Discrete Time-Crystalline Order in Cavity and Circuit QED Systems

Discrete time crystals are a recently proposed and experimentally observed out-of-equilibrium dynamical phase of Floquet systems, where the stroboscopic dynamics 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.

Figure 1

Cavity and circuit QED setups for realizing the DTC order. In the first (second) half of a Floquet period $T$, we switch on (off) the coupling $\lambda$ between photons and (artificial) atoms. For sufficiently large $\lambda$, almost persistent DTC order in the stroboscopic dynamics of a local observable is expected for an ensemble of a large number of atoms in an optical cavity, while transient DTC behavior can be observed for few superconducting qubits coupled to a microwave transmission line. Here $\kappa$ denotes the loss rate of (microwave) photons.

Figure 2

Dynamical phase diagram (top), typical stroboscopic dynamics (middle), and stroboscopic trajectories (bottom) of the atomic pseudospin for atom-photon coupling $\lambda =1$ and photon-loss rate $\kappa =0.05$. Top: As the detuning $\epsilon$ [see Eq. (3)] is varied, five different dynamical phases emerge: thermal ($T$, red), symmetric period doubling (normal DTC order, SD, blue), limit-cycle pair (LC, orange), period sextupling ($S$, purple), and asymmetric period doubling (AD, magenta). The phase boundaries are marked in white with resolution ${10}^{-3}$. Middle: Typical stroboscopic dynamics of $j_{\mu }\equiv ⟨{\widehat{J}}_{\mu }⟩/N$ [$\mu =x$ (solid blue), $y$ (dashed orange), $z$ (dotted black)] for the last 30 periods of the entire 5000-period evolution. Bottom: Full stroboscopic phase-space-point trajectories (light blue) and those of the last 200 periods (purple) projected on the pseudospin Bloch sphere.

Figure 3

(a) Stroboscopic dissipative dynamics of the scaled angular momenta of $j_x$ (solid), $j_y$ (dashed), and $j_z$ (dotted) in the two-qubit Dicke model with $\kappa =0.05$, $\epsilon =0.1$, and $\lambda =1$ (strong coupling). The inset shows quadratures, $x$ (dotted) and $p$ (solid). (b) Stroboscopic dynamics for isolated systems ($\kappa =0$, $\epsilon =0.1$, and $\lambda =1$) in the strong-coupling regime. (c) Stroboscopic dissipative dynamics in the weak-coupling regime ($\kappa =0.05$, $\epsilon =0.1$, and $\lambda =0.1$). (d) Stroboscopic unitary dynamics in the weak-coupling regime ($\kappa =0$, $\epsilon =0.1$, and $\lambda =0.1$). Only (a) shows a DTC transient. The initial state is always $|\Rightarrow ⟩\otimes |0⟩$, where $|\Rightarrow ⟩$ is the eigenstate of ${\widehat{J}}_x$ with eigenvalue $N/2(N=2)$ and $|0⟩$ is the photon vacuum.

Figure 4

(a) Typical Floquet-Lindblad spectrum of an open-system DTC. The DTC mode and the steady state (SS) locate at $-1+\delta$ and 1, respectively, with $\delta \sim O(e^{-cN})$. The other modes locate on a disk (shaded) with radius $r<1$ for $\forall N$, so their lifetime is bounded by a constant $-T/\ln r$. (b) Finite-size scaling for the lifetime $\tau =-T/\ln (1-\delta )$ of the DTC and the second longest-lived (SLL) modes in the Floquet-Lindblad-Landau model (5) for $\epsilon =0.02$ and $-0.05$.