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Journal of Mathematical Physics

Time-optimal control of quantum lambda systems in the KP configuration

Domenico D’Alessandro, Benjamin Sheller, Zhifei Zhu

Published May 1, 2020• 11 citations
Publisher

Abstract

We consider the problem of time-optimal control for a three-level quantum system where one level is coupled by the control field to the lowest two, which are not coupled to each other. A bound is assumed on the norm of the control at every time. Such a problem belongs to the class of KP sub-Riemannian problems for which we can perform a symmetry reduction and reduce to a Riemannian problem on the associated quotient space. We prove several properties of such a quotient space in our case, including the fact that it is an example of an almost-Kähler manifold that is not Kähler. We provide the explicit form of the optimal controls for any unitary transformation on the lowest two levels and discuss the geometric and practical implications of this result.

DOI
10.1063/5.0008034

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We consider the problem of time-optimal control for a three-level quantum system where one level is coupled by the control field to the lowest two, which are not coupled to each other. A bound is assumed on the norm of the control at every time. Such a problem belongs to the class of KP sub-Riemannian problems for which we can perform a symmetry reduction and reduce to a Riemannian problem on the associated quotient space. We prove several properties of such a quotient space in our case, including the fact that it is an example of an almost-Kähler manifold that is not Kähler. We provide the explicit form of the optimal controls for any unitary transformation on the lowest two levels and discuss the geometric and practical implications of this result.
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Sub-Riemannian Geodesics in SU(n)/S(U(n−1)×U(1))SU(n)/S(U(n-1) \times U(1))SU(n)/S(U(n−1)×U(1)) and Optimal Control of Three Level Quantum Systems
We study the time optimal control problem for the evolution operator of an n-level quantum system. For the considered models, the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. The resulting problem is a sub-Riemannian K-P problem, (as introduced in articles by U. Boscain etal. and by V. Jurdjevic), whose underlying symmetric space is SU(n)/S(U(n - 1) × U(1)). Following a method introduced by F. Albertini and D. D'Alessandro, we consider the action of S(U(n - 1) × U(1)) on SU(n) as a conjugation X → KXK-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 (A-systems). We reduce the latter problem to an integer quadratic optimization problem with linear constraints, which we solve completely for a specific set of final data.
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