%SourceDoc ./tesis.tex
\chapter{Photocount Statistics of Chaotic Lasers}
\label{ch:photo}


It has been known since the early days of the quantum theory of laser,
that the nonlinear interactions between the optical field and the
active medium in a single-mode laser leads to drastic changes in the
statistics of the photon field \cite{Sarg74,Hake85,Wall94,Mand95}:
Below the laser threshold, losses outweigh the gain by linear
amplification such that nonlinear saturation is negligible. Then the
photon statistics is well represented by a thermal distribution, and
the probability $P_n$ for the occupation of the mode with $n$ photons
decays as the power law $P_n \sim [\bar{n}/ (1+\bar{n})]^n$ ($\bar{n}$
is the mean photon number). Above threshold nonlinear interactions
stabilize the field intensity. The (relative) intensity fluctuations
are strongly reduced below the value found for a thermal distribution,
and far above threshold $P_n$ approaches the Poissonian distribution
$P_n = [\bar{n}^n/n!] \exp(-\bar{n})$ characteristic for a coherent
state.

Recent experiments on random lasers \cite{Cao03} as well as on laser
in chaotic resonator with irregularly shaped boundary \cite{Gmac98}
have dragged considerable interest on the characterization of the
properties of light emitted by these novel devises. Contrary to the
profusion of theoretical investigations
\cite{Been98,Been99,Been00,Mish01} on the linear optical regime of
such systems, theoretical studies on the nonlinear optical regime are
still scarce. In particular, little is known about the photocount
statistics above the laser threshold in random media. In this Chapter
we address that problem in the regime of single-mode lasing in an
almost closed chaotic resonator. This problem provide us with an
excellent opportunity to get familiar with the kind of mixing of ideas
and concepts coming from quantum optics and quantum chaos theory, that
the study of light in random media requires. We show that the chaotic
nature of the cavity modes gives rise to fluctuations of the
photocount {\em on top} of the quantum optical fluctuations known from
laser theory. Chaos-induced fluctuations are found when a single-mode
photodetection is performed over an {\em ensemble} of modes. The
ensemble may be obtained from a single resonator upon varying suitable
parameters or from different resonators with small variations in
shape. Two kind of ensembles are considered depending on the strength
of the pumping: An above threshold ensemble, in which the system gain
exceeds the ensemble average loss for the laser mode; and a below
threshold ensemble in which all lasers of the ensemble operates below
the laser threshold. The factorial moments of the photocount display
strong fluctuations from one mode to another. We evaluate numerically
the distribution of the mean photocount and show that for the case of
a chaotic cavity with a small number of escape channels this
distribution is broad, differing significantly from a Gaussian.

Most of the results presented in this chapter have been reported in
Ref.~\cite{Hack01}.


\section{Chaotic Lasers}

\begin{figure}[t]
  {\tiny
  \begin{center}
    \includegraphics[scale=.65]{./Figures/cav.eps}
    \caption{\label{fig:51}Sketch of chaotic laser cavity with a
      partly transmitting mirror. The cavity is connected to a
      waveguide and a frequency selective photodetector.}
  \end{center}}
\end{figure}
We consider a chaotic laser: a chaotic resonator with an irregular
shaped boundary homogeneously filled with an amplifying medium. In
this chapter we concentrate on the study of the simple case of laser
oscillation in a single cavity mode which is in resonance with the
amplifying medium at frequency $\go_{0}$. A sketch of the system is
shown in Fig.~\ref{fig:51}. The cavity has an opening and is coupled
to a waveguide that supports $M$ transverse modes at frequency
$\go_{0}$. A partly transmitting mirror reflects impinging radiation
with mean probability $R$ back into the resonator. The fraction
$T=1-R$ of radiation is transmitted and injected into the waveguide. A
frequency selective photodetector counts the transmitted photons with
efficiency $1$ at the other end of the waveguide. Due to the opening
the modes acquire finite widths $\Gamma$. We restrict ourselves to the
case of weak outcoupling to the outside world, defined by the
condition
\begin{equation}
\mean{\gG} \ll \Delta\go_{0}\,,
\end{equation} 
where $\Delta\go_{0}$ is the cavity mean mode separation. The cavity
then has a quasidiscrete spectrum and the mode wavefunctions can be
approximated by the eigenfunctions of the closed cavity $u(\vect{r})$
(for simplicity, in this chapter, we neglect the field polarization
and work with an scalar amplitude for the field). Since the mean
cavity escape rate $\mean{\Gamma} = MT \Delta \omega/2\pi$, in the weak
coupling regime the mirror transmission coefficient $T$ must be small,
$MT \ll 1$. The counting time $t$ of photons arriving at the
photodector will be assumed large enough that the radiation from
individual cavity modes can be resolved, $t \Delta \omega \gg 1$.

