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.

(1) |

(2) |

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

(3) |

(4) |

(5) |

The first two moments are given by

(6) |

(7) |

(8) |

The ensemble average for is

(9) |

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]^
^{a}^{b}L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge 1995. - [2]^
^{a}^{b}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.