Semiclassical Lindblad master equation for spin dynamics

Open microscopic systems, which interact with an environment, are prevalent in nature and experiments. The contact with the environment leads to energy loss, dissipation and fluctuation. If the relaxation time of the environment is short compared to the typical timescale of the system, a Markovian approximation can be used and quantum dynamics of the open microscopic system is governed by the Lindblad master equation (LME).

In general, the simulation of the LME in the thermodynamic limit takes extremely long time and is numerically demanding. However, classical behavior of open systems often facilitates to understand their quantum dynamics and the nonlinear phenomena observed in experiments. Here, we build a bridge to the classical phenomena with semi-classical master equations derived from the LME. To do so, we formulate the LME in the Wigner-Moyal formalism and derive its semi-classical limit for canonical and non-canonical Poisson brackets using the Moyal-star product. Semiclassical master equations can capture inherent quantum effects in the thermodynamic limit and allow us to identify connections between microscopic and macroscopic scales.