Lindblad Quantum Dynamics from Correlation Functions of Classical Spin Chains

The Lindblad quantum master equation is one of the central approaches to the physics of open quantum systems. In particular, boundary driving enables the study of transport, where a steady state emerges in the long-time limit, which features a constant current and a characteristic density profile. While the Lindblad equation complements other approaches to transport in closed quantum systems, it has become clear that a connection between closed and open systems exists in certain cases. Here, we build on this connection for magnetization transport in the spin-1/2 XXZ chain with and without integrability-breaking perturbations. Specifically, we study the question whether the time evolution of the open quantum system can be described on the basis of classical correlation functions, as generated by the Hamiltonian equations of motion for real vectors. By comparing to exact numerical simulations of the Lindblad equation, we find a good accuracy of such a description for a range of model parameters, which is consistent with previous studies on closed systems. While this agreement is an interesting physical observation, it also suggests that classical mechanics can be used to solve the Lindblad equation for comparatively large system sizes, which lie outside the possibilities of a quantum mechanical treatment. We also point out counterexamples and limitations for the quantitative extraction of transport coefficients. Remarkably, our classical approach to large open systems allows to detect superdiffusion at the isotropic point.