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Physical Review A

Nonlinear oscillator behavior in the Jaynes-Cummings model

Y.-T. Chough, H. J. Carmichael

Published Aug 1, 1996• 36 citations
Publisher

Abstract

The driven Jaynes-Cummings model shows collapses and revivals of the mean photon number on a time scale much longer than the revival time for the two-state inversion. The behavior is essentially that of a nonlinear quantum oscillator with Hamiltonian H=\ensuremath{\surd}(A${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$+\ensuremath{\alpha})(A^+\ensuremath{\alpha}). \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1709

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The driven Jaynes-Cummings model shows collapses and revivals of the mean photon number on a time scale much longer than the revival time for the two-state inversion. The behavior is essentially that of a nonlinear quantum oscillator with Hamiltonian H=\ensuremath{\surd}(A{\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}+\ensuremath{\alpha})(A^+\ensuremath{\alpha}). \textcopyright{} 1996 The American Physical Society.
The driven two-photon Jaynes-Cummings model (JCM) exhibits a collapse and revival phenomenon in its mean photon number on a time scale much larger than the periodic revival time for two-state inversion in the usual two-photon JCM. The time scale of revivals for these oscillations is much larger than the corresponding time scale for the driven one-photon JCM and can be explained with the Hamiltonian of type ${H}_{\ensuremath{\eta}}=\ifmmode\pm\else\textpm\fi{}\sqrt{{(J}_{\ensuremath{\eta}}^{\ifmmode\dagger\else\textdagger\fi{}}+\ensuremath{\alpha}{P}_{\ensuremath{\eta}}{)}^{2}{(J}_{\ensuremath{\eta}}+\ensuremath{\alpha}{P}_{\ensuremath{\eta}}{)}^{2}}$ constructed explicitly for this purpose. The effect of Stark shifts is also studied and it is observed that the dynamics is strongly influenced by such Stark shifts both at early stages as well as at late stages of the evolution but in different ways, and thus Stark shifts play a significant role in determining the dynamical evolution of the system.
A variant Jaynes-Cummings model (JCM) where the two-state atom coupled to the cavity mode is driven by a strong external field reveals many interesting behaviors. On resonance, a large scale oscillatory behavior of the mean cavity photon number is predicted. It varies sinusoidally with the amplitude of 2(/spl epsi//g)/sup 2/, centered at the same value, where /spl epsi/ is the (real) driving field amplitude and g the vacuum Rabi frequency. The period of the oscillation was found to be exactly twice the "revival time" of the atomic inversion, i.e., T/sub 0/=(4/spl pi//g)(/spl epsi//g). Furthermore, the oscillation then shows another kind of "collapse-and-revival" type behavior as time proceeds. It is very similar to that of the atomic inversion, but the size of oscillation and the time scales are fundamentally different. The "revival" time of the oscillation, for instance, is found to be exactly one half of the so-called "fractional revival" time of the atomic inversion. Our current interest is directed to the radiative behavior of the driven JCM in a more general situation where dissipation as well as detuning is introduced.
If the quantized field has initially a small mean number of photons (n\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}2), quantum inversion displays distinct revivals, provided the detuning between the field and the atom is large as compared to the coupling constant [(g/\ensuremath{\Delta})2{)}^{2})2\ensuremath{\ll}1]. However, the amplitudes of these revivals are very small. For long times, individual revivals partially overlap to form fractional revivals, and for even longer times superrevivals (revivals of the revivals) appear. These phenomena result from the beating of modes of the quantized field that are not nearest neighbors. The envelope of the electric field associated with the oscillating atomic dipole exhibits similar superstructures, but with a period twice that for the atomic inversion.
Collapse and revivals in the Jaynes-Cummings model are studied within the context of the Mollow transformation for a single-mode coherent-state cavity field. The Mollow-transformed Jaynes-Cummings Hamiltonian has two atom-field terms, one corresponding to a classical field and the other to a quantized field driving the atomic transition. As such it maps onto the complementary problem of an atom in an optical cavity, whose field is initially in the vacuum state, driven by an external classical field. Both problems have the same atomic-state dynamics. It is shown that the revivals can be associated with two distinct properties of the quantized field of the transformed Hamiltonian. Revivals occurring at even (odd) multiples of the revival time are correlated with field states that are close to the initial field state and for which the average energy in the field is a minimum (maximum). Using semiclassical dressed states, we are able to map the problem for the field states onto one involving two uncoupled oscillators driven by off-resonant fields, provided effects related to dispersion are neglected. Analytic expressions for the photon number distribution of the transformed cavity field, including dispersion, are obtained for the even revival times.