Shot noise

For large numbers, the Poisson distribution approaches a normal distribution about its mean, and the elementary events (photons, electrons, etc.) are no longer individually observed, typically making shot noise in actual observations indistinguishable from true Gaussian noise. 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:

${\displaystyle \mathrm {SNR} ={\frac {N}{\sqrt {N}}}={\sqrt {N}}.\,}$

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. However, when the other noise source is at a fixed level, such as thermal noise, or grows slower than ${\displaystyle {\sqrt {N}}}$, increasing N (the DC current or light level, etc.) can lead to dominance of shot noise.

Shot noise in electronic circuits consists of random fluctuations of DC current, which is due to electric current being the flow of discrete charges (electrons). Because the electron has such a tiny charge, however, shot noise is of relative insignificance in many (but not all) cases of electrical conduction. For instance 1 ampere of current consists of about 6.24×10¹⁸ electrons per second; even though this number will randomly vary by several billion in any given second, such a fluctuation is minuscule compared to the current itself. In addition, shot noise is often less significant as compared with two other noise sources in electronic circuits, flicker noise and Johnson–Nyquist noise. However, shot noise is temperature and frequency independent, in contrast to Johnson–Nyquist noise, which is proportional to temperature, and flicker noise, with the spectral density decreasing with increasing frequency. Therefore, at high frequencies and low temperatures shot noise may become the dominant source of noise.

With very small currents and considering shorter time scales (thus wider bandwidths) shot noise can be significant. For instance, a microwave circuit operates on time scales of less than a nanosecond and if we were to have a current of 16 nanoamperes that would amount to only 100 electrons passing every nanosecond. According to Poisson statistics the actual number of electrons in any nanosecond would vary by 10 electrons rms, so that one sixth of the time less than 90 electrons would pass a point and one sixth of the time more than 110 electrons would be counted in a nanosecond. Now with this small current viewed on this time scale, the shot noise amounts to 1/10 of the DC current itself.

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 ${\displaystyle f}$,

${\displaystyle S(f)=2e\vert I\vert \ ,}$

where ${\displaystyle e}$ is the electron charge, and ${\displaystyle I}$ is the average current of the electron stream. The noise spectral power is frequency independent, which means the noise is white. This can be combined with the Landauer formula, which relates the average current with the transmission eigenvalues ${\displaystyle T_{n}}$ of the contact through which the current is measured (${\displaystyle n}$ labels transport channels). In the simplest case, these transmission eigenvalues can be taken to be energy independent and so the Landauer formula is

${\displaystyle I={\frac {e^{2}}{\pi \hbar }}V\sum _{n}T_{n}\ ,}$

where ${\displaystyle V}$ is the applied voltage. This provides for

${\displaystyle S={\frac {2e^{3}}{\pi \hbar }}\vert V\vert \sum _{n}T_{n}\ ,}$

commonly referred to as the Poisson value of shot noise, ${\displaystyle S_{P}}$. 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)

${\displaystyle S={\frac {2e^{3}}{\pi \hbar }}\vert V\vert \sum _{n}T_{n}(1-T_{n})\ .}$

It was obtained in the 1990s by Khlus, Lesovik (independently the single-channel case), and Büttiker (multi-channel case).[2] This noise is white and is always suppressed with respect to the Poisson value. The degree of suppression, ${\displaystyle F=S/S_{P}}$, is known as the Fano factor. Noises produced by different transport channels are independent. Fully open (${\displaystyle T_{n}=1}$) and fully closed (${\displaystyle T_{n}=0}$) channels produce no noise, since there are no irregularities in the electron stream.

At finite temperature, a closed expression for noise can be written as well.[2] It interpolates between shot noise (zero temperature) and Nyquist-Johnson noise (high temperature).

