Discrete Time-Crystalline Order in Cavity and Circuit QED Systems

Abstract

Discrete time crystals are a recently proposed and experimentally observed out-of-equilibrium dynamical phase of Floquet systems, where the stroboscopic evolution of a local observable repeats itself at an integer multiple of the driving period. We address this issue in a driven-dissipative setup, focusing on the modulated open Dicke model, which can be implemented by cavity or circuit QED systems. In the thermodynamic limit, we employ semiclassical approaches and find rich dynamical phases on top of the discrete time-crystalline order. In a deep quantum regime with few qubits, we find clear signatures of a transient discrete time-crystalline behavior, which is absent in the isolated counterpart. We establish a phenomenology of dissipative discrete time crystals by generalizing the Landau theory of phase transitions to Floquet open systems.Introduction.-Phases and phase transitions of matter are key concepts for understanding complex many-body physics [1,2]. Recent experimental developments in various quantum simulators, such as ultracold atoms [3,4], trapped ions [5,6] and superconducting qubits [7,8], motivate us to seek for quantum many-body systems out of equilibrium [9][10][11], such as many-body localized phases [12][13][14][15][16][17] and Floquet topological phases [18][19][20][21][22][23][24][25].In recent years, much effort has been devoted to periodically driven (Floquet) quantum many-body systems that break the discrete time-translation symmetry (TTS) [26]. In contrast to the continuous TTS breaking [27][28][29] that has turned out to be impossible at thermal equilibrium [30,31], the discrete TTS breaking has been theoretically proposed [32][33][34][35][36] and experimentally demonstrated [37,38]. Phases with broken discrete TTS feature discrete time-crystalline (DTC) order characterized by periodic oscillations of physical observables with period nT , where T is the Floquet period and n = 2, 3, · · · . The DTC order is expected to be stabilized by many-body interactions against variations of driving parameters. Note that the system is assumed to be in a localized phase [33,34,36,37] or to have long-range interactions [38][39][40]. Otherwise, the DTC order only exists in a prethermalized regime [41,42] since the system will eventually be heated to a featureless infinite-temperature state due to persistent driving [43][44][45].While remarkable progresses are being made concerning the DTC phase, most studies focus on isolated systems. Indeed, as has been experimentally observed [37,38] and theoretically investigated [46], the DTC order in an open system is usually destroyed by decoherence. On the other hand, it is known that dissipation and decoherence can also serve as resources for quantum tasks such as quantum computation [47] and metrology [48]. From this perspective, it is natural to ask whether the DTC order exists and can even be stabilized in open systems [49]. Such a possibility has actually been pointed out in Ref. [41], but neither a detailed theoretical model nor a concrete experimental imp...

Topological time crystals

“…A crystalline structure in time is related to the periodic dynamics of a system. It has been shown that periodically driven quantum many-body systems can spontaneously self-reorganize their motion and start moving with a period different from the period of the driving [12][13][14] (see also [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]). This kind of spontaneous formation of a new crystalline structure in time is dubbed discrete or Floquet time crystals and has been already realized experimentally [30][31][32][33][34].…”

Discrete time-crystalline order in Bose–Hubbard model with dissipation

“…The extension of DTC to open quantum system with drive and dissipation attracts widespread research interest recently. Concrete models [12,13] shown that the DTC order can exist in open system with appropriately engineered drive and dissipation. This prediction was confirmed by the authors of [12] who shown that the modified Dicke model with the help of sufficiently strong atom-photon coupling and photon loss can exhibit time crystalline structure, and the same prediction was reported in [13] that a dissipative Rydberg model is also possible to possess time crystal order.…”

“…For DTCs in open systems, there exist some physical observableÔ with expectation value O(t), and ( ) ( ) = + O t O t nT shows sub-harmonic response to the driving field of period T. Here n is an integer and n2. The Dicke model exhibits a zero-temperature phase transition from a normal to a super-radiant phase [12,14] when the light-matter interaction increase. The parity symmetry of the Dicke model is spontaneously broken in the super-radiant phase [12,14].…”

“…The Dicke model exhibits a zero-temperature phase transition from a normal to a super-radiant phase [12,14] when the light-matter interaction increase. The parity symmetry of the Dicke model is spontaneously broken in the super-radiant phase [12,14]. Reference [12] utilizes this feature and drives the system in the superradiant phase to entail sub-harmonic dynamical responses.…”

“…The parity symmetry of the Dicke model is spontaneously broken in the super-radiant phase [12,14]. Reference [12] utilizes this feature and drives the system in the superradiant phase to entail sub-harmonic dynamical responses. The sub-harmonic response of [13] originates from the coexistence of two phases connected by first-order phase transition in dissipative Rydberg gas.…”