\documentclass{article}
\usepackage{amssymb,amsmath}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{eso-pic}
\usepackage{epstopdf}
\usepackage{type1cm}
\usepackage[utf8x]{inputenc}
\newcommand{\TeXForWeb}{\TeX$^4$Web}
\newcommand{\cref}[2][\relax]{\href{http://sciencewise.info/ontology/#2}{\ifx#1\relax#2\else#1\fi}}
\newcommand{\dref}[2][\relax]{\href{http://sciencewise.info/definitions/#2}{\ifx#1\relax#2\else#1\fi}}
\newcommand{\fileref}[2][\relax]{\href{http://sciencewise.info//definitions/grey-body_radiation_by_Carlo_Beenakker/#2}{\ifx#1\relax#2\else#1\fi}}
\title{Grey-body radiation by Carlo Beenakker}
\begin{document}
\maketitle
The emission of photons by matter in thermal equilibrium is not a series of
independent events. The textbook example is black-body radiation
\cite{Man95}: 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 photo\-detector
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,
\begin{equation}
P(n)\propto{n+\nu-1\choose n}\exp(-n\hbar\omega/k_{\rm B}T).\label{binomial}
\end{equation}
The binomial coefficient counts the number of partitions of $n$ bosons among
$\nu$ states. The mean photo\-count $\bar{n}=\nu f$ is proportional to the
Bose-Einstein function
\begin{equation}
f(\omega,T)=[\exp(\hbar\omega/k_{\rm B}T)-1]^{-1}.\label{BEfunction}
\end{equation}
In the limit $\bar{n}/\nu\rightarrow 0$, Eq.\ (\ref{binomial}) approaches the
\cref{Poisson distribution} $P(n)\propto\bar{n}^{n}/n!$ of independent photo\-counts.
The Poisson distribution has variance ${\rm Var}\,n=\bar{n}$ equal to its mean.
The negative-binomial distribution describes photo\-counts 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 \eqref{binomial}. A general expression for the photon statistics of grey-body radiation in terms of the scattering matrix was derived in Ref.\ \cite{Bee98}. The expression is simplest in terms of the generating function
\begin{equation}
F(\xi)=\ln\sum_{n=0}^{\infty}(1+\xi)^{n}P(n),\label{Fxidef}
\end{equation}
from which $P(n)$ can be reconstructed via
\begin{equation}
P(n)=\lim_{\xi\rightarrow -1}\frac{1}{n!}\frac{d^{n}}{d\xi^{n}}e^{F(\xi)}.
\end{equation}
The relation between $F(\xi)$ and $S$ is
\begin{equation}
F(\xi)=-\frac{t\delta\omega}{2\pi}
\ln{\rm Det}\,\bigl[{ 1}-({ 1}-SS^{\dagger})\xi f\bigr].\label{Fxilong}
\end{equation}
If the grey body is a chaotic resonator, \cref{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 $N$ \cite{Bee01}. In the large-$N$ limit, the eigenvalue density $\rho(\sigma)=\langle\sum_{n}\delta(\sigma-\sigma_{n})\rangle$ is known in closed-form \cite{Bee99}, which makes it possible to compute the ensemble average of arbitrary moments of $P(n)$.
The first two moments are given by
\begin{equation}
\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}.\label{firsttwomoments}
\end{equation}
For comparison
with black-body radiation we parameterize the variance in terms of the
effective number $\nu_{\rm eff}$ of degrees of freedom \cite{Man95},
\begin{equation}
{\rm Var}\,n=\bar{n}(1+\bar{n}/\nu_{\rm eff}),\label{nueffdef}
\end{equation}
with $\nu_{\rm eff}=\nu$ for a black body. Eq.\ \eqref{firsttwomoments} implies a \textit{reduced} number of degrees of freedom for grey-body radiation,
\begin{equation}
\frac{\nu_{\rm eff}}{\nu}=\frac{\bigl[\sum_{n}(1-\sigma_{n})\bigr]^{2}}{N\sum_{n}
(1-\sigma_{n})^{2}}\leq 1.\label{nueffrho}
\end{equation}
Note that the reduction occurs only for $N>1$.
\begin{figure}[!tb]
\centerline{
\includegraphics[width=20pc]{fig_nueff.png}
}
\caption{
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.\ \cite{Bee98}.
\label{fig_nueff}}
\end{figure}
The ensemble average for $N\gg 1$ is
\begin{equation}
\nu_{\rm eff}/\nu=(1+\gamma)^{2}(\gamma^{2}+2\gamma+2)^{-1},
\label{nueffcavity}
\end{equation}
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.\ \ref{fig_nueff} (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 \eqref{Fxilong} 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.\ \eqref{nueffcavity} can also be applied to an amplifying cavity, if we change $\gamma\mapsto -\gamma$. The result (blue dashed curve in Fig.\ \ref{fig_nueff}) 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.
\begin{thebibliography}{99}
\bibitem{Man95} L. Mandel and E. Wolf, \textit{Optical Coherence and
Quantum Optics}, Cambridge University Press, Cambridge 1995.
\bibitem{Bee98} C.W.J. Beenakker, Phys. Rev. Lett. \textbf{81} (1998) 1829.
\bibitem{Bee99} C.W.J. Beenakker, in \textit{Diffuse Waves in Complex Media}, edited by J.-P. Fouque, NATO Science Series C531, Kluwer, Dordrecht 1999 [arXiv:quant-ph/9808066].
\bibitem{Bee01} C.W.J. Beenakker and P.W. Brouwer, Physica E \textbf{9} (2001) 463.
\end{thebibliography}
\end{document}