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Journal of the Physical Society of Japan

A Microscopic Laser Theory. II. Numerical Calculations

Toshihico Arimitsu, Yuji Takada

Published Jul 1, 1984• 6 citations
Publisher

Abstract

Numerical calculations of the basic formulation are performed extensively, which is the most convenient formulation to study the time-dependent cooperative ( symmetry breaking ) behavior of the laser system from the microscopic and fully quantum-statistical-mechanical point of view. The threshold of the laser light is interpreted by the coherence of the electric field in the time domain, not by the intensity of the field, which stems from the dynamical symmetry breaking of the system.

DOI
10.1143/jpsj.53.2233

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Numerical calculations of the basic formulation are performed extensively, which is the most convenient formulation to study the time-dependent cooperative ( symmetry breaking ) behavior of the laser system from the microscopic and fully quantum-statistical-mechanical point of view. The threshold of the laser light is interpreted by the coherence of the electric field in the time domain, not by the intensity of the field, which stems from the dynamical symmetry breaking of the system.
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.
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.
After reviewing the interpretation of laser operation as a nonequilibrium Bose-Einstein condensation phase transition, we illustrate the novel features arising from the nonequilibrium nature of photon and polariton Bose-Einstein condensates recently observed in experiments. We then propose a quantitative criterion to experimentally assess the equilibrium versus nonequilibrium nature of a specific condensation process, based on fluctuation-dissipation relations. The power of this criterion is illustrated on two models which show very different behaviors. Historical and Conceptual Introduction The first introduction of concepts of nonequilibrium statistical mechanics into the realm of optics dates back to the early 1970s with pioneering works by Graham and Haken [1] and by DeGiorgio and Scully [2], who proposed a very insightful interpretation of the laser threshold in terms of a spontaneous breaking of the U( 1 ) symmetry associated with the phase of the emitted light. Similar to what happens to the order parameter at a second-order phase transition, such an optical phase is randomly chosen every time the device is switched on and remains constant for macroscopic times. Moreover, a long-range spatial order is established, as light emitted by a laser device above threshold is phase-coherent on macroscopic distances. While textbooks typically discuss this interpretation of laser operation in terms of a phase transition for the simplest case of a single-mode laser cavity, rigorously speaking this is valid only in spatially infinite systems. In fact, only in this case one can observe nonanalytic behaviors of the physical quantities at the transition point. In particular, the long-range order is typically assessed by looking at the long-distance behavior of the correlation function of the order parameter, which, for a laser, corresponds to the first-order spatial coherence of the emitted electric field, the spontaneous symmetry breaking is signaled by this quantity becoming nonzero (see Chapter 5). The average is taken on the stationary density matrix of the system. In order to be able to probe this long-distance behavior, experimental studies need devices with a spatially extended active region.
Preface. List of symbols. 1. Introduction. 2. Basic properties and types of lasers. 3. Laser resonators. 4. The intensity of laser light. Rate equations. 5. The basic equations of the semiclassical laser theory. 6. Applications of semiclassical theory. 7. Ultrashort pulses. 8. Instability hierachies of laser light. Chaos, and routes to chaos. 9. Optical bistability. 10. Quantum theory of the laser I. A first approach via quantum mechanical Langevin equations. Coherence, noise and photon-statistics. 11. Quantum theory of the laser II. A second approach via the density matrix equation and quantum classical correspondence. 12 A theoretical treatment of the two-photon laser. 13. The laser - trailblazer of synergetics. References. Subject index.