Some mathematical topics in control of quantum systems

Mathematical analysis of problems related to control of atomic, molecular, and nano-scale systems with quantum dynamics forms an active branch of modern mathematical physics with applications in physics, chemistry and biology. The dynamics of the controlled quantum system is governed either by Schrödinger equation (if the system is isolated from the environment) or by a master-equation (if the system interacts with an environment). In both cases the evolution equation depends on the control function. The goal of the optimal control is to find a control function which maximizes a desired objective functional. Some quantum control problems involve analysis of extrema of certain functions over Stiefel manifolds.
A significant progress in the mathematical analysis of quantum control has been obtained in recent years. In this talk we will discuss the following results of the author and colleagues.
1) Solution to the important problem of proving existence of second-order traps — critical controls where Hessian of the objective is negative semidefinite but which are not global maxima; jointly with D. J. Tannor, Weizmann Institute of Science.
2) Calculation of all critical points and proof of the absense of local maxima and minima for quantum control objectives over complex Stiefel manifolds appearing in a wide class of quantum control problems; these results have found applications in chemistry; jointly with R. Wu, D. Prokhorenko, H. Rabitz, Princeton University.
3) Solution to the problem of approximate realization of complete density matrix controllability for open quantum systems which involves a development of a method for approximate engineering of arnitrary pure and mixed quantum states (i.e., arbitrary density matrices).

References

1. A. Pechen, Phys. Rev. A, 84 (2011), 042106        
2. A. Pechen, D. J. Tannor, Phys. Rev. Lett., 106 (2011), 120402        
3. A. Pechen, D. Prokhorenko, R. Wu, H. Rabitz, J. Phys. A: Math. Gen., 41 (2008), 045205