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Journal of Radio Electronics

Model of an open quantum system for describing the operation of an erbium laser

I. V. Vovchenko, Kotelnikov IRE RAS, 125009, Russia, Moscow, Mokhovaya, 11 b.7, Evgenii S. Andrianov, A. V. Dorofeenko, Kotelnikov IRE RAS, 125009, Russia, Moscow, Mokhovaya, 11 b.7

Published Apr 1, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.30898/1684-1719.2024.4.6

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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.
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 have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
We derive the equation of motion for the quantum-mechanical radiation density matrix of a gas laser to lowest order in the dimensionless coupling constant. Our derivation is fully quantum mechanical and we can calculate the coherence properties from the radiation density matrix. Our model consists of NNN two-level systems interacting with radiation in a cavity in the presence of dissipation, pumping, and collisions. The method we use is a generalization of the Bogoliubov derivation of the kinetic equation for a small parameter. Our derivation holds for any physically realizable pump power, and near threshold reduces to Lamb's near-threshold theory. With the equation of motion for the radiation density matrix, we obtain solutions both for when the average field is nonzero and for when it is zero. The steady-state electromagnetic density is the same in both cases except for a small spontaneous-emission term. We show that the reason gas lasers do not satisfy rate equations is the existence of zeroth-order correlations between the internal atomic variables and atomic center-of-mass variables. It is these same zeroth-order correlations which are responsible for the Lamb dip. Our derivation includes collisions and reduces the calculations of their effect to quadrature.
"We propose to study the random laser problem. As a lasing system one considers an active gain medium interacting with a light field where the damping of the latter is treated in the spirit of open systems theory, considering a multimode case with the spectral overlap of the modes in the losses term [11]. We approach the problem by using the density matrix formalism, deriving the master equation for the light field in order to find out the detailed description of the laser like emission. The specific character of the field damping and gain effects essentially modifies the master equation in comparison to that of the ordinary laser and thus contributes to the properties of the resulting laser emission. The derivation of the density operator for the field is the main ingredient for describing the quantum statistical properties of the lasing and such a progress represents a great interest for the study of random laser systems."