Dissipative Dicke Time Quasicrystals

TL;DR

This work investigates time quasicrystals in the open Dicke model subjected to a Fibonacci quasi-periodic drive. Using mean-field dynamics in the thermodynamic limit, the authors identify a robust time-quasicrystal (TQC) phase via the quasi-crystalline fraction $f(\epsilon)$ and the decorrelator $\langle d \rangle$, and they show that long-lived TQC dynamics persist even for as few as two qubits, with the lifetime $\tau$ growing linearly with system size $N$. These results indicate that dissipative, quasi-periodically driven quantum systems can realize stable non-equilibrium phases and provide a platform for exploring new non-equilibrium behavior in open quantum matter. The study combines semi-classical analysis and exact diagonalization to map out the TQC regime under Fibonacci driving and highlights the role of dissipation in stabilizing time-quasi-crystalline order.

Abstract

We investigate the emergence of time quasicrystals (TQCs) in the open Dicke model, subjected to a quasi-periodic Fibonacci drive. TQCs are characterized by a robust sub-harmonic quasi-periodic response that is qualitatively distinct from the external drive. By directly analyzing the dynamics of the system in the thermodynamic limit, we establish the existence of TQC order in this system for a wide parameter regime. Remarkably, we demonstrate that this behavior persists even in the deep quantum regime with only two qubits. We systematically study the dependence of the TQC lifetime, $τ^{\ast}$, on the number of qubits and demonstrate that $τ^{\ast}$ increases monotonically with the system size. Our work demonstrates that quasi-periodically driven dissipative quantum systems can serve as a powerful platform for realizing novel non-equilibrium phases of matter.

Figures (4)

• Figure 1: Time quasicrystals in the open Dicke model: (a) A schematic representation of the open Dicke model dsecribing the dissipative dynamics of $N$ two-level atoms coupled to a cavity. (b) The Fibonacci drive protocol that is employed in this work. (c) The perfect Time quasi-crystal response at zero detuning ($\epsilon=0$) for $j_{\alpha} = \langle J_{\alpha}\rangle/N$ (d) The black dotted line corresponds to the Fourier transform(FT) of the system's response ($j_{x}$), and the blue solid line represents the FT of the drive. The FT of the response is shifted significantly from the drive, thereby demonstrating the existence of a TQC. (e) The phase-space trajectories projected on the pseudo-spin Bloch sphere for $\epsilon=0$. The TQC phase is associated with regular trajectories and consequently the system remains localized in two regions.
• Figure 2: Robustness of the TQC: (a) The normalized quasi-crystalline fraction $f / f_{\max}$ and the normalized time-averaged decorrelator $\langle d \rangle/\langle d \rangle_{\max}$ are employed to distinguish the TQC regime from the thermal regime. The blue solid circles represent the normalized quasi-crystalline fraction $f / f_{\max}$, while the black star-shaped markers represent the normalized time-averaged decorrelator $\langle d \rangle/ \langle d \rangle_{\max}$. The TQC phase is robust and it persists over a finite parameter regime. (b–c) illustrate the system’s dynamical response for representative values of $\epsilon$ in the TQC and the thermal phase, and (d–e) show the corresponding phase-space trajectories on the Bloch sphere. The TQC (thermal) phase is associated with regular (irregular) trajectories and consequent localization (delocalization) on the Bloch sphere.
• Figure 3: Time quasi-crystals in the deep quantum regime: The time-evolution of the two-qubit Dicke model for (a) $\epsilon=0$ , (b) $\epsilon=0.02$ and (c) $\epsilon=0.1$. Long-lived time quasi-crystal dynamics is observed both for (a) and (b).
• Figure 4: Linear growth of TQC lifetime with system size, $N$: For $\epsilon = 0$, the lifetime increases rapidly with system size. The inset shows the corresponding behavior for $\epsilon = 0.02$, where the lifetime also grows with system size but at a slower rate compared to the $\epsilon = 0$ case. These results indicate that a stable TQC phases emerges in this system in the thermodynamic limit.