Diverse efficiency of observable optimization for four-level quantum systems with higher-order traps

Abstract In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: double Λ like systems including Λ − Λ, V − V , and diamond-dash configurations, and a ladder type degenerate system with the chain interaction. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM) which are performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian—sharp increase of the optimization efficiency up to 100% when initial controls are quite far away from the null control for a V − V system, while much slower and less significant increase (and even small decrease) for a degenerate ladder type system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.