Dissipative Dicke Time Quasicrystals

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.

💡 Research Summary

In this work the authors explore a novel non‑equilibrium phase— a time quasicrystal (TQC)—in the open Dicke model when the light‑matter coupling λ is modulated according to a Fibonacci quasi‑periodic sequence. The open Dicke Hamiltonian describes N identical two‑level atoms collectively coupled to a single cavity mode, with photon loss at rate κ. When λ exceeds the critical value λc the system undergoes a Z₂ symmetry‑breaking transition. The drive protocol alternates λ between zero and λ·(1+r_n)/2, where r_n is the n‑th element of a binary Fibonacci sequence generated from the golden ratio. This creates a deterministic, aperiodic drive that is fundamentally different from the usual Floquet (periodic) protocols.

First, the authors take the thermodynamic limit (N→∞) and derive semiclassical mean‑field equations for the normalized collective spin components (j_x, j_y, j_z) and the cavity quadratures (x, p). These coupled nonlinear differential equations are integrated numerically with a fourth‑order Runge‑Kutta scheme. To identify a TQC they introduce two diagnostics: (i) the quasi‑crystalline fraction f(ε), defined as the relative weight of the sub‑harmonic peak at frequency ν₀ (the response for ε=0) in the Fourier spectrum, and (ii) the decorrelator ⟨d⟩, which measures the sensitivity of the trajectory to a tiny perturbation of the initial condition. A robust TQC regime is characterized by a large f and a small ⟨d⟩, whereas chaotic or thermal dynamics show the opposite behavior. Scanning the drive asymmetry ε, the loss rate κ and the coupling λ, the authors map out a sizable region of parameter space where f remains high and ⟨d⟩ stays low, establishing the existence of a stable TQC in the semiclassical limit.

Next, the study moves to the deep quantum regime with only a few qubits (N=2–6). Using exact diagonalization, the system is initialized in a product state with all spins polarized along +x and the cavity in vacuum. Time evolution of the normalized collective spin magnitude |j_x(t)| reveals long‑lived quasi‑periodic oscillations for ε=0 (perfect Fibonacci drive) and for a modest detuning ε=0.02, while a larger detuning ε=0.1 quickly destroys the oscillations. This demonstrates that the TQC is not an artefact of the mean‑field approximation but survives genuine quantum fluctuations even for the minimal case of two qubits.

A central quantitative result concerns the lifetime τ* of the TQC. By fitting the decay of |j_x| to an exponential form |j_x|≈A e^{−t/τ}, the authors extract τ for different system sizes. They find a linear scaling τ(N)≈α N+β, with a positive slope α, both for ε=0 and for small ε≠0 (the latter with a reduced slope). Consequently, τ grows without bound as N→∞, implying that the TQC becomes truly stable in the thermodynamic limit. This scaling is a direct consequence of the interplay between dissipation (which removes excess energy) and the quasi‑periodic drive (which prevents heating to an infinite‑temperature state).

The paper concludes by emphasizing several implications. First, it shows that quasi‑periodic driving can stabilize non‑thermal phases in open quantum systems, extending the concept of pre‑thermal time crystals beyond closed, high‑frequency Floquet settings. Second, the open Dicke model is experimentally relevant to cavity QED and circuit‑QED platforms, suggesting that the predicted TQC could be observed with current technology. Third, the linear increase of τ with N provides a clear route to arbitrarily long coherence times by scaling up the number of atoms or qubits. Finally, the authors outline future directions: incorporating non‑Markovian baths, adding short‑range atom‑atom interactions, exploring other aperiodic sequences (Thue‑Morse, Sturmian), and investigating the relationship between TQC order and quantum entanglement or entropy growth.

Overall, the study delivers a comprehensive theoretical demonstration that a dissipative, quasi‑periodically driven Dicke system hosts a robust time quasicrystal, bridges the gap between semiclassical and fully quantum regimes, and opens a promising pathway for experimental realization of exotic non‑equilibrium phases.