%SourceDoc ./tesis.tex
\chapter{Laser Equations with Overlapping Resonances}
\label{ch:LEQ}

In this chapter we employ the quantum description of the
electromagnetic field in open optical resonators developed in
\emph{Chapters \ref{chap:1}} and \emph{\ref{chp:F-D}} to study the
interaction of the field with a two-level atomic medium filling the
resonator.  The results are used to develop a quantum theory for
multi-mode fields in resonators with overlapping resonances. We derive
laser Lagenvin equations that generalize the standard laser theory
\cite{Sarg74,Hake84,Hake85} to account for lasing in weakly confining
cavities.

Before attempting a derivation of the laser equations, and in order to
gain some physical insight into the atom-field interaction, we study the
effect of the field on a single atom. As a simple but nontrivial
problem we consider the spontaneous emission of a two-level atom
inside a cavity. This problem has attracted considerable interest
\cite{Cohe92,Haro92}; and it was found that the cavity may drastically
modify the rate of spontaneous emission from its free
space value. The origin of the effect is the modification due to the cavity
of the local density of modes at the position of the atom.  Most
investigations of the spontaneous emission rate assumed cavities of
regular shape (see Ref.~\cite{Haro92} and references therein), but
recently
\cite{Lamp99,Lamp02,Misi97} also unstable and chaotic cavities were
addressed.  We show below that our system-and-bath Hamiltonian
reproduces the standard result for the atomic decay rate within the
Wigner--Weisskopf approximation. We express the result in terms of
left and right eigenmodes of a non-Hermitean matrix and demonstrate
that for chaotic resonators a statistical analysis of the decay rate
is possible using random matrix theory.

We consider resonators filled with a large number of two-level atoms
and derive generalized laser Langevin equations suitable for
overlapping modes. We present two set of equivalent laser Langevin
equations related by a linear transformation: One in terms of
orthogonal modes corresponding to the close cavity problem, and the
other in terms of the resonance modes of the empty open cavity. The
obtained multi-mode laser equations differ from the standard laser
equations in a fundamental feature: due to the resonator losses the
noise forces associated with different cavity modes are
correlated. This feature is characteristic of systems with overlapping
resonances and, as we show in \emph{Chapter \ref{ch:single-mode}}, is
responsible for the excess noise found in this kind of systems.

We use the multi-mode field equations to describe the laser operation
below the lasing threshold. In this regime the field amplitude is
small and nonlinear terms on the equations of motion might be
neglected; the laser then behaves like a linear amplifier. Using the
resulting linear set of equations we determine the laser threshold
which gives the value of pumping above which the linear description
breaks down. Then the input-output theory is utilized to evaluate the
laser power emission spectrum and express it in terms of the system
$S$ matrix. Finally, we consider the emission spectrum close to
threshold and show that for overlapping modes the linewidth of the
dominating contribution coming from the mode with the smallest loss is
enhanced by the Petermann excess noise factor.


\section{Atom-Field Interaction}
\label{sec:AF-Int}

Consider an open resonator of arbitrary shape filled with a large
number $\mathcal{N}$ of homogeneously broadened two-level atoms with
transition frequency $\nu$. The dipole strength of the atomic
transition is given by $\vect{d}= \la 0|e \vect{r}_{\text{el}} |1
\ra$, where $e$ is the elementary charge, $\vect{r}_{\text{el}}$
denotes the electron position operator, and $|0\ra$ and $|1\ra$ the
ground state and the excited state of the atom, respectively. To
describe the atoms we use the lowering and rising operators $\gs_{-} =
|0\ra \la 1|$ and $\gs_{+} = |1\ra \la 0|$, which introduce
transitions between the atomic levels $|0\ra$ and $|1\ra$, and the
Pauli $\gs_{z}$ matrix, $\gs_{z}= |1\ra \la 1| - |0\ra \la 0|$. These
operators obey the usual algebra for two-level systems
\begin{equation}
\gs_{\pm}^{2} = 0, \quad [\gs_{\pm},\gs_{z}] = \mp 2\gs_{\pm}, \quad
   [\gs_{+},\gs_{-}]_{+} = \openone, \quad \text{and} \quad
   [\gs_{+},\gs_{-}] = \gs_{z}.
\end{equation}
Each atom is located at a position $\vect{r}_{p}$ inside
the cavity and does not interact with any other atom; we use the index
$p$ to label the atoms.

