Journal of the Physical Society of Japan

Microscopic Theory of Laser Master Equation up to the Fourth Order. I

Toshihico Arimitsu, Tetsuo Tominaga

Published Oct 15, 1982• 8 citations

Abstract

A laser master equation is derived microscopically by using the time-convolution-less formulation of the damping theory up to the fourth-order terms with respect to the coupling constant between the laser field and the atoms, in order to investigate the quantum effect and the system size (number of the active atoms) dependence of the kinetic operator . The master equation is compared with the equation given by the time-convolution formalism which is essentially equal to the master equation derived by Haake.

DOI: 10.1143/jpsj.51.3102

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Based on the time-convolution-less formulation of the damping theory, time evolutions of the photon intensity and its fluctuation of a laser model with N identical two-level atom systems are investigated for various situations from a quantum-statistical mechanical point of view. The energy exchange effect between the photon field and atom systems on the time evolutions reveals to be well described by the formulation even the atom system is eliminated adiabatically.

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Journal of the Physical Society of Japan

Microscopic Theory of Laser Master Equation up to the Fourth Order. I

Toshihico Arimitsu, Tetsuo Tominaga

Published Oct 15, 1982• 8 citations

Publisher

Abstract

DOI: 10.1143/jpsj.51.3102

Related papers

Microscopic Theory of Laser Master Equation up to the Fourth Order. II

Tetsuo Tominaga, Toshihico Arimitsu

Journal of the Physical Society of JapanJan 1, 19845 citations

A Microscopic Laser Theory. I. Basic Formulation

Toshihico Arimitsu, Fumiaki Shibata

A basic formulation of the laser system is given in the form most convenient to study the time-dependent cooperative ( symmetry breaking ) behavior of the system in the far-from-equilibrium states from the microscopic and fully quantum-statistical-mechanical point of view. The quantum mechanical master equation, derived from a microscopic model Hamiltonian, is mapped onto a c-number function space with the use of the Boson and spin coherent states representations. The distribution function (mapped from the density operator) is expanded in terms of the Hermite polynomials and the spherical harmonics to obtain differential equations for the expansion coefficients with which time evolution of the laser system can be determined completely.A basic formulation of the laser system is given in the form most convenient to study the time-dependent cooperative ( symmetry breaking ) behavior of the system in the far-from-equilibrium states from the microscopic and fully quantum-statistical-mechanical point of view. The quantum mechanical master equation, derived from a microscopic model Hamiltonian, is mapped onto a c-number function space with the use of the Boson and spin coherent states representations. The distribution function (mapped from the density operator) is expanded in terms of the Hermite polynomials and the spherical harmonics to obtain differential equations for the expansion coefficients with which time evolution of the laser system can be determined completely.

Journal of the Physical Society of JapanMar 1, 198313 citations

Quantum Theory of a Laser Model

C. R. Willis

We derive the kinetic equations for the coupled single-particle density matrix $\ensuremath{\rho}$ and the electromagnetic density matrix $R$ to lowest order in the dimensionless coupling constant ${\ensuremath{\beta}}^{2}\ensuremath{\equiv}{(\frac{{\ensuremath{\omega}}_{L}}{{\ensuremath{\omega}}_{D}})}^{2}$. The laser frequency ${\ensuremath{\omega}}_{L}$ is ${(4\ensuremath{\pi})}^{\ensuremath{-}\frac{1}{2}}{(\mathfrak{N}{r}_{0}{\ensuremath{\lambda}}^{2})}^{\frac{1}{2}}{\ensuremath{\omega}}_{0}$ where $\mathfrak{N}$ is the number of two-level systems per unit volume, ${r}_{0}$ is the classical electron radius, $\ensuremath{\lambda}$ is the wavelength of the radiation, and $k{\ensuremath{\omega}}_{0}$ is the two-level energy difference. The Doppler frequency ${\ensuremath{\omega}}_{D}$ characterizes the center-of-mass motion. For gas lasers ${\ensuremath{\beta}}^{2}$ is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is ${\ensuremath{\beta}}^{2}$.We derive the kinetic equations for the coupled single-particle density matrix $\ensuremath{\rho}$ and the electromagnetic density matrix $R$ to lowest order in the dimensionless coupling constant ${\ensuremath{\beta}}^{2}\ensuremath{\equiv}{(\frac{{\ensuremath{\omega}}_{L}}{{\ensuremath{\omega}}_{D}})}^{2}$. The laser frequency ${\ensuremath{\omega}}_{L}$ is ${(4\ensuremath{\pi})}^{\ensuremath{-}\frac{1}{2}}{(\mathfrak{N}{r}_{0}{\ensuremath{\lambda}}^{2})}^{\frac{1}{2}}{\ensuremath{\omega}}_{0}$ where $\mathfrak{N}$ is the number of two-level systems per unit volume, ${r}_{0}$ is the classical electron radius, $\ensuremath{\lambda}$ is the wavelength of the radiation, and $k{\ensuremath{\omega}}_{0}$ is the two-level energy difference. The Doppler frequency ${\ensuremath{\omega}}_{D}$ characterizes the center-of-mass motion. For gas lasers ${\ensuremath{\beta}}^{2}$ is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is ${\ensuremath{\beta}}^{2}$.

