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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Is there a spontaneous breaking of the parity symmetry of the Jaynes - Cummings model without the rotating-wave approximation?

Chi‐Fai Lo, K L Liu, K.M. Ng, Piu-Hung Yuen

Published Dec 1, 1998• 8 citations
Publisher

Abstract

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.

DOI
10.1088/1355-5111/10/6/003

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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 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 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.
The eigenenergy spectrum of the Jaynes–Cummings (JC) model with and without the rotating-wave approximation (RWA) is investigated. The numerical analysis indicates that the non-RWA spectrum can only be approximated by the RWA in the range of sufficiently small coupling constant and detuning. In other region, the counter-rotating terms remarkably change the nature of the RWA energy spectrum. A simple expression with high accuracy for ground eigenenergy and eigenvector for non-RWA is available. The ground eigenvector for non-RWA shows that the ground state is not a dark state and very different from that of RWA which is a dark state.
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.