Fluctuation expansion in the quantum optimal control of one dimensional perturbed harmonic oscillator

Abstract

This work considers the quantum optimal control of one dimensional harmonic oscillator under linear control agents. The system's potential energy function contains a perturbation term which is bounded everywhere in the space variable's domain. We use the most preferred cost functional to construct the necessary equations. Equations are converted to a boundary value problem for a set of ordinary differential equations containing the expectation values of certain operators and the terms corresponding to the transitions between the states described by the wave and costate functions. Resulting equations involve certain undesired expectation values and transition terms. We use a recently developed scheme called fluctuation expansion to approximate these terms at the sharply localized wave and costate function limits. This enables us to construct n infinite number of ordinary differential equations and accompanying boundary conditions whose both halves are given at the beginning and end of the control. These equations are truncated and then the resulting boundary value problem is solved iteratively.