Generalized theory of nonlinear quantum optics

A quantum theory of nonlinear phenomena is considered in which account is taken of all contributions from electric and magnetic multipoles, without any multipolar approximation. Exact closed analytical expressions for the multipoles are first obtained from quantum electrodynamics by applying a generalization of the Power-Zienau transformation. The standard procedure is invoked to calculate general electromagnetic susceptibility tensors ${\chi }_{\mu \nu {\alpha }_1{\beta }_1\cdots {\alpha }_n{\beta }_n}^{\mathrm{}\left[n\right]\mathrm{}}$ (${\omega }_1, \dots , {\omega }_n$). Such a tensor is a shorthand for the electric and magnetic susceptibilities as well as mixed susceptibilities. By considering a relativistic theory based on the Dirac equation, the spin susceptibilities are automatically included. The formalism in this way allows for the computation of a wide range of electro- and magneto-optical effects to any order of perturbation and to any order of multipolar approximation.