We study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schrödinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation.

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平均场激光方程的基本性质

我们研究了在平均场近似下描述激光的非线性量子主方程。量子系统由一个单模光腔和两个能级原子组成，它们与水库相互作用。即，我们建立了所考虑的非线性算子方程的正则解的存在性和唯一性，并且我们得到了该解的概率表示，该解用平均场随机薛定谔方程表示。为此，我们找到了Gorini-Kossakowski-Sudarshan-Lindblad形式的非自治线性量子主方程的正则解，并证明了Gorini-Kossakowski-Sudarshan-Lindblad形式的非自治线性伴随量子主方程解的唯一性形式。此外，我们从平均场激光方程严格地获得了 Maxwell-Bloch 方程。