Macroscopic Quantum States and Quantum Phase Transition in the Dicke Model

The energy spectrum of Dicke Hamiltonians with and without the rotating wave approximation for an arbitrary atom number is obtained analytically by means of the variational method,in which the effective pseudo-spin Hamiltonian resulting from the expectation value in the boson-field coherent state is diagonalized by the spincoherent-state transformation.In addition to the ground-state energy,an excited macroscopic quantum-state is found corresponding to the south- and north-pole gauges of the spin-coherent states,respectively.Our results of ground-state energies in exact agreement with various approaches show that these models exhibit a zerotemperature quantum phase transition of the second order for any number of atoms,which was commonly considered as a phenomenon of the thermodynamic limit with the atom number tending to infinity.The critical behavior of the geometric phase is analyzed.