Inside the Jaynes-Cummings sum

TL;DR

This paper presents an exact integral representation of atomic inversion in the Jaynes-Cummings model, analyzing saddle point trajectories and their influence on collapse and revival phenomena in quantum optics.

Contribution

It introduces an exact integral form for atomic inversion and explores saddle point dynamics using Lambert functions, providing new insights into quantum revival behavior.

Findings

Saddle points follow Lambert function branches over time.

Different saddle point trajectories dominate at various times.

Revival phenomena are explained by multiple saddle point contributions.

Abstract

It is shown that the atomic inversion in the Jaynes-Cummings model has an exact representation as an integral over the Hankel contour. For a field in a coherent state, the integral is evaluated using the saddle point method. The trajectories of saddle points as a function of time are on the branches of the multi-valued Lambert function. All of them start at the initial moment of time, but make the maximum contribution to the inversion at different times. If the collapse and the first revival are clearly distinguished, then subsequent revivals are determined by the comparable contributions of several trajectories.