Quantum Optics. III. Two-Frequency Interactions

Abstract

A large number of "molecules," which are taken to be angular momentum oscillators of frequency $\nu$, are considered to interact through their dipole moment with two electromagnetic cavity modes of frequencies ${\omega }_1$ and ${\omega }_2$, respectively, where $\nu ={\omega }_1\pm {\omega }_2$. The behavior of the combined system is analyzed up to fourth order in perturbation theory. Phenomena known as two-photon emission or absorption, parametric amplification, various types of Raman emission, and frequency conversion are exhibited. The analytical method used is applicable to both a quantum-mechanical and a classical description of the modes and molecules; it displays explicitly the resulting differences. Expressions for the rate of energy change of the modes and molecules indicate resonance in second and fourth order. The second-order results are the same for the classical and quantum-mechanical treatments, while the fourth-order results show certain differences, which are examined in detail. The conditions under which each of the above phenomena occurs are discussed in a self-consistent manner, both classically and quantum-mechanically. It is shown that the second-order expression and the most significant part of the fourth-order expression can be derived very simply from a resonance interaction Hamiltonian that consists of a sum of products of three variables—two referring to the two modes, and the third to the molecules. The relationship between this resonance Hamiltonian and the fundamental dipole-moment Hamiltonian is examined, and the approximations by which the resonance Hamiltonian may be derived are considered. Arguments concerning the applicability of perturbation theory to steady-state situations are presented; it is pointed out that the second-order expressions for the energy transfer between molecules and field may be applicable to such a situation, in which case the classical and quantum-mechanical treatments are equivalent.