Solving and completing the Rabi-Stark model in the ultrastrong-coupling regime

In this work,we employ a unitary transformation with a suitable parameter to convert the quantum Rabi-Stark model into a Jaynes-Cummings-like model. Subsequently, we derive the analytical energy spectra in the ultrastrong coupling regime. The energy spectra exhibit a phenomenon known as spectral collapse, indicating the instability of the model due to the unboundedness of its energy from below at higher coupling parameters. To stabilize the Rabi-Stark model, we introduce a nonlinear photon-photon interaction term. We then compare the modified model with the original model in the classical oscillator (CO) limit. Interestingly, we observe a regular"staircase"pattern in the mean photon number of the ground state. This pattern exhibits a fixed slope and equal step width, which we determine analytically. Moreover, we analytically determine the phase boundary, which slightly differs from that in the original Rabi-Stark model. These findings offer insights into the investigation of those superradiant phase transitions that are unbounded from below due to the phenomenon of spectral collapse.