The interaction between the cavity electromagnetic field and the atoms
is described using two approximations \cite{Hake84,Sarg74}: The dipole
approximation and the rotating wave approximation. For the first
approximation we assume the size of the atoms to be much smaller than
the wavelength of the field, allowing us to take the field constant
over the atomic size. The second approximation amounts to neglecting
the rapidly oscillating nonresonant terms in the atom-field
interaction. The total Hamiltonian for the field and atoms then has
the form
\begin{equation}
\label{eq:at-fham}
H = H_{\text{SB}} + \sum_{p}\hbar\omega_0 \gs_{zp} + \hbar
\sum_{\lambda}\sum_{p} [ g_{\gl p} a_{\gl} \gs_{+p} + g_{\gl p}^{*}
a_{\gl}^{\dag} \gs_{-p}]. 
\end{equation} 
Here $H_{\text{SB}}$ is the system--and--bath Hamiltonian
\eqref{eq:hrwa}, the second term on the right hand side represents the
free Hamiltonian of the atoms, and the last term accounts for the
interaction between the atoms and the cavity modes with the coupling
amplitudes
\begin{equation}
\label{eq:gcoup}
g_{\lambda p} = -\rmi \frac{\nu}{\sqrt{2\hbar\go_{\gl}}}
\vect{d}\cdot\vect{u}_{\gl}(\vect{r}_p),
\end{equation}
where $\vect{u}_{\gl}$ is the wave function of the $\gl$th cavity
field mode and $\go_{\gl}$ its oscillation frequency.


\section{Atomic Spontaneous Emission}
\label{sec:atomic_decay}

We investigate the effect of the intracavity field on a single
atom\footnote{The result in this section have been reported in
Ref.~\cite{Vivi03}.}. Since the cavity resonances overlap we can
accurately describe the atomic spontaneous emission in the
perturbative regime \cite{Haro92} following a Wigner-Weisskopf
approach. We assume that initially the atom, which is located at
position $\vect{r}_{0}$ inside the cavity, is in the excited state
while there is no photon in the radiation field.  Hence, the state of
the total system at time $t=0$ is given by $|1,{\text{vac}}\rangle$,
where vac represents the vacuum state of the electromagnetic
field. Since the Hamiltonian \eqref{eq:at-fham} conserves the total
number of atom and field excitations, the time-dependent solution of
the Schr\"odinger equation can be written in the form
\begin{equation}
|\Phi (t)\rangle = c(t) |1,{\text{vac}}\rangle + \sum_{\lambda}
c_{\lambda}(t) |0,1_{\lambda}\rangle + \sum_m \int \! \rmd\omega \;
c_{m}(\omega,t) |0,1_{m}(\omega)\rangle, 
\end{equation}
where $c_{\lambda}(t)$ and $c_{m}(\omega,t)$ are, respectively, the
probability amplitude to find a single photon in the cavity mode
$\lambda$ and in channel $m$ with frequency $\omega$. The time
evolution of the amplitudes $c(t)$, $c_{\lambda}(t)$ and
$c_m(\omega,t)$ follows from the Schr\"odinger equation; an exact
solution can be obtained using Laplace transformation. However,
within the framework of the Wigner--Weisskopf approximation $c(t)$
decays exponentially
\begin{equation}
c(t)=\exp \left(- \rmi (\nu + \delta \nu)t
-\frac{\gamma_{\text{at}}}{2}t \right) c(0),
\end{equation}
where $\delta \nu$ is a frequency shift and
$\gamma_{\text{at}}$ the decay rate of the intensity $|c(t)|^2$. We
note that an exponential decay is only found if the local density of
modes is smooth on the scale of the atomic decay rate. The decay rate
is given by
\begin{equation}
\label{eq:gb}
\gamma_{\text{at}} = \lim_{\epsilon\to 0} \text{Re} \biggl[\frac{2}{\hbar^2}
\sum_{ij} d_i d^{*}_j \mathcal{C}_{ij}(\nu+i \epsilon) \biggr],
\end{equation}
where $i$, $j$ label the components of the dipole matrix element.
Here $C_{ij}(\nu +i \epsilon)$ is the Fourier transform of the
two time correlation function of the electric field
\begin{equation}
\label{eq:clap}
\mathcal{C}_{ij}(t-t^{\prime}) \equiv \Theta (t-t^\prime) 
\left\langle E_i^+(\vect{r}_0,t) E_j^-(\vect{r}_0,t^{\prime}) 
\right\rangle_{\text{vac}},
\end{equation}
$E^{\pm}$ denote the positive and negative frequency part of the
electric field, $\Theta(t-t^\prime)$ is the step function, and the
average $\langle \cdots \rangle_{\text{vac}}$ is the
quantum average over the initial state of the field.

