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arXiv:2006.13001 · Open Systems & Information Dynamics

Basic Properties of a Mean Field Laser Equation

Franco Fagnola, Carlos M. Mora

Published Sep 1, 2019• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s123016121950015x

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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.
We study the dynamics of the solution of a non-linear quantum master equation describing a simple laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and a set of two level atoms that are coupled to two reservoirs. First, we establish the existence of a unique regular stationary state for the non-linear evolution equation under consideration. Second, we examine the free interaction solutions, i.e., the solutions to the non-linear quantum master equation that coincide with unitary evolutions generated by the Hamiltonian resulting from neglecting the interactions between the laser mode, atoms and the bath. We obtain that a family of non-constant free interaction solutions is born at the regular stationary state as a relevant parameter, which is denoted by $C_\mathfrak{b} $, passes through the critical value 111. These free interaction solutions yield the periodic solutions of the Maxwell Bloch equations modeling our physical system in the framework of the semiclassical laser theory. Third, in case $ C_\mathfrak{b} < 1$ we deduce that the system converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. Thus, the quantum system has a Hopf bifurcation at $C_\mathfrak{b} =1 $.
Applying probabilistic techniques we study regularity properties of quantum master equations (QMEs) in the Lindblad form with unbounded coefficients; a density operator is regular if, roughly speaking, it describes a quantum state with finite energy. Using the linear stochastic Schrödinger equation we deduce that solutions of QMEs preserve the regularity of the initial states under a general nonexplosion condition. To this end, we develop the probabilistic representation of QMEs, and we prove the uniqueness of solutions for adjoint quantum master equations. By means of the nonlinear stochastic Schrödinger equation, we obtain the existence of regular stationary solutions for QMEs, under a Lyapunov-type condition.
Applying probabilistic techniques we study regularity properties of quantum master equations (QMEs) in the Lindblad form with unbounded coefficients; a density operator is regular if, roughly speaking, it describes a quantum state with finite energy. Using the linear stochastic Schrödinger equation we deduce that solutions of QMEs preserve the regularity of the initial states under a general nonexplosion condition. To this end, we develop the probabilistic representation of QMEs, and we prove the uniqueness of solutions for adjoint quantum master equations. By means of the nonlinear stochastic Schrödinger equation, we obtain the existence of regular stationary solutions for QMEs, under a Lyapunov-type condition.
A quantum mechanical system consisting of an electromagnetic mode interacting with a set of erbium atoms is considered. For the density matrix of this system, the Gorini-Kossakovsky-Sudarshan-Lindblad master equation was considered in the local approach. From this master equation, equations are obtained that describe the average population of the electromagnetic mode and erbium atoms. For these equations, laser radiation generation curves were obtained and a comparative analysis with rate equations was carried out. It is shown that the presence of terms responsible for spontaneous emission in the resulting equations leads to a smoothing of the laser radiation generation threshold.
Using two equivalent methods, the Heisenberg-Langevin and the master equation, we investigate the properties of a single-emitter nanolaser: an incoherently pumped single two-level system interacting with a single cavity mode of finite finesse. We are interested in a stationary generation regime when the mean number of photons inside the laser cavity is much larger than unity. In this case we linearize the Heisenberg-Langevin equations around the stationary classical solution and obtain analytical results for the linewidth, the quadrature fluctuation spectra, and the intracavity Mandel QQQ parameter. As the second method we use the master equation for the density operator of the laser field. Using the coherent-state representation of the density matrix we obtain the stationary solution for the Glauber PPP function of the laser mode and investigate the quantum statistics of the laser field and its threshold behavior. We compare the results of both methods with numerical simulations and find very good agreement. In the end we use our theoretical results for analyzing experiment with single-ion nanolasers.
The linear and nonlinear properties of propagation of an electromagnetic field through a medium may be derived by the effective elimination of the medium from the coupled Maxwell-Bloch equations. In this paper we suggest, on the other hand, the elimination of the field from the Maxwell-Bloch equations as a means to derive a nonlinear master equation for, e.g., an atomic sample in a laser field. This presents a straightforward and physically intuitive derivation of equations for ``nonlinear atom optics,'' obtained recently by other approaches.