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Quantum Optics Journal of the European Optical Society Part B

Dynamical renormalization of the Jaynes-Cummings model

Mark J. Fernée, P. D. Drummond

Published Feb 1, 1993• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0954-8998/5/1/001

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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.
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.
It is well known that, under the rotating-wave approximation, the Rabi Hamiltonian, which describes the interaction between a two-level atom and a single field mode, reduces to the Jaynes-Cummings model through exclusion of its counter-rotating term. In this work, it is found that, since photons in the mode are indistinguishable, there exist two conditions where the counter-rotating term can never be ignored. These conditions, all subject to the requirements that the atom-mode coupling is weak and the atom is initially in its ground state, are obtained in the stationary perturbation theory and confirmed through an analysis of the atom-mode system's evolution.
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.
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.