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Physical Review Letters

Macroscopic Quantum Fluctuations and First-Order Phase Transition in a Laser

Paul D. Lett, William R. Christian, Surendra Singh, L. Mandeļ

Published Dec 28, 1981• 56 citations
Publisher

Abstract

A homogeneously broadened two-mode ring laser has two metastable states in which one or the other mode intensity is zero. Quantum fluctuations cause the system to switch spontaneously, and at random times, between these states. The probability distribution of the light intensity of one laser mode has been measured, and found to exhibit two completely resolved peaks at zero and nonzero intensities, as predicted. This confirms the existence of a first-order laser phase transition.

DOI
10.1103/physrevlett.47.1892

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A homogeneously broadened two-mode ring laser has two metastable states in which one or the other mode intensity is zero. Quantum fluctuations cause the system to switch spontaneously, and at random times, between these states. The probability distribution of the light intensity of one laser mode has been measured, and found to exhibit two completely resolved peaks at zero and nonzero intensities, as predicted. This confirms the existence of a first-order laser phase transition.
The equations previously derived for the intensity fluctuations of a two-mode ring laser are applied to a homogeneously broadened laser. It is shown that the mean light intensity of the more lossy mode passes through a maximum and then tends to zero as the excitation is increased. The probability distribution of the intensity of each mode may exhibit two peaks, and spontaneous switching between the intensities associated with the peaks may occur. This is manifest in relative intensity fluctuations of the more lossy mode that can be much greater than unity. Estimates are given for the characteristic switching time.
With the help of a fast digitizer, the probability distribution of the light intensity of a single-mode dye laser has been measured photoelectrically, for various working points of the laser, from well-below to well-above threshold. In the region below threshold the laser exhibits short spiking and a double-peaked probability distribution with large value at zero intensity. A plot of most probable vs mean light intensity shows a discontinuity suggestive of a first-order phase transition. The experimental results are compared with Monte Carlo solutions of the equations of motion, with both additive spontaneous emission noise and multiplicative Ornstein-Uhlenbeck pump noise, and reasonable agreement is found.
The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\ensuremath{\xi}$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.
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.