Physical ReviewJul 8, 196622 citations

Quantum-optical master equations: An interaction picture

Berthold‐Georg Englert, M. Naraschewski, Axel Schenzle

Quantum-optical master equations---exemplified by the Jaynes-Cummings model with damping---are turned into numerical partial differential equations of first order for phase-space functions, which are generalizations of the Wigner function and its relatives. The time dependence of these phase-space functions originates solely in the atom-photon interaction; all other time dependences, in particular the dissipative contribution of the photon damping, are accounted for by the time-dependent operator bases to which the phase-space functions refer. The judicious choice of operator basis also effects the absence of second-order derivatives in the partial differential equation. Our first-order equations are hyperbolic and can be integrated conveniently along their characteristics. As an illustrative application we study how the Jaynes-Cummings revivals are affected by photon damping. We show how to handle squeezed reservoirs and how to apply the method to laser cooling.Quantum-optical master equations---exemplified by the Jaynes-Cummings model with damping---are turned into numerical partial differential equations of first order for phase-space functions, which are generalizations of the Wigner function and its relatives. The time dependence of these phase-space functions originates solely in the atom-photon interaction; all other time dependences, in particular the dissipative contribution of the photon damping, are accounted for by the time-dependent operator bases to which the phase-space functions refer. The judicious choice of operator basis also effects the absence of second-order derivatives in the partial differential equation. Our first-order equations are hyperbolic and can be integrated conveniently along their characteristics. As an illustrative application we study how the Jaynes-Cummings revivals are affected by photon damping. We show how to handle squeezed reservoirs and how to apply the method to laser cooling.

Physical Review ASep 1, 199443 citations

Quantum statistical approach to the theory of random lasers

Vitalie Eremeev, Miguel Orszag

"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.""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."

Revista Mexicana de FísicaJan 1, 20110 citations

Reduction of a Boson Subsystem up to the Lowest Order

Toshihico Arimitsu

Derivation of a master equation is given in the case where a Boson subsystem can be eliminated besides reservoirs by using the time-convolution-less formulation of the damping theory within the conventional treatment, up to the lowest order expansion of the “kinetic coefficient” with respect to the coupling constant between the Boson subsystem and the remaining ones. Useful diagramatic expansion of Liouville operators, formula for the elimination of Boson operators and algebraic formulae of the spin coherent representation are given in Appendices.Derivation of a master equation is given in the case where a Boson subsystem can be eliminated besides reservoirs by using the time-convolution-less formulation of the damping theory within the conventional treatment, up to the lowest order expansion of the “kinetic coefficient” with respect to the coupling constant between the Boson subsystem and the remaining ones. Useful diagramatic expansion of Liouville operators, formula for the elimination of Boson operators and algebraic formulae of the spin coherent representation are given in Appendices.

Journal of the Physical Society of JapanFeb 1, 19827 citations

We derive the kinetic equations for the coupled single-particle density matrix $\ensuremath{\rho}$ and the electromagnetic density matrix

R

to lowest order in the dimensionless coupling constant ${\ensuremath{\beta}}^{2}\ensuremath{\equiv}{(\frac{{\ensuremath{\omega}}_{L}}{{\ensuremath{\omega}}_{D}})}^{2}$. The laser frequency ${\ensuremath{\omega}}_{L}$ is ${(4\ensuremath{\pi})}^{\ensuremath{-}\frac{1}{2}}{(\mathfrak{N}{r}_{0}{\ensuremath{\lambda}}^{2})}^{\frac{1}{2}}{\ensuremath{\omega}}_{0}$ where

\mathfrak{N}

is the number of two-level systems per unit volume,

{r}_{0}

is the classical electron radius, $\ensuremath{\lambda}$ is the wavelength of the radiation, and $k{\ensuremath{\omega}}_{0}$ is the two-level energy difference. The Doppler frequency ${\ensuremath{\omega}}_{D}$ characterizes the center-of-mass motion. For gas lasers ${\ensuremath{\beta}}^{2}$ is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is ${\ensuremath{\beta}}^{2}$.