Quantum Control Landscapes Beyond the Dipole Approximation: Controllability, Singular Controls, and Resources

We investigate the control landscapes of closed n-level quantum systems beyond the dipole approximation by including a polarizability term in the Hamiltonian. The latter term is quadratic in the control field. Theoretical analysis of singular controls is presented, which are candidates for producing landscape traps. The results for considering the presence of singular controls are compared to their counterparts in the dipole approximation (i.e., without polarizability). A numerical analysis of the existence of traps in control landscapes for generating unitary transformations beyond the dipole approximation is made upon including the polarizability term. An extensive exploration of these control landscapes is achieved by creating many random Hamiltonians which include terms linear and quadratic in a single control field. The discovered singular controls are all found not to be local optima. This result extends a great body of recent work on typical landscapes of quantum systems where the dipole approximation is made. We further investigate the relationship between the magnitude of the polarizability and the fluence of the control resulting from optimization. It is also shown that including a polarizability term in an otherwise uncontrollable dipole coupled system removes traps from the corresponding control landscape by restoring controllability. We numerically assess the effect of a polarizability term on a known example of a particular three-level Λ-system with a second order trap in its control landscape. It is found that the addition of the polarizability removes the trap from the landscape. The general practical control implications of these simulations are discussed.

On quasi-singular controls in discrete systems with delay

Сconsider an optimal control problem described by a system of differential controls with a delaying argument and a multipoint performance functional under the assumption that the control domain is convex. A number of integral and multipoint necessary optimality conditions in the case of degeneration of the linearized maximum condition are established.

Necessary conditions for quasi-singular controls in the stochastic optimal control problem with delayed argument

Рассмотрена задача оптимального управления, математические модели которых задаются нелинейными стохастическими дифференциальными уравнениями Ито с запаздывающим аргументом и диффузными компонентами, позволяющими учитывать действующие на систему случайные возмущения непрерывной природы. В предположении выпуклости области допустимого управления получено линеаризованное необходимое условие оптимальности. Исследован квазиособый случай. Описаны общие необходимые условия оптимальности квазиособых управлений. Рассмотрены частные случаи.

Necessary optimality condition for the singular controls in an optimal control problem with nonlocal conditions

In this paper, we continue investigation of the problem considered in our earlier works. The paper deals with an optimal control problem for an ordinary differential equation with integral boundary conditions that generalizes the Cauchy problem. The problem is investigated the case when Pontryagin?s maximum principle is degenerate. Moreover, the second order optimality conditions are derived for the considered problem.