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arXiv:2307.07838 · Physica Scripta

Inside the Jaynes–Cummings sum

S. I. Pavlik

Published Dec 13, 2023• 1 citation
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Abstract

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.

DOI
10.1088/1402-4896/ad156c

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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.
The existence of revivals in the atomic inversion of the Jaynes-Cummings model (JCM) has been pinpointed on the numerical display of its expectation value. However, until recently its individual identification, isolation, and analytical representation was successful if the initial photon statistics allowed the use of saddle point techniques. 1 The recent introduction of a new analytical method,2 free of such restriction, and its successful use with thermal JCM revivals allows us to study the mixed thermal coherent JCM revivals, drawing a direct relation to the saddle point technique. We compare our results with previous ones of the pure thermal case,2 and the coherent limiting cases, in particular with the systematic technique proposed by Fleischhauer and Schleich.
We investigate the phenomenon of quantum revivals in the Jaynes-Cummings model for an arbitrary quantized field mode. With the help of the Poisson summation formula, we cast the infinite sum determining the atomic inversion into an infinite sum of integrals. Each integral, when evaluated using the method of stationary phase, yields under appropriate conditions one revival. We present simple approximate analytical expressions for these revivals and illustrate this general technique by the examples of a coherent and a highly squeezed state. The oscillatory photon distribution of the latter creates slightly different Rabi frequencies which give rise to a beat note; that is, echos in the revivals. We obtain the photon statistics of the quantized field by ``measuring'' the atomic collapse of a single revival--a technique which might be applicable in the realm of the one-atom maser.
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 (n) in the Jaynes-Cummings summation formula in a Taylor series about the mean photon number: n=n¯[1+12n−n¯n¯−18(n−n¯n¯)2+…]. The analytical expression of Eberly ef al. is recovered using standard sums and integrals and this Taylor expansion to second order.
In this paper, we give an analytical treatment to study the behavior of the collapse and the revival of the Rabi oscillations in the Jaynes–Cummings model (JCM). The JCM is an exactly soluble quantum mechanical model, which describes the interaction between a two-level atom and a single cavity mode of the electromagnetic field. If we prepare the atom in the ground state and the cavity mode in a coherent state initially, the JCM causes the collapse and the revival of the Rabi oscillations many times in a complicated pattern in its time-evolution. In this phenomenon, the atomic population inversion is described with an intractable infinite series. (When the electromagnetic field is resonant with the atom, the nth term of this infinite series is given by a trigonometric function for [Formula: see text], where t is a variable of the time.) According to Klimov and Chumakov's method, using the Abel–Plana formula, we rewrite this infinite series as a sum of two integrals. We examine the physical meanings of these two integrals and find that the first one represents the initial collapse (the semi-classical limit) and the second one represents the revival (the quantum correction) in the JCM. Furthermore, we evaluate the first- and second-order perturbations for the time-evolution of the JCM with an initial thermal coherent state for the cavity mode at low temperature, and write down their correction terms as sums of integrals by making use of the Abel–Plana formula.
We present a new approach to the study of the function of atomic inversion in the model of interaction of a single two-level atom with a single mode of the quantized electromagnetic field in the coherent state in an ideal resonator. The approach suggested is based on an application of certain number-theoretic results to the approximation of the trigonometric sums of a special form, in particular of the functional equation of the Jacobi theta function. New asymptotic formulae for the atomic inversion are found. The asymptotics that we obtain make it possible to predetermine the details of the behavior of the inversion on various time intervals depending on the parameters of the system.