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arXiv:cond-mat/0001157 · Physical Review A

Photonic excess noise and wave localization

C. W. J. Beenakker, M. Patra, Piet W. Brouwer

Published Mar 30, 2000• 12 citations
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Abstract

This is a theory for the effect of localization on the super-Poissonian noise of radiation propagating through an absorbing disordered waveguide. Localization suppresses both the mean photon current $\ifmmode \bar{I}\else \={I}\fi{}$ and the noise power P, but the Fano factor $P/\ifmmode \bar{I}\else \={I}\fi{}$ is found to remain unaffected. For strong absorption the Fano factor has the universal value $1+\frac{3}{2}f$ (with f the Bose-Einstein function), regardless of whether the waveguide is long or short compared to the localization length.

DOI
10.1103/physreva.61.051801

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This is a theory for the effect of localization on the super-Poissonian noise of radiation propagating through an absorbing disordered waveguide. Localization suppresses both the mean photon current \ifmmodeIˉ\elseIˉ\fi\ifmmode \bar{I}\else \={I}\fi{}\ifmmodeIˉ\elseIˉ\fi and the noise power P, but the Fano factor P/\ifmmodeIˉ\elseIˉ\fiP/\ifmmode \bar{I}\else \={I}\fi{}P/\ifmmodeIˉ\elseIˉ\fi is found to remain unaffected. For strong absorption the Fano factor has the universal value 1+32f1+\frac{3}{2}f1+23​f (with f the Bose-Einstein function), regardless of whether the waveguide is long or short compared to the localization length.
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.
A second-quantized theory of the radiation field is used to study the origin of the excess noise observed in gain-guided amplifiers. We find that the reduction of the signal-to-noise ratio is a function of the length of the amplifier, and thus the enhancement of the noise is a propagation effect arising from longitudinally inhomogeneous gain of the noise rather than from an excess of local spontaneous emission. We confirm this conclusion by showing that the microscopic rate of spontaneous emission into a given non-power-orthogonal cavity mode is not enhanced by the Petermann factor. In addition, we illustrate the difficulties associated with photon statistics for this and other open systems by showing that no acceptable family of photon-number operators corresponds to a set of non-power-orthogonal cavity modes.
We analyze how nonclassical features of squeezed radiation (in particular, the sub-Poissonian noise) are degraded when it is transmitted through an amplifying or absorbing medium with randomly located scattering centra. Both the cases of direct photodetection and of homodyne detection are considered. Explicit results are obtained for the dependence of the Fano factor (the ratio of the noise power and the mean current) on the degree of squeezing of the incident state, on the length and the mean free path of the medium, the temperature, and on the absorption or amplification rate.
Shot noise of electrons that are transmitted with probability TTT through a quantum point contact (biased at a voltage V0{V}_{0}V0​) leads to a fluctuating current that in turn emits radiation in the microwave regime. By calculating the Fano factor FFF for the case where only a single channel contributes to the transport, it has been shown that the radiation produced at finite frequency ${\ensuremath{\omega}}_{0}$ close to $e{V}_{0}/\ensuremath{\hbar}$ and at low temperatures is nonclassical with sub-Poissonian statistics (F<1F<1F<1). The origin of this effect is the fermionic nature of the electrons producing the radiation, which reduces the probability of simultaneous emission of two or more photons. However, the Fano factor, being a time-averaged quantity, offers only limited information about the system. Here, we calculate the second-order coherence ${g}^{(2)}(\ensuremath{\tau})$ for this source of radiation. We show that due to the interference of two contributions, two photon processes (leading to bunching) are completely absent at zero temperature for T=50T=50\phantom{\rule{0.16em}{0ex}}%T=50. At low temperatures, we find a competition of the contribution due to Gaussian current-current fluctuations (leading to bunching) with the one due to non-Gaussian fluctuations (leading to antibunching). At slightly elevated temperatures, the non-Gaussian contribution becomes suppressed, whereas the Gaussian contributions remain largely independent of temperature. We show that the competition of the two contributions leads to a nonmonotonic behavior of the second-order coherence as a function of time. As a result, ${g}^{(2)}(\ensuremath{\tau})$ obtains a minimal value for times ${\ensuremath{\tau}}^{*}\ensuremath{\simeq}{\ensuremath{\omega}}_{0}^{\ensuremath{-}1}$. Close to this time, the second-order coherence remains below 1 at temperatures where the Fano factor is already above 1. We identify realistic experimental parameters that can be used to test the sub-Poissonian nature of the radiation.
In an earlier work [P. J. Bardroff and S. Stenholm, Phys. Rev. A 60, 2529 (1999)], we have derived a fully quantum-mechanical description of excess noise in strongly damped lasers. This theory is used here to derive the corresponding quantum Langevin equations. Taking the semiclassical limit of these we are able to regain the starting point of Siegman's treatment of excess noise [Phys. Rev. A 39, 1253 (1989)]. Our results essentially constitute a quantum derivation of his theory and allow some generalizations.