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arXiv:cond-mat/9905024 · Modern Physics Letters B

PHOTON SHOT NOISE

C. W. J. Beenakker, M. PATRA

Published May 10, 1999• 25 citations

Abstract

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.

DOI: 10.1142/s0217984999000439

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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.

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.

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.

A general theory is presented for the photodetection statistics of coherent radiation that has been amplified by a disordered medium. The beating of the coherent radiation with the spontaneous emission increases the noise above the shot-noise level. The excess noise is expressed in terms of the transmission and reflection matrices of the medium, and evaluated using the methods of random-matrix theory. Intermode scattering between N propagating modes increases the noise figure by up to a factor of N, as one approaches the laser threshold. Results are contrasted with those for an absorbing medium.

Random point processes are often classified as bunched, unbunched, or antibunched, depending on the value assumed by their variance-to-mean ratio (the Fano factor); in the low-degeneracy limit, the Fano factor of a Bose-Einstein process approaches the Poisson value of unity. This fact probably added strength to the conjecture that in the limit of zero degeneracy a Bose-Einstein process reduces to a memoryless Poisson process. After considering some statistics related both to counting and to arrival times, we have experimentally verified that a Bose-Einstein process retains its correlation properties down to values of the degeneracy parameter of the order of 0.01. It also follows that the two most common definitions of photon bunching, the one based on the Fano factor and the one based on the correlation function, are not equivalent at low degeneracy levels.

The shot noise which occurs in photoelectric detection is calculated utilizing fully quantum mechanical procedures. The results which are obtained in this paper include the effects of photon bunching and they differ considerably from classical results which are obtained on the assumption of Poisson statistics. In some cases the Poisson-based classical results lead to shot-noise powers that are in error by as much as an order of magnitude. The results reveal the existence of a relative minimum in the plot of shot noise vs input signal to noise ratio. This suggests the possibility of optimum input signal to noise ratios within certain restricted ranges of the input signal to noise ratio. A plot of the shot noise which is in excess of that predicted by Poisson-based statistics vs input signal to noise ratio reveals a maximum at an input signal to noise ratio of approximately 1.5. This suggests that the enhancement of the Hanbury-Brown and Twiss effect that is obtained when coherent radiation is superposed upon chaotic radiation will be maximized at a coherent-to-chaotic ratio of approximately 1.5.