We derive the semi-classical Lindblad master equation in phase space for both canonical and non-canonical Poisson brackets using the Wigner–Moyal formalism and the Moyal star-product. The semi-classical limit for canonical dynamical variables, i.e. canonical Poisson brackets, is the Fokker–Planck equation, as derived before. We generalize this limit and show that it holds also for non-canonical Poisson brackets. Examples are gyro-Poisson brackets, which occur in spin ensembles, systems of recent interest in atomic physics and quantum optics. We show that the equations of motion for the collective spin variables are given by the Bloch equations of nuclear magnetization with relaxation. The Bloch and relaxation vectors are expressed in terms of the microscopic operators: the Hamiltonian and the Lindblad functions in the Wigner–Moyal formalism.

我们使用 Wigner-Moyal 形式主义和 Moyal 星积推导出正则和非正则泊松括号在相空间中的半经典 Lindblad 主方程。规范动力学变量的半经典极限，即规范泊松括号，是前面推导出来的福克-普朗克方程。我们概括了这个限制并表明它也适用于非规范泊松括号。例子是陀螺-泊松括号，它出现在自旋系综、原子物理学和量子光学最近感兴趣的系统中。我们表明集体自旋变量的运动方程由具有弛豫的核磁化的布洛赫方程给出。Bloch 和松弛向量用微观算子表示：Wigner-Moyal 形式主义中的哈密顿量和 Lindblad 函数。