As for the active medium, we allow for $\mathcal{N}$ active two-level
atoms with transition frequency $\go_{0}$. The $p$-th atom ``sees''
the field mode through a coupling constant $g_p$ proportional to both
the atomic dipole matrix element $d$ and the value of the lasing mode
at the location of the atom, $g_p \propto d\, u(\vect{r}_{p})$. We
only consider the simplest situation in which the characteristic times
for atomic pump and losses are short compared to the mean life time of
a photon in the cavity, so that we can adiabatically eliminate the
medium variables from the equations of motion for the system.


\subsection{Single-Mode Laser}

A laser oscillation in a single-mode is characterized by three
parameters comprising the effect of the atoms on the field mode: $A$,
$B$, $C$ that characterize the linear gain, the nonlinear saturation,
and the total loss, respectively. The first two parameters depend on
the atomic-field coupling constants as \cite{Sarg74,Wall94,Mand95}
\begin{equation}
\label{eq:ln-g}
A \sim \sum_{p=1}^{\mathcal{N}} g_{p}^2, \qquad B \sim
\sum_{p=1}^{\mathcal{N}} g_{p}^4 \, ,
\end{equation} 
while the total loss rate
\begin{equation}
C = \Gamma+\kappa
\end{equation}
is the sum of the photon escape rate $\Gamma$, due to the cavity
opening, and the absorption rate $\kappa$, that accounts for all other
loss mechanisms of the the radiation field inside the cavity. While
here $\gk$ may be considered fixed, both the coupling amplitudes
$g_{p}$ and the photon escape rate $\gG$ depend on the resonator mode,
and inasmuch as the resonator mode represents wave chaos these two
quantities become random numbers. We may then ask for the effect of
the chaotic wavefunction nature on the parameters $A$, $B$, and $C$;
this problem will be considered in the following section.

As an illustration of the way these three parameters determined the
laser state, we write down the steady--state photon number
distribution for a single-mode laser \cite{Sarg74,Wall94,Mand95}
\begin{equation}
\label{eq:distin}
P_{n} = \mathsf{N} \frac{ [A n_s / C]^{n+n_s}}{(n+n_s)!}\,,
\end{equation}
which gives the probability to find $n$ photons at a time $t$ in the
laser field. The symbol $\mathsf{N}$ represents a normalization
constant and the nonlinear saturation $B$ enters through the so-called
saturation photon number $n_{s} = A/B$. Two limits of the distribution
\eqref{eq:distin} sufficiently far from threshold will be of relevance
for us: The below threshold limit, $A < C$, in which the photon number
distribution can be shown to reduce to a thermal distribution. And the
above threshold limit, $A > C$, where the distribution
\eqref{eq:distin} is well approximated by a Poissonian distribution
characteristic of a coherent state. The mean photon number in this two
limits is then given by
\begin{equation}
\label{eq:nmean}
\mean{n}=\begin{cases}
	\frac{A}{C-A} & \text{for $A<C$}\,, \\
	n_{s}\frac{A-C}{C} & \text{for $A>C$}\,.
	\end{cases}
\end{equation}


\subsection{Chaotic Modes}

If the cavity is weakly coupled to the outside world we may use the
modes of the closed cavity to describe the electromagnetic field
inside the resonator. Let $u_{\lambda}(\vect{r})$ be the amplitude of
the closed cavity mode at frequency $\go_{\lambda}$, normalized
according to $\int \! \rmd \vect{r} \, |u_{\lambda}(\vect{r})|^{2}=1
$. In a chaotic cavity the amplitude $u_{\lambda}(\vect{r})$ at the
point $\vect{r}$ behaves like a random Gaussian variable, and is
uncorrelated with the amplitude at any other point provided it lies
further apart than an optical wave length $\gl$
\cite{Deus95,Alt95,Alt97}. The chaotic modes $u_{\gl}(\vect{r})$ are then
mimicked by a random superposition of plane waves with wave number
$k_{\lambda}=\go_{\gl}/c$ \cite{Berr77,Haak01,Urbi03}.

