Photon statistics ($p_n=\mathrm{Pr}(N=n)$)

Photon statistics is the probability distribution of the number operator in specified field modes, or the distribution of registered counts in a specified gate. Those distributions agree only for an appropriate ideal measurement. Mode selection, losses, and detector dead time matter. For a one-mode density matrix, $p_n=⟨n|\rho |n⟩$ andThese two sums already distinguish preparations with the same expected value.

For a coherent state, write $a|\alpha ⟩=\alpha |\alpha ⟩$ and $|\alpha ⟩={\sum }_n{c}_n|n⟩$ in number states. The annihilation operator gives $c_{n+1}\sqrt{n+1}=\alpha c_n$, so $c_n=c_0{\alpha }^n/\sqrt{n!}$. Normalization yields $|c_0{|}^2=e^{-|\alpha {|}^2}$. Thus photon statistics follows a Poisson distribution with $\mu =|\alpha {|}^2$, and the normal ordering of every moment gives $⟨(a^{†}{)}^k{a}^k⟩={\mu }^k$. This factorization is the defining virtue of coherent states for optical calculations.^[ref]

As an original comparison, choose $\mu =2$. Then $p_0=e^{-2}$, $p_1=p_2=2e^{-2}$, $\mathrm{Var}N=2$, and $⟨N(N-1)⟩=4$. A two-photon Fock state also has mean two but has $\mathrm{Var}N=0$ and pair moment two. ConsequentlyIn particular, $|1⟩$ cannot supply a pair. Nevertheless, photon statistics alone does not determine a quantum state: averaging $|\alpha ⟩⟨\alpha |$ over a uniform quantum phase removes its off-diagonal entries while retaining exactly the same $p_n$ and these equal-time moments.

For a single mode in thermal equilibrium, its quantum harmonic oscillator energy ladder gives a geometric distributionIts probability generating function is $F(z)=(1-q)/(1-qz)$. Differentiating twice gives $F^{\prime \prime }(1)=2q^2/(1-q{)}^2=2{\mu }^2$, hence $g^{(2)}(0)=2$. At the chosen mean $\mu =2$, $q=2/3$, $p_0=1/3$, $p_1=2/9$, $p_2=4/27$, and $\mathrm{Var}N=6$. Rare high occupations increase the pair count even though the mean matches the other preparations. The geometric weights also arise from a Gaussian mixture of coherent states.^[ref]

Single-mode selection is essential to this factor of two. If $M$ independent equally populated thermal modes are collected together, let $N={\sum }_{j=1}^M{N}_j$ with $⟨N_j⟩=\mu /M$. Adding their variances gives $\mathrm{Var}N=\mu +{\mu }^2/M$, hence $g^{(2)}(0)=1+1/M$. Averaging over many unresolved modes can make thermal pair counts approach the independent-count reference.

Photon bunching is enhanced near-coincident detection relative to an appropriate independent-count reference. It need not require nonclassical light. In a classical intensity model with nonnegative instantaneous light intensity $I$, ideal detection with a conditional Poisson distribution givesA fluctuating intensity therefore bunches counts. For a stationary classical intensity with finite second moment, the Cauchy-Schwarz inequality further gives $⟨I(t)I(t+\tau )⟩\le ⟨I^2⟩$. A zero-delay correlation smaller than one at a later delay cannot be explained by this model.

Photon antibunching means suppression of close pairs compared with pairs at some later delay: $g^{(2)}(0)<g^{(2)}(\tau )$ for some $\tau$. If correlations decay to the independent-count reference, $g^{(2)}(0)<1$ is a sufficient signature. It violates the classical intensity bounds in photon bunching. For the specified single-mode number measurement, second-order optical coherence also shows that $g^{(2)}(0)<1$ is equivalent to $\mathrm{Var}N<\mu$. This variance criterion concerns that mode or gate; it must not be applied unchanged to arbitrarily long integrated counting intervals.