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Quantum Optics Journal of the European Optical Society Part B

The Jaynes-Cummings model thermal revivals

Gabriel Arroyo-Correa, J. J. Sánchez-Mondragón

Published Dec 1, 1990• 22 citations
Publisher

Abstract

Consistent with the standard interpretation of the revivals in the Jaynes-Cummings models the time evolution interference among the quantized field states, the authors have developed an exact mathematical analysis of the series describing the evolution of the atomic inversion when the quantum electrodynamic field is initially prepared in a thermal state. A complete and exact functional analysis to isolate the revivals is carried out.

DOI
10.1088/0954-8998/2/6/001

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Mixed thermal-coherent revivals in the Jaynes-Cummings model

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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.

Annual Meeting Optical Society of America•Jan 1, 1991•0 citations

APPLICATION OF ABEL–PLANA FORMULA FOR COLLAPSE AND REVIVAL OF RABI OSCILLATIONS IN JAYNES–CUMMINGS MODEL

Hiroo Azuma

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 $\\sqrt{n} t$, 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.

•Jan 1, 2010•5 citations

Analytic approximation for Jaynes-Cummings revivals

H. J. Carmichael

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.

Annual Meeting Optical Society of America•Jan 1, 1988•0 citations
Consistent with the standard interpretation of the revivals in the Jaynes-Cummings models the time evolution interference among the quantized field states, the authors have developed an exact mathematical analysis of the series describing the evolution of the atomic inversion when the quantum electrodynamic field is initially prepared in a thermal state. A complete and exact functional analysis to isolate the revivals is carried out.
Exact solutions in the form of infinite series are well known for the Jaynes-Cummings model describing the interaction of a two-level atom with a single-mode field prepared in a coherent state. While these solutions have led to the discovery of many interesting properties of the Jaynes-Cummings model as a function of time, the nature of the solutions is such that it is not a simple matter to make statements of a general nature about the properties of the system. In this paper it is shown that by taking infinite time averages of the series expressions for the atomic inversion and inversion correlation, squeezing, and antibunching, simple closed-form expressions are obtained from which general properties of the Jaynes-Cummings model can be readily inferred. In particular, new conditions are obtained under which the field is significantly squeezed on average, and further information is obtained on the long-time behavior of the revivals in the atomic inversion.
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.
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 sqrtnt\\sqrt{n} tsqrtnt, 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.
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.