At this point we determined the consequences that considering a
chaotic wavefunction may have on the laser parameters $A$, $B$, and
$C$. The coupling amplitudes $g_{p}$, being proportional to the mode
wavefunction, behave like random Gaussian variables. It follows from
Eq.~\eqref{eq:ln-g} for the linear gain and the nonlinear saturation
that for a large number of atoms, $\mathcal{N}\gg 1$, the
distributions for the parameters $A$ and $B$ become sharp due to
central limit theorem. The total loss rate $C$, on the other hand,
depends on the photon escape rate, $\gG = \sum_{i=1}^{M}
|\gga_{i}|^{2}$. The latter, like the $g_{p}$, is also a random
variable, as will be revealed presently. The outcoupling amplitude
$\gga_{i}$ of the field mode into the $i$th transverse channel of
the waveguide are independent random quantities with Gaussian
statistics, since they represent the local fluctuations of the
resonator mode across the outcoupling mirror. In fact, their
randomness constitutes the principle effect of the wave chaos within
the resonator on the ensemble fluctuations of the laser output (see
Eq.~\eqref{eq:bound}). The Gaussian statistics of the $\gga$'s implies
that the distribution of $\gG$ over an \emph{ensemble of modes} is the
$\chi^2_\nu$ distribution
\begin{equation}
\label{eq:distgam}
P(\Gamma) = \mathsf{A}_\nu \Gamma^{\nu / 2 -1} \exp(-\nu \Gamma / 2
\mean{\Gamma})
\end{equation}
well known from random--matrix theory \cite{Guhr98}.  Here, $\nu =
\beta M$ is an integer and $\mathsf{A}_\nu$ a normalization constant. The
value of the parameter $\beta$ depends on whether the system is
time-reversal invariant ($\beta=1$) or whether time-reversal
invariance is broken ($\beta=2$) \cite{Guhr98,Haak01}. The special case
$M=\beta=1$ is known as the Porter--Thomas distribution. We have thus
shown that in a single-mode laser the effects of the chaotic nature
of the wave functions are contained in the fluctuations of the loss
$C$ given by Eq.~\eqref{eq:distgam}.


\section{Photodetection Statistics}

A photocount experiment gives a statistical measure of the arrival
frequency of photons at the photodetector in a given time interval. As
is usually the case, the photodetector is placed outside the
cavity. Thus, although the photons registered at the detector
correspond to the state of the field outside the resonator, we may use
the input--output relations to relate the results of a photocount
experiment to the intracavity field.

\subsection{Input-Output Relation}

The photons arriving at the photodector correspond to the field in the
wave guide and do not provide direct information about the state of
the field inside the cavity. In order to relate the measurements made
by the photodector to the state of the field inside the cavity, we use
the input--output theory of Gardiner and Collet
\cite{Gard85,Gard00,Wall94} (see also \emph{Chapter \ref{chp:F-D}} for a
detail discussion). Waves entering and leaving the waveguide are
described by $M$ annihilation and creation operators
$b_{i}^{\text{in}}$, $b_{i}^{\text{in}\dagger}$, $b_{i}^{\text{out}}$,
$b_{i}^{\text{out}\dagger}$, which obey the commutation relations
\begin{equation}
[b_i(t),b_j^\dagger(t^\prime)] = \delta_{ij}\delta(t-t^\prime), \qquad
[b_i(t),b_j(t^\prime)] =  0 \,.
\end{equation}
Here, $i,j=1,\ldots,M$ label the transverse modes of the waveguide and
$b=b^{\text{in}}$ or $b=b^{\text{out}}$. The boundary condition
\begin{align}
\label{eq:bound}
b_{i}^{\text{in}} (t) + b_{i}^{\text{out}}(t)=\gamma_i a(t)\, ,
\end{align}
connects incoming and outgoing radiation in each transverse mode
with the annihilation operator $a$ of the cavity mode. While the
boundary condition (\ref{eq:bound}) results from assuming a linear
coupling between the waveguide and the cavity field, {\em no}
restriction is imposed on the intracavity dynamics.  Therefore, the
input-output relation (\ref{eq:bound}) holds both below and above
threshold. 

