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.
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.
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.
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.
The Q function in the undamped Jaynes-Cummings model for a coherent initial state in the rotating-wave approximation is investigated. By applying the method of steepest decent and saddle-point integration in the limit of high photon numbers, closed analytic expressions for the Q function at different times are given, from which analytic expressions for the Rabi frequency, revival times, envelope shapes of the inversion, beat frequencies, etc. are derived. As Eiselt and one of us have shown [J. Eiselt and H. Risken, Phys. Rev. A 43, 346 (1991)], the Q function first splits into two peaks which counterrotate in the complex plane and then collide on the opposite side of the plane. This behavior of the Q function leads to an interpretation of the collapses and revivals of the Rabi oscillations and a better understanding of expressions for the Rabi frequency, revival times, envelope shapes of the inversion, beat frequencies, etc. The results are generalized to N-photon processes; in particular, the cases N=2 and 4 are treated in some detail.
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.
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, 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 Jaynes-Cummings model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probabilities for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive, only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further non-classical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics. The relative simplicity of the JCM and the ease with which it can be extended through analytic expressions or numerical computation continues to motivate attention, as evidenced by the growing abundance of publications. We here present an overview of the theory of the JCM and some of the many extensions and generalizations that have appeared.