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Physical Review

Quantum Statistical Dynamics of Laser Amplifiers

A. E. Glassgold, Dennis Holliday

Published Sep 13, 1965• 56 citations
Publisher

Abstract

A quantum statistical procedure for studying interacting radiation fields is developed and applied to a model laser operating as a linear amplifier. The method utilizes an expansion of the density matrix in terms of a complete set of orthogonal operators originally used by Weyl. One mode of the laser field is considered to interact with a pumping mechanism and with a loss mechanism. A general solution for the density matrix is obtained and then applied to various initial conditions. It is found that a classical statistical interpretation of the results is frequently appropriate.

DOI
10.1103/physrev.139.a1717

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A quantum statistical procedure for studying interacting radiation fields is developed and applied to a model laser operating as a linear amplifier. The method utilizes an expansion of the density matrix in terms of a complete set of orthogonal operators originally used by Weyl. One mode of the laser field is considered to interact with a pumping mechanism and with a loss mechanism. A general solution for the density matrix is obtained and then applied to various initial conditions. It is found that a classical statistical interpretation of the results is frequently appropriate.
The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the PPP representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and nnn-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative PPP representation for the density operator for a single mode of oscillation.
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.
We study in detail the influence of pumping statistics on the laser dynamics. We apply the technique of quantum Langevin operators and generalize the corresponding noise operators to incorporate the statistical properties of the pump mechanism. These equations are then used to derive expressions for the phase and intensity fluctuations of lasers with various pump statistics. We find that a reduction of pump noise can lead to a significant squeezing of the photon number noise below the shot-noise limit. This is in complete agreement with our previous analysis, which used a density-operator approach.
"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."
In Chap. 17 we developed the quantum theory of the laser obtaining in particular expressions for the laser photon statistics and linewidth. In this chapter we discuss and amplify those results in terms of measurement theory. Section 18-1 provides an overview of measurement theory as it applies to the quantum electric field. The discussion leads naturally into a specific discussion of the measurement of photon statistics and the laser linewidth. In Sec. 18-2 a heuristic model of a photodetector is presented with reveals how the photo­ electron statistics resulting from incident radiation is related to the photon statistics of that radiation, t In Sec. 18-3 we discuss the laser linewidth with the use of a simple two level atom beam much like that of Sec. 16-1. Our present discussion utilizes the density matrix technique throughout rather than the density matrix-state vector combination of Sec. 16-1. The laser spectrum “seen” by the atomic beam has, in fact, the same Lorentzian profile given in Sec. 17-3. In Sec. 18-4 the laser linewidth is discussed in terms of a quantum version of the classical correlation function <£(* -f t )E(t)y. In spite of the fact that two time correlations are apparently required, we show that a single time and solutions to the density operator equation of motion are sufficient. This reduction is an example of the so-called Onsager regression hypothesis, which shows how to proceed from single-time to multitime correlation functions.