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Soviet Journal of Quantum Electronics

Evolution of a statistical distribution of photons in a laser radiation field

S. A. Reshetnyak, Sergey Kharchev, Leonid A. Shelepin

Published Dec 31, 1983• 0 citations
Publisher

Abstract

Decoupling of equations for the diagonal and nondiagonal elements of the density matrix of laser radiation is used in an analysis of the interaction of the radiation field with an active medium. An algorithm for the calculation of the eigenvalues and eigenvectors is proposed for the diagonal elements describing the radiation field statistics. The first minimal (in respect of the magnitude) eigenvalues are calculated and nonequilibrium distribution functions are found for the dynamics of a transition from a vacuum to an equilibrium state; the results agree well with those obtained in machine (numerical) experiments. The equations for the nondiagonal elements of the density matrix are used to find the characteristic times of changes in these elements and to determine the emission line width.

DOI
10.1070/qe1983v013n12abeh004985

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Decoupling of equations for the diagonal and nondiagonal elements of the density matrix of laser radiation is used in an analysis of the interaction of the radiation field with an active medium. An algorithm for the calculation of the eigenvalues and eigenvectors is proposed for the diagonal elements describing the radiation field statistics. The first minimal (in respect of the magnitude) eigenvalues are calculated and nonequilibrium distribution functions are found for the dynamics of a transition from a vacuum to an equilibrium state; the results agree well with those obtained in machine (numerical) experiments. The equations for the nondiagonal elements of the density matrix are used to find the characteristic times of changes in these elements and to determine the emission line width.
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.
It is shown that, in general, the interaction between relaxing quantum systems and radiation is described by non-Markov transport equations for the density matrix ρ p and the relaxation matrix depends on the properties of the radiation. The usual Markov-type equations for ρ p are obtained from these equations if γτ c ≪1 and (∊ 2 +4 V 2 ) 1/2 τ c ≪1 (γ is the width of the investigated transition in a quantum system, rc is the correlation time of the perturbations causing relaxation, e is the frequency detuning, and V is the energy of the interaction between the system and the radiation field). When the second inequality is disobeyed, the relaxation process deviates strongly from the Markov type. In the case of small values of V , this results in a non-Lorentzian profile of the absorption lines characterized by a steep descent in the wings. In the range of high values of V , when 2 V τ c 1, the whole relaxation process becomes greatly modified: the dependence of the relaxation coefficients on V becomes strong and a coupling appears between the relaxations of the diagonal and the nondiagonal elements of the density matrix. This is manifested, in particular, in the saturation effect: the term describing saturation in the expression for the polarizability changes considerably and the expression for the line width loses the component due to the adiabatic (elastic) perturbation.
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 RRR 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 RRR 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 RRR to answer time-dependent questions about quantities, such as the line shape, that depend on phase.
A quantum statistical analysis of an optical maser is presented in generalization of the recent semiclassical theory of Lamb. Equations of motion for the density matrix of the quantized electromagnetic field are derived. These equations describe the irreversible dynamics of the laser radiation in all regions of operation (above, below, and at threshold). Nonlinearities play an essential role in this problem. The diagonal equations of motion for the radiation are found to have an apparent physical interpretation. At steady state, these equations may be solved via detailed-balance considerations to yield the photon statistical distribution ${\ensuremath{\rho}}_{n,n}$. The resulting distribution has a variance which is significantly larger than that for coherent light. The off-diagonal elements of the radiation density matrix describe the effects of phase diffusion in general and provide the spectral profile ${| E(\ensuremath{\omega}) |}^{2}$ as a special case. A detailed discussion of the physics involved in this paper is given in the concluding sections. The theory of the laser adds another example to the short list of solved problems in irreversible quantum statistical mechanics.
In Chapters 5 and 7 we presented a treatment of laser physics in which the light is described as a classical Maxwell field while the lasing medium is described as a collection of atoms whose dynamic evolution is governed by the Schrödinger equation. This semi-classical theory of laser behavior is sufficient to describe a rich variety of phenomena. However, there are many questions which require a fully quantized theory of the radiation. For example, the photon statistics and linewidth of the laser can be properly understood only via the full quantum theory of a laser. The laser linewidth is an important quantity. For example, it determines the fundamental limit of operation of an active ring laser gyroscope. The first fully quantized derivation of the laser linewidth general enough to include even the semiconductor laser linewidth problem utilized a quantum noise operator approach, and is presented in chapter 12. The photon statistical distribution for the laser is of interest for several reasons. Historically, it was initially thought by some that the statistical photon distribution should be a Bose–Einstein distribution. A little reflection shows that this can not be, since the laser is operating far from thermodynamic equilibrium. However, a different paradigm recognizes many atoms oscillating in phase produce what is essentially a classical current, and this would generate a coherent state; the statistics of which is Possionian. But, for example, the photon statistics of a typical Helium–Neon laser is substantially different from a Possionian distribution.