Bulletin de la Classe des sciences

Derivation of kinetic equation, up to order λ6, for the interaction between radiation and a two-level system. Part I

Françoise Hénin, Michel De Haan

Published Jan 1, 1968• 0 citations

Abstract

In the bare particle representation, the kinetic operator Ψ(z) and its first derivatives play a very important role in the asymptotic equation for the evolution of the diagonal elements of the density matrix. Explicit expressions for these operators are derived, up to order λ6, for the interaction between radiation and a two level system. The transformation to the dressed particle distribution function will be given in a next paper.

DOI: 10.3406/barb.1968.62171

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Derivation of kinetic equation, up to order λ6, for the interaction between radiation and a two-level system. Part II

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Quantum Theory of a Laser Model

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Kinetic equations for the density matrix of two-level particles in a strong field

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On the basis of the Green’s function method for nonequilibrium systems, the kinetic equation for two-level particles in a strong field is derived. In the binary approximation, all diagrams are expressed in terms of the reducible vertex part, whose calculation is reduced to the problem of two-channel scattering and finding the corresponding scattering amplitudes.

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Bulletin de la Classe des sciences

Derivation of kinetic equation, up to order λ6, for the interaction between radiation and a two-level system. Part I

Françoise Hénin, Michel De Haan

Published Jan 1, 1968• 0 citations

Publisher

Abstract

DOI: 10.3406/barb.1968.62171

Related papers

Derivation of kinetic equation, up to order λ6, for the interaction between radiation and a two-level system. Part II

Françoise Hénin, Michel De Haan

Bulletin de la Classe des sciencesJan 1, 19680 citations

Comparison between kinetic theory and wave function formalism in interaction between a two level system and radiation

Françoise Hénin

The meaning of perturbed states in the kinetic theory is first discussed. It is shown that the canonical form of the density matrix for a particle in a perturbed level does not correspond to a pure case, in the bare particle representation. The problem of spontaneous emission by an atom in its unperturbed excited state is then discussed. We first show that the results of the kinetic theory can also be obtained by the level shift operator formalism if one goes beyond the usual approximation. It is also shown that these results give a modified Lorentzian shape. Finally, we consider the resonant scattering of a photon. The solution of the kinetic equation gives us the transition probability per unit time. The frequency dependence is given by a modified Lorentz function, analogous to that obtained for the spontaneous emission line.The meaning of perturbed states in the kinetic theory is first discussed. It is shown that the canonical form of the density matrix for a particle in a perturbed level does not correspond to a pure case, in the bare particle representation. The problem of spontaneous emission by an atom in its unperturbed excited state is then discussed. We first show that the results of the kinetic theory can also be obtained by the level shift operator formalism if one goes beyond the usual approximation. It is also shown that these results give a modified Lorentzian shape. Finally, we consider the resonant scattering of a photon. The solution of the kinetic equation gives us the transition probability per unit time. The frequency dependence is given by a modified Lorentz function, analogous to that obtained for the spontaneous emission line.

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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 Theory of a Gas Laser. II

C. R. Willis

We generalize the method of deriving a kinetic equation for a gas laser developed in the first paper of this series to include radiation-matter correlations. As a consequence, we are able to show that the first Born approximation with asymptotic conditions which contain radiation-matter correlations is sufficient to explain the nonthermal photon distribution observed in photon-counting experiments. Our derivation includes the spatial and velocity effects of the motion of the gas atoms. After adiabatically eliminating the matter variables, we obtain a master equation for the radiation density matrix $R$ with nonlinear coefficients. Our radiation master equation for stationary atoms without spatial effects is the same as the generalized Fokker-Planck equation of the Langevin-noise-equation treatment of lasers. We show that the generalized Fokker-Planck equation is the result of the first-Born-approximation treatment of the j\ifmmode\cdot\else\textperiodcentered\fi{}A interaction and is not dependent upon the microscopic structure of the reservoirs. In the first Born approximation, the diagonal and off-diagonal matrix elements of $R$ do not interact with each other. Consequently, an initially diagonal radiation density matrix remains diagonal. However, we show that even though the stationary state is diagonal it is necessary to know the propagator of the off-diagonal part of $R$ to answer time-dependent questions about quantities, such as the line shape, that depend on phase.We generalize the method of deriving a kinetic equation for a gas laser developed in the first paper of this series to include radiation-matter correlations. As a consequence, we are able to show that the first Born approximation with asymptotic conditions which contain radiation-matter correlations is sufficient to explain the nonthermal photon distribution observed in photon-counting experiments. Our derivation includes the spatial and velocity effects of the motion of the gas atoms. After adiabatically eliminating the matter variables, we obtain a master equation for the radiation density matrix $R$ with nonlinear coefficients. Our radiation master equation for stationary atoms without spatial effects is the same as the generalized Fokker-Planck equation of the Langevin-noise-equation treatment of lasers. We show that the generalized Fokker-Planck equation is the result of the first-Born-approximation treatment of the j\ifmmode\cdot\else\textperiodcentered\fi{}A interaction and is not dependent upon the microscopic structure of the reservoirs. In the first Born approximation, the diagonal and off-diagonal matrix elements of $R$ do not interact with each other. Consequently, an initially diagonal radiation density matrix remains diagonal. However, we show that even though the stationary state is diagonal it is necessary to know the propagator of the off-diagonal part of $R$ to answer time-dependent questions about quantities, such as the line shape, that depend on phase.

Physical ReviewJan 10, 196814 citations

Kinetic equations for the density matrix of two-level particles in a strong field

Е. А. Титов

Optics and SpectroscopyOct 1, 20042 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 ${\ensuremath{\beta}}^{2}$.

R

with nonlinear coefficients. Our radiation master equation for stationary atoms without spatial effects is the same as the generalized Fokker-Planck equation of the Langevin-noise-equation treatment of lasers. We show that the generalized Fokker-Planck equation is the result of the first-Born-approximation treatment of the j\ifmmode\cdot\else\textperiodcentered\fi{}A interaction and is not dependent upon the microscopic structure of the reservoirs. In the first Born approximation, the diagonal and off-diagonal matrix elements of

R

do not interact with each other. Consequently, an initially diagonal radiation density matrix remains diagonal. However, we show that even though the stationary state is diagonal it is necessary to know the propagator of the off-diagonal part of

R

to answer time-dependent questions about quantities, such as the line shape, that depend on phase.