The electric field is connected with the canonical momentum field
through the relation ${\bf E}(\vect{r},t) = -c {\bm \Pi}
(\vect{r},t)$. Inside the cavity both can be expanded in terms of the
cavity modes using Eq.~\eqref{eq:pfin}. Substitution into
Eq.~\eqref{eq:clap} reduces the field correlation function to a sum
over the Green functions of the cavity modes,
\begin{subequations}
\begin{align}
\label{eq:2tcorr}
\mathcal{C}_{ij}(\tau) & = \rmi \sum_{\lambda \lambda^{\prime}}
\frac{\hbar \sqrt{\omega_{\lambda} \omega_{\lambda^{\prime}}}}{2}
u_{\lambda i}(\vect{r}_0) u^{*}_{\lambda^\prime j}(\vect{r}_0)
G_{\lambda \lambda^{\prime}} (\tau),\\
\label{eq:green1}
G_{\lambda \lambda^{\prime}}(\tau) & \equiv -\rmi \Theta(\tau) \langle
a_{\lambda}(\tau) a^{\dagger}_{\lambda^{\prime}}(0)
\rangle_{\text{vac}}.
\end{align}
\end{subequations}
To compute the Green functions, we differentiate Eq.~\eqref{eq:green1}
with respect to $\tau$ and use the equations of motion \eqref{eq:aRWA}
of the cavity operators $a_\lambda$. This yields the equations of
motion of the Green functions
\begin{equation}\label{eq:geqm}
\dot{G}_{\lambda \lambda^{\prime}} (\tau) =  \delta(\tau) G_{\lambda
\lambda^\prime} (0) -\rmi \omega_{\lambda} G_{\lambda
\lambda^{\prime}} (\tau) - \frac{1}{\pi} \int \! \rmd\go
\int\limits_{0}^{\gt} \! \rmd t \, \rme^{-\rmi\go  (\gt - t)} [\gG(\go)
G(t)]_{\gl \gl'} \, .   
\end{equation}
There is no contribution from the noise term in Eq.~\eqref{eq:aRWA} as
$\langle b_m(\omega,0) a^{\dagger}_{\lambda}(0) \rangle_{\textrm{vac}}
= 0$. The initial condition at $\tau=0$ is $G_{\lambda
  \lambda^{\prime}}(0) = -\rmi \delta_{\lambda
  \lambda^{\prime}}$. Equation \eqref{eq:geqm} is readily solved by
Fourier transformation. The result is
\begin{equation}\label{eq:gfre}
G (\omega) = D^{-1}(\omega),
\end{equation}
where the non-Hermitean matrix $D$ was defined in Eq.~\eqref{eq:D}.
Substituting the result into the Fourier transform of
Eq.~\eqref{eq:2tcorr}, we obtain the field correlation function in the
frequency domain,
\begin{equation}
\label{eq:ct}
\mathcal{C}_{ij}(\omega) = \frac{\rmi\hbar \nu}{2} \sum_{\lambda
  \lambda^{\prime}} u_{\lambda i}(\vect{r}_0) u^{*}_{\lambda^\prime
  j}(\vect{r}_0) \left[D^{-1}(\omega) \right]_{\lambda
  \lambda^{\prime}},
\end{equation}
where we again made use of the rotating wave approximation to replace
$\sqrt{\omega_{\lambda} \omega_{\lambda^\prime}} \simeq \nu$. The
decay rate follows upon substitution of Eq.~\eqref{eq:ct} into
Eq.~\eqref{eq:gb},
\begin{equation}
\label{eq:gct}
\gamma_{\text{at}} = -\frac{\nu}{\hbar} \text{Im} \biggl[ \sum_{ij} d_i
  d^{*}_j \sum_{\lambda \lambda^{\prime}} u_{\lambda i}(\vect{r}_0)
  [D^{-1}(\nu)]_{\lambda \lambda^{\prime}}
  u^{*}_{\lambda^\prime j}(\vect{r}_0) \bigg] .
\end{equation}
The sum over modes may be simplified in the eigenbasis of the
non-Hermitean matrix $D$. In this basis, the double sum over the
mode functions $\vect{u}_\lambda$ reduces to a summation over the
left and right eigenmodes of the wave equation of the open cavity,
\begin{subequations}
\label{eq:gfin}
\begin{align}
\label{eq:gfina}
\gamma_{\text{at}} & = \frac{\pi \nu d^2}{\hbar} \rho
(\nu,\vect{r}_0), \\ 
\rho (\nu,\vect{r}_0) &= \frac{1}{\pi}
\text{Im} \left[\sum_l \frac{ L_l^* (\vect{r}_0,\nu) R_l
    (\vect{r}_0,\nu) }{\Omega_l - \nu - \rmi
    \kappa_l} \right].
\end{align}
\end{subequations}
Here, $\rho (\nu,\vect{r}_0)$ is the local density of modes at the
position of the atom, $L_l$, $R_l$ denote the component along ${\bf
  d}$ in the left and right mode $l$, and $\Omega_l$, $\kappa_l$ are
the mode frequency and the mode broadening.

