Systems which can spontaneously reveal periodic evolution are dubbed time crystals. This is in analogy with space crystals that display periodic behavior in configuration space. While space crystals are modeled with the help of space periodic potentials, crystalline phenomena in time can be modeled by periodically driven systems. Disorder in the periodic driving can lead to Anderson localization in time: the probability for detecting a system at a fixed point of configuration space becomes exponentially localized around a certain moment in time. We here show that a three-dimensional system exposed to a properly disordered pseudoperiodic driving may display a localized-delocalized Anderson transition in the time domain, in strong analogy with the usual three-dimensional Anderson transition in disordered systems. Such a transition could be experimentally observed with ultracold atomic gases.
Systems which can spontaneously reveal periodic evolution are dubbed time crystals. This is in analogy with space crystals that display periodic behavior in configuration space. While space crystals are modeled with the help of space periodic potentials, crystalline phenomena in time can be modeled by periodically driven systems. Disorder in the periodic driving can lead to Anderson localization in time: the probability for detecting a system at a fixed point of configuration space becomes exponentially localized around a certain moment in time. We here show that a three-dimensional system exposed to a properly disordered pseudoperiodic driving may display a localized-delocalized Anderson transition in the time domain, in strong analogy with the usual three-dimensional Anderson transition in disordered systems. Such a transition could be experimentally observed with ultracold atomic gases.
Particles in space periodic potentials constitute standard models for investigation of crystalline phenomena in solid state physics. Time periodicity of periodically driven systems is a close analogue of space periodicity of solid state crystals. There is an intriguing question if solid state phenomena can be observed in the time domain. Here we show that wave-packets localized on resonant classical trajectories of periodically driven systems are ideal elements to realize Anderson localization or Mott insulator phase in the time domain. Uniform superpositions of the wave-packets form stationary states of a periodically driven particle. However, an additional perturbation that fluctuates in time results in disorder in time and Anderson localization effects emerge. Switching to many-particle systems we observe that depending on how strong particle interactions are, stationary states can be Bose-Einstein condensates or single Fock states where definite numbers of particles occupy the periodically evolving wave-packets. Our study shows that non-trivial crystal-like phenomena can be observed in the time domain.
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.
Abstract By analogy with the formation of space crystals, crystalline structures can also appear in the time domain. While in the case of space crystals we often ask about periodic arrangements of atoms in space at a moment of a detection, in time crystals the role of space and time is exchanged. That is, we fix a space point and ask if the probability density for detection of a system at this point behaves periodically in time. Here, we show that in periodically driven systems it is possible to realize topological insulators, which can be observed in time. The bulk-edge correspondence is related to the edge in time, where edge states localize. We focus on two examples: Su-Schrieffer-Heeger model in time and Bose Haldane insulator which emerges in the dynamics of a periodically driven many-body system.
Understanding how a system's behavior extrapolates beyond 3D is a fundamental question in physics, spanning topics from unification theories to critical phenomena. In statistical physics, fluctuations' strength is highly sensitive to dimensionality, affecting phase transitions. In low dimensions, phase transitions are suppressed, while high-dimensional systems exhibit simpler mean-field behavior. In some cases, like the Anderson localization-delocalization transition in disordered media, criticality remains non-trivial even in dimensions larger than three, presenting challenges to existing frameworks. In this work, using a periodically-driven ultracold atomic gas to engineer disorder and synthetic dimensions, we experimentally observe a phase transition between localized and delocalized phases. The results display three key features of the 4D transition: 1) observables follow d=4 critical scale invariance, 2) critical exponents match numerical predictions for the 4D Anderson transition, and 3) they agree with Wegner's relation in 4D. These findings provide a new avenue for exploring complex critical phenomena in higher dimensions.
Time crystals are quantum many-body systems that, due to interactions between particles, are able to spontaneously self-organize their motion in a periodic way in time by analogy with the formation of crystalline structures in space in condensed matter physics. In solid state physics properties of space crystals are often investigated with the help of external potentials that are spatially periodic and reflect various crystalline structures. A similar approach can be applied for time crystals, as periodically driven systems constitute counterparts of spatially periodic systems, but in the time domain. Here we show that condensed matter problems ranging from single particles in potentials of quasicrystal structure to many-body systems with exotic long-range interactions can be realized in the time domain with an appropriate periodic driving. Moreover, it is possible to create molecules where atoms are bound together due to destructive interference if the atomic scattering length is modulated in time.
A discrete time crystal is an out-of-equilibrium phase of matter characterized by the spontaneous breaking of discrete time-translation symmetry. Using extensive numerical simulations based on matrix-product-state methods, we provide evidence for discrete time-crystalline behavior in strongly disordered spin chains with Heisenberg interactions, including the isotropic point, subject to periodic driving. Starting from a many-body localized regime, we observe that rotations induced by delta kicks produce a pronounced subharmonic response at half the drive frequency in spin observables. We investigate the stability of this response against rotation-angle errors through entanglement entropy, quantum Fisher information, short-range spin correlations, and restricted-control ergotropy. Increasing the rotation error reveals an intermediate dynamical regime separating the time-crystalline and Floquet-localized responses. In this regime, most observables exhibit signatures of weakly correlated dynamics reminiscent of Anderson localization.