Analysis And Design For Spin 1/2 Systems

This thesis is mainly concerned with the matrix representations of adjoint and anti-adjoint opera-tors for multi-spin 1/2 systems as well as feedback control problems for single-spin 1/2 systems. Thetype of quantum system considered plays an important role not only in the field of physics but also inthe development of quantum control and quantum information processing. Motivated by some recentadvances in the areas which have rendered the introduction of a series of constructive mathematicaltechniques, the primary results obtained in this paper include the following three parts.In the first part, the feedback stabilization problem for single-spin 1/2 systems is discussed froma control theoretic perspective. The noninvasive nature of the bulk measurement allows in principlefor a fully unitary and deterministic closed loop. The Lyapunov-based feedback design presented doesnot require spins that are selectively addressable. With this method, it is possible to obtain approachesto design controllers for single-spin 1/2 systems.In the second part, a multi-index transformation mapping is introduced which establishes therelationship between the spaces gl(2, C)^(?)n and gl(2ⁿ, C). Matrix representations of adjoint as well asanti-adjoint operators on gl(2, C)^(?)n have been defined based on the introduced mapping. Formulasto compute the matrices of adjoint as well as anti-adjoint operators in multi-spin 1/2 systems areobtained by making use of a result on Lie brackets of tensor product matrices. These results not onlyreveal the relationship between the matrices of adjoint as well as anti-adjoint operators on gl(2, C)^(?)nto gl(2ⁿ, C) and matrices of adjoint as well as anti-adjoint operators on gl(2, C) to gl(2, C), but alsoprovide algorithms for computing the matrices of adjoint as well as anti-adjoint operators in multi-spin1/2 systems. Some illustrative examples are also given to show the processes of these computations.In the final part, the dynamics of two-spin 1/2 systems is described in terms of the matrices ofadjoint and anti-adjoint operators for two-spin 1/2 systems. These matrices play an important rolein representing the dynamics of two-spin 1/2 systems. The vectors e and h introduced in this partcan be regarded as the coordinates of the density operatorρand the Hamiltonian H on the basis of{λ_j1j2}j₁,j₂∈I4, respectively. By making use of the coordinates e and h, the density operator equation(Liouville-von Neumann Equation) can be transformed into a form of coordinate differential equation.It can be seen that the dynamic behavior of the coordinate of the density operator can be completelycharacterized by ?, h as well as the matrices {adλ_j1j2}j₁,j₂∈I4 of the adjoint operators adλ_j1j2(·) intwo-spin 1/2 systems and that the matrices {adλ_j1j2}j₁,j₂∈I4 play a central role in characterizing thedynamic behavior of the coordinate ? of the density operatorρ. Based on the coordinate differentialequation obtained, the control and filtering problems of two-spin 1/2 systems can be investigated in the framework of theory of nonlinear control systems.