Observable-preserving control of quantum dynamics over a family of related systems

Quantum control aims at the manipulation of atomic- and molecular-scale dynamics phenomena. An important objective in this regard is the understanding of dynamical control within a family of related quantum systems. To explore this issue, diffeomorphic changes in the system Hamiltonian H(s,t) are introduced by scanning over a homotopy parameter s and then monitoring the control field response needed to maintain the value of a specified target observable. This operation is implemented through a procedure referred to as diffeomorphic modulation under observable-response-preserving homotopy (D-MORPH). The governing D-MORPH differential equation determining the control laser field E(s,t) is shown to explicitly allow for innumerable solutions, with each characterized by the choice of an arbitrary function f(s,t) of s and time t. The presence of f(s,t) in the D-MORPH differential equation makes clear the origin of multiple control fields that produce the same observable objective. A stable algorithm is presented for practical execution of D-MORPH with the only criterion that the Hamiltonian H(s,t) permit reaching the objective over the full domain of s being sampled. Both analytic and numerical examples are presented to illustrate the D-MORPH concept.