A simple, analytic solution of the Rabi model using multiple-scales

TL;DR

This paper presents a simple, analytic solution to the semiclassical Rabi model that accounts for counterrotating terms, providing accurate descriptions of system evolution beyond the rotating-wave approximation within specific parameter regimes.

Contribution

It introduces an approximate, analytic solution for the Rabi model that includes counterrotating effects and offers a criterion for the validity of the RWA.

Findings

Solution accurately describes Bloch vector evolution

Applicable when Rabi frequency is up to half the field frequency

Provides a criterion for RWA validity

Abstract

We considered the semiclassical Rabi model, that is, a two-level system interacting with a single-mode, classical field. In resonance and for an arbitrary initial state of the system, we obtained a simple, approximate, and analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/2 (1/4) of the angular frequency of the field and 1 (10) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and allows one to describe the evolution of the Bloch vector of the system in terms of a \textit{slow-precessional} and a \textit{fast-nutational} motion similar to that of a symmetric top with one point fixed. Finally, the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA.