Instabilities of the Dicke model with field interactions

The instability in the Dicke model with a rotating-wave approximation for finite N atoms is investigated in terms of the Berry phase and the fidelity. It can be identify by the discontinuity of these quantities as a function of the atom-field coupling parameter. Involving an additional field interaction, it is observed that the instability is not eliminated even for strong interaction of the bosonic fields, contrarily to the previous results limited to the minimal coupling regions.