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. 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. Such a light source can be modeled by,[1]
E(x,t)=E0\sin(kx-\omegat+\phi)
\omega
\phi
The analogue in quantum mechanics is the coherent state
|\alpha\rangle=
infty | |
\sum | |
n=0 |
{\alpha | |
n}{\sqrt{n!}} |
| ||||
e |
2
|n\rangle
Pn
n
Pn=
{{\left\vert\alpha\right\vert | |
2n |
The above result is a Poissonian distribution with
{\Deltan}2=\langlen\rangle
Light that is governed by super-Poissonian statistics exhibits a statistical distribution with variance
{\Deltan}2>\langlen\rangle
Thermal light can be modeled as a collection of
N
j
Ej(t)=E0e-i\omega
i\phij(t) | |
e |
\phij(t)
N
E(t)=
N | |
\sum | |
j=1 |
Ej(t)=
N | |
\sum | |
j=1 |
E0e-i\omega
i\phij(t) | |
e |
After pulling out all the variables that are independent of the summation index
j
\beta(t)=
N | |
\sum | |
j=1 |
i\phij(t) | |
e |
=b(t)ei\Phi
\beta(t)
b(t)=\left\vert\beta\right\vert
ei\Phi
\beta(t)
N
N
\beta(t)
p(\beta(t))=\left[
1 | |
\sqrt{2\pi\sigma2 |
=
1 | |
2\pi\sigma2 |
| ||||
e |
2\right)}{2\sigma2
After
N
\langle{\left\vert\beta(t)\right\vert}2\rangle=2\sigma2=N
\langle\left\vert\beta(t)\right\vert\rangle=0
N
p(\beta(t))=
1 | |
\piN |
| ||||
e |
{N}}
With the probability distribution above, we can now find the average intensity of the field (where several constants have been omitted for clarity)
\langleI(t)\rangle=\langleE*(t)E(t)\rangle
={E0
The instantaneous intensity of the field
I(t)
I(t)=E*(t)E(t)=
2 | |
E | |
0 |
{\left\vert\beta(t)\right\vert}2
Because the electric field and thus the intensity are dependent on the stochastic complex variable
\beta(t)
I
I+dI
p(I(t))dI=p(I(\beta(t)))d2\beta
where
d2\beta
d2\beta=2\pi\left\vert\beta(t)\right\vertd\left\vert\beta(t)\right\vert
The above intensity distribution can now be written as
p(I(t))=p(\left\vert\beta(t)\right\vert)2\pi\left\vert\beta(t)\right\vert{\left(
dI | |
d\left\vert\beta(t)\right\vert |
\right)}-1
=
1 | |
\piN |
| ||||
e |
{N}}2\pi\left\vert\beta(t)\right\vert{\left(
2 | |
2E | |
0 |
\left\vert\beta(t)\right\vert\right)}-1
=
1 | ||||||||
|
| ||||||||||||
e |
{N
2 | |
E | |
0 |
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. The formula describes the probability of observing n photon counts and is given by
P(n)=
infty | |
\int | |
0 |
{\left(\epsilonI\right) | |
n |
The factor
\epsilon=
η | |
h\nu |
η
η=1
I
I=\iint\limitsA
t+\tau | |
\int | |
t |
I(x,y,t')dt'dxdy
On substituting the intensity probability distribution of thermal light for P(I), Mandel's formula becomes
P(n)=
infty | |
\int | |
0 |
{\left(\epsilonI\right) | |
n |
Using the following formula to evaluate the integral
infty | |
\int | |
0 |
xne-axdx=
n! | |
an+1 |
\left(n=0,1,2,..a>0\right)
The probability distribution for n photon counts from a thermal light source is
P(n)=
1 | |
\left(\langlen\rangle+1\right) |
{\left(
\langlen\rangle | |
\langlen\rangle+1 |
\right)}n
where
\langlen\rangle=\epsilonI
{\Deltan}2=\langlen\rangle+{\langlen\rangle}2
In contrast with the Poisson distribution for a coherent light source, the Bose-Einstein distribution has
{\Deltan}2>\langlen\rangle
Light that is governed by sub-Poisson statistics cannot be described by classical electromagnetic theory and is defined by
{\Deltan}2<\langlen\rangle