We present a density matrix based time dependent projection operator formalism to calculate the beyond mean-field dynamics of systems with non-Markovian local baths and one-to-all interactions. Such models encapsulate the physics of condensed phase systems immersed in optical cavities. We use this method, combined with tensor network influence functionals, to study the dynamics of the Dicke model coupled to non-Markovian local dephasing baths at zero temperature, which has a superradiant phase transition in the mean-field limit. The method corrects a spurious initial state dependence found in the mean-field dynamics and describes the emergence of new time scales which are absent in the mean-field dynamics. Our formalism, based on density matrices, is applicable to other quantum optical systems with one-to-all interactions at finite temperatures.
We present a density matrix based time dependent projection operator formalism to calculate the beyond mean-field dynamics of systems with non-Markovian local baths and one-to-all interactions. Such models encapsulate the physics of condensed phase systems immersed in optical cavities. We use this method, combined with tensor network influence functionals, to study the dynamics of the Dicke model coupled to non-Markovian local dephasing baths at zero temperature, which has a superradiant phase transition in the mean-field limit. The method corrects a spurious initial state dependence found in the mean-field dynamics and describes the emergence of new time scales which are absent in the mean-field dynamics. Our formalism, based on density matrices, is applicable to other quantum optical systems with one-to-all interactions at finite temperatures.
We investigate nonequilibrium phase transitions for driven atomic ensembles interacting with a cavity mode and coupled to a Markovian dissipative bath. In the thermodynamic limit and at low frequencies, we show that the distribution function of the photonic mode is thermal, with an effective temperature set by the atom-photon interaction strength. This behavior characterizes the static and dynamic critical exponents of the associated superradiance transition. Motivated by these considerations, we develop a general Keldysh path-integral approach that allows us to study physically relevant nonlinearities beyond the idealized Dicke model. Using standard diagrammatic techniques, we take into account the leading-order corrections due to the finite number N of atoms. For finite N, the photon mode behaves as a damped classical nonlinear oscillator at finite temperature. For the atoms, we propose a Dicke action that can be solved for any N and correctly captures the atoms' depolarization due to dissipative dephasing.
The Dicke model famously exhibits a phase transition to a superradiant phase with a macroscopic population of photons and is realized in multiple settings in open quantum systems. In this paper, we study a variant of the Dicke model where the cavity mode is lossy due to the coupling to a Markovian environment while the atomic mode is coupled to a colored bath. We analytically investigate this model by inspecting its low-frequency behavior via the Schwinger-Keldysh field theory and carefully examine the nature of the corresponding superradiant phase transition. Integrating out the fast modes, we can identify a simple effective theory allowing us to derive analytical expressions for various critical exponents including the dynamical exponent. We find excellent agreement with previous numerical results when the non-Markovian bath is at zero temperature; however, contrary to these studies, our low-frequency approach reveals that the same exponents govern the critical behavior when the colored bath is at finite temperature unless the chemical potential is zero. Furthermore, we show that the superradiant phase transition is classical in nature, while it is genuinely nonequilibrium. We derive a fractional Langevin equation and conjecture the associated fractional Fokker-Planck equation that captures the system's long-time memory as well as its nonequilibrium behavior. Finally, we consider finite-size effects at the phase transition and identify the finite-size scaling exponents, unlocking a rich behavior in both statics and dynamics of the photonic and atomic observables.
We consider the Dicke model, describing an ensemble of N quantum spins interacting with a cavity field, and study how the coupling to a non-Markovian environment with power-law spectrum changes the physics of superradiant phase transition. Quite remarkably we find that dissipation can induce, rather than suppress, the ordered phase, a result which is in striking contrast with both thermal and Markovian quantum baths. We interpret this dissipation-induced superradiance as a genuine dissipative quantum phase transition that exists even at finite N due to the coupling with the bath modes and whose nature and critical properties strongly depend on the spectral features of the non-Markovian environment.
The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.
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 Sc such that the superradiant phase transition occurs only for S>Sc. 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/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.
The interplay between many-body interactions and controlled dissipation provides a rich setting for nonequilibrium quantum phases. We explore the superradiant phase transitions in a Rydberg-interacting open Dicke model with cavity loss and local dissipation. We find that the nonlinear feedback from atomic interactions stabilizes the atomic collective excited state and induces resonant collective motion, which can be further converted into robust superradiant oscillations with the inclusion of local dissipation. Beyond the mean-field approximation, we prove this system can support the finite-size local dissipative temporal order, i.e., continuous time crystal, from evidences of the decreasing damping of a finite-frequency Liouvillian mode and the linear-size scaling of quantum fluctuations. The correlation between the atomic colletive dynamics and the cavity photon number makes our findings easy to detect from output optical signal. By bridging interacting atomic matter and driven-dissipative cavity QED, our Rydberg-interacting Dicke model provides a unique spin-1/2 platform for exploring dynamical phase transitions and macroscopic temporal order in open quantum matter.