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Journal of Modern Optics

Cavity field spectrum for multiphoton Jaynes-Cummings model

Kuniaki Matsuo, H. Iwasawa

Published Nov 1, 1996• 4 citations
Publisher

Abstract

The field spectra in an ideal cavity for the multiphoton Jaynes-Cummings model are studied. The analytical expression for the spectrum is obtained from the finite double-Fourier transform of the two-time field correlation function. The spectral differences between the initial coherent and initial thermal states are discussed, and the comparisons between the field spectra and atomic emission spectra for k = 2, 3 and 4 are presented. It is shown that the kth power of the photon annihilation operator and atomic lowering operator are subjected to identical forms of a second-order differential equation, in which the coefficients consist of the constants of motion. The field spectrum in the cavity and the emission spectrum of the atom for the initial vacuum state are compared.

DOI
10.1080/09500349608232895

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The field spectra in an ideal cavity for the multiphoton Jaynes-Cummings model are studied. The analytical expression for the spectrum is obtained from the finite double-Fourier transform of the two-time field correlation function. The spectral differences between the initial coherent and initial thermal states are discussed, and the comparisons between the field spectra and atomic emission spectra for k = 2, 3 and 4 are presented. It is shown that the kth power of the photon annihilation operator and atomic lowering operator are subjected to identical forms of a second-order differential equation, in which the coefficients consist of the constants of motion. The field spectrum in the cavity and the emission spectrum of the atom for the initial vacuum state are compared.
The emission spectra of an atom in a cavity for the multiphoton Jaynes–Cummings model are investigated. Using Eberly's formula for the time-dependent physical spectrum, the analytical expression for the atomic emission spectrum is obtained from the two-time dipole-moment correlation function of the atom. The appearance of two strong peaks in the spectrum is explained in detail, and the relationship between the initial photon distributions and the shapes of the spectra are examined. The fundamental differences in the atomic emission spectra for the two-photon and multiphoton models are specified. Furthermore, the role of the detector filter is thoroughly studied. It is explicitly shown that the atomic emission spectra can be obtained by averaging over the convolution integrals between the Fourier transform of the atomic dipole-moment correlation function associated with a pure number state and the detector filter function on frequency domain. The comparisons of the spectra for a coherent initial field to those for a thermal initial field are presented.
The time-dependent two-photon emission spectra of a two-level atom interacting with a single-mode radiation field in an ideal cavity have been studied. The expressions of the system operators as a function of time have been derived based on the two-photon Jaynes-Cummings model and Heisenberg equation. As a result, the theoretical curves of the time-dependent two-photon emission spectra have been given. The features of two-photon emission spectra depend not only on the photon statistical distribution of the initial field, such as the thermal field, the squeezed vacuum field, and the coherent field, but also on the passband width of the filter detector \ensuremath{\Gamma}. The two-photon emission spectra exhibit multi- peaked structures when \ensuremath{\Gamma}(two-photon coupling constant).
Atomic emission and cavity field spectra of the Jaynes-Cummings model are analytically compared. We show that the two spectra are in general different, except for the special case where the interaction is resonant, the cavity is initially empty and the retarded time of one-photon emission is considered in calculations of the spectra. In this case, the two spectra are the same. We also show that this result comes from the fact that the two-time autocorrelation function of the cavity field is not directly proportional to that of the atom.
Atomic emission and cavity field spectra of the Jaynes-Cummings model are analytically compared. We show that the two spectra are in general different, except for the special case where the interaction is resonant, the cavity is initially empty and the retarded time of one-photon emission is considered in calculations of the spectra. In this case,the two spectra are the same. We also show that this result comes from the fact that the two-time autocorrelation function of the cavity field is not directly proportional to that of the atom.
The authors have investigated the spectrum of the emitted light by a single atom interacting with a single mode of the radiation in an ideal cavity via the intensity-dependent coupling (the Jaynes-Cummings model with intensity-dependent coupling see B. Buck, Phys. Lett., vol.A81, p.132 (1981)). From the authors' results it follows that in the resonant case the two-peaked structure of the spectrum associated with the vacuum-field Rabi splitting is preserved for any initial state of the radiation field (e.g. number state, thermal state, coherent state and the squeezed vacuum) with an arbitrary average number of photons.