Quantum observable homotopy tracking control

This paper presents a new tracking method where the target observable 〈O (s,T) 〉 at the final dynamical time T follows a predefined track P (s) with respect to a homotopy tracking variable s≥0. The procedure calculates the series of control fields E (s,t) required to accomplish observable homotopy tracking by solving a first-order differential equation in s for the evolution of the control field. Controls produced by this technique render the desired track for all s without encountering field singularities. This paper also extends the technique to the case where the field-free Hamiltonian and dipole moment operator change with s in order to explore the control of new physical systems along the track. Several simulations are presented illustrating the various uses for this quantum tracking control technique.