Photon statistics
Photon statistics is the theoretical and experimental study of the statistical distributions produced in photon counting experiments, which use photodetectors to analyze the intrinsic statistical nature of photons in a light source.
In these experiments, light incident on the photodetector generates photoelectrons and a counter registers electrical pulses generating a statistical distribution of photon counts.
Low intensity disparate light sources can be differentiated by the corresponding statistical distributions produced in the detection process.
Three regimes of statistical distributions can be obtained depending on the properties of the light source: Poissonian, super-Poissonian, and sub-Poissonian.
[1] The regimes are defined by the relationship between the variance and average number of photon counts for the corresponding distribution.
Both Poissonian and super-Poissonian light can be described by a semi-classical theory in which the light source is modeled as an electromagnetic wave and the atom is modeled according to quantum mechanics.
In contrast, sub-Poissonian light requires the quantization of the electromagnetic field for a proper description and thus is a direct measure of the particle nature of light.
In classical electromagnetic theory, an ideal source of light with constant intensity can be modeled by a spatially and temporally coherent electromagnetic wave of a single frequency.
photons using the Born rule, which gives The above result is a Poissonian distribution with variance
[2] Using the intensity distribution together with Mandel's formula[3] which describes the probability of the number of photon counts registered by a photodetector, the statistical distribution of photons in thermal light can be obtained.
oscillators is After pulling out all the variables that are independent of the summation index
Because the oscillators are uncorrelated, the phase of the superposed field will be random.
It represents the sum of the uncorrelated phases of the oscillators which models the intensity fluctuations in thermal light.
On the complex plane, it represents a two dimensional random walker with
a random walker has a Gaussian probability distribution.
Thus, the joint probability distribution for the real and imaginary parts of the complex random variable
results in With the probability distribution above, we can now find the average intensity of the field (where several constants have been omitted for clarity) The instantaneous intensity of the field
is given by Because the electric field and thus the intensity are dependent on the stochastic complex variable
This infinitesimal element can be rewritten as The above intensity distribution can now be written as This last expression represents the intensity distribution for thermal light.
The last step in showing thermal light satisfies the variance condition for super-Poisson statistics is to use Mandel's formula.
[3] The formula describes the probability of observing n photon counts and is given by The factor
is the intensity incident on an area A of the photodetector and is given by[4] On substituting the intensity probability distribution of thermal light for P(I), Mandel's formula becomes Using the following formula to evaluate the integral
The probability distribution for n photon counts from a thermal light source is where
Light that is governed by sub-Poisson statistics cannot be described by classical electromagnetic theory and is defined by
[1] The advent of ultrafast photodetectors has made it possible to measure the sub-Poissonian nature of light.
Recently researchers have shown that sub-Poissonian light can be induced in a quantum dot exhibiting resonance fluorescence.
[5] A technique used to measure the sub-Poissonian structure of light is a homodyne intensity correlation scheme.
[6] In this scheme a local oscillator and signal field are superimposed via a beam splitter.
The superimposed light is then split by another beam splitter and each signal is recorded by individual photodetectors connected to correlator from which the intensity correlation can be measured.
Evidence of the sub-Poissonian nature of light is shown by obtaining a negative intensity correlation as was shown in.
