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Physical Review A

Conditions under which the Jaynes-Cummings model becomes inapplicable

Wei Guo

Published Sep 17, 2009• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.80.033828

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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.
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.
We have investigated vacuum Rabi oscillation of an atom coupled with single-mode cavity field exactly, and compared the results with that of the Jaynes–Cummings (J–C) model. The results show that for resonant case, there is no Rabi oscillation for an atom. For small detuning and weak coupling case, the probability for the atom in excited state oscillates against time with different frequencies and amplitudes from that of the J-C model. It exhibits that the counter-rotating wave interaction could significantly effect the dynamic behaviour of the atom, even under the condition in which the RWA is considered to be justified.
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.
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.