Sub-Riemannian Geodesics on SU(n)/S(U(n-1)xU(1)) and Optimal Control of Three Level Quantum Systems

TL;DR

This paper investigates the optimal control of n-level quantum systems using sub-Riemannian geometry, providing explicit solutions for three-level systems and reducing the problem to an integer quadratic optimization.

Contribution

It introduces a symmetry reduction approach for sub-Riemannian problems on SU(n)/S(U(n-1) x U(1)) and explicitly solves the three-level case with a quadratic optimization method.

Findings

Explicit optimal control for three-level quantum systems derived.

Reduction of the control problem to an integer quadratic optimization.

Structural analysis of the quotient space as a stratified space.

Abstract

We study the time optimal control problem for the evolution operator of an n-level quantum system from the identity to any desired final condition. For the considered class of quantum systems the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. From a mathematical perspective, such a problem is a sub-Riemannian K-P problem, whose underlying symmetric space is SU(n)/S(U(n-1) x U(1)). Following the method of symmetry reduction, we consider the action of S(U(n-1) xU(1)) on SU(n) as a conjugation X ---> AXA^{-1}. This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels (Lambda-systems). We solve this latter problem by reducing it to an integer quadratic optimization problem with linear constraints.