Generalized adiabatic approximation to the quantum Rabi model

TL;DR

This paper introduces the generalized adiabatic approximation (GAA) for the quantum Rabi model, improving accuracy in predicting level crossings and spectrum across broader parameters by leveraging a novel connection to Laguerre polynomials.

Contribution

The GAA provides a new, more accurate approximation for the quantum Rabi model by correctly predicting level crossings and spectrum, extending the applicability of the adiabatic approximation.

Findings

GAA predicts the exact exceptional spectrum.

GAA approximates the regular spectrum more accurately.

GAA is effective over a larger parameter regime.

Abstract

The quantum Rabi model (QRM) describes the interaction between a two-level system (qubit) and a quantum harmonic oscillator. In the limit where the qubit frequency is smaller than the harmonic frequency, the QRM can be well approximated by the adiabatic approximation (AA). The AA is widely used due to its simplicity and explicit physical interpretation. However, the level crossings in the spectrum of the QRM predicted by the AA are determined by the zeros of Laguerre polynomials, which deviate from the exact points. We propose a new approximation to the QRM that predicts the level crossings correctly. This is done by exploiting a surprising connection between isolated exact solutions to the QRM and the Laguerre polynomials in the AA. We thus refer to this approach as the generalized adiabatic approximation (GAA). By construction, the GAA always predicts the exact exceptional spectrum and approximates the regular spectrum remarkably well in a much larger parameter regime than the AA. This generalized approach offers a framework to deal with the family of Rabi-type light-matter interaction models in a simple but accurate manner.