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Journal of Physics A Mathematical and General

The phase transition in a modified Dicke model

B. V. Thompson

Published Jan 1, 1975• 15 citations
Publisher

Abstract

Extra terms which describe the phonon absorption and emission associated with Brillouin scattering are added to the usual Dicke Hamiltonian. The general effect on the Dicke phase transition is an enhancement of the photon numbers. When an effective phonon coupling constant B exceeds a certain value, the transition is of the first-order type. The condition epsilon / lambda 2 <1 found by Wang and Hioe (1973) to be necessary for a phase transition to occur, is no longer necessary if the constant B is large enough and in such cases the transition is again first order in type.

DOI
10.1088/0305-4470/8/1/019

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Extra terms which describe the phonon absorption and emission associated with Brillouin scattering are added to the usual Dicke Hamiltonian. The general effect on the Dicke phase transition is an enhancement of the photon numbers. When an effective phonon coupling constant B exceeds a certain value, the transition is of the first-order type. The condition epsilon / lambda 2 <1 found by Wang and Hioe (1973) to be necessary for a phase transition to occur, is no longer necessary if the constant B is large enough and in such cases the transition is again first order in type.
The Dicke model of superradiance studied in a previous paper by Wang and Hioe is generalized (i) to include an interaction energy of the form ${a}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\sigma}}^{+}+a{\ensuremath{\sigma}}^{\ensuremath{-}}$, where $a,{a}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${\ensuremath{\sigma}}^{\ensuremath{-}},{\ensuremath{\sigma}}^{+}$ are the usual photon and spin operators, (ii) to include the spatial variation of the electromagnetic field, and (iii) to include the possibility of small classical motions of atoms from their equilibrium positions. The free energies of these generalized models are calculated exactly in the thermodynamic limit, and their phase-transition properties are presented.
The effect of the antiresonant atom-field interactions on the phase-transition property of the Dicke model is studied. A phase transition is shown to exist when the following inequality is satisfied between the coupling constant $\ensuremath{\lambda}$ and the ratio of the atomic transition frequency to the field frequency $\ensuremath{\epsilon}$: ${\ensuremath{\lambda}}^{2}>\frac{\ensuremath{\epsilon}}{4}$. Relative to the Dicke-model result the necessary value of ${\ensuremath{\lambda}}^{2}$ is thus smaller by a factor of \textonequarter{}. In addition it is established that the critical temperature is higher than in the Dicke model. A further significant difference between the two models is reflected in the properties of the "photon statistics." In the present model an additional polynomial temperature dependence in the powers of the average number of photons per atom is found. It is concluded that the effect of the antiresonant terms is not negligible concerning the thermodynamic properties of the system, in contrast to the evidence for many time-dependent properties studied previously.
The Dicke model generalized to the case where the dependence of the coupling constant on individual atoms is taken into account is investigated. It is shown that the generalized Dicke model exhibits a quantum phase transition from the normal phase to the superradiant phase in the thermodynamic limit, as well as the standard Dicke model. The mean photon number, the atomic inversion, the lowest excitation energy, and the entanglement entropy between the field and the atoms are evaluated analytically or numerically, and the critical behavior is examined. As a result, it turns out that the critical behavior in the generalized model is essentially the same as that in the standard model. This implies universality in the Dicke model.
Abstract In this article, we use the spin coherent state transformation and the ground state variational method to theoretically calculate the ground function. In order to consider the influence of the atom-atom interaction on the extended Dicke model’s ground state properties, the mean photon number, the scaled atomic population and the average ground energy are displayed. Using the self-consistent field theory to solve the atom-atom interaction, we discover the system undergoes a first-order quantum phase transition from the normal phase to the superradiant phase, but a famous Dicke-type second-order quantum phase transition without the atom-atom interaction. Meanwhile, the atom-atom interaction makes the phase transition point shift to the lower atom-photon collective coupling strength.