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Journal of the Optical Society of America B

Stochastic and probabilistic equations for three- and four-level lasers: tutorial

Colin J. McKinstrie

Published Feb 3, 2020• 7 citations
Publisher

Abstract

In this tutorial, Langevin stochastic equations and Markov probability equations are used to model electron- and photon-number fluctuations in three- and four-level lasers. Equations are derived for the moments of the electron and photon numbers (means, variances and correlations). Both approaches produce the same moment equations. In the Langevin approach, the moments of the noise terms must be specified by other means, whereas in the Markov approach, they are determined self-consistently and satisfy the shot-noise rule: For each process that is modeled by the rate equations, the driving terms in the variance equations equal the moduli of the associated terms in the mean equations. The driving terms in the correlation equations have the same magnitudes as the variance terms, but can be positive or negative, depending on whether the changes in the electron and photon numbers are correlated or anti-correlated, respectively. Formulas are derived for the relative intensity noise and its spectrum. The consequences of these results for three- and four-level lasers are discussed.

DOI
10.1364/josab.379976

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In this tutorial, Langevin stochastic equations and Markov probability equations are used to model electron- and photon-number fluctuations in three- and four-level lasers. Equations are derived for the moments of the electron and photon numbers (means, variances and correlations). Both approaches produce the same moment equations. In the Langevin approach, the moments of the noise terms must be specified by other means, whereas in the Markov approach, they are determined self-consistently and satisfy the shot-noise rule: For each process that is modeled by the rate equations, the driving terms in the variance equations equal the moduli of the associated terms in the mean equations. The driving terms in the correlation equations have the same magnitudes as the variance terms, but can be positive or negative, depending on whether the changes in the electron and photon numbers are correlated or anti-correlated, respectively. Formulas are derived for the relative intensity noise and its spectrum. The consequences of these results for three- and four-level lasers are discussed.
In this tutorial, a classical Langevin equation, a classical Markov equation, and a quantum Langevin equation are used to model the amplification and attenuation of an optical signal. Equations are derived for the photon-number moments (mean and variance). All three approaches produce the same moment equations. In the first approach, the moments of the noise sources must be specified (using physical insight), whereas in the second and third approaches, they are determined self-consistently and satisfy the shot-noise rule: The number variance is driven at a rate that equals the sum of the moduli of the rates in the mean equation.
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.
The fluctuations of the constituent populations in a multi-level system are considered as applied to a three and four level semiconductor model. It is shown that under certain conditions the fluctuations of the electron numbers in the conduction band can be super-poisson, defined as having a variance greater than the mean number. The general autocorrelation function for a three level system is obtained. For certain limits, this correlation function is the sum of two damped sinusoidal terms. This would indicate there is an oscillatory interaction between the population numbers. The photon distribution as a function of position within a one-dimensional active medium is derived. When two partially reflecting mirrors are situated at both ends of the medium, the stationary photon distribution obtained is a function of the cavity amplification and the mirror reflectivities. The distribution of photo-electrons emitted from a detector of area A and resolving time T when illuminated by an incident light beam is derived. By using a binomial rather tha a deterministic quantum efficiency an additional term is obtained in the autocorrelation function. The resulting spectral density of the photo-current fluctuations is shown to be the sum of a Poisson particle noise and wave interference term. Several examples of different spectral line shapes are discussed. Also considered is an intensity modulated light beam.
The basic equations governing noise phenomena are derived from first principles and applied to examples in optical communications. Quantum noise arises from two sources, the momentum fluctuations of electrons at optical frequencies and the uncertainty-related fluctuations of the electromagnetic field. Shot noise results from the beating of the noise sources with the signal field. In high-gain amplifiers, the spontaneous-emission noise dominates shot noise and results in a noise figure of at least 3 dB. It is shown explicitly how, at high power, both the laser field and the laser noise source become classical. The increase in noise in lasers with open cavities is not due to enhanced spontaneous emission as once thought, but to single-pass amplification. The noise fields and spontaneous currents have Gaussian distributions, while nonlasing modes have exponential photon-number distributions. Low-frequency intensity fluctuations arise from the electric current driving the laser and can be sub-Poissonian, in contrast to shot noise, which has a Poissonian distribution. The calculational tools are a wave equation for the field operator and a rate equation for the carrier-number operator, each containing spontaneous current noise sources. The correlation functions of these sources are determined by the fluctuation-dissipation theorem. [S0034-6861(96)00503-X]
We have studied the squeezing and statistical properties of the light produced by degenerate three level lasers. In order to carry out the analysis, we have obtained the c-number Langevin equations associated with the normal order using the pertinent master equations. Applying the Q-functions for degenerate three-level laser, we have determined the mean and variance of the photon number as well as the photon number distribution for degenerate three-level laser. It has been found that the mean photon number and quadrature squeezing increase with linear gain coefficient. Keywords: Degenerate, Laser light, Squeezing, Quantum Noise, Photon
The amplitude and phase noise spectra of the internal and external fields of a laser oscillator are calculated using the quantum mechanical Langevin equations. The outgoing field spectra are demonstrated to be different from the internal field spectra. The outgoing photon flux\hat{N}from a laser oscillator operating at far above the threshold has a spectral density equal to the classical shot noise level,2\langle\hat{N}\rangle. On the other hand, the internal photon number spectrum is Lorentzian, and the variance is equal to the average photon number,\langle\hat{N}\rangle. The difference stems from the quantum mechanical interference between the transmitted internal field and the reflected zero-point fluctuation at the output coupling mirror. The relation between the present analysis and the previous noise theory of a laser is also discussed.