Quantum Rabi-Stark model: Solutions and exotic energy spectra. (arXiv:1811.04431v2 [quant-ph] UPDATED)

The quantum Rabi-Stark model, where the linear dipole coupling and the
nonlinear Stark-like coupling are present on an equal footing, are studied
within the Bogoliubov operators approach. Transcendental functions responsible
for the exact solutions are derived in a compact way, much simpler than
previous ones obtained in the Bargmann representation. The zeros of
transcendental functions reproduce completely the regular spectra. In terms of
the explicit pole structure of these functions, two kinds of exceptional
eigenvalues are obtained and distinguished in a transparent manner. Very
interestingly, a first-order quantum phase transition indicated by level
crossing of the ground state and the first excited state is induced by the
positive nonlinear Stark-like coupling, which is however absent in any previous
isotropic quantum Rabi Models. When the absolute value of the nonlinear
coupling strength is equal to twice the cavity frequency, this model can be
reduced to an effective quantum harmonic oscillator, and solutions are then
obtained analytically. Below a critical coupling, infinite discrete spectra
accumulates into a point from below. While approaching the critical coupling,
the spectra collapse is observed.

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