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.

中文翻译：

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$SU(n)/S(U(n-1) \times U(1))$ 中的亚黎曼测地线和三能级量子系统的最优控制


我们研究了n级量子系统演化算子的​​时间最优控制问题。对于所考虑的模型，控制将所有能级耦合到给定的能级，并假定受欧几里得范数限制。由此产生的问题是亚黎曼 KP 问题（如 U. Boscain 等人和 V. Jurdjevic 的文章中所介绍），其基础对称空间为 SU(n)/S(U(n - 1) × U( 1））。遵循 F. Albertini 和 D. D'Alessandro 引入的方法，我们将 S(U(n - 1) × U(1)) 对 SU(n) 的作用视为共轭 X → KXK -1 。这使我们能够进行对称性约简并在商空间上考虑问题。我们对这样一个具有分层空间结构的商空间给出了明确的描述。我们证明了给定结构的亚黎曼问题的几个性质。我们针对三能级量子系统的情况得出了显式最优控制，其中所需的操作位于最低的两个能级（A 系统）。我们将后一个问题简化为具有线性约束的整数二次优化问题，并针对一组特定的最终数据完全求解。