Equations \eqref{eq:gfin} are the final result for the decay rate.
They describe spontaneous emission not only in cavities with
quasi-discrete modes but also in unstable resonators with strongly
overlapping modes. In the latter case, the left eigenfunctions of the
cavity may differ strongly from the corresponding right
eigenfunctions. Our result agrees with the decay rate derived in
Refs.~\cite{Lamp02,Brow02} using a field expansion in terms of
non--orthogonal modes.  Equations \eqref{eq:gfin} have recently
\cite{Misi97} been used to calculate the distribution
$P(\gamma_{\text{at}})$ of decay rates for a two--level atom inside
a chaotic cavity. The local density of modes also determines the
photo-dissociation rate of small molecules with chaotic internal
dynamics \cite{Fyod98}.

Our derivation of the decay rate was based on the system-and-bath
Hamiltonian \eqref{eq:hrwa} which describes the cavity field dynamics
in the rotating wave approximation.  This approximation is valid for
the (typical) case in which the broadening of the resonator modes is
much smaller than the atomic transition frequency.  When the mode
broadening is of the order of the transition frequency, one can still
compute the decay rate in the Wigner-Weisskopf approximation provided
the coupling between the field and the atom is sufficiently
small. Then Eq.~\eqref{eq:gfina} still holds, but the local density of
states must be evaluated using the complete Hamiltonian
\eqref{eq:hlast} for the field. The calculation is done in Appendix
\ref{app:C}.


