Nonlinear quantum optics

Abstract

This chapter begins by reviewing the semi-classical theory of weakly nonlinear optics, in which the atomic polarization is a power series in the field amplitude. After replacing the classical amplitudes by field operators, the expansion for the classical electromagnetic energy becomes the Hamiltonian. The third- and fourth-order classical terms (respectively representing three- and four-wave mixing) become three- and four-photon interaction terms. The three-photon terms describe sum-frequency generation and spontaneous down conversion. Down conversion is the principal modern source of entangled photons pairs. It has also been applied to yield absolute measurements of the quantum efficiency of detectors and as a source of hyper-entangled photons. The four-photon interaction terms describe frequency tripling, three-photon down conversion, and photon-photon scattering. The description of an experimental demonstration of photon-photon scattering in a rubidium vapor is followed by a discussion of Kerr media.