Lasers and Laser Light — Question 8

Question 8

Photon-number fluctuations distinguish coherent and fixed-number light

For one selected optical mode, coherent light has a Poisson photon distribution with mean $\bar n$. Define $g^{(2)}(0)=\langle n(n-1)\rangle/\bar n^2$. Compare with an exact N-photon state, for which n is always N.

1. Find the variance and $g^{(2)}(0)$ for a coherent state.

2. Find the same quantities for $N=4$ and compare with coherent light of mean four.

Question 8 - Solution

Photon-number fluctuations distinguish coherent and fixed-number light

(a) Method and calculation

$$\text{Poisson: }\mathrm{Var}(n)=\bar n,\quad\langle n^2\rangle=\bar n^2+\bar n.$$$$\boxed{g^{(2)}_{\mathrm{coh}}(0)=[\langle n^2\rangle-\bar n]/\bar n^2=1}.$$

(b) Interpretation and further calculation

$$\text{Fixed }N=4:\quad\boxed{\mathrm{Var}(n)=0,\quad g^{(2)}(0)=N(N-1)/N^2=3/4}.$$$$\bar n=4\text{ coherent}:\quad\boxed{\mathrm{Var}(n)=4,\quad g^{(2)}(0)=1}.$$

Physical check

Equal mean energy does not imply equal counting statistics. The coherent-state model is an idealization of laser light; technical noise can change measured fluctuations.