The mixed quantum Rabi model

TL;DR

This paper derives exact analytical solutions for the mixed quantum Rabi model with one- and two-photon interactions, revealing spectral properties, energy limits, and effective regimes that facilitate access to deep-strong coupling in quantum systems.

Contribution

It provides the first exact solutions for the mixed quantum Rabi model using Bogoliubov operators and introduces G-functions that describe its spectra and exceptional eigenvalues.

Findings

Exact solutions via G-functions and determinants.

Energy levels collapse to a negative diverging limit.

Weak two-photon coupling mimics an effective one-photon model.

Abstract

The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of $4 \times 4$ determinants responsible for the exact solutions are derived. These so-called $G$-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of $G$-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.