Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry.The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization.Here, we consider a simple model for a onedimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied.We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator.Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions.Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry.The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization.Here, we consider a simple model for a onedimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied.We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator.Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions.Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.

PhysicsJan 18, 201720 citations

Colloquium : Quantum and classical discrete time crystals

Michael P. Zaletel, Mikhail D. Lukin, Christopher R. Monroe +3 more

The spontaneous breaking of time-translation symmetry has led to the discovery of a new phase of matter: the discrete time crystal. Discrete time crystals exhibit rigid subharmonic oscillations that result from a combination of many-body interactions, collective synchronization, and ergodicity breaking. This Colloquium reviews recent theoretical and experimental advances in the study of quantum and classical discrete time crystals. The breaking of ergodicity is focused upon as the key to discrete time crystals and the delaying of ergodicity as the source of numerous phenomena that share many of the properties of discrete time crystals, including the ac Josephson effect, coupled map lattices, and Faraday waves. Theoretically, there is a diverse array of strategies to stabilize time-crystalline order in both closed and open systems, ranging from localization and prethermalization to dissipation and error correction. Experimentally, many-body quantum simulators provide a natural platform for investigating signatures of time-crystalline order; recent work utilizing trapped ions, solid-state spin systems, and superconducting qubits are reviewed. Finally, this Colloquium concludes by describing outstanding challenges in the field and a vision for new directions on both the experimental and theoretical fronts.The spontaneous breaking of time-translation symmetry has led to the discovery of a new phase of matter: the discrete time crystal. Discrete time crystals exhibit rigid subharmonic oscillations that result from a combination of many-body interactions, collective synchronization, and ergodicity breaking. This Colloquium reviews recent theoretical and experimental advances in the study of quantum and classical discrete time crystals. The breaking of ergodicity is focused upon as the key to discrete time crystals and the delaying of ergodicity as the source of numerous phenomena that share many of the properties of discrete time crystals, including the ac Josephson effect, coupled map lattices, and Faraday waves. Theoretically, there is a diverse array of strategies to stabilize time-crystalline order in both closed and open systems, ranging from localization and prethermalization to dissipation and error correction. Experimentally, many-body quantum simulators provide a natural platform for investigating signatures of time-crystalline order; recent work utilizing trapped ions, solid-state spin systems, and superconducting qubits are reviewed. Finally, this Colloquium concludes by describing outstanding challenges in the field and a vision for new directions on both the experimental and theoretical fronts.

arXiv:2305.08904Reviews of Modern PhysicsJul 7, 2023193 citations

Realizing discrete time crystal in an one-dimensional superconducting qubit chain

Huikai Xu, Jing-Ning Zhang, Jiaxiu Han +5 more

Floquet engineering, i.e. driving the system with periodic Hamiltonians, not only provides great flexibility in analog quantum simulation, but also supports phase structures of great richness. It has been proposed that Floquet systems can support a discrete time-translation symmetry (TTS) broken phase, dubbed the discrete time crystal (DTC). This proposal, as well as the exotic phase, has attracted tremendous interest among the community of quantum simulation. Here we report the observation of the DTC in an one-dimensional superconducting qubit chain. We experimentally realize long-time stroboscopic quantum dynamics of a periodically driven spin system consisting of 8 transmon qubits, and obtain a lifetime of the DTC order limited by the coherence time of the underlying physical platform. We also explore the crossover between the discrete TTS broken and unbroken phases via various physical signatures. Our work extends the usage of superconducting circuit systems in quantum simulation of many-body physics, and provides an experimental tool for investigating non-equilibrium dynamics and phase structures.Floquet engineering, i.e. driving the system with periodic Hamiltonians, not only provides great flexibility in analog quantum simulation, but also supports phase structures of great richness. It has been proposed that Floquet systems can support a discrete time-translation symmetry (TTS) broken phase, dubbed the discrete time crystal (DTC). This proposal, as well as the exotic phase, has attracted tremendous interest among the community of quantum simulation. Here we report the observation of the DTC in an one-dimensional superconducting qubit chain. We experimentally realize long-time stroboscopic quantum dynamics of a periodically driven spin system consisting of 8 transmon qubits, and obtain a lifetime of the DTC order limited by the coherence time of the underlying physical platform. We also explore the crossover between the discrete TTS broken and unbroken phases via various physical signatures. Our work extends the usage of superconducting circuit systems in quantum simulation of many-body physics, and provides an experimental tool for investigating non-equilibrium dynamics and phase structures.

