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Shot Noise

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Definition: quantum-limited intensity noise

German: Schrotrauschen

Categories: fluctuations and noise, quantum optics

How to cite the article; suggest additional literature

A fundamental limit to the optical intensity noise as observed in many situations (e.g. in measurements with a photodiode or a CCD camera) is given by shot noise. This is a quantum noise effect, related to the discreteness of photons and electrons. Originally, it was interpreted as arising from the random occurrence of photon absorption events in a photodetector, i.e. not as noise in the light field itself. Intensity noise at the shot noise level is obtained when the probability for an absorption event per unit time is constant and not correlated with former events. However, the existence of amplitude-squeezed light, which exhibits intensity noise below the shot noise level (sub-Poissonian intensity noise), proves that shot noise must be interpreted as a property of the light field itself, rather than as an issue of photodetection only – although a photodetector may be blamed for shot noise if it requires optical attenuation, which raises the shot noise level of the relative intensity (→ relative intensity noise).

Intensity noise at the shot noise level is obtained e.g. for a so-called coherent state, which may be approximated by the output of a laser at high noise frequencies. At lower noise frequencies, laser noise is normally much higher due to relaxation oscillations, mode hopping, excess pump noise, and other phenomena. The intensity noise of a simple incandescent lamp is fairly close to the shot noise level. Noise below the shot noise level is obtained for amplitude-squeezed light, which can be obtained e.g. by transforming an original coherent state with the help of nonlinear interactions.

Measurements at the Shot Noise Level

balanced homodyne setup

Figure 1: Setup for balanced optical homodyne measurement.

Photodetectors with high quantum efficiency and appropriate electronic circuitry are required for obtaining sub-shot noise sensitivity of intensity noise measurements. A common configuration is that of a balanced homodyne detector (Figure 1) containing two photodetectors, where a beam splitter sends 50% of the optical power to each detector, and the sum and difference of the photocurrents are obtained electronically. Whereas the sum of the photocurrents is the same as for using all light on a single detector, the difference signal provides a reference for the shot noise level. The article on optical heterodyne detection gives more details.

A severe challenge can come from thermal noise in the electronics, particularly when the photocurrent is converted to a voltage in a small resistor, as is often required for achieving a high detection bandwidth. Also, the full optical power needs to be detected, i.e. the measurement cannot be done on an attenuated beam. Otherwise, the optical attenuation adds additional quantum noise. (The finite quantum efficiency of the detector has the same kind of effect.) If the full optical power is too high for a single detector, a possible method is to use beam splitters for distributing the power on several photodetectors, and to combine the photocurrents.

Sub-shot-noise Electric Currents and Optical Noise

Note that an electric current with noise below the shot noise level can be obtained very easily, e.g. by connecting a quiet voltage source to a resistor. The reason for this is that electrons, being equally charged particles, experience a mutual repulsion, which gives them a natural tendency to “line up”, i.e. to pass a conductor with more regular than just random distances between them.

Efficient laser diodes, operated at low temperatures, can convert sub-shot-noise electric currents into light with intensity noise below the shot noise level (→ amplitude-squeezed light). Surprisingly, the degree of squeezing is not even limited by the quantum efficiency of the laser diode.

Various optical nonlinearities can be used to generate light with quantum noise below the shot noise limit. This can be squeezed light, where one quadrature component is below the shot noise level, or light exhibiting certain quantum correlations.

Important Equations

The one-sided power spectral density of the optical power in the case of shot noise is

power spectral density of shot noise

which is proportional to the average power and the photon energy hν, and is independent of the noise frequency (i.e., shot noise is “white noise”). As the power of a modulation signal with a given relative modulation amplitude scales with the square of the average power, the relative intensity noise decreases with increasing optical power. In the formula for the power spectral density of the relative intensity noise at the shot noise limit, one would divide by the average power, rather than multiplying with it.

An often quoted equation for the shot noise in an electric current is

shot noise of a current

where e is the elementary charge. This formula indicates the variance of the current for an average current I and a measurement bandwidth Δf. The equation corresponds to a one-sided power spectral density

shot noise of a current

of the photocurrent.

Bibliography

[1]N. Campbell, “The study of discontinuous phenomena”, Proc. Cambr. Phil. Soc. 15, 117 (1909)
[2]W. Schottky, “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern”, Ann. Physik 57, 541 (1918)
[3]E. N. Gilbert and H. O. Pollak, “Amplitude distribution of shot noise”, Bell Syst. Tech. J. 39, 333 (1960)
[4]C. M. Caves, “Quantum limits on noise in linear amplifiers”, Phys. Rev. D 26 (8), 1817 (1982)
[5]H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection”, Opt. Lett. 8 (3), 177 (1983)
[6]W. H. Richardson et al., “Squeezed photon-number noise and sub-Poissonian electrical partition noise in a semiconductor laser”, Phys. Rev. Lett. 66 (22), 2867 (1991)
[7]G. Brida et al., “Experimental realization of sub-shot-noise quantum imaging”, Nature Photon. 4, 227 (2010)

(Suggest additional literature!)

See also: intensity noise, relative intensity noise, quantum noise, photodetectors, optical heterodyne detection, coherent states, standard quantum limit, photons, squeezed states of light, Spotlight article 2007-04-16, Spotlight article 2009-11-13

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