Level sets of quantum control landscapes

A controlled quantum system possesses a search landscape defined by the observable value as a functional of the control field. Within the search landscape, there exist level sets of controls giving the same observable value. This paper focuses on level sets of the transition probability P_i¿f. For transition probabilities 0 < P_i¿f < 1, a first order diffeomorphic modulation observable response preserving homotopy (D-MORPH) algorithm is utilized to investigate level sets. At the top of the control landscape, P_i¿f = 1, a second order D-MORPH algorithm is presented that can explore the perfect control level set. D-MORPH is utilized to identify level set members that exhibit certain desirable secondary characteristics, e.g., minimal pulse fluence. Numerical simulations for finite level systems are presented to illustrate the variety of control behavior found across level set members.