In the linear regime below threshold, one can eliminate the cavity
operators from Eq.\ (\ref{eq:bound}) and express the outgoing
radiation in terms of the incoming radiation, the intracavity noise,
and the $S$-matrix \cite{Wall94,Been98,Been99}
(cf. Eq.~\eqref{eq:slinm}). One thus obtains the photocount
fluctuations \cite{Been98,Been99} through the $S$-matrix statistics
known for chaotic scattering.


\subsection{Photocount Distribution}

For times $t$ larger than $1/\Delta\go_{0}$ the photodector performs a
frequency resolved measurement and individual modes of the cavity can
be identify. We assume that all outgoing modes in the waveguide are
detected with unit efficiency by the photodector. The probability
$p(n)$ that $n$ photons are counted in the time interval $t$ is given
by \cite{Glau63,Kell64,Mand95}
\begin{align} 
\label{eq:count}
p(n) & = \frac{1}{n!} \la :W^n \rme^{-W}: \ra \,, \\
\label{eq:W} 
W &= \int_{0}^{t} \rmd t' \,  \sum_{i=1}^M b_{i}^{\text{out} \dagger}
(t') b_{i}^{\text{out}} (t')\,. 
\end{align}
Here $\la \cdots \ra$ denote the quantum steady--state average and the
colons demand normal and a certain time ordering. The photocount
distribution may be evaluated using the state of the field inside the
cavity. For this we use the input-output relation \eqref{eq:bound},
which for the case of vacuum input, allow us to write Eq.~\eqref{eq:W}
as
\begin{equation}
\label{eq:W2}
W = \gG I \,,
\end{equation}
where $I$ is the integrated cavity field intensity
\begin{equation}
\label{eq:intens}
I   =   \int_0^t \! \rmd t^\prime \, a^\dagger (t^\prime) a(t^\prime)\, .
\end{equation} 
We characterize $p(n)$ through its factorial moments
\begin{equation}
\label{eq:fact1}
\mu_r = \sum_{n=0}^\infty n(n-1)\cdots(n-r+1)p(n).
\end{equation}
Combining Eqs.~\eqref{eq:count}, \eqref{eq:W2}, and \eqref{eq:fact1}
one obtains for the factorial moments
\begin{equation}
\label{eq:fact2}
\mu_r = \Gamma^r \langle: I^r :\rangle\, .
\end{equation}
Clearly, the first moment alias mean photocount, $\mu_1=\Gamma
t \mean{n}$, is a purely static quantity as it is
proportional to the stationary mean photon number inside the cavity .


\section{Factorial Moments Distribution}

In what follows we will show that for an ensemble of chaotic cavity
modes the moments of the photocount distribution become random
variables, and evaluate numerically the distribution of the mean
photocount.  The factorial moments $\mu_{r}$ specify the output
statistics of a single cavity mode with the escape rate $\Gamma$. For
an ensemble of chaotic modes the escape rate is a random quantity and
due to Eq.~\eqref{eq:fact2} the $\mu_r$ become also random quantities.
Their distribution is given by
\begin{equation}
\label{eq:main}
\mathcal{P}(\mu_r) = \int \! \rmd \Gamma \, P(\Gamma)\, \delta(\mu_r -
\Gamma^r \langle: I^r \! :\rangle)\, .
\end{equation}
Note that the right hand side involves a twofold average, the
quantum optical average (represented by the brackets $\langle \cdots
\rangle$) and the ensemble average over the cavity modes
(represented by the integral with the probability distribution
$P(\Gamma)$). We emphasize that the quantum optical average
$\langle: I^r \! :\rangle$ depends on $\Gamma$ through the total
loss rate $C$. We now discuss the result \eqref{eq:main} in
various limiting cases.


