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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,
 (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
 (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
 (3)
from which $P(n)$ can be reconstructed via
 (4)
The relation between $F(\xi )$ and $S$ is
 (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
 (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],
 (7)
with $\nu _{\rm eff}=\nu$ for a black body. Eq. (6) implies a reduced number of degrees of freedom for grey-body radiation,
 (8)
Note that the reduction occurs only for $N>1$.
Figure 1: Effective number of degrees of freedom as a function of normalized absorption or amplification rate in a chaotic cavity (inset). The black-body limit for absorbing systems (red, solid line) and the laser threshold for amplifying systems (blue, dashed line) are indicated by arrows. Adapted from Ref. [2].

The ensemble average for $N\gg 1$ is
 (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.
References:
• [1]^ab L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge 1995.
• [2]^ab C.W.J. Beenakker, Phys. Rev. Lett. 81 (1998) 1829.
• [3]^ C.W.J. Beenakker, in Diffuse Waves in Complex Media, edited by J.-P. Fouque, NATO Science Series C531, Kluwer, Dordrecht 1999 [arXiv:quant-ph/9808066].
• [4]^ C.W.J. Beenakker and P.W. Brouwer, Physica E 9 (2001) 463.
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