Phase Transitions in Open Dicke Model: a degenerate perturbation theory approach

TL;DR

The paper addresses how open-system Dicke physics and phase transitions are affected by degenerate steady states. It introduces degenerate perturbation theory across total-spin subspaces to model the effects of homogeneous local dephasing and local decay, showing that a critical total spin $S_c$ governs the presence of the superradiant phase for a given coupling, and that the steady state becomes a distribution $p(S)$ over subspaces with width $\sigma \sim 1/\sqrt{N}$. The key finding is that infinitesimal dephasing destroys the transition by concentrating weight below $S_c$, whereas infinitesimal decay can restore it by shifting weight above $S_c$, with the coupling matrix providing a low-order, moment-based route to predict these effects. The work connects quantum Rabi-like behavior with many-body Dicke physics in open systems and provides a broadly applicable framework for degenerate steady-state dynamics using only the first two moments, enabling analysis of very large $N$ and offering insights into Liouvillian spectra and transient dynamics.

Abstract

We study the steady-state behavior of the open Dicke model, which describes the collective interaction of $N$ spin-$1/2$ particles with a lossy, quantized cavity mode and exhibits a superradiant phase transition above a critical light-matter coupling. While the standard model conserves total spin, Kirton and Keeling \cite{PhysRevLett.118.123602} demonstrated that even infinitesimal homogeneous local dephasing destroys this phase transition, and that local atomic decay can restore it. We analyze this interplay using degenerate perturbation theory across subspaces of fixed total spin, $S$. For coupling strengths above the threshold, there exists a critical spin value $S_c$ such that the superradiant phase transition occurs only for $S>S_c$. The perturbative approach captures how weak dephasing and decay induce mixing between different $S$-subspaces, yielding a steady-state spin distribution whose width scales as $1/\sqrt{N}$. This framework requires only the first and second moments and can be implemented via different methods that can yield these two moments (for example, the 2nd-cumulant approach), circumventing the need for full density matrix calculations. These results bridge the quantum Rabi model and Dicke physics, elucidate the roles of dephasing and decay in collective quantum effects, and apply broadly to open quantum systems with degenerate steady states.

Figures (15)

• Figure 1: The cluster of $3$ eigenvalues with the least negative real parts under the dephasing perturbation, leading to the long-term behavior under investigation. The atom number $N = 4$, with $g = 0.9$, $\omega_c = 1$, $\omega_0 = 0.5$, and $\kappa = 1$. The perturbation strength $\Gamma_\phi$ and the eigenvalues $\lambda$ are scaled to $\kappa$. The results from DPT-DM agree well with the exact diagonalization of the Liouvillian matrix.
• Figure 2: The steady-state probability distribution $p(S)$ under the combined perturbation parameterized by $f = 0, 0.2, 0.4, 0.6, 0.8, 1$ in Eq. (\ref{['eq:overall_pert']}) obtained from DPT-DM for $N = 40$. The parameters are $g = 0.9$, $\omega_c = 1$, $\omega_0 = 0.5$, and $\kappa = 1$. The superradiant phase transition happens for $\Tilde{S} > 0.3858$ (the region to the right of the vertical black solid line).
• Figure 3: Expectation value of the normalized total spin as a function of $f$. Obtained from DPT-DM for $N = 40$ atoms, the parameters are the same as Fig. \ref{['fig:prob_dist_f']}.
• Figure 4: Scaled probability distribution for pure local dephasing ($f = 1$, the peaks on the left) and pure local decay ($f = 0$, the peaks on the right) obtained from DPT-MF2 with $N = 50, 100, 200, 1000$. The parameters are the same as Fig. \ref{['fig:prob_dist_f']}. The black solid line indicates $\Tilde{S}_c$ and the blue solid line refers to $\langle \Tilde{S} \rangle = 0.7891$ evaluated from MF1 with $f = 0$.
• Figure 5: Relation between the logarithm of $\sigma$ and $N$ for the curves with $f = 0$ in Fig. \ref{['fig:pop_dist_scale_N']}. The linear relation, indicated by the red straight line from fitting the symbols, infers the power law between them. This relation holds for different ranges of N: $100 \sim 1000$ with increment of $100$, $1000 \sim 10000$ with step of $1000$ and a single point at $N = 100000$.