Control Landscape Topology for Nonlinear Quantum Dynamics

Abstract

Manipulation of a quantum system can be viewed in the framework of optimal control theory, with the system possessing a control landscape defined as the physical observable as a functional of the controls. Such analyses have thus far been limited to linear quantum dynamics. This thesis aims to extend analysis of the control landscape topology to nonlinear quantum dynamics, considering in particular the objective of steering the system from an initial state to a final target state for finite-level quantum systems. Under basic assumptions that the control is unconstrained, that the final state is reachable from the initial state, and that the endpoint map from control space to state space is surjective, it is shown that the landscape critical points (where the slopes vanish) for nonlinear quantum dynamics only appear as the global maximum and minimum; thus, the landscape is free of traps that could impede optimization algorithms in search for the global maximum. Moreover, the landscape Hessian (the second-order derivative) at the global maximum has finite rank, implying that the optimal solutions are robust to noise due to the presence of a level set of optimal controls that preserve the value of the maximum. Extensive numerical simulations on the Gross-Pitaevskii equation in a particle-in-a-box basis, using an external electric field and the Feshbachtuned scattering length as controls, confirm the trap-free nature of the landscape and the Hessian rank analysis, as well as demonstrate the feasibility of applying standard landscape exploration techniques to nonlinear Schrëodinger equations. These results are a generalization of previous findings for linear Schrëodinger equations, and show promise for successful control of a wide range of nonlinear quantum dynamics.