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. At low counts, so-called shot noise may be observed due to the random Poisson nature of photon emission. 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. 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.

Poissonian 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. Such a light source can be modeled by,

E ( x , t ) = E 0 sin ⁡ ( k x − ω t + ϕ ) {\displaystyle E(x,t)=E_{0}\sin(kx-\omega t+\phi )}

where ω {\displaystyle \omega } is the frequency of the field and ϕ {\displaystyle \phi } is a time independent phase shift. The analogue in quantum mechanics is the coherent state

| α ⟩ = ∑ n = 0 ∞ α n n ! e − | α | 2 2 | n ⟩ {\displaystyle |\alpha \rangle =\sum _{n=0}^{\infty }{\frac {{\alpha }^{n}}{\sqrt {n!}}}e^{\frac {{-\left\vert {\alpha }\right\vert }^{2}}{2}}|n\rangle }

By projecting the coherent state onto the Fock state | n ⟩ {\displaystyle |n\rangle } , we can find the probability P n {\displaystyle P_{n}} of finding n {\displaystyle n} photons using the Born rule, which gives

P n = | α | 2 n n ! e − | α | 2 = ⟨ n ⟩ n n ! e − ⟨ n ⟩ {\displaystyle P_{n}={\frac {{\left\vert \alpha \right\vert }^{2n}}{n!}}e^{{-\left\vert \alpha \right\vert }^{2}}={\frac {{\langle n\rangle }^{n}}{n!}}e^{-\langle n\rangle }}

The above result is a Poissonian distribution with variance Δ n 2 = ⟨ n ⟩ {\displaystyle {\Delta n}^{2}=\langle n\rangle } which is a distinct feature of the coherent state.

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