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Acta Physica Sinica (Overseas Edition)

Higher-order correction for rotating wave approximation, Rabi transformation, and its applications in the Jaynes-Cummings model

Wang XiaoGuang, Sun Changpu

Published Dec 1, 1996• 2 citations
Publisher

Abstract

Taking into account the consistence of the quantum adiabatic approximation and the rotating wave approximation in essence, we propose a method to study the interaction between atom and radiation field in a quantized cavity systematically. By using the exact solution of Jaynes-Cummings model as the lowest-order approximation, the effect of high frequency terms on the dynamics of atom-cavity field system is studied analytically.

DOI
10.1088/1004-423x/5/12/001

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Taking into account the consistence of the quantum adiabatic approximation and the rotating wave approximation in essence, we propose a method to study the interaction between atom and radiation field in a quantized cavity systematically. By using the exact solution of Jaynes-Cummings model as the lowest-order approximation, the effect of high frequency terms on the dynamics of atom-cavity field system is studied analytically.
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.
In this paper, we have investigated the time evolution of the phase operator of the radiation field in the Jaynes-Cummings model with and without the rotating-wave approximation, making use of the phase formalism of Barnett and Pegg [J. Mod. Opt. 36, 7 (1989); Phys. Rev. A 39, 1665 (1989)]. We verify by an analytical method that the atomic Rabi oscillation leads to phase dissipation of the radiation field, and also that the effect of the virtual-photon field can exhibit quantum fluctuations in the atom-field-coupling system and lead to a frequency shift of the radiation field.
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 Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by extended Swain's ansatz with the conserved parity. The analytical approximations are then performed at different levels. The well-known rotating-wave approximation is naturally covered in the present zero and first approximations. The effect of the counter rotating-wave term emerges clearly in the second order approximation. The concise analytical expressions are given explicitly and can be applicable up to the ultra-strong coupling regime. The preliminary application to the vacuum Rabi splitting is shown to be very successful.
The Jaynes-Cummings model (JCM) is studied in the absence of the rotating-wave approximation (RWA) by a coherent-state expansion technique. In comparison with the previous paper in which the coherent-state expansion was performed only to the third order, we carry out in this paper a complete expansion to demonstrate exactly the dynamics of the JCM without the RWA. Our study gives a systematic method to solve the non-RWA problem, which would be useful in various physical systems, e.g., in a system with an ultracold trapped ion experiencing the running waves of lasers.
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.