论文标题
边界处的量子控制
Quantum Control at the Boundary
论文作者
论文摘要
该论文提出并证明了通过修改其边界条件而不是依靠外部磁场的作用来控制量子系统状态的非标准方法的可行性。量子控制基础的标准方法是使用外部场来操纵系统的标准方法。由于以这种方式控制量子系统时，出现了一些技术困难，这是由于操纵少量粒子在维持量子相关性的同时操纵系统的并发症。结果，系统需要保持在非常低的温度，并且必须非常快地执行相互作用。边界方法处的量子控制与标准量完全不同。通过操纵系统的边界条件来实现控制，而不是通过直接通过外部场与量子系统进行控制。量子系统的频谱，例如一个在盒子中移动的电子，取决于施加的边界条件。因此，这种边界条件的修改会修改系统的状态，允许其操纵并最终控制。这种相互作用较弱，这使人们期望它可能有助于维持量子相关性。为了在边界方法显示量子控制的生存力，引入了量子控制系统（量子Grah的概括）上的边界控制系统家族。在能够解决可控性问题之前，解决了与时间有关的边界条件的Schrödinger方程的存在问题。使用T. Chambrion等人的可控性结果证明了研究系统的近似可控性。 （2009年）和稳定结果构成了本文的另一个原始贡献。
This dissertation presents and prove the viability of a non-standard method for controlling the state of a quantum system by modifying its boundary conditions instead of relying on the action of external fields. The standard approach to quantum control bases on the use of an external field to manipulate the system. Some technological difficulties appear when controlling a quantum system in this way, due to the complications of manipulating a system made of few particles while maintaining the quantum correlations. As a consequence the systems need to be kept at very low temperatures and the interactions have to be performed very fast. The Quantum Control at the Boundary approach is radically different to the standard one. Instead of seeking the control of the quantum system by directly interacting with it through an external field, the control is achieved by manipulating the boundary conditions of the system. The spectrum of a quantum system, for instance an electron moving in a box, depends on the boundary conditions imposed on it. Hence, a modification of such boundary conditions modifies the state of the system allowing for its manipulation and, eventually, its control. This kind of interaction is weaker, which makes one to expect that it may help maintaining the quantum correlations. For showing the viability of the Quantum Control at the Boundary method, a family of boundary control systems on Quantum Circuits (a generalization of quantum grahs) is introduced. Before being able to address the problem of controllability, the problem of existence of solutions for the Schrödinger equation with time-dependent boundary conditions is addressed. The approximate controllability of the systems under study is proven using a controllability result by T. Chambrion et al. (2009) and a stability result which constitutes another original contribution of this dissertation.
