We deal with the dynamical system properties of a Gorini–Kossakowski–Sudarshan–Lindblad equation with mean-field Hamiltonian that models a simple laser by applying a mean-field approximation to a quantum system describing a single-mode optical cavity and a set of two-level atoms, each coupled to a reservoir. We prove that the mean-field quantum master equation has a unique regular stationary solution. In case a relevant parameter \(C_\mathfrak {b} \), i.e., the cavity cooperative parameter, is less than 1, we prove that any regular solution converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. We obtain that a locally exponential stable limit cycle is born at the regular stationary state as \(C_\mathfrak {b} \) passes through the critical value 1. Then, the mean-field laser equation has a Poincaré–Andronov–Hopf bifurcation at \(C_\mathfrak {b} =1 \) of supercritical-like type. Namely, we derive rigorously, at the level of density matrices—for the first time—the transition from a global attractor quantum state, where the light is not emitted, to a locally stable set of coherent quantum states producing coherent light. Moreover, we establish the local exponential stability of the limit cycle in case a relevant parameter is between the first and second laser thresholds appearing in the semiclassical laser theory. Thus, we get that the coherent laser light persists over time under this condition. In order to prove the exponential convergence of the quantum state, as the time goes to \(+ \infty \), we develop a new technique for proving the exponential convergence in open quantum systems that is based on a new variation of constant formula, which is obtained by combining probabilistic techniques with classical arguments from the semigroup theory. Furthermore, applying our main results we find the long-time behavior of the von Neumann entropy, the photon number statistics, and the quantum variance of the quadratures.

我们处理具有均场哈密顿量的Gorini-Kossakowski-Sudarshan-Lindblad方程的动力学系统特性，该方程通过对描述单模光学腔和一组两个的量子系统的量子系统应用均场近似来建模简单的激光级原子，每个原子都耦合到一个容器。我们证明了平均场量子主方程具有唯一的正则平稳解。如果相关参数\（C_ \ mathfrak {b} \），即腔合作参数小于1，我们证明了任何正则解都以指数形式快速收敛到平衡，因此正则稳态是一个整体渐近稳定的平衡解。我们得到一个局部指数稳定极限环在正态静止状态下产生\（C_ \ mathfrak {b} \）穿过临界值1。然后，平均场激光方程在类似于超临界的\（C_ \ mathfrak {b} = 1 \）处具有Poincaré–Andronov–Hopf分叉类型。也就是说，我们是在密度矩阵级别上严格地得出的，这是第一次，从不发光的全局吸引子量子态到局部稳定的一组相干量子态的过渡，产生相干光。此外，在相关参数介于半经典激光理论中出现的第一激光阈值和第二激光阈值之间的情况下，我们建立了极限环的局部指数稳定性。因此，我们得出相干激光在这种情况下会随时间持续存在。为了证明量子态的指数收敛性，随着时间的流逝，\（+ \ infty \），我们基于常数公式的新变体，开发了一种用于证明开放量子系统中指数收敛的新技术，该常数变体是通过将概率技术与半群论的经典论证相结合而获得的。此外，利用我们的主要结果，我们发现了冯·诺依曼熵的长期行为，光子数统计和正交的量子方差。