Functional methods in the generalized Dicke model

Abstract

The Dicke model describes an ensemble of N identical two-level atoms (qubits) coupled to a single quantized 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, g1 and g2 for rotating and counter-rotating terms, respectively. In the limit N → ∞, the thermodynamic of the fermion Dicke model can be analyzed using the path integral approach with the functional method. The system exhibits a second-order phase transition from normal to superradiance 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 for the general case (g1 ≠ 0 and g2 ≠ 0). There is a quantum critical behavior when the coupling constants g1 and g2 satisfy , where ω0 is the frequency of the mode of the field and Ω is the energy gap between the energy eigenstates of the qubits. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations, in the case g1 ≠ 0 and g2 = 0, recovering the well-known results. Second, the case g1 = 0 and g2 ≠ 0 is studied. In this last case, it is possible to have a superradiant phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at the critical coupling , and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.