The connection between the critical properties of the Dicke model, subject to equilibrium and nonequilibrium boundary conditions, is explicitly exhibited. The Langevin equations generated by the Dicke Hamiltonian lead to a single nonlinear order-parameter equation that characterizes the stationary states of the system. We prove the existence of two essentially identical manifolds of stationary states, one for equilibrium, the other for nonequilibrium boundary conditions. These manifolds can each be derived from a potential obtained from the nonlinear order-parameter equations. In the equilibrium case the potential is the free energy of the system; in the nonequilibrium case it can be identified with the so-called "laser potential." The reduced-density operator for the field and atomic subsystems factors into a geometric and a physical part. The geometric part is determined by the system signal; the physical part, by the system noise. This is exhibited explicitly with an example taken from the theory of photoelectron counting. The identification of the stationary-state manifolds for the Dicke model subject to equilibrium and nonequilibrium conditions with the same geometric (cusp) manifold allows a real analytic continuation of model properties from the equilibrium configuration to the nonequilibrium dissipative regime. Three types of stability are considered: static, dynamical, and structural. Structural-stability considerations lead to strong conclusions about the effects of additional perturbations on the Dicke model.
The connection between the critical properties of the Dicke model, subject to equilibrium and nonequilibrium boundary conditions, is explicitly exhibited. The Langevin equations generated by the Dicke Hamiltonian lead to a single nonlinear order-parameter equation that characterizes the stationary states of the system. We prove the existence of two essentially identical manifolds of stationary states, one for equilibrium, the other for nonequilibrium boundary conditions. These manifolds can each be derived from a potential obtained from the nonlinear order-parameter equations. In the equilibrium case the potential is the free energy of the system; in the nonequilibrium case it can be identified with the so-called "laser potential." The reduced-density operator for the field and atomic subsystems factors into a geometric and a physical part. The geometric part is determined by the system signal; the physical part, by the system noise. This is exhibited explicitly with an example taken from the theory of photoelectron counting. The identification of the stationary-state manifolds for the Dicke model subject to equilibrium and nonequilibrium conditions with the same geometric (cusp) manifold allows a real analytic continuation of model properties from the equilibrium configuration to the nonequilibrium dissipative regime. Three types of stability are considered: static, dynamical, and structural. Structural-stability considerations lead to strong conclusions about the effects of additional perturbations on the Dicke model.
The Dicke Hamiltonian is linearized by expanding the field and atomic shift operators about disposable c- number parameters. When these parameters are chosen to be the expectation values of the corresponding operators with respect to the linearized Hamiltonian, the difference between the free energy of the original Hamiltonian and the free energy of the linearized Hamiltonian is minimized. This difference (per particle) vanishes in the thermodynamic limit; so the equilibrium statistical mechanics of the system is described by the linearized Hamiltonian. The disposable parameters are order parameters of the linearized Hamiltonian. The field and atom order parameters obey a system of coupled nonlinear equations characteristic of meanfield theories. These equations determine the critical temperature and the ordered-state behavior. The thermodynamic state of each subsystem is a statistical superposition of thermal noise and a coherent state produced by the complementary subsystem acting as a classical source. The classical driving terms are the order parameters. The order parameter and coherent-state parameter are equal for the field subsystem but not for the atomic subsystem. The ordered state is characterized by an enhanced condensation of the atomic subsystem into the state of maximum cooperation number r. This enhancement is attributed entirely to the Starksplitting of the atomic energy levels due to the classical driving field. A different but equivalent system of coupled nonlinear order-parameter equations is derived from the order-parameter equations of motion in thermodynamic equilibrium in the thermodynamic limit. This alternative form of the coupled self-consistent equations leads to a very simple method for locating the critical temperature and determining the ordered-state behavior of interacting systems. This method is illustrated by determining the gap equations for three model Hamiltonians. These describe (a) interaction of spin- j systems with a single mode of the radiation field, (b) interaction of two-level atoms with a finite number of modes of the radiation field, and (c) interaction of two-level atoms with one or two modes of the radiation field through double photon absorption and emission processes.
