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Physical Review A

Limitations to the realization of the Jaynes-Cummings model with electric-dipole coupling

R. J. C. Spreeuw, J. P. Woerdman

Published Oct 1, 1991• 2 citations
Publisher

Abstract

We show that the Jaynes-Cummings model cannot be realized when using a strongly driven electric-dipole transition. The limitations are inherent in electric-dipole coupling and are much more severe than those recently discussed by Crisp [Phys. Rev. A 43, 2430 (1991)]. The level shifts associated with the violation of the rotating-wave approximation are roughly equal for circular and linear polarizations.

DOI
10.1103/physreva.44.4765

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We show that the Jaynes-Cummings model cannot be realized when using a strongly driven electric-dipole transition. The limitations are inherent in electric-dipole coupling and are much more severe than those recently discussed by Crisp [Phys. Rev. A 43, 2430 (1991)]. The level shifts associated with the violation of the rotating-wave approximation are roughly equal for circular and linear polarizations.
The interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic field is usually analyzed in the dipole approximation by applying the rotating-wave approximation (RWA) to obtain the exactly solvable Jaynes-Cummings model (JCM). In this paper, it is shown that for a spin-1/2 particle, at rest in a constant magnetic field and for a system that obeys the electric dipole \ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1 selection rules, a choice of a perturbing field that consists of circularly polarized photons can make the RWA unnecessary for the solution of the quantum-electrodynamic (QED) problem. For these cases, the only approximation necessary to obtain an exact solution of the QED problem is to neglect the spontaneous emission of photons with a sense of polarization opposite that of the perturbing field. The matrix elements corresponding to these ``virtual'' processes are independent of the number of photons n in the applied field and become relatively small, compared with those that correspond to energy-conserving processes, as n becomes large.However, a two-level system that obeys the \ensuremath{\Delta}m=0 selection rule will require the RWA in order to reduce it to the JCM, regardless of the choice of polarization of the perturbing field. For this case, the emission of the photons that do not conserve energy is an induced process that corresponds to matrix elements that increase as \ensuremath{\surd}n+1 . It is thus concluded that, provided the right choice of selection rules and polarization of the applied field is made, the JCM provides a more accurate description of some systems than the accuracy of the RWA. Furthermore, it is concluded that an experimental test of the validity of the RWA can be made by performing the same experiments on an electric dipole \ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1 transition, first with circularly polarized light and then with linearly polarized light of the same intensity. Any difference in response of the atomic system for these two experiments can be identified with terms that are neglected in the RWA. This experimental technique may prove useful for observing the optical Bloch-Siegert shift and for investigating the failure of the RWA and the chaotic quantum behavior predicted by Milonni, Ackerhalt, and Galbraith [Phys. Rev. Lett. 50, 966 (1983)].
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.
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.