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Physical Review A

Coupled order-parameter treatment of the Dicke Hamiltonian

Robert Gilmore, C. M. Bowden

Published May 1, 1976• 41 citations
Publisher

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.

DOI
10.1103/physreva.13.1898

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The Dicke Hamiltonian is linearized by expanding the field and atomic shift operators about disposable ccc- 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 rrr. 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- jjj 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.
Two-level atoms interacting with a one-mode cavity field at zero temperature have order parameters which reflect the presence of a quantum phase transition at a critical value of the atom–cavity coupling strength. Two popular examples are the number of photons inside the cavity and the number of excited atoms. Coherent states provide a mean field description, which becomes exact in the thermodynamic limit. Employing symmetry-adapted (SA) SU (2) coherent states the quantum crossover, precursor of the critical behavior, can be described for a finite number of atoms. A variation after projection treatment, involving a numerical minimization of the SA energy surface, associates the quantum crossover with a discontinuity in the order parameters, which originates from competition between two local minima in the SA energy surface. Although this discontinuity is not present in finite systems, it provides a good description of 1/ N effects in the 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.
A system of N two-level atoms interacting with a resonant single-mode quantum field (the Dicke model) is described using the operator method for solving the Schrödinger equation. The spectrum of states is calculated without recourse to the rotating-wave approximation and the assumption regarding smallness of the linear sizes of the system as compared to the radiation wavelength. Analytical approximate expressions are obtained for the energy spectrum. These expressions approximate the energy spectrum over the entire range of Hamiltonian parameters in the normal and collective states of the system and make it possible to calculate the thermodynamic characteristics.
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 model describes an ensemble of N identical two-level atoms (qubits) coupled to a single mode of a bosonic field. The fermion Dicke model should be obtained by changing the atomic pseudo-spin operators by a linear combination of Fermi operators. The generalized fermion Dicke model is defined introducing different coupling constants between the single mode of the bosonic field and the reservoir. In the thermodynamic limit, the fermion Dicke model can be analized using the path integral approach with functional method. The system exhibits a second order phase transition from normal to superrandiance at some critical temperature with the presence of a condensate. We evaluate the critical transition temperature and present the spectrum of the collective bosonic excitations. There is quantum phantum critical behavior when the coupling constants satisfy an especific condition. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations in the case using the rotating-wave approximation, recovering the well know results. Second, the case only considering virtual processes. In the last case, it is possible to have a superradiance phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at some critical coupling, and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.
A system of N two-level atoms, Tavis-Cummings Dicke (TC-Dicke) model, interacting with a one-mode electromagnetic radiation field in the presence of the Stark shifts is studied, which is expected to predict new phenomena that are not explored in the original TC-Dicke model. We obtained the potential energy surface of the system using a trial state the direct product of coherent states in each subspace. In the frame of mean-field approaches, the variational energy is evaluated as the expectation value of the Hamiltonian for this state. The order of the quantum phase transitions is determined explicitly and numerically. We estimate the ground-state energy and the macroscopic excitations in the superradiant phase. Moreover, we investigated the critical properties of the TC-Dicke model in the classical spin limit and coherent state. We observed that in the thermodynamic limit, the energy surface takes a simple form a direct description of the phase transition. Moreover, it is found that when the microwave amplitude changes the new phase transition occurs with the Stark shift. The analytical solutions and numerical results, which appear in this paper are agreement with our paper which published recently in Int. J. Mod. Phys. B when we studied the same model using a different coherent state.