Abstract The Dicke model describes the coupling between a quantized cavity field and a large ensemble of two‐level atoms. When the number of atoms tends to infinity, this model can undergo a transition to a superradiant phase, belonging to the mean‐field Ising universality class. The superradiant transition was first predicted for atoms in thermal equilibrium and was recently realized with a quantum simulator made of atoms in an optical cavity, subject to both dissipation and driving. This progress report offers an introduction to some theoretical concepts relevant to the Dicke model, reviewing the critical properties of the superradiant phase transition and the distinction between equilibrium and nonequilibrium conditions. In addition, it explains the fundamental difference between the superradiant phase transition and the more common lasing transition. This report mostly focuses on the steady states of atoms in single‐mode optical cavities, but it also mentions some aspects of real‐time dynamics, as well as other quantum simulators, including superconducting qubits, trapped ions, and using spin–orbit coupling for cold atoms. These realizations differ in regard to whether they describe equilibrium or nonequilibrium systems.
We investigate the critical dynamics of the n-component relaxational models C and D, which incorporate the coupling of a nonconserved and conserved order parameter S, respectively, to the conserved energy density rho, under nonequilibrium conditions by means of the dynamical renormalization group. Detailed balance violations can be implemented isotropically by allowing for different effective temperatures for the heat baths coupling to the slow modes. In the case of model D with conserved order parameter, the energy density fluctuations can be integrated out, leaving no trace of the nonequilibrium perturbations in the asymptotic regime. For model C with scalar order parameter, in equilibrium governed by strong dynamic scaling (z(S)=z(rho)), we find no genuine nonequilibrium fixed point either. The nonequilibrium critical dynamics of model C with n=1 thus follows the behavior of other systems with nonconserved order parameter wherein detailed balance becomes effectively restored at the phase transition. For n> or =4, the energy density generally decouples from the order parameter. However, for n=2 and n=3, in the weak dynamic scaling regime (z(S)< or =z(rho)) entire lines of genuine nonequilibrium model C fixed points emerge to one-loop order, which are characterized by continuously varying static and dynamic critical exponents. Similarly, the nonequilibrium model C with spatially anisotropic noise and n<4 allows for continuously varying exponents, yet with strong dynamic scaling. Subjecting model D to anisotropic nonequilibrium perturbations leads to genuinely different critical behavior with softening only in subsectors of momentum space and correspondingly anisotropic scaling exponents. Similar to the two-temperature model B (randomly driven diffusive systems) the effective theory at criticality can be cast into an equilibrium model D dynamics, albeit incorporating long-range interactions of the uniaxial dipolar or ferroelastic type.
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.
The authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.
In this thesis, we study the nonequilibrium dynamics of the time dependent single mode Dicke model in the thermodynamic limit in the context of the Liouville-von Neumann canonical method. We report an exact solution for this model under a non-equilibrium situation, generated by time dependent control parameters. We compute the correlations: in the atom-field ground state by means of the density operator for the pure state of the universe, and the corresponding reduced density operator for the atoms when the field is in resonance with the atomic ensemble. We obtain a closed relationship between the correlations in the ground state, the localisation of the atom¿s Wigner function in phase space, and the localisation of the wave function of the universe. As a measure of bipartite entanglement, we calculate the linear entropy from which we study the atom-field entanglement dynamics. In particular, we find a strong relationship between the stability of the auxiliary dynamical parameters, the reported entanglement, and the dynamics of certain physical observables. Furthermore, we use the Liouville-von Neumann technique to study the influence of quantum and thermal fluctuations on the dynamics of the normal and the superradiant phase. In addition, we calculate the density matrix for the quantum mechanical evolution of an initial thermal equilibrium state in the case of an instantaneous atom-field coupling and time independent ω and ω0.