\section{Laser Langevin Equations}
\label{sec:LLE}

We now extend our theory to many atoms and consider nonlinear
effects. The resulting laser model of open cavities with overlapping
resonances is a generalization of the theory of Hacken \cite{Hake84}
for lasing in cavities with isolated resonances. The laser Langevin
equations are obtained by accounting for the atom-field interaction in
the electric--dipole and rotating wave approximations. Additionally,
we take the field losses in the Markovian approximation and neglect
the frequency dependence of the outcoupling amplitudes
$\mathcal{W}(\go)$. To model pumping and losses for the atoms (e.g.,
due to nonradiative transitions out of the excited level) we couple
them to external pumping and loss baths. This model leads to the
following set of coupled Heisenberg-Langevin equations
\begin{subequations}
\label{eq:leqom}
\begin{align}
\dot{a}_{\lambda}(t) &= -\frac{\rmi}{\hbar} \sum_{\lambda^{\prime}}
\mathcal{H}_{\lambda \lambda^{\prime}} a_{\lambda^{\prime}}(t) - \rmi
\sum_{p} g_{\lambda p}^{*} \sigma_{-p}(t) + F_{\lambda}(t), \\
\dot{\sigma}_{-p}(t) &= -(\rmi\nu + \gamma_\bot ) \sigma_{-p}(t)
+ \rmi \sum_{\lambda} g_{\lambda p} a_{\lambda}(t) \sigma_{zp}(t) +
F_{\bot p}(t), \\ 
\dot{\sigma}_{zp}(t) &= \gamma_{\|} (\gL_{p} -
\sigma_{zp}(t)) + 2\rmi \sum_\lambda [g_{\lambda p}^{*}
  a^{\dagger}_{\lambda}(t) \sigma_{-p}(t) - g_{\lambda p}
  \sigma_{+p}(t) a_{\lambda}(t)] + F_{\|p}(t),
\end{align}
\end{subequations}
where $\gl$ labels the close resonator modes and $p$ the atoms. The
dynamics of the resonator field is damped by the coupling to the
external radiation field and is described by the non-Hermitian matrix
$\mathcal{H}$ defined in Eq.~\eqref{eq:Heff0}. Likewise, the atomic
polarization and inversion are damped with damping amplitudes
$\gga_{\bot}$ and $\gga_{\|}$. The atomic polarization oscillates with
frequency $\nu$, and $\gL_{p}$ is a parameter for the pump strength
which is assumed to be time independent. The coupling amplitudes
$g_{\gl p}$ between atoms and field are defined in
Eq.~\eqref{eq:gcoup}. Finally, the operators $F_{\gl}$, $F_{\bot p}$,
and $F_{\| p}$ represent quantum noise due to the coupling to external
baths. The presence of such noise terms is a consequence of the
fluctuation-dissipation theorem.

In order to complete the equations of motion \eqref{eq:leqom} the
statistics of the Langevin force operators should be specify. All
noise operators are treated as Gaussian random variables with zero mean
and are therefore fully defined by their second-order moments. The
nonzero correlations functions of the field Langevin force $F_{\gl}$
are specified in Eq.~\eqref{eq:f-2cf}. In this thesis we are mainly
interested in the optical domain where $\hbar \nu \gg
kT$. Hence, we take the cavity bath to be at temperature $T=0$ so that
the average number of thermal photons $n_{\text{th}}$ in the cavity is
zero. According to the simple relation between the field noise
operator and the bath-input, $F_{\gl}=-2\pi\rmi
\sum_{m} \mathcal{W}_{\gl m} b_{m}^{\text{in}}$, the above assumption
corresponds to the situation in which there is no external
illumination.

The correlation functions for the atomic noise operators $F_{\bot p}$
and $F_{\| p}$ are obtained using the generalized Einstein
relations and correspond to the ones used in the standard laser
theory \cite{Sarg74,Hake84}. The correlation functions different from
zero are written down in the Appendix \ref{app:3B}.

