Spectral structure of two-mode Rabi–Stark model

Abstract

The analytical solutions of the two-mode Rabi–Stark model (tmRSM) are obtained by using the Bogoliubov operators approach in Lie algebra space, which fit the exact numerical results well. The structure of the energy spectra is related to many fundamental physics characters such as symmetry, quantum phase transition (QPT), spectral collapse etc. In this paper, the spectral structure of tmRSM is discussed analytically. The regular energy spectra are given by the zeros of the G-function, and the poles appearing in the G-function are responsible for the exceptional solutions. The double degenerate exceptional solutions could be predicted by discussing the divergence of the coefficients in the G-function. If the numerator and denominator of vanish, the lowest double degenerate exceptional solutions for the nth energy levels would be located, including the first-order QPT point, the corresponding energy () is independent of the coupling strength and the energy level, even independent of the Bargmann index q. While, the nondegenerate exceptional solutions can be reproduced by the nondegenerate exceptional G-functions, the results show that more nondegenerate exceptional solutions would be found in the subspace with larger q. Then, the regular solution and two kinds of exceptional Juddian solutions of tmRSM are accurately located. The spectral collapse energy are dependent on the strength of Stark coupling and the frequency of two-level system, and Stark coupling could results in the limit of E 0 pole line is higher than that of E n pole lines, which may cause more energy levels separate from the collapse energy.