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
) that fully absorbs any incident radiation in
propagating modes within a frequency interval
around
. A photodetector counts the emission of
photons in this frequency interval during a long time
. The probability distribution
is given by the negative-binomial distribution with
degrees of freedom,
The binomial coefficient counts the number of partitions of
bosons among
states. The mean photocount
is proportional to the Bose-Einstein function
In the limit
, Eq. (
1) approaches the
Poisson distribution of independent photocounts. The Poisson distribution has variance
equal to its mean. The negative-binomial distribution describes photocounts that occur in ``bunches'', leading to an increase of the variance by a factor
.
By definition, a black body has scattering matrix
, because all incident radiation is absorbed. If the absorption is not strong enough, some radiation will be transmitted or reflected and
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
from which
can be reconstructed via
The relation between
and
is
If the grey body is a chaotic resonator,
random matrix theory can be used to determine the sample-to-sample statistics of
and thus of the photocount distribution. What is needed is the distribution of the socalled ``scattering strengths''
, which are the eigenvalues of the matrix product
. All
's are equal to zero for a black body and equal to unity in the absence of absorption. The distribution function
is known exactly for weak absorption (Laguerre orthogonal ensemble) and for a few small values of
[4]. In the large-
limit, the eigenvalue density
is known in closed-form
[3], which makes it possible to compute the ensemble average of arbitrary moments of
.
The first two moments are given by
For comparison with black-body radiation we parameterize the variance in terms of the effective number
of degrees of freedom
[1],
with
for a black body. Eq.
(6) implies a
reduced number of degrees of freedom for grey-body radiation,
Note that the reduction occurs only for
.
The ensemble average for
is
with
the product of the absorption rate
and the mean dwell time
of a photon in the cavity in the absence of absorption. (The cavity has a mean spacing
of eigenfrequencies.) As shown in Fig.
1 (red solid curve), weak absorption reduces
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
now describes the degree of population inversion of a two-level system, with
for complete inversion (empty lower level, filled upper level). The scattering strengths
for an amplifying system are
, and in fact one can show that
upon changing
(absorption rate
amplification rate). As a consequence, Eq.
(9) can also be applied to an amplifying cavity, if we change
. The result (blue dashed curve in Fig.
1) is that the ratio
decreases with increasing
— vanishing at
. 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.