  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.