The main difference between our Langevin equations and the
independent-oscillator equations of the standard laser theory
\cite{Hake84,Sarg74} is the coupling among the modes by the matrix
$\mathcal{H}$. As discussed before, this coupling is due to the
leakage of energy from the cavity through the opening, and its most
important consequence is that the Langevin forces for different modes
are correlated. In Sec.~\ref{sec:nonorth} we showed that the field
cavity dynamics simplifies if one uses the modes of the open cavity to
represent the field. In terms of such modes the equations of motion
decouple ($\mathcal{H}$ becomes diagonal). The correlation between the
noise for different modes, however, does not disappear and can be
interpreted as a consequence of the nonorthogonality of the open
cavity modes. The laser equations in terms of the operators $d$,
associated with the modes of the open cavity, are obtained from
Eqs.~\eqref{eq:leqom} with the help of the linear transformation
\eqref{eq:d-def}. They read
\begin{subequations}
\label{eq:leqnonH}
\begin{align}
\dot{d}_{k}(t) &= -\rmi\go_{k} d_{k}(t) - \rmi \int \! d\vect{r}\,
g_{k}^{L*}(\vect{r}) S_{-}(\vect{r},t) + F_{k}(t), \\
\dot{S}_{-}(\vect{r},t) &= -(\rmi\nu + \gamma_\bot )
S_{-}(\vect{r},t) + \rmi \sum_{k} \, g_{k}^{R}(\vect{r}) d_{k}(t)
S_{z}(\vect{r},t) + F_{\bot}(\vect{r},t), \\ 
\dot{S}_{z}(\vect{r},t)
&= \gamma_{\|} (\gL(\vect{r}) - S_{z}(\vect{r},t)) \nonumber \\
& \quad + 2\rmi \sum_{k}
[g_{k}^{R*}(\vect{r}) d^{\dagger}_{k}(t) S_{-}(\vect{r},t) -
  g_{k}^{R}(\vect{r}) S_{+}(\vect{r},t) d_{k}(t)] +
F_{\|}(\vect{r},t),
\end{align}
\end{subequations}
where $\go_{k}$ are the complex eigenvalues of $\mathcal{H}/\hbar$
defined in Eq.~\eqref{eq:cplx-f}. Additionally, we now describe the
atoms in terms of the atomic polarization and inversion densities
operators
\begin{subequations}
\label{eq:atomac}
\begin{align}
S_{-}(\vect{r},t) & = \sum_p \sigma_{-p}(t)\,
\delta(\vect{r}-\vect{r}_p),\\ 
S_z(\vect{r},t) & = \sum_p \sigma_{zp}(t)\,
\delta(\vect{r}-\vect{r}_p), 
\end{align}
\end{subequations}
and the pump strength density 
\begin{equation}
\gL(\vect{r})= \sum_p \gL_{p}(t)\, \delta(\vect{r}-\vect{r}_p).
\end{equation}
Since the linear transformation \eqref{eq:d-def} is not unitary there
are two coupling amplitudes for each laser mode $k$,
\begin{equation}
\label{eq:glr}
g_{k}^{R}(\vect{r}) = -\rmi \sqrt{\frac{\nu}{2\hbar}}
\vect{d}\cdot\vect{R}_{k}(\vect{r})\quad \text{and} \quad g_{k}^{L}(\vect{r}) =
-\rmi \sqrt{\frac{\nu}{2\hbar}}
\vect{d}\cdot\vect{L}_{k}(\vect{r}),
\end{equation}
associated with the $k$th right and left mode, $\vect{R}_{k}$, and
$\vect{L}_{k}$, respectively. The field Langevin forces $F_{k}$ are
related with the input field through Eq.~\eqref{eq:nonf-in}, their
second order correlation functions are given by
Eq.~\eqref{eq:cffs-non}, while the correlation functions for the
atomic Langevin density forces $F_{\|}(\vect{r},t) = \sum_p
F_{\|p}(t)\, \delta(\vect{r}-\vect{r}_p) $ and $F_{\bot}(\vect{r},t) =
\sum_p F_{\bot}(t)\, \delta(\vect{r}-\vect{r}_p)$ are given in
Appendix \ref{app:3B}. The proposed multi-mode laser equations
\eqref{eq:leqnonH} describe the laser
phenomena in resonators with overlapping resonances. In the limiting
case of weak damping they reduce to the standard laser equations
\cite{Hake84}; then the set of left and right modes coincide and form
an orthonormal basis, the operators $d$ reduce to the operators $a$,
and the coupling amplitudes $g^{L/R}$ equals the coupling amplitude
$g$ for orthogonal modes. Similar laser equations for nonorthogonal
modes were introduce recently by Cheng and Siegman
\cite{Chen03}. Earlier, Langevin laser equations in terms of
non-orthogonal transverse modes of unstable cavities were deduced by
Dutra et al \cite{Dutr99,Dutr00c} using $c$-number representation
instead of operators.

\section{Below Threshold Operation}
\label{sec:laser-bt} 

Before dealing with the complete non-linear dynamics of the laser
equations we consider the operation below the laser threshold. This
regime has previously been considered for one-dimensional cavities
with arbitrary large couplings to the external field \cite{Pras92}, in
semiclassical studies of excess quantum noise \cite{Exte01}, and in
statistical studies of on the Petermann factor in chaotic cavities by
Beenakker and coworkers \cite{Patr00a,Frah00,Scho00}. In this section
we estimate the laser threshold and evaluate the linear amplifier
emission spectrum. Close to threshold the emission spectrum is
dominated by the resonance with the smallest loss and its linewidth is
seen to be enhanced by the Petermann excess noise factor as compare
with its Schawlow--Townes value \cite{Scha58}.

