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Physical Review A

Application of the coupled cluster method to the Jaynes-Cummings model without the rotating-wave approximation

R. F. Bishop, NEIL J. DAVIDSON, R. M. Quick, DAVID M. VAN DER WALT

Published Dec 1, 1996• 30 citations
Publisher

Abstract

We investigate the Rabi Hamiltonian (or the Jaynes-Cummings model without the rotating-wave approximation), describing the coupling of a single mode of electromagnetic radiation to a two-level system, by means of the coupled cluster method. We find strong evidence for a second-order quantum phase transition. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.r4657

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We investigate the Rabi Hamiltonian (or the Jaynes-Cummings model without the rotating-wave approximation), describing the coupling of a single mode of electromagnetic radiation to a two-level system, by means of the coupled cluster method. We find strong evidence for a second-order quantum phase transition. \textcopyright{} 1996 The American Physical Society.
Recently, Bishop and co-workers (1996 Phys. Rev. A 54 R4657) applied the coupled-cluster method to study the ground state of the Jaynes - Cummings model without the rotating-wave approximation and reported strong evidence for a second-order quantum phase transition which was believed to be caused by spontaneous breaking of the parity symmetry of the system. In the present work we have re-investigated this conjecture via examining the nature of the exact (numerical) ground state of the system. Our analysis has indicated that the ground state has definite positive parity and there is no spontaneous breaking of the parity symmetry of the system.
The Rabi Hamiltonian, which describes the interaction of a single mode of electromagnetic radiation with a two-level system, is one of the fundamental models of quantum optics. It is also of wider interest as it provides a. generic model for the interaction of bosons and fermions. To allow for a systematic analysis of the strong-coupling behaviour, we have applied the coupled cluster method (CCM) to the Rabi Hamiltonian to calculate its spectrum. We find strong evidence for the existence of a somewhat subtle quantum phase transition.
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.
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 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.