Floquet time spirals and stable discrete-time quasicrystals in quasiperiodically driven quantum many-body systems

Abstract

We analyse quasi-periodically driven quantum systems that can be mapped exactly to periodically driven ones and find Floquet Time Spirals in analogy with spatially incommensurate spiral magnetic states. Generalising the mechanism to many-body systems we discover that a form of discrete time-translation symmetry breaking can also occur in quasi-periodically driven systems. We construct a discrete time quasi-crystal stabilised by many-body localisation. Crucially, it persists also under perturbations that break the equivalence with periodic systems. As such it provides evidence of a stable quasi-periodically driven many-body quantum system which does not heat up to the featureless infinite temperature state.The dynamics of a quantum system governed by a generic time-dependent Hamiltonian is generally hard to analyse because to reach a certain point in time the entire history of evolution is required. Hence, simulations quickly become computationally unfeasible, especially for many-body systems. There are some exceptions such as periodically driven (PD) systems, i.e. whose Hamiltonian is periodic in time H(t + T ) = H(t), for which Floquet theory significantly reduces the effort to simulate the dynamics by decomposing the wave function to Floquet modes periodic over one time period T up to phases. PD systems are of immense recent interest as they have been exploited extensively for engineering sought after Hamiltonians [1-5], controlling quantum systems [6][7][8][9][10][11][12][13][14]and realising non-equilibrium quantum phases [15][16][17]. For instance, the proper combination of periodic driving and many-body localisaton [18] gave birth to the discovery of discrete time crystals (DTCs) [19][20][21][22][23][24][25][26], a new quantum phase with broken discrete time translation symmetry.Here, we address the intriguing question whether quasi-periodically driven (QPD) quantum systems can show features similar to time-crystalline behaviour of periodically driven ones? Normally it is expected -apart for special integrable systems with momentum conservation [27] -that QPD systems generically heat up to featureless infinite-temperature states [28,29]. Hence, a related important question we address is whether there are interacting QPD quantum systems with a stable nontrivial steady state? Of course, generally such questions are hard to answer because Floquet theory, which provides an elegant framework for the analysis of PD quantum systems normally does not generalise to aperiodic time-dependencies. Here, we address the above questions by studying a class of QPD systems which can be efficiently treated within Floquet theory. This allows us to find QPD steady states with time-crystalline behaviour at special fine-tuned points whose stability to generic perturbations is studied with exact diagonalisation.The main idea is that special cases of originally QPD Hamiltonian H can be mapped exactly to a periodic(a) (b) Δθ 2Δθ FIG. 1. (a) Mapping a quasi-periodic magnetic spiral to a periodic ferromagnet. (b)Illustration of the tw...

Long-Lived Interacting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems

“…[64], as well as that of Ref. [66], and we address the stability of such patterns when using smooth driving.…”

“…Interesting dynamical behavior related to topology in fewbody or noninteracting scenarios have also been reported [59][60][61][62][63]. Many-body quasiperiodically driven quantum systems with interactions have received increasing attention recently [64][65][66]. Our approach is distinguished from previous studies in that we explicitly establish the stability of the nonequilibrium phases we discuss, by rigorously providing a bound on their lifetimes.…”

Fine structure of heating in a quasiperiodically driven critical quantum system

“…The latter finding demonstrates that even a heating regime can support nontrivial emergent structures as a system is driven towards the infinite-temperature fixed point. Our system breaks translation symmetry in space via a smooth deformation of hopping parameters, rather than short-range correlated disorder, and in time due to a quasiperiodic drive, which has also been in the focus of several other recent works that study (the absence of) heating [12][13][14][15][16].…”

Time crystals in a shaken atom-cavity system

“…We use experimentally relevant parameters for N a = 60 × 10 3 , ω rec = 2π ×3.55 kHz, κ = 2π ×4.50 kHz, and ∆ 0 = −2π ×0. 36 Hz based on the set up in Ref. [49].…”

“…To this end, we study the dynamics of atoms inside an optical resonator when the standing wave potential produced by a retroreflected pump beam in the transverse direction is periodically shaken. Incommensurate time crystals have dominant subharmonic response at a noninteger multiple of the drive, meaning Ô (t) = Ô (t + mT ) with a noninteger m > 1 [32][33][34][35][36][37][38][39]. In Fig.…”