Below threshold the atomic and field operators are basically driven by
the noise forces. The field exhibits big amplitude and phase
fluctuations. In the steady-state the field is weak as it is generated
only from spontaneous emission \cite{Hake84,Hake85}. The photonic
population on the electromagnetic modes is then characterized by a
thermal distribution and, inasmuch as the field intensity is small,
the non-linear interaction in Eq.~\eqref{eq:leqom} between the field
operators and the atomic inversion can be neglected. The inversion
population $S_z$ is then well approximated by its steady-state value
\begin{equation}
S_z(\vect{r},t) = \gL(\vect{r}),
\end{equation}
determined just by the pumping process. Hence the field equations of
motions \eqref{eq:leqom} reduce to the linear coupled equations
\begin{subequations}
\label{eq:lleq-non}
\begin{align}
\dot{d}_{k}(t) &= -\rmi\go_{k} d_{k}(t) - \rmi \int \! d\vect{r}\,
g_{k}^{L*}(\vect{r}) S_{-}(\vect{r},t) + F_{k}(t), \\
\dot{S}_{-}(\vect{r},t) &= -(\rmi\nu + \gamma_\bot )
S_{-}(\vect{r},t) + \rmi \gL \sum_{k} \, g_{k}^{R}(\vect{r})
d_{k}(t) + F_{\bot}(\vect{r},t).
\end{align}
\end{subequations}
Here we assumed that the atomic medium fills the cavity uniformly and that
the pumping is isotropic, so that $\gL(\vect{r})=\gL$. This
linear system of equations can be solved immediately by Fourier
transformation. Introducing the operators in the frequency domain
through
\begin{align}
d_{k}(t) &= \frac{1}{2\pi} \int \! \rmd\go \, \rme^{-\rmi\go t}
d_{k}(\go), \\
S_{-}(\vect{r},t) &= \frac{1}{2\pi} \int \! \rmd\go \,
\rme^{-\rmi\go t} S_{-}(\vect{r},\go),
\end{align}
and in a similar fashion the noise operators, we may solve the
resulting set of algebraic equations. To keep the notation simple the
same symbol is used for both members of the Fourier transformed pair,
the partners will be distinguished through their arguments. Provided
all the poles of the resolvent lay in the lower half of the complex
plane, the steady--state solution for the field is given by
\begin{equation}
\label{eq:d-lnw}
d(\go) = \rmi D^{-1}(\go) G(\go),
\end{equation}
where we combined the mode operators and the noise operators to
$N$-component vectors. The noise vector $G(\go)$ has the elements
\begin{align}
\label{eq:G-lamp1}
G_{k}(\go) &= F_{k}(\go) + F_{\bot k}(\go)\\ 
\intertext{with}
F_{\bot k}(\go) &= -\frac{\rmi}{\gga_{\bot} + \rmi(\nu - \go)}\int\!
\rmd\vect{r} g_{k}^{L*}(\vect{r}) F_{\bot}(\vect{r},t).
\end{align}
The inverse of the $N\times N$ resolvent matrix $D^{-1}(\go)$ has
elements
\begin{equation}
\label{eq:D-lamp}
D_{kl}(\go)= \delta_{kl} \left(\go - \go_{k} - \frac{\rmi V g^{2}\gL}{
  \gga_{\bot} + \rmi (\nu - \go)} \right).
\end{equation}
Here $V$ is the cavity volume, $g^{2}= \nu|\vect{d}|^2/(2\hbar V)$, and we used
\begin{equation}
\int \! d\vect{r}\, g_{k}^{L*}(\vect{r}) g_{l}^{R}(\vect{r}) = V g^2
\delta_{kl},
\end{equation}
which is evaluated using the definitions \eqref{eq:glr} and the
biorthogonality of left and right modes of the open cavity. The
diagonal form of $D(\go)$ in this representation is a consequence of our
assumption of uniform gain over the cavity volume. The spatially
uniform population inversion does not affect the spatial profile of
the cold cavity modes and hence the eigenvectors of $\mathcal{H}$ and
$D(\go)$ coincide. The complex eigenfrequencies of oscillation,
notwithstanding, differ according to Eq.~\eqref{eq:D-lamp}.

