Analytical solutions for the Rabi model

TL;DR

This paper introduces an analytical approach to solve the Rabi model exactly by transforming it into a Jaynes-Cummings-like model, enabling explicit calculation of energy spectra and wavefunctions.

Contribution

The authors develop a unitary transformation that maps the Rabi model to a solvable form, providing exact solutions and explaining experimental phenomena.

Findings

Analytical energy spectra and wavefunctions obtained.

Results agree with numerical simulations across parameter ranges.

Explains experimental observations of Bloch-Siegert shift.

Abstract

The Rabi model that describes the fundamental interaction between a two-level system with a quantized harmonic oscillator is one of the simplest and most ubiquitous models in modern physics. However, this model has not been solved exactly because it is hard to find a second conserved quantity besides the energy. Here we present a unitary transformation to map this unsolvable Rabi model into a solvable Jaynes-Cummings-like model by choosing a proper variation parameter. As a result, the analytical energy spectrums and wavefunctions including both the ground and the excited states can be obtained easily. Moreover, these explicit results agree well with the direct numerical simulations in a wide range of the experimental parameters. In addition, based on our obtained energy spectrums, the recent experimental observation of Bloch-Siegert in the circuit quantum electrodynamics with the ultrastrong coupling can be explained perfectly. Our results have the potential application in the solid-state quantum information processing.