\subsubsection{Large Number of Escape Channels}

We first consider the case $M \gg 1$ of many transverse modes in the
waveguide. The diameter of the waveguide is then much larger than the
wavelength of the cavity mode. A simple saddle point argument shows
that $P(\Gamma)$ for large $M$ approaches a Gaussian distribution with
mean $\mean{\Gamma} \sim M$ and standard deviation $\Delta \Gamma =
\mean{\Gamma}/\sqrt{\beta M}$ \cite{Guhr98}. The relative fluctuations
are small, $\Delta \Gamma / \mean{\Gamma} \sim 1/\sqrt{M} \to 0$ for
$M \to \infty$, and the same is true for the fluctuations of all
factorial moments. For cavities with large outcoupling mirrors we thus
recover the sharp factorial moments one is used to from non-chaotic
resonators.


\subsubsection{Vanishing Photocount}

Second, we investigate the limit of vanishing photocount, $\mu_r \to
0$.  According to Eq.~\eqref{eq:fact2}, this is the weak--coupling
limit $\Gamma \to 0$ for which the photons in the cavity field can
hardly escape into the waveguide. The laser dynamics becomes
independent of the outcoupling loss as the total cavity loss $C$ is
fully dominated by the absorption loss, $\gk > \mean{\gG}$. All cavity
field moments become independent of $\Gamma$, and Eq.~\eqref{eq:fact2}
reduces to $\mu_r \sim \Gamma^r$.  Substitution into
Eq.~\eqref{eq:main} yields the power-law behavior
\begin{equation}
\mathcal{P}(\mu_r) \sim \mu_{r}^{{(\beta M) / (2r)}-1} ,
\end{equation}
for $\mu_r \to 0$.  A special case of this result is $M=\beta=1$ for
which the distribution of the first and second moment diverge as
$\mu_1^{-1/2}$ and $\mu_2^{-3/4}$ for $\mu_1$, $\mu_2 \to 0$,
respectively.

\subsubsection{Short Time Regime}

The third case is the short--time regime $t \ll t_c$ where $t_c$ is
the correlation time of the intensity fluctuations of the cavity
mode.  During counting intervals that short the field intensity
cannot vary appreciably. Therefore, the integrated intensity $I$
becomes the product of $t$ with the instantaneous photon number
$\hat{n}=a^\dagger a$ in the cavity. The factorial moments of the
output take the simple form
\begin{equation}
\label{eq:fact3}
\mu_r = (\Gamma t)^r \langle \hat{n}^{(r)}\rangle \, .
\end{equation}
That is, the factorial moments of the photocount distribution become
proportional to the factorial moments $\langle \hat{n}^{(r)}\rangle$
of the photon number distribution. To obtained Eq.~\eqref{eq:fact3} we
used $\langle \hat{n}^{(r)}\rangle = \langle :\!\hat{n}^r\! :\rangle$
\cite{Mand95}, which is easily proved using the optical equivalence
theorem.


