Discrete Time-Crystalline Order in Cavity and Circuit QED Systems

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems Zongping Gong,1 Ryusuke Hamazaki,1 and Masahito Ueda1, 2

arXiv:1708.01472v2 [cond-mat.stat-mech] 9 Aug 2017

1

Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 2 RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: August 10, 2017)

Discrete time crystals are a recently proposed and experimentally observed dynamical phase of out-of-equilibrium 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 is readily implemented by cavity and circuit QED systems. In the thermodynamic limit, we employ semiclassical approaches and find unexpectedly rich dynamical phases with different types of discrete time-crystalline order, which can be well explained by bifurcation theory. In a deep quantum regime with few qubits, we find clear signatures of a transient discrete time-crystalline order, which is absent in the isolated counterpart. Our work generalizes the notion of time crystals to open systems and proposes experimental implementation of discrete time-crystalline order with cold atoms and superconducting qubits under driving and dissipation.

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], have enabled us to seek for quantum many-body systems out of equilibrium [9–11]. These artifical quantum systems exhibit novel out-of-equilibrium phases, such as manybody localized phases [12–16] and Floquet topological phases [17–21]. In recent years, much effort has been devoted to periodically driven (Floquet) quantum many-body systems that break the discrete time-translation symmetry (TTS) [22]. In contrast to the continuous TTS breaking [23–25] that has turned out to be impossible at thermal equilibrium [26, 27], the discrete TTS breaking has been theoretically [28–31] and experimentally [32, 33] demonstrated. Phases with broken discrete TTS feature discrete timecrystalline (DTC) order characterized by 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 detuning in driving parameters. Note that the system is assumed to be in a localized phase [29–32] or to have long-range interactions [33–35]. Otherwise, the DTC order only exists in a prethermalized regime [36, 37] since the system will eventually be heated to a featureless state due to persistent driving [38–40]. While remarkable progresses are being made concerning the DTC phase, most studies focus on isolated systems. Indeed, as has been experimentally observed [32, 33] and theoretically addressed [41], 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 [42] and metrology [43]. From this perspective, it is natural to ask whether the DTC order exists and can even be stabilized in open sys-

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