Coupled order-parameter treatment of the Dicke Hamiltonian

Abstract

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.