\subsection{Mean Photocount}

\begin{figure}[t]
  \begin{center}
    \includegraphics[scale=.8]{./Figures/Photocount_AboveTh.eps}
    \caption{\label{fig:mean_above}Distribution $\mathcal{P}(\mu_1 /
      \mu_{\text{max}})$ as a function of the dimensionless mean
      photocount $\mu_1 / \mu_{\text{max}}$ for $\beta=1$ in an
      ensemble above the laser threshold on average. We consider two
      set of laser parameters. Rates are given in units of $A \equiv
      1$, the nonlinearity is $B= 0.005$: (a) $\kappa= 0.7$,
      $\mean{\Gamma} = 0.2$, and (b) $\kappa= 0.7$, $\mean{\Gamma} =
      0.02$. The inset shows $\mu_1$ as a function of $\Gamma /A$. The
      function maximum occurs at $\Gamma^{*}\sim 0.13$. The arrow
      indicates the threshold--value of $\Gamma$ below which lasing
      takes place.}
  \end{center}
\end{figure}
We restrict now our analysis to the distribution of the mean
photocount $\mu_{1}$. According to Eq.~\eqref{eq:fact2}, the
calculation of the mean photocount reduces to the evaluation of the
steady-state average
\begin{equation}
\mu_1 = \gG t \mean{n}\,,
\end{equation} 
for arbitrary counting time $t$. We calculated $\mu_1$ numerically as
a function of $\Gamma$ using the distribution \eqref{eq:distin}.
Substituting the result $\mu_1(\Gamma)$ into Eq.~\eqref{eq:main} and
carrying out the integration over $\Gamma$, we obtained
$\mathcal{P}(\mu_1)$. The results for time-reversal invariant cavities
connected to a waveguide supporting $M=1,2,3,6,10$ transverse modes
are plotted in Figs.~\ref{fig:mean_above} and \ref{fig:mean_below},
for $\nu=M$ and different sets of laser parameters. The distribution
for cavities with broken time-reversal invariance\footnote{the case
  $\beta=2$ can be realized in microwave cavities containing
  magnetized ferrites} and $M'$ escape channels are obtained from
those for time-reversal invariance by making $M'=M/2$ with $M$ even in
Figs.~\ref{fig:mean_above} and \ref{fig:mean_below}. All distributions
in Fig.~\ref{fig:mean_above} correspond to $A> \mean{C}$, i.e., to
lasers above threshold in the ensemble average. By contrast, $A<
\kappa$ for the distributions of Fig.~\ref{fig:mean_below}; all lasers
of those ensembles are below threshold irrespective of the escape rate
$\Gamma$. While for a large number of the escape channels the
distributions get sharper around
$\mu_{1}(\bar{\Gamma})/\mu_{\text{max}}$, in agreement with our
previous analysis for $M\gg 1$, we note that for small number of
escape channels the distributions are strongly non--Gaussian. In
particular, for $M=1$ they are all peaked as $\mu_1^{-1/2}$ at small
mean photocount, in accord with the general argument presented above
for the asymptotics at $\mu_1 \to 0$. Two further features spring to
the eye and demand explanation: First, above threshold but not below
we encounter an additional peak at maximum photocount; second, for
certain cases (Figs.~\ref{fig:mean_above}(a) and
\ref{fig:mean_below}(a)) the distribution $\mathcal{P}(\mu_1)$
displays a shoulder for submaximal $\mu_1$.
\begin{figure}[t]
  \begin{center}
    \includegraphics[scale=.8]{./Figures/Photocount_BelowTh.eps}
    \caption{\label{fig:mean_below}Distribution ${\cal P}(\mu_1 /
      \mu_{\text{max}})$ as a function of the dimensionless mean
      photocount $\mu_1 / \mu_{\text{max}}$ for $\beta=1$ in an
      ensemble bellow threshold. We consider two sets of laser
      parameters. Rates are given in units of $A \equiv 1$, the
      nonlinearity is $B= 0.005$: (a) $\kappa= 2.0$, $\mean{\Gamma} =
      4.0$, and (b) $\kappa= 2.0$, $\mean{\Gamma} = 0.5$. The inset
      shows $\mu_1$ as a function of $\Gamma /A$.}
  \end{center}
\end{figure}

The origin of these structures lies in the $\Gamma$ dependence of the
mean photocount $\mu_1$. That dependence is depicted in the insets in
Figs.~\ref{fig:mean_above} and \ref{fig:mean_below}, and seen to be
qualitatively different above and below threshold.  While $\mu_1$
increases monotonically with $\Gamma$ in the below threshold case, it
develops a maximum at an intermediate value of $\Gamma$ when the laser
is above threshold.  This behavior can be understood from the simple
analytic expressions
\begin{equation}
\label{eq:cases}
\frac{\mu_1}{t}=\begin{cases}
 \Gamma n_s\frac{A-C}{C}  & \text{for $C=\Gamma+\kappa<A$},\\
 \frac{\Gamma A}{C - A} & \text{for $ C =\Gamma+\kappa>A$}.
\end{cases}
\end{equation}
that follow from the photon number average results \eqref{eq:nmean}
sufficiently far from threshold. According to Eq.~\eqref{eq:cases},
$\mu_1$ rises linearly with $\Gamma$ out of the origin and approaches
an asymptotic plateau for very large $\Gamma$, as visible in the
insets in Figs.~\ref{fig:mean_above} and \ref{fig:mean_below}. The rise to the
plateau is monotonic when $A<\kappa$ since the below-threshold case of
\eqref{eq:cases} applies for all values of $\Gamma$; the maximum
photocount $\mu_{\text{max}}$ is then just the plateau value. The
laser of Fig.~\ref{fig:mean_above} is above threshold for small
$\Gamma$ but below for large $\Gamma$. The maximum photocount
$\mu_{\text{max}}$ then arises for an intermediate value $\Gamma^*$. A
simple argument can be employed to determine the border between the
below- and the above-threshold ensemble. The argument follows from the
observation that $\mu_1(\Gamma)$ approaches its plateau value from
above for the above-threshold ensemble while the plateau is approached
from below when the ensemble is below threshold.  From the lower case
of Eq.~\eqref{eq:cases}, which becomes exact for $\Gamma\to\infty$,
this yields the threshold condition $A = \kappa$. To estimate the
value of $\Gamma^*$ we may employ the above-threshold case of
Eq.~\eqref{eq:cases} and find $\Gamma^*=\sqrt{A\kappa}-\kappa$.

