FIELD QUANTIZATION IN DIELECTRIC MEDIA AND THE GENERALIZED MULTIPOLARHAMILTONIAN

Canonical quantization of the electromagnetic (EM) field is carried ou t for the situation where the total charge and current densities are t he sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent ele ctric permittivity and magnetic permeability functions, and neutral, s tationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of t he vector potential in terms of mode functions determined from a gener alized Helmholtz equation are chosen as independent generalized coordi nates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the farmer also occurs in the mode function orthogonality and nor malization conditions. The quantum Hamiltonian is derived in a general ized multipolar form rather than the minimal coupling form obtained in other work, The radiative energy is the sum of quantum harmonic oscil lator terms, one for each mode. The modes are independent in the prese nt case of exact mode functions associated with the spatially dependen t electric permittivity and magnetic permeability, there being no dire ct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative a tom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operat or, plus the Hermitian adjoint. This form has been widely used in stud ies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The re sults apply to the theoretical treatment of numerous quantum optical e xperiments involving such interactions in the presence of passive, los sless, dispersionless, linear classical optics devices such as resonat or cavities, lenses, beam splitters, and so on. An illustrative applic ation of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given.