Photon antibunching generally refers to a light field with photons more equally spaced than a coherent laser field, a signature being a measured two-time correlation suppressed below that of a coherent laser field. More specifically, it can refer to sub-Poissonian photon statistics, that is a photon number distribution for which the variance is less than the mean. A coherent state, as output by a laser far above threshold, has Poissonian statistics yielding random photon spacing; while a thermal light field has super-Poissonian statistics and yields bunched photon spacing. In the thermal (bunched) case, the number of fluctuations is larger than a coherent state; for an antibunched source they are smaller.
Explanation The variance of the photon number distribution is
V n = ⟨ Δ n 2 ⟩ = ⟨ n 2 ⟩ − ⟨ n ⟩ 2 = ⟨ ( a † a ) 2 ⟩ − ⟨ a † a ⟩ 2 . {\displaystyle V_{n}=\langle \Delta n^{2}\rangle =\langle n^{2}\rangle -\langle n\rangle ^{2}=\left\langle \left(a^{\dagger }a\right)^{2}\right\rangle -\langle a^{\dagger }a\rangle ^{2}.}
Using commutation relations, this can be written as
V n = ⟨ ( a † ) 2 a 2 ⟩ + ⟨ a † a ⟩ − ⟨ a † a ⟩ 2 . {\displaystyle V_{n}=\langle {(a^{\dagger }})^{2}a^{2}\rangle +\langle a^{\dagger }a\rangle -\langle a^{\dagger }a\rangle ^{2}.}
This can be written as
V n − ⟨ n ⟩ = ⟨ ( a † ) 2 a 2 ⟩ − ⟨ a † a ⟩ 2 . {\displaystyle V_{n}-\langle n\rangle =\langle (a^{\dagger })^{2}a^{2}\rangle -\langle a^{\dagger }a\rangle ^{2}.}
The second-order intensity correlation function (for zero delay time) is defined as
g ( 2 ) ( 0 ) = ⟨ ( a † ) 2 a 2 ⟩ ⟨ a † a ⟩ 2 . {\displaystyle g^{(2)}(0)={{\langle (a^{\dagger })^{2}a^{2}\rangle } \over {\langle a^{\dagger }a\rangle ^{2}}}.}
This quantity is basically the probability of detecting two simultaneous photons, normalized by the probability of detecting two photons at once for a random photon source. Here and after we assume stationary counting statistics. Then we have
1 ( ⟨ n ⟩ ) 2 ( V n − ⟨ n ⟩ ) = g ( 2 ) ( 0 ) − 1. {\displaystyle {{1} \over {(\langle n\rangle )^{2}}}(V_{n}-\langle n\rangle )=g^{(2)}(0)-1.}
Then we see that sub-Poisson photon statistics, one definition of photon antibunching, is given by g ( 2 ) ( 0 ) < 1 {\displaystyle g^{(2)}(0)<1} . We can equivalently express antibunching by Q < 0 {\displaystyle Q<0} where the Mandel Q parameter is defined as
Q ≡ V n ⟨ n ⟩ − 1. {\displaystyle Q\equiv {\frac {V_{n}}{\langle n\rangle }}-1.}
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