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Physical Review A

Asymptotic approximations for theQfunction in the Jaynes-Cummings model

C. A. Miller, J. Hilsenbeck, H. Risken

Published Oct 1, 1992• 34 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.46.4323

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arXiv:1106.2191•Jun 11, 2011•0 citations
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.
A splitting in the Q distribution functions for the Jaynes-Cummings model is explained, based on the interpretations of the density matrix. It has been shown [J. Eiselt and H. Risken, Phys. Rev. A 43, 346 (1991)] that the Q function splits into two peaks, which counterrotate in phase space and that the collision on the opposite side of the space and the behavior of the Q function are closely connected to the collapse of the Rabi oscillations. This paper reveals the origin of the splitting of the Q distribution functions in terms of the interpretation of the density-matrix formalism. The effects of the initial atomic state on the Q distribution functions and on the quadrature fields are emphasized. It will be shown that the nonrotating Q function is responsible both for the collapse and revival of the Rabi oscillations and for the high-frequency components of the quadrature fields. In this analysis it is assumed that the initial light field of the model is a coherent state and that an atom is prepared in a coherent superposition of the excited and ground states. Three-dimensional figures of the Q distribution functions are presented as a function of the interaction time.
The Jaynes-Cummings model with cavity damping is investigated in the rotated-wave approximation. First we introduce six appropriate combinations of the matrix elements of the density operator, which are still operators with respect to the light field. With the help of the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)] the equations of motion for the density operator transform to six coupled partial-differential equations. By expanding the quasiprobability distributions into two suitable sets, we obtain six tridiagonally coupled differential equations for the expansion coefficients, which are solved by a Runge-Kutta method. Starting with an initial coherent state of the cavity field and the atom in its upper state, we find that the initially one-peak quasiprobability function splits into two peaked functions counterrotating in the complex plane and, depending on the damping constant, spiraling into the origin. Revivals of the inversion oscillation are found for those times, when the two peaks collide. The time dependence of the inversion and the intensity as well as some special distributions of interest are also discussed.
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.
The Jaynes–Cummings model (JCM) is used to give a detailed analysis of the long-time behavior of a two-level atom interacting with a (Glauber) coherent-state field. This work is based on the earlier work of J. H. Eberly and collaborators, who drew attention to the collapse and revival of the Rabi oscillations of the atomic inversion. We use the three components S1, S2, and S3 of the pseudospin vector S to analyze the dynamics of the atom within the rotating-wave approximation and exact resonance. We show that, at the times when the revivals peak, the variables σ = |S| and η = S3/σ become independent of (1) the transition dipole moment of the atom, (2) the intensity of the coherent-state field, and (3) the (common) frequency of the atom and the field. Thus these special values of σ and η are a universal feature of the JCM that should be observed in experiments with any two-level atom in an arbitrarily intense (but resonant) coherent-state field. The derivation of the above results was facilitated by transforming into a rotating coordinate frame in which S1 = 0 so that S2 and S3 are the only dynamical variables. The two-dimensional nature of S in this frame makes it easy to visualize the dynamical changes in the atom. The dynamics of the JCM is clarified by comparison with the closely related Buck–Sukumar model. The relationship of the phenomena discussed here to photon echoes and to quench echoes (in condensed media) is pointed out.
The effects of counter-rotating terms (CRTs) on Rabi splitting and the dynamic evolution of atomic population in the Jaynes—Cummings model are studied with a coherent-state approach. When the coupling strength increases, the Rabi splitting becomes of multi-Rabi frequencies for the initial state of an excited atom in a vacuum field, and the collapses and revivals gradually disappear, and then reappear with quite good periodicity. Without the rotating-wave approximation (RWA), the initial excited state contains many eigenstates rather than two eigenstates under the RWA, which results in the multi-peak emission spectrum. An analytical approximate solution for the strong coupling regime is obtained, which gives a new oscillation frequency and explains the recovery of collapses and revivals due to the equal energy spacing.
he effect of the counter-rotating terms in the Jaynes-Cummings model is investigated with an extended coherent-state approach. The counter-rotating terms greatly modify the vacuum Rabi splitting. Two peaks with different heights in the weak coupling regime and more than two peaks in the intermediate coupling regime are predicted. The collapses and revivals in the evolution of the atomic population inversion disappear in the intermediate coupling regime, but reappear in the strong coupling regime. This reappearance is similar to that under the rotating-wave approximation, attributed to the summation of periodic cosine functions of the evolution.