Abstract

Recent interest in the collapse and revival of Rabi oscillations in the Jaynes-Cummings model was stimulated by a paper in 1980 by Eberly et al.¹ These authors presented an analytic expression for the inversion of a two-level atom interacting with a single mode of the radiation field taken to be initially in a coherent state. Their analytic expression agreed well with a numerical evaluation of the standard summation formula for the inversion. Since the publication of this analytic result various discussions of its proof have appeared. Existing versions of the proof all convert the summation formula into an integral and evaluate this integral using the method of stationary phase.² This approach is mathematically intricate and leaves one with little physical intuition for the approximations that are made. I offer an alternative proof, which, in compact form, takes only a few lines. The proof is built up in a series of approximations based on an expansion of the troublesome square root $\left(\sqrt{n}\right)$ in the Jaynes-Cummings summation formula in a Taylor series about the mean photon number:

$\sqrt{n}=\sqrt{\overline{n}}\left[1+\frac{1}{2}\frac{n-\overline{n}}{\overline{n}}-\frac{1}{8}{\left(\frac{n-\overline{n}}{\overline{n}}\right)}^2+\dots \right].$

The analytical expression of Eberly ef al. is recovered using standard sums and integrals and this Taylor expansion to second order.

© 1988 Optical Society of America