Solving and Completing the Rabi-Stark Model in the Ultrastrong Coupling Regime

TL;DR

This paper analytically solves the Rabi-Stark model in the ultrastrong coupling regime, addresses spectral collapse by introducing nonlinear interactions, and explores phase transitions and ground state properties.

Contribution

It provides an analytical solution to the Rabi-Stark model, introduces a stabilization method, and analyzes phase boundaries and ground state features.

Findings

Derives analytical energy spectra showing spectral collapse.

Introduces nonlinear photon-photon interaction to stabilize the model.

Identifies a staircase pattern in the ground state's mean photon number.

Abstract

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.