Relation between the equilibrium and nonequilibrium critical properties of the Dicke model
Robert Louis Gilmore, Lorenzo M. Narducci · Physical Review A · 1978
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.