Assessing optimality and robustness of control over quantum dynamics

Abstract

This work presents a general framework for assessing the quality and robustness of control over quantum dynamics induced by an optical field $\mathcal{E}(t).$ The control process is expressed in terms of a cost functional, including the physical objectives, penalties, and constraints. The first variations of such cost functionals have traditionally been utilized to create designs for the controlling electric fields. Here, the second variation of the cost functional is analyzed to explore (i) whether such solutions are locally optimal, and (ii) their degree of robustness. Both issues may be assessed from the eigenvalues of the stability operator S whose kernel $K(t,\ensuremath{\tau})$ is related to $\ensuremath{\delta}\mathcal{E}(t)/\ensuremath{\delta}\mathcal{E}(\ensuremath{\tau}){|}_{c}$ for $0<t,$ $\ensuremath{\tau}<~T,$ where $T$ is the target control time. Here $c$ denotes the constraint that the field satisfies the optimal control dynamical equations. The eigenvalues \ensuremath{\sigma} of S satisfying $\ensuremath{\sigma}<1$ assure local optimality of the control solution, with $\ensuremath{\sigma}=1$ being the critical value separating optimal solutions from false solutions (i.e., those with negative second variational curvature of the cost functional). In turn, the maximally robust control solutions with the least sensitivity to field errors also correspond to $\ensuremath{\sigma}=1.$ Thus, sufficiently high sensitivity of the field at one time $t$ to the field at another time \ensuremath{\tau} (i.e., $\ensuremath{\sigma}>1$) will lead to a loss of local optimality. An expression is obtained for a bound on the stability operator, and this result is employed to qualitatively analyze control behavior. From this bound, the inclusion of an auxiliary operator (i.e., other than the target operator) is shown to act as a stabilizer of the control process. It is also shown that robust solutions are expected to exist in both the strong- and weak-field regimes.