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Journal of the Physical Society of Japan

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

Tetsuo Tominaga, Toshihico Arimitsu

Published Jan 1, 1984• 5 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.53.93

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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.
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.
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.
The interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.
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.