Global Stabilization of N-dimensional Quantum Spin Systems via Continuous Feedback

In this paper, a stabilization problem of quantum spin systems in general dimension under continuous measurement is considered and it is shown that the global stabilization at an eigenstate by continuous inputs is possible. Quantum states of spin systems under continuous measurement by mutual interference with laser beams can be continuously estimated by quantum filtering and with its information, the intension of magnetic field, which is applied to atoms, can be controlled. Because of the structural symmetry of the nonlinear dynamics, this stabilization problem by continuous feedback has been considered to be hard and recently a switching rule was introduced which attains the global stability. Our proposing control rule is the sum of two terms: a term which attracts the quantum states to a target and the other term which draw apart from the other equilibrium points. The proof is done by the strict analysis on the sample paths of solutions of a stochastic differential equation. We also show numerical examples to demonstrate the efficiency of the proposing control rule.