Effect of antiresonant atom-field interactions on phase transitions in the Dicke model

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 $\lambda$ and the ratio of the atomic transition frequency to the field frequency $\epsilon$: ${\lambda }^2>\frac{\epsilon }{4}$. Relative to the Dicke-model result the necessary value of ${\lambda }^2$ is thus smaller by a factor of ¼. 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.