Modeling of Quantum Control Systems; Examples
Quantum systems whose dynamics depends on one or more control functions will be named quantum control systems. In a typical situation the objects of control are charged particles while the control functions are appropriately shaped electromagnetic ﬁelds. Therefore, to derive the models, in this chapter we describe elements of the theory of ﬁeld-particle interaction. We start with the classical theory of electrodynamics and show how to obtain the quantum mechanical description through a process called canonical quantization. We present physical examples of (ﬁnite dimensional) quantum control systems of interest in applications and we highlight the fact that such systems have a common mathematical structure, as bilinear systems whose evolution is determined by an element of the Lie group of unitary matrices. In a parallel to classical control theory, we identify a state, an input, and an output in our description of quantum control systems.