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Modern Physics Letters B

JAYNES-CUMMINGS SPECTRA BY LINEARIZATION WITH CLIFFORD VARIABLES

Maria Luisa Rastello

Published Oct 20, 1992• 0 citations
Publisher

Abstract

The effect of rotating-wave and counter-rotating terms in the energy spectrum of a single two-level atom interacting with a quantized field is investigated within a fermionic linearization scheme involving Clifford variables. After reproducing the Jaynes-Cummings results in the rotating-wave approximation, the counter-rotating terms are shown to determine an energy spectrum sensitive to low photon-number conditions.

DOI
10.1142/s0217984992001228

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The effect of rotating-wave and counter-rotating terms in the energy spectrum of a single two-level atom interacting with a quantized field is investigated within a fermionic linearization scheme involving Clifford variables. After reproducing the Jaynes-Cummings results in the rotating-wave approximation, the counter-rotating terms are shown to determine an energy spectrum sensitive to low photon-number conditions.
We have investigated the eigenenergy spectrum of the two-photon Jaynes-Cummings (JC) model with and without the rotating-wave approximation (RWA). Our analysis has indicated that the counter-rotating term dramatically changes the nature of the RWA energy spectrum and that the non-RWA spectrum can be approximated by the RWA spectrum only in the range of a sufficiently small coupling constant. Furthermore, unlike the one-photon counterpart, the two-photon JC model without the RWA is well defined only if the coupling parameter is below a certain critical value. As a result, the dynamics of the two-photon JC model without the RWA is significantly different from its RWA counterpart. For instance, the counter-rotating term can dramatically enhance the field squeezing effect. Besides, we would expect that the quantum dynamics of the two-photon JC model without the RWA is qualitatively different from that of the usual one-photon case.
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.
In this paper we have investigated the eigen energy spectrum of the k-photon Jaynes-Cummings (JC) model without the rotating-wave approximation (RWA). Our analysis shows that for k > 2 the k-photon JC model without the RWA does not have eigenstates in the Hilbert space spanned by the photon number states, i.e. the model becomes ill-defined. Hence, the k-photon JC model without the RWA is qualitatively different from the RWA counterpart which is valid for all values of k and the coupling parameter.
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)].
In this paper we have investigated the eigenenergy spectrum of the nonlinear k -quantum Jaynes-Cummings (JC) model without the rotating-wave approximation (RWA). Our analysis has shown that the nonlinear k -quantum JC model without the RWA does not have eigenstates in the Hilbert space spanned by the number states, i.e. the model becomes ill-defined. Hence, the nonlinear k -quantum JC model without the RWA is qualitatively different from the RWA counterpart which is valid for all values of k and the coupling parameter. In other words, the counter-rotating terms do drastically alter the nature of the system.
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.