arXiv:2108.00942Aug 2, 20219 citations

Time crystals: Analysis of experimental conditions

Krzysztof Giergiel, Arkadiusz Kosior, Peter Hannaford +1 more

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.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.

arXiv:1805.05718Physical Review AJul 12, 201871 citations

Time-crystalline behavior in an engineered spin chain

Robin Schäfer, Götz Silvester Uhrig, Joachim Stolze

Time crystals break the discrete time translational invariance of an external periodic drive by oscillating at an integer multiple of the driving period. In addition to this fundamental property, other aspects are often considered to be essential characteristics of a time crystal, such as the presence of disorder or interactions, robustness against small variations of system parameters, and the free choice of the initial quantum state. We study a finite-length polarized XX spin chain engineered to display a spectrum of equidistant energy levels without drive and show that it keeps a spectrum of equidistant Floquet quasienergies when subjected to a large variety of periodic driving schemes. Arbitrary multiples of the driving period can then be reached by adjusting parameters of the drive, for arbitrary initial states. This behavior is understood by mapping the XX spin chain with $N+1$ sites to a single large spin with $S=N/2$ invoking the closure of the group SU(2). Our simple model is neither intrinsically disordered nor is it an interacting many-body system (after suitable mapping), and it does not have a thermodynamic limit in the conventional sense. It does, however, show controllable discrete time translational symmetry breaking for arbitrary initial states and a degree of robustness against perturbations, thereby carrying some characteristic traits of a discrete time crystal.Time crystals break the discrete time translational invariance of an external periodic drive by oscillating at an integer multiple of the driving period. In addition to this fundamental property, other aspects are often considered to be essential characteristics of a time crystal, such as the presence of disorder or interactions, robustness against small variations of system parameters, and the free choice of the initial quantum state. We study a finite-length polarized XX spin chain engineered to display a spectrum of equidistant energy levels without drive and show that it keeps a spectrum of equidistant Floquet quasienergies when subjected to a large variety of periodic driving schemes. Arbitrary multiples of the driving period can then be reached by adjusting parameters of the drive, for arbitrary initial states. This behavior is understood by mapping the XX spin chain with $N+1$ sites to a single large spin with $S=N/2$ invoking the closure of the group SU(2). Our simple model is neither intrinsically disordered nor is it an interacting many-body system (after suitable mapping), and it does not have a thermodynamic limit in the conventional sense. It does, however, show controllable discrete time translational symmetry breaking for arbitrary initial states and a degree of robustness against perturbations, thereby carrying some characteristic traits of a discrete time crystal.

arXiv:1904.12328Physical review. B./Physical review. BNov 5, 201910 citations

Discrete time crystals enabled by Floquet strong Hilbert space fragmentation

Ling-Zhi Tang, Xiao Li, Z. D. Wang +1 more

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 $π$ -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.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 $π$-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.

quant-pharXiv:2512.14182Physical review. B./Physical review. BDec 17, 20253 citations

N+1

sites to a single large spin with

S=N/2

invoking the closure of the group SU(2). Our simple model is neither intrinsically disordered nor is it an interacting many-body system (after suitable mapping), and it does not have a thermodynamic limit in the conventional sense. It does, however, show controllable discrete time translational symmetry breaking for arbitrary initial states and a degree of robustness against perturbations, thereby carrying some characteristic traits of a discrete time crystal.

π

-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.