Based on this understanding of $\mu_1(\Gamma)$ one can appreciate the
above-threshold peak of ${\cal P}(\mu_1)$ at $\mu_1=\mu_{\rm
  max}$. Substituting $\mu_1(\Gamma)= \mu_{\text{max}} +
\frac{1}{2}\mu''(\Gamma^*)(\Gamma-\Gamma^*)^2$ in the photocount
distribution \eqref{eq:main} we find that ${\cal P} (\mu_1)$ has a
square--root singularity $|\mu_1 - \mu_{\text{max}}|^{-1/2}$ which is
precisely the peak depicted in Fig.~\ref{fig:mean_above}. This peak,
contrary to the peak at $\mu_{1}$, is independent of the number of
channels on the waveguide, and is a genuine signature of the laser
nonlinearity. We note that in Fig.~\ref{fig:mean_above}(a) the width
of the peak at $\nu_{\text{max}}$ is enhanced with increasing number
of channels. The opposite happens in Fig.~\ref{fig:mean_above}(b),
where for $M>2$ the peak cannot be resolved. The explanation for this
behavior again follows from our previous analysis for the general
case $M\gg 1$. For increasing $M$ the distribution $P(\Gamma)$
narrows around $\mean{\Gamma}$ and is enhanced for values of $\Gamma$
close to $\mean{\Gamma}$, on the contrary, values of $\Gamma$ far
from $\mean{\Gamma}$ are strongly suppressed. Thus in
Fig.~\ref{fig:mean_above}(a), where $\mean{\Gamma} \sim \Gamma^{*}$,
definition \eqref{eq:main} implies an enhancement of the probability
of $\mu_{1}$ around the peak at $\mu_{\text{max}}$; while in
Fig.~\ref{fig:mean_above}(b) the small mean value of $\mean{\Gamma}$
accounts for the narrowing of $\mathcal{P}(\mu_{1})$ around
$\mu_{\text{max}}$.    

Clearly, no such peak can arise in the below--threshold
case of Fig.~\ref{fig:mean_below} as the photocount increases
monotonically with $\Gamma$. For the set of parameters in
Fig.~\ref{fig:mean_below} the mean photocount is well approximated by
the second line in Eq.~\eqref{eq:cases} and we can analytically
evaluate $\mathcal{P}(\mu_{1})$. Defining
$y=\mu_{1}/\mu_{\text{max}}$, we obtain
\begin{equation}
\label{eq:below}
\mathcal{P}(y)=\mathsf{B}_{\nu} \frac{y^{\nu/2 -1}}{(1-y)^{\nu/2+1}}
\exp\left(-\frac{\nu}{2\beta} \frac{y}{1-y}\right)\,.
\end{equation}
Here $\mathsf{B}_{\nu}$ is a normalization constant. The analytical
curves obtained using \eqref{eq:below} do not differed appreciably
from the numerical results, therefore, for clarity, in
Fig.~\ref{fig:mean_below} we have only plotted the numerical results
for $\mathcal{P}(y)$. The suppression of the peak at
$\mu_{\text{max}}$ can be appreciated in Eq.~\eqref{eq:below}, where
for $y\to 1$ ($\mu \to \mu_{\text{max}}$) we see that $\mathcal{P}(y)$
vanishes.

