ASSESSING OPTIMALITY AND ROBUSTNESS OF CONTROL OVER QUANTUM DYNAMICS

This work presents a general framework for assessing the quality and r obustness of control over quantum dynamics induced by an optical field epsilon(t). The control process is expressed in terms of a cost funct ional, 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 st ability operator S whose kernel K(t, tau) is related to delta epsilon( t)/delta epsilon(tau)\(c) for 0<t, tau less than or equal to T, where T is the target control time. Here c denotes the constraint that the f ield satisfies the optimal control dynamical equations. The eigenvalue s sigma of S satisfying sigma<1 assure local optimality of the control solution, with sigma=1 being the critical value separating optimal so lutions from false solutions (i.e., those with negative second variati onal curvature of the cost functional). In turn, the maximally robust control solutions with the least sensitivity to field errors also corr espond to sigma=1. Thus, sufficiently high sensitivity of the field at one time t to the field at another time tau(i.e., 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 qualitativel y analyze control behavior. From this bound, the inclusion of an auxil iary operator (i.e., other than the target operator) is shown to act a s a stabilizer of the control process. It is also shown that robust so lutions are expected to exist in both the strong-and weak-field regime s.