  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.

Quantum Gases