We consider the problem of population transfer optimal control for a quantum Lambda system where the control couples pairwise only the lowest two energy levels to the highest level. The cost to be minimized expresses a compromise between minimizing the energy of the control and the average population in the highest level (occupancy), which is the one mostly subject to decay. Such a problem admits a group of symmetries, that is, a Lie group acting on the state space, which leaves dynamics, cost and initial and final conditions unchanged. By identifying a splitting of the tangent bundle into a vertical (tangent to the orbits) and horizontal (complementary) subspace at every point (a ), we develop a technique. In this setting, the problem reduces to a problem on the sphere for which we derive several properties and provide a practical method for the solution. We also describe an explicit numerical example.

中文翻译：

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具有占用成本的量子 lambda 系统的最优控制

我们考虑量子 Lambda 系统的群体转移最优控制问题，其中控制仅将最低的两个能级与最高能级成对耦合。要最小化的成本表示最小化控制能量与最高级别（占用率）的平均人口之间的折衷，这是最容易衰减的。这样的问题承认一组对称性，即作用于状态空间的李群，这使得动力学、成本以及初始和最终条件保持不变。通过识别切束在每个点 (a ) 处分裂为垂直（与轨道相切）和水平（互补）子空间，我们开发了一种技术。在这种情况下，问题简化为球体上的问题，我们为此推导了几个属性并提供了解决方案的实用方法。我们还描述了一个明确的数值示例。