• Tunnel junction is characterized by low transmission in all transport channels, therefore the electron flow is Poissonian, and the Fano factor equals one.
• Quantum point contact is characterized by an ideal transmission in all open channels, therefore it does not produce any noise, and the Fano factor equals zero. The exception is the step between plateaus, when one of the channels is partially open and produces noise.
• A metallic diffusive wire has a Fano factor of 1/3 regardless of the geometry and the details of the material.[3]
• In 2DEG exhibiting fractional quantum Hall effect electric current is carried by quasiparticles moving at the sample edge whose charge is a rational fraction of the electron charge. The first direct measurement of their charge was through the shot noise in the current.[4]

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. Take the previous example in which an average of 100 electrons go from point A to point B every nanosecond. During the first half of a nanosecond we would expect 50 electrons to arrive at point B on the average, but in a particular half nanosecond there might well be 60 electrons which arrive there. This will create a more negative electric charge at point B than average, and that extra charge will tend to repel the further flow of electrons from leaving point A during the remaining half nanosecond. Thus the net current integrated over a nanosecond will tend more to stay near its average value of 100 electrons rather than exhibiting the expected fluctuations (10 electrons rms) we calculated. This is the case in ordinary metallic wires and in metal film resistors, where shot noise is almost completely cancelled due to this anti-correlation between the motion of individual electrons, acting on each other through the coulomb force.

However this reduction in shot noise does not apply when the current results from random events at a potential barrier which all the electrons must overcome due to a random excitation, such as by thermal activation. This is the situation in p-n junctions, for instance.[5][6] A semiconductor diode is thus commonly used as a noise source by passing a particular DC current through it.

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.[7]

Shot noise is distinct from voltage and current fluctuations expected in thermal equilibrium; this occurs without any applied DC voltage or current flowing. These fluctuations are known as Johnson–Nyquist noise or thermal noise and increase in proportion to the Kelvin temperature of any resistive component. However both are instances of white noise and thus cannot be distinguished simply by observing them even though their origins are quite dissimilar.

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]

${\displaystyle \sigma _{i}={\sqrt {2qI\,\Delta f}}}$

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.

For a current of 100 mA, measuring the current noise over a bandwidth of 1 Hz, we obtain

${\displaystyle \sigma _{i}=0.18\,\mathrm {nA} \;.}$

If this noise current is fed through a resistor a noise voltage of

${\displaystyle \sigma _{v}=\sigma _{i}\,R}$

would be generated. Coupling this noise through a capacitor, one could supply a noise power of

${\displaystyle P={\frac {1}{2}}qI\,\Delta fR.}$

to a matched load.

The flux signal that is incident on a detector is calculated as follows, in units of photons:

$${\displaystyle P={\frac {\Phi \,\Delta t}{\frac {hc}{\lambda }}}}$$

where c is the speed of light, and h is the Planck constant. Following Poisson statistics, the photon noise is calculated as the square root of the signal:

$${\displaystyle S={\sqrt {P}}}$$

The SNR for a CCD camera can be calculated from the following equation:[9]

$${\displaystyle \mathrm {SNR} ={\frac {I\cdot QE\cdot t}{\sqrt {I\cdot QE\cdot t+N_{d}\cdot t+N_{r}^{2}}}},}$$

where:

• I = photon flux (photons/pixel/second),
• QE = quantum efficiency,
• t = integration time (seconds),
• N_d = dark current (electrons/pixel/sec),
• N_r = read noise (electrons).

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. This is therefore another consequence of discretization, in this case of the energy in the electromagnetic field in terms of photons. 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. Of course there are other mechanisms of noise in optical signals which often dwarf the contribution of shot noise. When these are absent, however, optical detection is said to be "photon noise limited" as only the shot noise (also known as "quantum noise" or "photon noise" in this context) remains.

Shot noise is easily observable in the case of photomultipliers and avalanche photodiodes used in the Geiger mode, where individual photon detections are observed. However the same noise source is present with higher light intensities measured by any photo detector, and is directly measurable when it dominates the noise of the subsequent electronic amplifier. Just as with other forms of shot noise, the fluctuations in a photo-current due to shot noise scale as the square-root of the average intensity:

${\displaystyle (\Delta I)^{2}\ {\stackrel {\mathrm {def} }{=}}\ \langle \left(I-\langle I\rangle \right)^{2}\rangle \propto I.}$

The shot noise of a coherent optical beam (having no other noise sources) is a fundamental physical phenomenon, reflecting quantum fluctuations in the electromagnetic field. 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. What the semiclassical theory does not predict, however, is the squeezing of shot noise.[11] Shot noise also sets a lower bound on the noise introduced by quantum amplifiers which preserve the phase of an optical signal.