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Physical Review A

Revivals made simple: Poisson summation formula as a key to the revivals in the Jaynes-Cummings model

Michael Fleischhauer, W. P. Schleich

Published May 1, 1993• 108 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.47.4258

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Analytic approximation for Jaynes-Cummings revivals

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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.

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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.

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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.

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The phenomenon of atomic population trapping in the Jaynes-Cummings model is analysed from a dressed-state point of view. A general condition for the occurrence of partial or total trapping from an arbitrary, pure initial atom-field state is obtained in the form of a bound to the variation of the atomic inversion. More generally, it is found that in the presence of initial atomic or atom-field coherence the population dynamics is governed not by the field's initial photon distribution, but by a ‘weighted dressedness’ distribution characterizing the joint atom-field state. In particular, individual revivals in the inversion can be analytically described to good approximation in terms of that distribution, even in the limit of large population trapping. This result is obtained through a generalization of the Poisson Summation Formula method for an analytical description of revivals developed by Fleischhauer and Schleich (1993, Phys. Rev A, 47, 4258).

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A resummation formula for collapse and revival in the Jaynes–Cummings model

Anatolii Alekseevich Karatsuba, Ekatherina Anatol'evna Karatsuba

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.

Journal of Physics A Mathematical and Theoretical•Apr 27, 2009•20 citations
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.
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.
The phenomenon of atomic population trapping in the Jaynes-Cummings model is analysed from a dressed-state point of view. A general condition for the occurrence of partial or total trapping from an arbitrary, pure initial atom-field state is obtained in the form of a bound to the variation of the atomic inversion. More generally, it is found that in the presence of initial atomic or atom-field coherence the population dynamics is governed not by the field's initial photon distribution, but by a ‘weighted dressedness’ distribution characterizing the joint atom-field state. In particular, individual revivals in the inversion can be analytically described to good approximation in terms of that distribution, even in the limit of large population trapping. This result is obtained through a generalization of the Poisson Summation Formula method for an analytical description of revivals developed by Fleischhauer and Schleich (1993, Phys. Rev A, 47, 4258).
When the displacement and squeeze parameters for a squeezed coherent state are chosen appropriately, the squeezed state has an oscillatory photon number distribution P(n). Schleich and Wheeler1 have shown that the oscillations arise from the interference between contributions to P(n) from two regions in phase space where an elongated error ellipse, characterizing the squeezed coherent state, overlaps the annuli characterizing the number states. We study the Jaynes-Cummings model for the interaction of a two-level atom with a single mode of the radiation field when the radiation field is initially in such a squeezed coherent state. Revivals of the atomic inversion occur which have a quite different appearance from those produced when the radiation field is initially In a coherent state.2 Each revival is followed by a series of echoes—a phenomenon we call ringing revivals. We discuss the relationship between these echoes and the oscillations in P(n). There is a one-to-one correspondence between echoes and the peaks in P(n); however, interference between the contributions to the inversion from different peaks in P(n) plays an important role in forming the ringing structure of the revivals.
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.