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In this paper, we give an analytical treatment to study the behavior of thecollapse and the revival of the Rabi oscillations in the Jaynes-Cummings model(JCM). The JCM is an exactly soluble quantum mechanical model, which describesthe interaction between a two-level atom and a single cavity mode of theelectromagnetic field. If we prepare the atom in the ground state and thecavity mode in a coherent state initially, the JCM causes the collapse and therevival of the Rabi oscillations many times in a complicated pattern in itstime-evolution. In this phenomenon, the atomic population inversion isdescribed with an intractable infinite series. (When the electromagnetic fieldis resonant with the atom, the $n$th term of this infinite series is given by atrigonometric function for $\sqrt{n}t$, where $t$ is a variable of the time.)According to Klimov and Chumakov's method, using the Abel-Plana formula, werewrite this infinite series as a sum of two integrals. We examine the physicalmeanings of these two integrals and find that the first one represents theinitial collapse (the semi-classical limit) and the second one represents therevival (the quantum correction) in the JCM. Furthermore, we evaluate the firstand second-order perturbations for the time-evolution of the JCM with aninitial thermal coherent state for the cavity mode at low temperature, andwrite down their correction terms as sums of integrals by making use of theAbel-Plana formula.

Application of Abel-Plana formula for collapse and revival of Rabi oscillations in Jaynes-Cummings model