Stabilizing open quantum system dynamics and entanglement with quantum reservoir engineering by direct minimization

Non-equilibrium steady states are important in quantum many-body physics, statistical physics, and quantum information science, due to their properties in long time limit, which allows us to reconstruct the dissipative dynamics, and access the information of quantum fluctuations. Weak interactions between system and environment in open quantum systems can result in stationary states, and tuning of the system Hamiltonian and environmental interactions provides a way to engineer the properties of those states. Here, we present an optimization approach to determine steady states of open quantum systems whose dynamics are governed by the Lindblad master equation via semi-definite programming in Hilbert space. Additionally, we show how to determine optimal steady-state properties of Lindbladian dissipations, including finding optimal entanglement in various models under experimentally relevant parameter regimes. We demonstrate our results in a variety of physical models, including Dicke, Kitaev chain, and others.