An approximation formula for the Bloch-Siegert shift of the Rabi model

TL;DR

This paper introduces an explicit analytical approximation formula for the Bloch-Siegert shift in the Rabi model, valid across all driving regimes, improving upon previous numerical and perturbative methods.

Contribution

It develops a novel perturbation expansion extrapolation technique to derive a universal analytical formula for the Bloch-Siegert shift in the Rabi model.

Findings

Excellent agreement with numerical results across all regimes

Valid for both weak and strong driving conditions

Provides a simple analytical tool for practical use

Abstract

So far the Bloch-Siegert shift of the Rabi model has only been calculated numerically or by means of perturbation theory valid in either the weak or strong driving regime only. Recently Yan, L\"u, and Zheng [Phys.~Rev.~A {\bf 91}, 053834 (2015)] showed how to reduce the problem to solving a system of three nonlinear equations. Here, we pursue an alternative approach based on a perturbation expansion extrapolation technique. We are thus able to derive an explicit analytical approximation formula for the Bloch-Siegert shift of the Rabi model which is valid for all parameter regimes from weak to strong driving. Comparison with numerically exact results reveals an excellent agreement over the entire driving-strength range.