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 $K\hbox{--}P$ problem, (as introduced in articles by U. Boscain and by V. Jurdjevic), whose underlying symmetric space is $SU(n)/S(U(n-1) \times U(1))$. Following a method introduced by F. Albertini and D. D'Alessandro, we consider the action of $S(U(n-1) \times U(1))$ on $SU(n)$ as a conjugation $X \rightarrow KXK^{-1}$. 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 ($\Lambda$-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级量子系统的最优控制

我们研究了进化算子的时间最优控制问题 $n$级量子系统。对于所考虑的模型，控制将所有能级耦合到给定的能级，并假设受欧几里德范数的限制。由此产生的问题是一个子黎曼$K\hbox{--}P$ 问题，（在 U. Boscain 和 V. Jurdjevic 的文章中介绍），其潜在的对称空间是 $SU(n)/S(U(n-1) \times U(1))$. 遵循 F. Albertini 和 D. D'Alessandro 介绍的方法，我们考虑$S(U(n-1) \times U(1))$ 上 $SU(n)$ 作为共轭 $X \rightarrow KXK^{-1}$. 这允许我们进行对称约简并在商空间上考虑问题。我们对这种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。我们为三级量子系统的情况导出了明确的最优控制，其中所需的操作是在最低的两个能级（$\Lambda$-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，我们针对一组特定的最终数据完全解决该问题。