A solution ansatz for Lindblad master equation with a dynamical system theory approach

We study interacting quantum systems coupled to a dissipative Markovian environment. We use an ansatz for the solution of Lindblad master equation of the composite system, which splits the time evolution of the system into two parts, unitary evolution of the wave function of the free system and time evolution of the density matrix coefficients of the composite system mainly controlled by the Lindblad operators. We show that the obtained system of first order linear differential equations, autonomous or otherwise, governing the density matrix coefficients can be treated using the machinery of dynamical system theory to reveal all the information about the system dynamics and its asymptotic behavior, fixed points (steady states) and limiting oscillatory behavior (limit cycle), with minimal amount of work. For the sake of illustration, we apply this technique to an interacting spin-1/2 XXZ system coupled to a dissipative environment, which demonstrates the main aspects of the approach.