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Conference on Information Sciences and Systems

Quantum Noise in Linear Amplifiers Revisited

Alfonso García Martínez

Published Mar 1, 2006• 0 citations
Publisher

Abstract

This paper presents a model for radiation as a photon gas. For each frequency, the photon distribution is a mixture of Bose-Einstein and Poisson distributions, respectively for thermal noise and the useful signal. Poisson (shot) noise contributes with noise at all frequencies. When applied to the operation of linear amplifiers and attenuators, the model gives a natural interpretation to the noise figure in terms of the uncertainty in the amplification process itself.

DOI
10.1109/ciss.2006.286406

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This paper presents a model for radiation as a photon gas. For each frequency, the photon distribution is a mixture of Bose-Einstein and Poisson distributions, respectively for thermal noise and the useful signal. Poisson (shot) noise contributes with noise at all frequencies. When applied to the operation of linear amplifiers and attenuators, the model gives a natural interpretation to the noise figure in terms of the uncertainty in the amplification process itself.
Two types of noise are discussed that may become important when light frequencies are used for communication purposes. At such high frequencies a quantum of energy is not negligible compared to the total transmitted energies (low number of quanta!). The fluctuations caused by the nonzero size of a quantum of energy will predominate over thermal noise, and the usual amplifier noise encountered at microwave (and lower) frequencies. The two types of processes producing quantum noise that we intend to discuss are shown in Pig. 1. They are: The radiative attenuation noise introduced when a signal, phase-and amplitude-control1ed at the high-power transmitter end, is highly attenuated through the inverse square law radiative attenuation. Such a process would take place in the use of optical frequencies for energy transmission over large distances. The noise introduced in linear amplification. objective of this paper is to show how these forms of noise can be analyzed using simple statistics and classical electromagnetic theory with the aid of only a few quantum-theoretical results that have a simple physical meaning.
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]
The quantum noise in a linear amplifier is shown to be thermal noise. The theory of linear amplifiers is applied first to the simplest, single or double oscillator model of an amplifier, and then to linear model of an amplifier with continuous fields and input and outputs. Finally it is shown that the thermal noise emitted by black holes first demonstrated by Hawking, and of dumb holes (sonic and other analogs to black holes), arises from the same analysis as for linear amplifiers. The amplifier noise of black holes acting as amplifiers on the quantum fields living in the spacetime surrounding the black hole is the radiation discovered by Hawking. For any amplifier, that quantum noise is completely characterized by the attributes of the system regarded as a classical amplifier, and arises out of those classical amplification factors and the commutation relations of quantum mechanics.
A recent theory is reviewed for the shot noise of coherent radiation propagating through a random medium. The Fano factor [Formula: see text] (the ratio of the noise power and the mean transmitted current) is related to the scattering matrix of the medium. This is the optical analogue of Büttiker's formula for electronic shot noise. Scattering by itself has no effect on the Fano factor, which remains equal to 1 (as for a Poisson process). Absorption and amplification both increase the Fano factor above the Poisson value. For strong absorption [Formula: see text] has the universal limit [Formula: see text], with f the Bose–Einstein function at the frequency of the incident radiation. This is the optical analogue of the one-third reduction factor of electronic shot noise in diffusive conductors. In the amplifying case the Fano factor diverges at the laser threshold, while the signal-to-noise ratio [Formula: see text] reaches a finite, universal limit.
In recent experiments, quantum effects have become visible in the device noise of hot electron bolometers. Modelling attempts have been made to asses this additional noise [1] Here a model is presented, where the bolometer is associated with a photon counting operator. Since the local oscillator and the radio frequency signal are treated as photon flows, a stochastic fluctuation of the absorbed power in time is caused by the graininess of the incoming radiation. Applying counting operators to the downconverted signal and using a simple Poisson distribution for the presence or absence of a photon, a closed form expression for the quantum noise of the HEB is found that is similar to the relations given in [1]. Quantum noise is a substantial part in the mixer noise at larger THz frequencies and dominates the mixer noise of NbN phonon cooled HEB at frequencies around 8THz and above. Quantum noise contributions are found to be subject to the same electrothermal feedback process as fluctuation and thermal noise are. A way to minimize quantum noise contribution is one is achieved by a correct matching of the active part of the bolometer to the antenna. On the other side, for frequencies above the quasiparticle bandgap, the whole HEB must be matched to the antenna for optimum performance so the best mixer performance lies in between these extrema.
The phase of an optical field inside a linear amplifier is widely known to diffuse with a diffusion coefficient that is inversely proportional to the photon number. The same process occurs in lasers which limits its intrinsic linewidth and makes the phase uncertainty difficult to calculate. The most commonly used simplification is to assume a narrow photon-number distribution for the optical field (which we call the small-noise approximation). This relies on (i) the input to have a good signal-to-noise ratio, and (ii) that such a signal-to-noise ratio can be maintained throughout the amplification process. Here we ask: How many photons must be present in the input to a quantum linear amplifier for the phase noise to be amenable to a small-noise analysis? We show that for a linear amplifier which is signal-to-noise optimised (i.e. an amplifier most favourable to the small-noise approximation), the small-noise approximation breaks down for a coherent-state input with only a few photons on average. Interestingly, when the input strength is increased to tens of photons, the small-noise approximation can be seen to perform much better and the process of phase diffusion permits a small-noise analysis. This demarcates the limit of the small-noise assumption in linear amplifiers as such an assumption is less true for an amplifier which does not have its signal-to-noise performance optimised. In order to explore this limit we propose a description of phase diffusion that goes beyond the small-noise approximation which we call the inverse-number expansion. The small-noise approximation can then be understood as the lowest-order contribution to the inverse-number expansion while phase diffusion with only a few-photon input can be captured by the inclusion of higher-order terms.