Using and Optimizing Time-Dependent Decoherence Rates and Coherent Control for a Qutrit System

We consider an open qutrit system in which the evolution of the density matrix $\rho (t)$ is governed by the Gorini–Kossakowski–Sudarshan–Lindblad master equation with simultaneous coherent (in the Hamiltonian) and incoherent (in the dissipation superoperator) controls. To control the qutrit, we propose to use not only coherent control but also generally time-dependent decoherence rates which are adjusted by the so-called incoherent control. In our approach, the incoherent control makes the decoherence rates time-dependent in a specific controlled manner and within a clear physical mechanism. We consider the problem of maximizing the Hilbert–Schmidt overlap between the final state $\rho (T)$ of the system and a given target state $\rho _{\textup {target}}$, as well as the problem of minimizing the squared Hilbert–Schmidt distance between these states. For both problems, we perform their realifications, derive the corresponding Pontryagin functions, adjoint systems (with two variants of transversality conditions for the two terminal objectives), and gradients of the objectives, and adapt the one-, two-, and three-step gradient projection methods. For the problem of maximizing the overlap, we also adapt the regularized first-order Krotov method. In the numerical experiments, we analyze first the operation of the methods and second the obtained control processes, in respect of considering the environment as a resource via incoherent control.