  Reservoir engineering has proven to be a practical approach to control open quantum systems, preserving quantum coherence by appropriately manipulating the reservoir and system-reservoir interactions. In this context, for systems comprised of different parts, it is common to describe the dynamics of a subsystem of interest by making an adiabatic elimination of the remaining components of the system. This procedure often leads to an effective master equation for the subsystem that is not in the well-known Gorini-Kossakowski-Lindblad-Sudarshan form (here called standard Lindblad form). Instead, it has a more general structure (here called generalized Lindblad form), which explicitly reveals the dissipative coupling between the various components of the subsystem. Moreover, for systems weakly coupled to a reservoir, the presence of counter-rotating terms in the interaction Hamiltonian or the nonstationarity of the reservoir state guarantees that the master equation describing the system of interest will be of the generalized Lindblad form. In this work, we present a set of dynamical equations for the first and second moments of the canonical variables for linear systems, bosonic and fermionic, described by generalized Lindblad master equations. Our method is efficient and allows one to obtain analytical solutions for the steady state. Further, we include as a review some covariance matrix methods for which our results are particularly relevant, paying special attention to those related to the measurement of entanglement. Finally, we prove that the Duan criterion for entanglement is also applicable to fermionic systems. 
Generalized Lindblad master equations in quantum reservoir engineering