Quantum-theoretical description of coherent four-wave mixing

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Abstract
The problem of coherent four-wave mixing is treated quantum theoretically in the dressed-atom picture using a set of "angular-momentum-like" operators. The phases of the radiation fields are considered by including a space-dependent phase factor in the raising (lowering) field operators. All possible transition pathways are considered, and their contributions to the total transition probability are added coherently. It is shown that interferences between the different pathways play an essential role and that the phases of the radiation fields should be properly handled. The result of this calculation is a directional, time-dependent transition probability that automatically expresses the phase-matching requirements of a coherent four-wave mixing process.