The emission of photons by matter in thermal equilibrium is not a series of independent events. The textbook example is black-body radiation [1]: Consider a system in thermal equilibrium (temperature $T$) that fully absorbs any incident radiation in $N$ propagating modes within a frequency interval $\delta \omega$ around $\omega$. A photodetector counts the emission of $n$ photons in this frequency interval during a long time $t\gg 1/\delta \omega$. The probability distribution $P(n)$ is given by the negative-binomial distribution with $\nu =Nt\delta \omega /2\pi$ degrees of freedom,

$$P(n)\propto {n+\nu -1\choose n}\exp (-n\hbar \omega /k_{\rm B}T).$$

(1)

The binomial coefficient counts the number of partitions of $n$ bosons among $\nu$ states. The mean photocount $\bar {n}=\nu f$ is proportional to the Bose-Einstein function

$$f(\omega ,T)=[\exp (\hbar \omega /k_{\rm B}T)-1]^{-1}.$$

(2)

In the limit $\bar {n}/\nu \rightarrow 0$, Eq. (1) approaches the Poisson distribution$P(n)\propto \bar {n}^{n}/n!$ of independent photocounts. The Poisson distribution has variance ${\rm Var}\,n=\bar {n}$ equal to its mean. The negative-binomial distribution describes photocounts that occur in ``bunches'', leading to an increase of the variance by a factor $1+\bar {n}/\nu$.
By definition, a black body has scattering matrix $S=0$, because all incident radiation is absorbed. If the absorption is not strong enough, some radiation will be transmitted or reflected and $S$ will differ from zero. Such a ``grey body'' can still be in thermal equilibrium, but the statistics of the photons which its emits will differ from the negative-binomial distribution (1). A general expression for the photon statistics of grey-body radiation in terms of the scattering matrix was derived in Ref. [2]. The expression is simplest in terms of the generating function

$$F(\xi )=\ln \sum _{n=0}^{\infty }(1+\xi )^{n}P(n),$$

(3)

from which $P(n)$ can be reconstructed via

$$P(n)=\lim _{\xi \rightarrow -1}\frac {1}{n!}\frac {d^{n}}{d\xi ^{n}}e^{F(\xi )}.$$

(4)

The relation between $F(\xi )$ and $S$ is

$$F(\xi )=-\frac {t\delta \omega }{2\pi } \ln {\rm Det}\,\bigl [{ 1}-({ 1}-SS^{\dagger })\xi f\bigr ].$$

(5)

If the grey body is a chaotic resonator, random matrix theory can be used to determine the sample-to-sample statistics of $S$ and thus of the photocount distribution. What is needed is the distribution of the socalled ``scattering strengths'' $\sigma _{1},\sigma _{2},\ldots \sigma _{N}$, which are the eigenvalues of the matrix product $SS^{\dagger }$. All $\sigma _{n}$'s are equal to zero for a black body and equal to unity in the absence of absorption. The distribution function $P(\{\sigma _{n}\})$ is known exactly for weak absorption (Laguerre orthogonal ensemble) and for a few small values of $N$[4]. In the large-$N$ limit, the eigenvalue density $\rho (\sigma )=\langle \sum _{n}\delta (\sigma -\sigma _{n})\rangle$ is known in closed-form [3], which makes it possible to compute the ensemble average of arbitrary moments of $P(n)$.
The first two moments are given by

$$\bar {n}=\nu f\frac {1}{N}\sum _{n=1}^{N}(1-\sigma _{n}),\;\;{\rm Var}\,n=\bar {n}+\nu f^{2}\frac {1}{N}\sum _{n=1}^{N}(1-\sigma _{n})^{2}.$$

(6)

For comparison with black-body radiation we parameterize the variance in terms of the effective number $\nu _{\rm eff}$ of degrees of freedom [1],

$${\rm Var}\,n=\bar {n}(1+\bar {n}/\nu _{\rm eff}),$$

(7)

with $\nu _{\rm eff}=\nu$ for a black body. Eq. (6) implies a reduced number of degrees of freedom for grey-body radiation,

$$\frac {\nu _{\rm eff}}{\nu }=\frac {\bigl [\sum _{n}(1-\sigma _{n})\bigr ]^{2}}{N\sum _{n} (1-\sigma _{n})^{2}}\leq 1.$$

(8)

Note that the reduction occurs only for $N>1$.

The ensemble average for $N\gg 1$ is

$$\nu _{\rm eff}/\nu =(1+\gamma )^{2}(\gamma ^{2}+2\gamma +2)^{-1},$$

(9)

with $\gamma =\sigma \tau _{\rm dwell}$ the product of the absorption rate $\sigma$ and the mean dwell time $\tau _{\rm dwell}\equiv 2\pi /N\delta$ of a photon in the cavity in the absence of absorption. (The cavity has a mean spacing $\delta$ of eigenfrequencies.) As shown in Fig. 1 (red solid curve), weak absorption reduces $\nu _{\rm eff}$ by up to a factor of two relative to the black-body value.
So far we have discussed thermal emission from absorbing systems. The general formula (5) can also be applied to amplified spontaneous emission, produced by a population inversion of the atomic levels in the cavity. The factor $f$ now describes the degree of population inversion of a two-level system, with $f=-1$ for complete inversion (empty lower level, filled upper level). The scattering strengths $\sigma _{n}$ for an amplifying system are $>1$, and in fact one can show that $\sigma _{n}\mapsto 1/\sigma _{n}$ upon changing $\sigma \mapsto -\sigma$ (absorption rate $\mapsto$ amplification rate). As a consequence, Eq. (9) can also be applied to an amplifying cavity, if we change $\gamma \mapsto -\gamma$. The result (blue dashed curve in Fig. 1) is that the ratio $\nu _{\rm eff}/\nu$ decreases with increasing $\gamma =|\sigma |\tau _{\rm dwell}$ — vanishing at $\gamma =1$. This is the laser threshold, which we discuss next.