Optimal Control of One Dimensional Quantum Harmonic Oscillator Under an External Field With Quadratic Dipole Function and Penalty on Momentum: Construction of the Linearised Field Amplitude Integral Equation

Abstract

In this work, the optimal control of an harmonic oscillator is considered. The external field is assumed to be weak and hence is represented by only dipole interaction. The dipole function is taken as a second degree polynomial in spatial coordinate. The objective term is composed of the expectation value of algebraic position operator. Penalty term related operator is taken as momentum. Some specific structures for spatial dependence is assumed and temporal equations are obtained for unknowns. The equations represent forward and backward evolution. The connection is provided by an algebraic equation coming from field amplitude related equation. The key unknown is the field amplitude since its determination leads us to evaluate all unknowns without remarkable difficulty. The purpose is not to determine field amplitude at the most general case but to construct an equation which involves this unknown only. The equation obtained via the linearisation of the field amplitude dependence is shown to be a linear integral equation. We do not attempt to solve it but discuss how to use appropriate methods for the solution. The equation is analytically solved at the zero interaction time limit. The real comprehensive implementation is left for future work. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)