Quantum phase transitions and quantum chaos in generalized Dicke and Jahn-Teller polaron model and finite-size effects

The Dicke model extended to two bosons of different frequencies or equivalent generalized Jahn-Teller lattice model are shown to exhibit a spontaneous quantum phase transition between the polaron-modified "quasi-normal" and squeezed "radiation" phase with the transition point dependent on the frequencies. In a finite lattice a mixed domain of coexistence of the quasi-normal and modified radiation phase is created within the quasi-normal phase domain. There occurs a field-directed oscillation-assisted tunneling (hopping). The field is driven by simultaneous squeezing and polaron-dressing of the collective boson level mode due to the additional boson mode. In a finite lattice in the radiation domain there occurs a sequence of local tunnelings (oscillations) between two minima of a local potential weakly coupled to two assisting oscillations. The "radiation" phase reveals itself as an almost ideal instanton--anti-instanton gas phase. The correlations among the energy levels mediated by the additional mode in the mixed domain considerably reduce the level repulsions. As a consequence, the Wigner level spacing probability distribution of the two-boson Dicke model is non-universally reduced from the Wigner to the semi-Poisson and asymptotically to the Poisson distribution of level spacings. The correlations cause a suppression of the coherence of the radiation phase as finite-size effect. Possible applications of the present theory are suggested.