Heisenberg-picture operator perturbation theory

Abstract

We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schrödinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.