Metastability in the Dissipative Quantum Rabi Model

TL;DR

The paper addresses how weak spin relaxation affects the dissipative phase transition in the quantum Rabi model, showing that the superradiant phase becomes metastable due to strong perturbations from spin quantum jumps. It employs mean-field theory, higher-order cumulant expansions, and exact numerical methods—including Husimi $Q$-representation, quantum trajectories, and Liouvillian spectral analysis—to demonstrate finite lifetimes of symmetry-breaking states and to characterize the metastable manifold. A finite-size scaling analysis confirms that the Liouvillian gap remains finite in the thermodynamic limit, establishing metastability of the parity-breaking states when $\gamma>0$. The work highlights fundamental differences between equilibrium and non-equilibrium phase transitions and provides a framework applicable to open quantum systems, with potential experimental verification in trapped ions, NMR, and superconducting circuits.

Abstract

The dissipative quantum Rabi model exhibits rich non-equilibrium physics, including a dissipative phase transition from the normal phase to the superradiant phase. In this work, we investigate the stability of the superradiant phase in the presence of a weak spin relaxation. We find that even a weak spin relaxation can render the superradiant phase to a superradiant metastable phase, in which symmetry-breaking states are stable only for a finite time. This arises because each spin-jump induced by relaxation applies as a strong perturbation to the system, potentially driving the system from a symmetry-breaking state to the symmetry-preserving saddle point with finite probability, before it eventually relaxes back to a symmetry-breaking state. Such dynamical processes lead to a finite lifetime of the symmetry-breaking states and restore the symmetry in the steady state. To substantiate these results, we combine mean-field and cumulant expansion analyses with exact numerical simulations. The lifetime of the symmetry-breaking states are analyzed in finite-size systems, and the conclusions are extrapolated to the thermodynamic limit via finite-size scaling. Our findings establish the metastable nature of the symmetry-breaking states in the dissipative quantum Rabi model and reveal the complexity of the dissipative phase transition beyond their equilibrium counterpart. The mechanisms uncovered here can be generalized to a broad class of open quantum systems, highlighting fundamental distinctions between equilibrium phase transitions and steady-state phase transitions.

Figures (8)

• Figure 1: (a) Schematic diagram of the dissipative quantum Rabi model under a weak spin relaxation and (b) its corresponding phase diagram. Here, NP denotes the normal phase; SP denotes the superradiant phase; and SMP denotes the superradiant metastable phase. When $\gamma=0$, the system undergoes a phase transition from the NP to the SP at the critical point $\lambda_c$. When $\gamma\neq0$, this critical point extends into a critical line, and the SP transforms into the SMP. The inset shows the typical Husimi $Q$-representation of the bosonic mode in different phases.
• Figure 2: High order cumulants $|\langle \hat{x}^n\rangle_c|$ obtained from solving the steady-state equations of cumulants truncated at $6$-th order. Odd order cumulants vanish in the NP as $\langle \hat{x}^n\rangle_c=0$ holds due to the parity symmetry. Parameters are $\omega_0=1.0$, $\kappa/\omega_0=0.5$, $\gamma/\omega_0=0.05$, $\Omega/\omega_0=1200$. For NP case, $\lambda=0.6\sqrt{{\omega_0 \Omega}/{2}}$, and for SMP case, $\lambda=1.4\sqrt{{\omega_0 \Omega}/{2}}$.
• Figure 3: The evolution of the steady-state solution in the NP (a) and SMP (b) under strong perturbations within the mean-field framework. Phenomenological interpretations are shown in the top panels. For the NP, we consider a quench $\langle \hat{x}\rangle=r\cos 2\theta$, $\langle \hat{p}\rangle= r \sin 2\theta$ of the parity-preserving solution. For the SMP, we consider a quench $\langle \hat{\sigma}_x\rangle=\cos \theta \langle \hat{\sigma}_x \rangle_{s}+\sin \theta \langle \hat{\sigma}_z \rangle_{s}$ and $\langle \hat{\sigma}_z\rangle=\cos \theta \langle \hat{\sigma}_z \rangle_{s}-\sin \theta \langle \hat{\sigma}_x \rangle_{s}$ of one parity-breaking solution. We set $r=10$ and $\theta=\frac{\pi}{7},\frac{2\pi}{7}, \cdots, \frac{6\pi}{7}$. Other parameters are same as those in Fig. \ref{['fig:cumulant']}.
• Figure 4: Husimi $Q$-representation of the bosonic mode in the NP (a) and SMP (b). Panel (c) shows a magnified view of (b), where the low-intensity regions are emphasized by rescaling the color map. Here $x\equiv \sqrt{2} \rm{Re}[\alpha]$ and $p\equiv \sqrt{2} \rm{Im}[\alpha]$. Parameters are same as those in Fig. \ref{['fig:cumulant']}.
• Figure 5: Principal components of the steady state in the SMP. Panels (a) and (b) reveal two parity-breaking components, while (c) and (d) show a spiral and a sigmoid components, respectively. Parameters are the same as those in Fig. \ref{['fig:cumulant']}.