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Journal of Lightwave Technology

Gain, Loss and the Shot-Noise Rule

Colin J. McKinstrie

Published Jan 22, 2020• 2 citations
Publisher

Abstract

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.

DOI
10.1109/jlt.2020.2968720

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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.
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.
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]
For a quantum system describable in a Markoffian way, via a set of variables $\mathrm{a}={{a}_{1}, {a}_{2}, \ensuremath{\cdots}}$, we show that the Langevin noise sources ${F}_{\ensuremath{\mu}}$ in the operator equations of motion $\frac{d{a}_{\ensuremath{\mu}}}{\mathrm{dt}}={A}_{\ensuremath{\mu}}(\mathrm{a})+{F}_{\ensuremath{\mu}}$ possess second moments $〈{F}_{\ensuremath{\mu}}(t){F}_{\ensuremath{\nu}}(u)〉=2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathrm{a}, t)〉\ensuremath{\delta}(t\ensuremath{-}u)$. The diffusion coefficients ${D}_{\ensuremath{\mu}\ensuremath{\nu}}$ can be determined from a knowledge of the mean equations of motion via the (exact) time-dependent Einstein relation $2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}〉=\ensuremath{-}〈{A}_{\ensuremath{\mu}}{a}_{\ensuremath{\nu}}〉\ensuremath{-}〈{a}_{\ensuremath{\mu}}{A}_{\ensuremath{\nu}}〉+\frac{d〈{a}_{\ensuremath{\mu}}(t){a}_{\ensuremath{\nu}}(t)〉}{\mathrm{dt}}$, where 〈〉〈 〉〈〉 represents a reservoir average. The sources ${F}_{\ensuremath{\mu}}$, ${F}_{\ensuremath{\nu}}$ do not commute with one another, and as a result the commutation rules of the ${a}_{\ensuremath{\mu}}$ are shown to be preserved in time. The mean motion and diffusion coefficients are calculated for a harmonic oscillator, and for a set of atomic levels. We prove that two dynamically coupled systems (e.g., field and atoms) have uncorrelated Langevin forces if they are coupled to independent reservoirs. Radiation-field-atom coupling adds no new noise sources. We thus obtain simply the maser model including noise sources used in Quantum Noise V. Direct calculations of the mean motion and fluctuations in a system coupled to a reservoir yield relationships in agreement with the Einstein relation. For reservoirs violating time reversal, anomalous frequency shifts are found possible that violate the Ritz combination principle since $\ensuremath{\Delta}{\ensuremath{\omega}}_{12}+\ensuremath{\Delta}{\ensuremath{\omega}}_{23}+\ensuremath{\Delta}{\ensuremath{\omega}}_{31}$ need not vanish.
In this paper the quantum noise properties of phase-insensitive and phase-sensitive parametric processes are studied. Formulas for the field-quadrature and photon-number means and variances are derived, for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios associated with the direct and homo-dyne detection of optical signals. The consequences of the aforementioned formulas are described for frequency conversion, amplification, monitoring, and transmission through sequences of attenuators and amplifiers.
In this paper we start from our Quantum Noise IV model of a homogenously broadened maser and specialize to the case in which the number of photons and the population difference are the slowest changing variables. In spite of the high coherence of the system, we find that by eliminating all other variables adiabatically we obtain the usual rate equations plus a set of noise sources with precisely those moments appropriate to shot noise. Thus we justify McCumber's model for amplitude noise in a laser, and after making a quasi-linear approximation obtain his amplitude spectrum. Comparison with exact calculations of Hempstead and Lax for the rotating wave van der Pol oscillator suggests that quasi-linear methods are accurate except within a factor 10 (in photon number) of threshold.
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.