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Physical Review A

Comment on the relationship between an exact two-photon solution and a two-photon vector model

D. Grischkowsky, Richard G. Brewer

Published Apr 1, 1977• 19 citations
Publisher

Abstract

The exact pulse solution of Brewer and Hahn for optical two-photon transitions is shown to reduce in the adiabatic limit to the two-photon vector model used by Grischkowsky, Loy, and Liao. The connection between the two approaches is explained and the complementary relationship between the two papers is discussed.
DOI
10.1103/physreva.15.1789

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Recently, in a very interesting article [Phys. Rev. A 56, 796 (1997)], Bhattacherjee has discussed the influence of nonlinearity in one-photon processes on the relationship between field and dipole squeezing in the two-level thermal Jaynes-Cummings model. Here we correct Eqs. (14)--(17) of that paper and exactly show that the squeeze period and duration in the case of $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}=0$ and $\ensuremath{\lambda}\ensuremath{\ne}0$ are inversely proportional to the nonlinear parameter $\ensuremath{\lambda}.$
Two-photon resonantly enhanced parametric generation processes have generally been described using time-dependent perturbation theory. In this paper we show that a theory of two-photon coherent effects can be used to derive and explain these nonlinear mixing processes. Our technique makes use of the adiabatic following (AF) approximation to obtain solutions to a vector model describing the two-photon resonance. We show that the usual results for the nonlinear susceptibilities correspond to the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ vector of Feynman, Vernon, and Hellwarth adiabatically following the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\gamma}}}$ vector in the small-angle limit. Consequently, the theory allows a natural extension to large angles, and power-dependent nonlinear susceptibilities are obtained. We then use these AF results for the polarization to study the propagation of pulses nearly resonant with a two-photon transition, and we demonstrate that the pulse reshaping is due to the two related effects of a nonlinear pulse velocity and self-phase modulation.
The possible generation of photon echoes is discussed for a system where the incident radiation resonantly drives two-photon transitions. Use of the adiabatic-vector-model approximation of Grischkowsky, Loy, and Liao allows echo behavior to be calculated in a multilevel system by methods similar to those previously employed for two-level atoms. We find that observation of the double-quantum photon echo requires the presence of a nonresonant probe field at the time of formation of the echo pulse. The echo then radiates at the sum and difference of the probe and two-photon frequencies. As an example, we consider a possible experimental arrangement for the detection of double-quantum photon echoes in Rb vapor.