Fully differentiable optimization protocols for non-equilibrium steady states

In the case of quantum systems interacting with multiple environments, the time-evolution of the reduced density matrix is described by the Liouvillian. For a variety of physical observables, the long-time limit or steady state (SS) solution is needed for the computation of desired physical observables. For inverse design or optimal control of such systems, the common approaches are based on brute-force search strategies. Here, we present a novel methodology, based on automatic differentiation, capable of differentiating the SS solution with respect to any parameter of the Liouvillian. Our approach has a low memory cost, and is agnostic to the exact algorithm for computing the SS. We illustrate the advantage of this method by inverse designing the parameters of a quantum heat transfer device that maximizes the heat current and the rectification coefficient. Additionally, we optimize the parameters of various Lindblad operators used in the simulation of energy transfer under natural incoherent light. We also present a sensitivity analysis of the SS for energy transfer under natural incoherent light as a function of the incoherent-light pumping rate.