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Physical Review A

Phase transitions in the open Dicke model: A degenerate-perturbation-theory approach

Wenqi Tong, Hadiseh Alaeian, F. Robicheaux

Published Nov 20, 2025• 1 citation
Publisher

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 [Phys. Rev. Lett. 118, 123602 (2017)] 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 second-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.

DOI
10.1103/l55z-kjwt

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

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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.

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We study the steady-state behavior of the open Dicke model, which describes the collective interaction of NNN spin-1/21/21/2 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 [Phys. Rev. Lett. 118, 123602 (2017)] 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, SSS. For coupling strengths above the threshold, there exists a critical spin value Sc{S}_{c}Sc​ such that the superradiant phase transition occurs only for S>ScS>{S}_{c}S>Sc​. The perturbative approach captures how weak dephasing and decay induce mixing between different SSS subspaces, yielding a steady-state spin distribution whose width scales as 1/N1/\sqrt{N}1/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 second-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.
We study the steady-state behavior of the open Dicke model, which describes the collective interaction of NNN spin-1/21/21/2 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, SSS. For coupling strengths above the threshold, there exists a critical spin value ScS_cSc​ such that the superradiant phase transition occurs only for S>ScS>S_cS>Sc​. The perturbative approach captures how weak dephasing and decay induce mixing between different SSS-subspaces, yielding a steady-state spin distribution whose width scales as 1/N1/\sqrt{N}1/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.
The Dicke model describes the collective behavior of a subwavelength-size ensemble of two-level atoms (i.e., spin-1/2) interacting identically with a single quantized radiation field of a cavity. Across a critical coupling strength it exhibits a zero-temperature phase transition from the normal state to the superradiant phase where the field is populated and the collective spin acquires a nonzero x component, which can be imagined as ferromagnetic ordering of the atomic spins along x. Here we introduce a variant of this model where two subwavelength-size ensembles of spins interact with a single quantized radiation field with different strengths. Subsequently, we restrict ourselves to a special case where the coupling strengths are opposite (which is unitarily equivalent to equal-coupling strengths). Because of the conservation of the total spin in each ensemble individually, the system supports two distinct superradiant states with x-ferromagnetic and x-ferrimagnetic spin ordering, coexisting with each other in a large parameter regime. The stability and dynamics of the system in the thermodynamic limit are examined using a semiclassical approach, which predicts nonstationary behaviors due to the multistabilities. At the end, we also perform small-scale full quantum-mechanical calculations, with results consistent with the semiclassical ones.
We study a generalization of a recently introduced Dicke trimer model [Phys. Rev. Lett. 128, 163601 (2022); Phys. Rev. Res. 5, L042016 (2023)], which allows for cavity losses and unbalanced light-matter interactions (in which rotating and counter-rotating terms can be tuned independently). In the original description of a Dicke trimer, three Dicke models are coupled in a ring topology via a complex photon hopping whose complex phase describes a synthetic magnetic field threading the loop. This original model features several intriguing equilibrium phases and critical phenomena such as frustrated superradiance, two-critical scalings in the frustrated superradiant phase, and finite critical fluctuations in the anomalous normal phase. Here, we find that in the extreme unbalanced limit, where only rotating terms are present, the U(1) symmetry of the Tavis-Cummings model is restored, qualitatively altering the critical phenomena in the superradiant phase due to the presence of a zero-energy mode. To analyze this general regime, we develop a semiclassical theory based on a requantization technique. This theory also provides further physical insight on recently reported anomalous finite critical fluctuations in the time-reversal broken regime. Moving to the open-Dicke case, by introducing local dissipation to the cavities, we observe the emergence of a rich range of nonequilibrium phases characterized by trivial and nontrivial dynamical signatures. In the former case, we identify, when time-reversal symmetry is present, a new stationary phase that features superradiant states in two of the three cavities and a normal state in the other cavity. In the latter case, we observe the emergence of dynamical phases in which the system exhibits superradiant oscillations, characterized by periodic or chaotic phase space patterns. The landscape of transitions associated with these dynamical phases features a wide range of qualitatively different behaviors such as Hopf bifurcations (followed by period-doubling cascades or quasiperiodic oscillations), anomalous Hopf bifurcations (with burst-oscillation-like post-bifurcation dynamics), collisions between basins of attraction (associated with different symmetry-broken equilibria), and exterior crises (featuring transient chaotic dynamics). We highlight how the two-critical-scalings feature of the closed model is robust under dissipation (with doubled critical exponents) while the phenomenon of anomalous finite critical fluctuations becomes a mean-field scaling (as a consequence of Hopf bifurcations of the equilibria featuring the normal state) in the open model.
Interactions among emitters provide a powerful means of controlling collective light--matter phenomena, yet their role in superradiant criticality has not been thoroughly investigated. Here we construct a minimal model that can yield analytical insights --- a Dicke model with two interacting atoms coupled to a single mode cavity --- to study such interaction effects. We show that interaction changes the phase boundary, and may even completely suppress the atom-photon coupling threshold for superradiance and change the universality class of the phase transition. We further study the dissipative phase transition and quantum dynamics in the presence of dissipation channels such as photon loss and spin relaxation. Our results demonstrate important roles played by the atom-atom interaction and how it can be engineered to control atom-photon coupling.
We explore the steady-state critical behavior of the finite-size open Dicke model—a model that incorporates spontaneous emission decay of the collective atomic spin states and decay of the cavity field. From the perspective of quantum information theory, we can often better characterize the quantum phase transition. In this paper, we characterize the super-radiant phase transition of the steady state of the open Dicke model by numerically calculating the quantum Fisher information (QFI). We calculate the QFI for the atomic state and the cavity field state, as well as their derivatives. We find that the QFI of the cavity field state is more sensitive to atomic decay, and is suppressed more severely in the presence of atomic decay. In contrast, the QFI of the atomic state is less sensitive to the photon loss of the cavity field.
The Dicke model is a staple of theoretical cavity quantum electrodynamics (cavity QED), describing the interaction between an ensemble of atoms and a single radiation mode of an optical cavity. It is studied both quantum mechanically and semiclassically for two-level atoms, and demonstrates a rich variety of dynamics such as phase transitions, phase multistability, and chaos. In this work we explore an open, spin-1 Dicke model with independent co and counterrotating coupling terms as well as a quadratic Zeeman shift enabling control over the atomic energy-level structure. We investigate the stability of operator and moment equations under two approximations and show the system undergoes phase transitions. To compliment these results, we relax the aforementioned approximations and investigate the system semiclassically. We show evidence of phase transitions to steady-state and oscillatory superradiance in this semiclassical model, as well as the emergence of chaotic dynamics. The varied and complex behaviors admitted by the model highlights the need to more rigorously map its dynamics.