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Journal of Modern Optics

The Effect of the Counter-rotating Terms on the FieldQ-function in the Jaynes-Cummings Model

Simon J. D. Phoenix

Published Apr 1, 1991• 16 citations
Publisher

Abstract

The fully-coupled Jaynes-Cummings model contains both counter-rotating and rotating wave terms. It is usual to solve for the rotating wave dynamics exactly and to include the counterrotating contributions as a perturbation. In this paper we choose an alternative approach in which the counterrotating terms are treated exactly and the rotating wave interaction is included as a perturbation. We find that, under the influence of the counterrotating terms, the field Q-function undergoes an oscillatory bifurcation in phase space. The cycle-averaged splitting of the Q-function is found to be of the same order in the interaction as the Bloch-Siegert shift.

DOI
10.1080/09500349114550691

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The fully-coupled Jaynes-Cummings model contains both counter-rotating and rotating wave terms. It is usual to solve for the rotating wave dynamics exactly and to include the counterrotating contributions as a perturbation. In this paper we choose an alternative approach in which the counterrotating terms are treated exactly and the rotating wave interaction is included as a perturbation. We find that, under the influence of the counterrotating terms, the field Q-function undergoes an oscillatory bifurcation in phase space. The cycle-averaged splitting of the Q-function is found to be of the same order in the interaction as the Bloch-Siegert shift.
We have investigated the eigenenergy spectrum of the two-photon Jaynes-Cummings (JC) model with and without the rotating-wave approximation (RWA). Our analysis has indicated that the counter-rotating term dramatically changes the nature of the RWA energy spectrum and that the non-RWA spectrum can be approximated by the RWA spectrum only in the range of a sufficiently small coupling constant. Furthermore, unlike the one-photon counterpart, the two-photon JC model without the RWA is well defined only if the coupling parameter is below a certain critical value. As a result, the dynamics of the two-photon JC model without the RWA is significantly different from its RWA counterpart. For instance, the counter-rotating term can dramatically enhance the field squeezing effect. Besides, we would expect that the quantum dynamics of the two-photon JC model without the RWA is qualitatively different from that of the usual one-photon case.
In this paper, we investigate various dynamical properties of the Jaynes–Cummings (JC) model beyond the rotating wave approximation (RWA). We first show that in the absence of the RWA, the JC Hamiltonian can be transformed into an intensity-dependent Hamiltonian. Corrections produced by the counter-rotating terms (CRTs) appear in the first order as the intensity-dependent detuning, i.e. the dynamical Stark shift and in the second order as the intensity-dependent atom–field coupling. Then, by determining the atom–field wavefunction evolution, we study the effects of CRTs on various dynamical properties of the JC model, including the atomic population inversion, atomic dipole squeezing, atomic entropy squeezing, photon counting statistics, quadrature field squeezing, field entropy squeezing and quantum phase properties of the cavity field.
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.
The effect of counter-rotating terms on the Jaynes-Cummings model of a two-level atom is studied. The technique of dynamical renormalization is used to generate a hybrid gauge which exists between the d.E and p.A gauges. The resulting truncated Hamiltonian including counter-rotating type terms has an interaction that couples the atomic inversion to the field intensity. This Hamiltonian is exactly soluble, resulting in a phase-dependent inversion, together with simultaneous frequency and refractive-index shifts.
The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by an extended Swain ansatz with conserved parity. Analytical approximations are then performed at different levels. The well-known rotating-wave approximation (RWA) is naturally covered in the present zeroth- and first-order approximations. A first-order correction to the RWA can be obtained in a second-order approximation, by which the effect of the counter-rotating-wave term emerges clearly. Concise analytical expressions are given explicitly and can be applicable up to the ultrastrong-coupling regime. A preliminary application to vacuum Rabi splitting is shown to be very successful.