The shoulders at submaximal photocount are caused by amplified
spontaneous emission below the laser threshold. Formally, the
shoulders arise from the asymptotic plateau of $\mu_1$ for large
$\Gamma$. The definition \eqref{eq:main} immediately implies enhanced
probability for photocounts $\mu_1$ in the vicinity of that plateau.
Note that the shoulder is invisible for the distributions shown in
Fig.~\ref{fig:mean_above}(b), and it disappear for increasing $M$ in
Fig.~\ref{fig:mean_above}(a). In the first case, large values of
$\Gamma$ are strongly suppressed by the small mean value
$\mean{\Gamma}$; this suppression is however a general feature of the
distribution $\mathcal{P}(\mu_{1})$ for increasing number of channels $M$,
explaining the disappearance of the shoulder in the second case.
Further note that the shoulders for all curves in
Fig.~\ref{fig:mean_below} with $M\sim 1$ lie closer to
$\mu_{\text{max}}$ than for the curves in Fig.~\ref{fig:mean_above}(a)
since $\mu \to \mu_{\text{max}}$ coincides with $\Gamma \to \infty$ in
the regime below threshold.

In contrast to the mean photocount, which can be expressed in terms
of the stationary distribution of the laser, all $\mu_r$ with $r \ge
2$ involve dynamical information through correlation functions with
$r-1$ time arguments; their behavior will not be discussed here.

\section{Discussion and Outlook}

In this chapter we derived the photocount statistics of the radiation
emitted from a laser inside a chaotic resonator in the regime of
single-mode laser. The new ingredient added to the standard laser
theory came from the chaotic nature of the cavity modes. We considered
two possible kind of laser ensembles: One corresponding to laser
operation on the ensemble average, the other corresponding to all
lasers on the ensemble being below threshold. The spatial fluctuations
of the resonator eigenfunctions induced a strong mode-to-mode
fluctuations of the laser emission. However, we showed that the
probability density for each factorial moment depends only on general
symmetries of the ensemble and on four parameters describing the laser
dynamics; these are the coefficients of linear gain, mean escape loss,
absorption loss and saturation of the amplifying medium. The
distribution of the mean photocount over the \emph{ensemble of modes}
was shown to change qualitatively at the lasing transition, and
display up to three peaks above the lasing threshold.

We now compare our results with related fluctuation phenomena in
other areas of physics and discuss possible experimental tests. In
nuclear physics the Porther--Thomas distribution describes
level--width fluctuations in neutron scattering \cite{Guhr98}. The
amplitude fluctuations of Coulomb blockade oscillations in
semiconductor quantum dots are also of the Porther--Thomas type
\cite{Jala92}. In both cases, as well as for the chaotic lasers
studied in this chapter, the fluctuations result from the chaotic
nature of wavefunctions.  However, in chaotic lasers new interesting
features arise due to the interplay of wave chaos with the nonlinear
dynamics of the laser. As a consequence of this interplay, the
distribution of the mean photocount can strongly deviate from the
Porter--Thomas distribution.  To test the predicted mode-to-mode
fluctuations experimentally, one must study the photocount statistics
for an ensemble of chaotic modes. It seems feasible to generate such
ensembles in tunable lasers e.g.\ by shape variations, or by the
injection and displacement of artificial scatterers in the case of
microwave cavities.

Finally let us mention that in contrast to the wealth of results for
random linear amplifiers, little is known about the photon statistics
of random media above the lasing threshold. Our study addressed this
problem for the simple case of single-mode laser in a chaotic cavity
with isolated modes. The observed generic operation regime of random
lasers, and the one expected for lasers in chaotic resonators,
corresponds nevertheless to multi-mode lasing. Recently Patra
\cite{Patr02} showed numerically that in this regime, due to the mode
competition, the statistics of the laser emission change: a laser mode
above threshold will show an increment on the photon number
fluctuations depending on the total number of lasing modes operating
above threshold for a given value of pump. In a related work,
Mishchenko \cite{Mish03} considered a multi-mode laser in a chaotic
cavity with overlapping resonances, which allowed him to described the
field through a uniform density of photons over the cavity volume
employing a transfer equation. Although his analysis is restricted
to the below threshold case, the nonlinear character of the equations
of motion for the field and the active medium are kept, allowing for
the evaluation of corrections to previous results for linear
amplifiers close to the laser threshold \cite{Been98,Been99}.

