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Physical Review A

Stochastic pump effects in lasers

Ronald F. Fox, Glenn E. James, Rajarshi Roy

Published Nov 1, 1984• 30 citations
Publisher

Abstract

We examine from first principles the effects of a stochastic pump on the output fluctuations of a laser. The Hamiltonian for a four-level molecule interacting with a stochastic semiclassical pump field and quantized laser field is used to obtain density-matrix equations of motion. Noise from the pump field and from spontaneous emission is introduced systematically, as opposed to the ad hoc inclusion of random forces in the semiclassical equations. The relationship of this theory with previous quantum and semiclassical equations is described. New equations are derived on which computer simulations are performed to compare our theoretical results to experimental data for a single-mode dye laser.

DOI
10.1103/physreva.30.2482

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We examine from first principles the effects of a stochastic pump on the output fluctuations of a laser. The Hamiltonian for a four-level molecule interacting with a stochastic semiclassical pump field and quantized laser field is used to obtain density-matrix equations of motion. Noise from the pump field and from spontaneous emission is introduced systematically, as opposed to the ad hoc inclusion of random forces in the semiclassical equations. The relationship of this theory with previous quantum and semiclassical equations is described. New equations are derived on which computer simulations are performed to compare our theoretical results to experimental data for a single-mode dye laser.
Measurements of the photon-number fluctuations in a single-mode dye jet laser are compared with solutions of the equations of motion with colored pumping and white spontaneous-emission noise. A search of parameter space for best agreement between theory and experiment shows that the experimental results cannot be well modeled by a theory with white pumping noise, although the correlation time is small, but can be modeled with colored noise. This emphasizes the importance of not treating the pumping fluctuations as δ correlated, even for time-independent quantities.
The random nature of spontaneous emission leads to unavoidable fluctuations in a laser's output. This is often included through random Langevin forces in laser rate equations, but this approach falls short for nanolasers. In this paper, we show that the laser quantum noise can be quantitatively computed for a very broad class of lasers by starting from simple and intuitive rate equations and merely assuming that the number of photons and excited electrons only takes discrete values. While the approach has seen previous success, we here derive it rigorously from an open quantum system master equation, whereas it was previously introduced only on phenomenological grounds. We further show that in the many-photon limit, the model simplifies to Langevin equations. We perform an extensive comparison of different approaches for computing quantum noise in lasers, identifying the best approach for different system sizes, ranging from nanolasers to macroscopic lasers, and different levels of excitation, i.e., cavity photon number. In particular, we show that below the laser threshold, stochastic fluctuations in the numerical solution to the Langevin equations can drive populations to unphysical negative values, requiring the introduction of population bounds, which in turn skew the noise statistics, leading to inaccuracies. The Laser Markov Chain model, on the other hand, is accurate for all pump values and laser sizes when collective emitter effects are excluded.
Recent theoretical analyses of the single-mode laser with colored-noise pump fluctuations have allowed the detailed prediction of the short-time behavior of the intensity autocorrelations of such a laser. In particular, a rounding off of the intensity correlation function on the time scale of the correlation time of the pump fluctuations has been predicted. We present results from experiments performed on a single-mode dye-laser system that clearly confirm these predictions. Numerical simulations of the laser Langevin equations including the colored-noise pump fluctuations are used to fit the experimental results.
We consider a laser model consisting of a single four-level or three-level atom, an optical cavity, and an incoherent pump. Results for photon statistics for varying pump levels are obtained using a quantum trajectory algorithm. In particular, we calculate the mean photon number, Fano factor (which is the variance over the mean). We examine that the behavior of the single-atom device as \ensuremath{\beta}, the fraction of spontaneous emission into the lasing mode, is varied. Typical values considered for \ensuremath{\beta} are $0.01<\ensuremath{\beta}<1.0.$ We find that for large enough \ensuremath{\beta}, lasing action, with properties similar to those predicted by semiclassical theories that factorize atom-field correlations and use a small-noise approximation, can occur. Squeezing can occur as \ensuremath{\beta} is increased. There is no evidence of a sharp phase transition from weakly excited thermal light to coherent light at a particular pump power. This is consistent with work on many-atom lasers with \ensuremath{\beta} values in the range considered here. As \ensuremath{\beta} is increased, the output goes from quasithermal light to coherent and finally to squeezed light, progressing into a fully quantum-mechanical regime. We also consider the effects of cavity damping and spontaneous emission rates on these results.
We use a probability-theory approach to study the laser noise's effects on laser-atom interactions. We consider the case where the atom is described by a two-level system without spontaneous emission and the laser has both intensity and frequency noises. A stochastic differential equation is established based on the Schr\"odinger equation of the laser-atom interaction in the semiclassical picture. We then analyze the equation using the path-integral technique to the first order of a perturbation approach. Because of the presence of laser noises, the atom wave function at a given time is a random variable. Therefore we construct a stochastic process characterizing its time evolution. We also provide the theoretical description for the experimental realization of measuring the laser linewidth by driving a narrow atomic transition, and establish the connection between the laser noise's roles in the laser-atom interaction and laser linewidth measurement by beat signal.