Multiple solutions in the tracking control of quantum systems

This paper demonstrates the existence of multiple solutions at each time point in tracking control of quantum systems. These solutions are shown to arise from the nonlinear dependence of the short-time propagators U(t+δt,t) on the control field. The multiplicity of solutions depends on the parameters of the controlled system and the nature of the imposed track. Multiple solutions necessitate that a choice be made at each time point, resulting in an exponentially expanding space of distinct control fields that maintain the prescribed track. This behavior is illustrated by application to a small model system. The presence of multiple tracking control fields is consistent with behavior observed from quantum control landscape theory.