers in diamond, and quantum computers.However, in real physical systems, we cannot avoid dissipation stemming from coupling to an external environment.Nevertheless, the effects of dissipation on DTCs have not been fully explored yet.This thesis reports the author's two original studies on dissipative DTCs.First, we formulate a symmetry-based theory of dissipative DTCs.To this end, we introduce Floquet dynamical symmetry (FDS) governing the long-time dynamics of dissipative Floquet systems.The FDS protects the quantum coherence of the system from dissipation and dephasing and leads to two types of time crystals: the discrete time crystal (DTC) and discrete time quasicrystal (DTQC), which are different in periodicity.We show that these time crystals appear in spin-S Bose-and Fermi-Hubbard models driven by an ac field and that their periodicity can be tuned by adjusting the field strength.Since these time crystals rely on the FDS alone, they appear even in the presence of perturbations as long as the FDS exists.We also elucidate that the FDS explains the symmetry structure behind the conventional DTCs proposed in the previous studies.Second, we investigate dissipative DTCs in solid-state materials by considering a dissipative quantum Ising model periodically driven by π-pulses.While the previous studies on dissipative DTCs have focused on well-controllable artificial quantum systems, we show that realistic dissipation in solids does not destroy but can stabilize the DTC order.Using the time-dependent mean-field theory, we analyze the DTC phase transition and determine critical exponents, including a dynamical one.We also find a novel reentrant phase transition in varying the pulse interval.This transition is due to a nontrivial interplay of the periodic drive and dissipation and has no equilibrium counterparts in the corresponding undriven system.Furthermore, to demonstrate the rigidity (stability) of the DTC, we consider imperfect πpulses, showing that the DTC is robust against minor imperfections and finding that DTQC appears for major imperfections.These results uncover a general symmetry aspect of dissipative DTCs and lay the foundation for realizing DTCs in material science.
Floquet (periodically driven) systems can give rise to unique nonequilibrium phases of matter without equilibrium analogs. The most prominent example is the realization of discrete time crystals. An intriguing question emerges: What other novel phases can manifest when the constraint of time periodicity is relaxed? In this study, we explore quantum systems subjected to a quasiperiodic drive. Leveraging a strongly interacting spin ensemble in diamond, we identify the emergence of long-lived discrete time quasicrystals. Unlike conventional time crystals, time quasicrystals exhibit robust subharmonic responses at multiple incommensurate frequencies. Furthermore, we show that the multifrequency nature of the quasiperiodic drive allows for the formation of diverse patterns associated with different discrete time quasicrystalline phases. Our findings demonstrate the existence of nonequilibrium phases in quasi-Floquet settings, significantly broadening the catalog of novel phenomena in driven many-body quantum systems.
The classification of phases of matter has been revolutionised by topological order and non equilibrium phenomena such as time crystals. However, a unified framework connecting temporal order, topology, and dissipation remains elusive. Here we introduce the concept of "Time-Quasicrystals"—non-equilibrium topological phases that exhibit quasi-periodic temporal order protected by dissipative dynamics. These phases emerge in driven dissipative quantum systems through the interplay of Floquet engineering and non-local dissipation, leading to an effective non-Hermitian Floquet Hamiltonian with topological edge modes in the frequency domain. We demonstrate three hallmarks of Time-Quasicrystals: (1) fractal-like temporal correlations with multiple incommensurate frequencies, (2) a quantised topological invariant robust against disorder, and (3) algebraically decaying memory of phase perturbations. We provide explicit experimental realisations in Rydberg atom arrays and circuit quantum electrodynamics platforms. Our work establishes Time-Quasicrystals as a bridge between topological phases, time crystals, and quantum simulation, opening pathways to dissipative protected quantum memory and novel non-equilibrium quantum phenomena.
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 250000 driving periods.
Motivated by experimental observations of time-symmetry breaking behavior in a periodically driven (Floquet) system, we study a one-dimensional spin model to explore the stability of such Floquet discrete time crystals (DTCs) under the interplay between interaction and the microwave driving. For intermediate interactions and high drivings, from the time evolution of both stroboscopic spin polarization and mutual information between two ends, we show that Floquet DTCs can exist in a prethermal time regime without the tuning of strong disorder. For much weak interactions the system is a symmetry-unbroken phase, while for strong interactions it gives its way to a thermal phase. Through analyzing the entanglement dynamics, we show that large driving fields protect the prethermal DTCs from many-body localization and thermalization. Our results suggest that by increasing the spin interaction, one can drive the experimental system into optimal regime for observing a robust prethermal DTC phase.
Time crystals, a phase showing spontaneous breaking of time-translation symmetry, has been an intriguing subject for systems far away from equilibrium. Recent experiments found such a phase in both the presence and the absence of localization, while in theories localization by disorder is usually assumed a priori. In this work, we point out that time crystals can generally exist in systems without disorder. A series of clean quasi-one-dimensional models under Floquet driving are proposed to demonstrate this unexpected result in principle. Robust time crystalline orders are found in the strongly interacting regime along with the emergent integrals of motion in the dynamical system, which can be characterized by level statistics and the out-of-time-ordered correlators. We propose two cold atom experimental schemes to realize the clean Floquet time crystals, one by making use of dipolar gases and another by synthetic dimensions.