Dynamics of an atomic electron and its electromagnetic field in a cavity

Nonperturbative analytical and numerical methods are presented for the solution of the coupled nonlinear Maxwell-Schroedinger equations. The theory has been derived within the Hamiltonian or canonical formalism. The canonical approach to dynamics, starting from the Maxwell and Schroedinger Lagrangians with a Lorenz gauge fixing term, yields a set of first order Hamilton equations. They form a well-defined initial value problem. The Maxwell-Schroedinger equations of motion are then represented in a spatial basis of Gaussian functions. In the limit of a complete basis this representation is exact. For any choice of finite basis it provides an approximate system of dynamical equations that can be integrated in time and made systematically more accurate by enriching the basis. The basis form of the theory has been implemented numerically and is used to investigate the dynamics of a single nonrelativistic spinless one electron atom interacting with the electromagnetic modes of a cavity