Quantum Control for Trapped Particles at Matter Surface

In this theoretic work, trapped particles at matter surface to be considered as target system of quantum control. At the framework of variational method in Hilbert space, it would be quite interesting for us to explore particles which is trapped via optical lattice or other kinds of constraints at a matter surface (metal, crystal). The aim of this task is to survey theoretical control for quantum particles as they are appeared and trapped at matter surface (cf. [1]). The physical background of this work is laying on the specified particles motion or reaction under a certain chemical surface. As is well known, one can move a particle at surface smoothly through a point force above it or according to a proper angle, such quantum mechanical motion had already been achieved by the IBM team several years ago. At the viewpoint of quantum control, what is theoretic support? Can we make these control theoretically, computationally or experimentally? In fact, free trapped particles had been considered by scientists and researchers at worldwide scale. The most exciting things in this study is to take particle as target as it constrained on a surface. Theoretically, this work is to describe quantum control system consisting of time-varying Schrodinger equation at physical constraints condition. Then to apply control theory to quantum system of trapped particles, find and characterize optimal quantum control. Further, to compose optimality system (Euler-Lagrange system). Comprising of control free trapped particle, this work is focusing on control taking place at matter surface (on it particle is trapped), that is, try to discuss the external force constrain (e.g. optical lattice) and surface constrain are acting at particle together. Amazing result is desired in control of different multi-forces as control inputs, what would be happened as a particle changing its position, displacement or status under trapped situation? can we make a trapped chemical quantum well, or a physical optical lattice which worked using external force? what is extension of such kind of works at a variety of fields? whether the general quantum control is worked in this case? It is the purpose to solve these mysteries in this work, and report the initial conclusion of theoretic aspect for trapped particle at matter surface.