State transfer and maintenance for non-Markovian open quantum systems in a hybrid environment via Lyapunov control method

This paper concerns state transfer in a non-Markovian two-level open quantum system coupled to a hybrid bath composed of bosons and fermions based on the Lyapunov control method. First, the differential equation of the decay rate factor in the hybrid environment has been achieved and after that, the effect of bath parameters on the system characteristics is investigated to not only disclose the behavior of system dynamics but also reveal that how the bath parameters affect the coherence and purity. Next, a control field has been introduced based on the Lyapunov theory where a trace distance-based exponential form has been suggested as a Lyapunov function. Moreover, a limitation is defined on control laws in order to have control fields with upper and lower bound that is suitable from the practical point of view. Finally, considering bounded and unbounded control fields, numerical simulations have been illustrated for different situations including the state transfer between two eigenstates, between two superposition states, and from a mixed state to a superposition state. The results have been compared with a control field which has been designed by considering the conventional trace distance as a Lyapunov function. The simulation outcomes demonstrate that the proposed control field governs the system state to the desired state with more accuracy and faster convergence rate rather than the conventional method.