Shot noise

In electronics shot noise originates from the discrete nature of electric charge.

In a statistical experiment such as tossing a fair coin and counting the occurrences of heads and tails, the numbers of heads and tails after many throws will differ by only a tiny percentage, while after only a few throws outcomes with a significant excess of heads over tails or vice versa are common; if an experiment with a few throws is repeated over and over, the outcomes will fluctuate a lot.

Shot noise exists because phenomena such as light and electric current consist of the movement of discrete (also called "quantized") 'packets'.

Consider light—a stream of discrete photons—coming out of a laser pointer and hitting a wall to create a visible spot.

The concept of shot noise was first introduced in 1918 by Walter Schottky who studied fluctuations of current in vacuum tubes.

[1] Shot noise may be dominant when the finite number of particles that carry energy (such as electrons in an electronic circuit or photons in an optical device) is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are significant.

The term can also be used to describe any noise source, even if solely mathematical, of similar origin.

The magnitude of shot noise increases according to the square root of the expected number of events, such as the electric current or intensity of light.

Thus shot noise is most frequently observed with small currents or low light intensities that have been amplified.

Since the standard deviation of shot noise is equal to the square root of the average number of events N, the signal-to-noise ratio (SNR) is given by: Thus when N is very large, the signal-to-noise ratio is very large as well, and any relative fluctuations in N due to other sources are more likely to dominate over shot noise.

Because the electron has such a tiny charge, however, shot noise is of relative insignificance in many (but not all) cases of electrical conduction.

With very small currents and considering shorter time scales (thus wider bandwidths) shot noise can be significant.

The result by Schottky, based on the assumption that the statistics of electrons passage is Poissonian, reads[2] for the spectral noise density at the frequency

This can be combined with the Landauer formula, which relates the average current with the transmission eigenvalues

In the simplest case, these transmission eigenvalues can be taken to be energy independent and so the Landauer formula is where

This is a classical result in the sense that it does not take into account that electrons obey Fermi–Dirac statistics.

The correct result takes into account the quantum statistics of electrons and reads (at zero temperature) It was obtained in the 1990s by Viktor Khlus, Gordey Lesovik (independently the single-channel case), and Markus Büttiker (multi-channel case).

While this is the result when the electrons contributing to the current occur completely randomly, unaffected by each other, there are important cases in which these natural fluctuations are largely suppressed due to a charge build up.

In other situations interactions can lead to an enhancement of shot noise, which is the result of a super-poissonian statistics.

For example, in a resonant tunneling diode the interplay of electrostatic interaction and of the density of states in the quantum well leads to a strong enhancement of shot noise when the device is biased in the negative differential resistance region of the current-voltage characteristics.

Since shot noise is a Poisson process due to the finite charge of an electron, one can compute the root mean square current fluctuations as being of a magnitude[8] where q is the elementary charge of an electron, Δf is the single-sided bandwidth in hertz over which the noise is considered, and I is the DC current flowing.

Following Poisson statistics, the photon noise is calculated as the square root of the signal:

where: In optics, shot noise describes the fluctuations of the number of photons detected (or simply counted in the abstract) because they occur independently of each other.

In the case of photon detection, the relevant process is the random conversion of photons into photo-electrons for instance, thus leading to a larger effective shot noise level when using a detector with a quantum efficiency below unity.

Only in an exotic squeezed coherent state can the number of photons measured per unit time have fluctuations smaller than the square root of the expected number of photons counted in that period of time.

In optical homodyne detection, the shot noise in the photodetector can be attributed to either the zero point fluctuations of the quantised electromagnetic field, or to the discrete nature of the photon absorption process.

[10] However, shot noise itself is not a distinctive feature of quantised field and can also be explained through semiclassical theory.