Theory of four-wave mixing using an amplitude approach

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Abstract
A theory of four-wave mixing is developed using an amplitude approach in the Schrödinger picture. It is shown that the four-wave mixing signal produced when three fields are incident on an ensemble of two-level atoms can be viewed as arising from constructive interference of the signals emitted at the various atomic sites. Fourteen distinct amplitudes contribute to the four-wave mixing signal. Some of these amplitudes involve final states in which one or two atoms are excited, and some involve final states in which photons are emitted into modes other than a phase-matched mode of interest. The amplitude approach has the advantage of clearly showing the atom-field correlations. It helps one to understand the underlying physical mechanisms responsible for creation of the four-wave mixing signal, as well as features in the line shape resulting from collisions and effects relating to atomic recoil on the absorption or emission of radiation. These so-called collision- and recoil-induced resonances are calculated explicitly using the amplitude approach, and an interpretation of the results in terms of the interference between different amplitudes is given. Our results, calculated for a finite interaction time between the atoms and the fields, are compared with analogous calculations in which an effective ground-state decay time is assumed. It is shown that the results differ if the atom-field interaction time is not the longest relevant time parameter in the problem.