For a given pumping value a description of the dynamics in the
time domain is recovered from Eq.~\eqref{eq:d-lnw} by contour
integration with a path closed in the lower half of the complex plane.
In the linear regime the dynamics is then determined by the resonances
$\bar{\go}_{k} = \bar{\gO}_{k}-\rmi \bar{\gk}_{k}$ of the filled
cavity. The cavity resonances are complex frequencies satisfying
\begin{equation}
\label{eq:D-res}
\det [D(\bar{\go})] = 0,
\end{equation}
which are easily evaluated using the explicit form \eqref{eq:D-lamp}
for the elements of $D(\go)$. For $\gL = 0$ all resonances of the
filled cavity lay in the lower half of the complex plane and coincide
with the empty cavity resonances, $\bar{\go}_{k} = \go_{k}$. As the
pump increases the gain experienced by the resonance modes grows, and
all resonances move towards the real axis. Additionally a frequency
pulling by the active medium takes place and all resonances experience
a shift towards the center of the Lorentzian gain profile. Resonances
closer to the center of the gain profile move upward faster than those
far from the it. The linear solution (\ref{eq:d-lnw}) breaks down when
the first resonance reaches the real axis, and a steady--state solution
no longer exists. Figure \ref{fig:res-mov} illustrates schematically
the resonance shift as function of the pump.
\begin{figure}[t]
  \begin{center}
    \includegraphics[scale=.9]{./Figures/resonances_belowTh.eps}
    \caption{\label{fig:res-mov}(a) Schematic illustration of the
    resonances (crosses) of the linear amplifier in the complex
    frequency plane. With increasing pump the resonances move towards
    the real axis as indicated by the arrows. The dashed line
    represents the center of the gain spectrum. (b) The lasing
    threshold is reached when the first resonance hits the real axis.}
    \end{center}
\end{figure}


\subsection{Laser Threshold}

The solution \eqref{eq:d-lnw} to the linear equations
\eqref{eq:lleq-non} gives an accurate description of the laser below
threshold as long as the amplification of the noise is small; then the
modes losses dominate and a steady--state solution exits. However, if
the pump strength is increased such that the gain in the system
overcomes the losses for any of the oscillating modes, an instability
in the mode dynamics arises, the nonlinear terms neglected in going
from Eq.~\eqref{eq:leqnonH} to Eq.~\eqref{eq:lleq-non} become relevant
and the linear theory breaks down. The minimum value of the pumping
parameter $\gL$ for which this happens defines the laser threshold.

The $k$-mode will start to lase when its resonance reaches the real
axis. Thus in order to find its threshold value one must look for real
solutions, $\bar{\go}_{k} = \bar{\go}_{k}^{*}=\bar{\gO}_{k}$, of
Eq.~\eqref{eq:D-res}. Using the definition \eqref{eq:D-lamp} of the
matrix $D(\go)$, one must solve
\begin{equation}
\label{eq:res-cond}
\go - \go_{k} - \frac{\rmi V g^{2}\gL}{\gga_{\bot} + \rmi (\nu -
\go)} = 0
\end{equation}
for real $\go$. From the real part of this equation we obtain the
resonance oscillating frequency
\begin{equation}
\bar{\gO}_{k} =  \frac{\gga_{\bot}\gO_{k} +
\gk_{k}\nu}{\gga_{\bot}+\kappa_{k}}, 
\end{equation}
which for the lasing mode mode will determined the laser
frequency. The imaginary part of Eq.~\eqref{eq:res-cond} yields the
value of pumping necessary for the $k$-resonance to reach the real axis,
\begin{equation}
\label{eq:km-th}
\gL_{k} = \frac{\gk_{k} \gga_{\bot}}{V g^{2}} \left[ 1 +
  \left(\frac{\nu-\gO_{k}}{\gga_{\bot}+\gk_{k}}\right)^2 \right].
\end{equation}
The laser threshold is given by the minimum value of all $\gL_{k}$,
\begin{equation}
\label{eq:lth}
\gL_{\text{th}} = \min_k \gL_{k},
\end{equation}
and the corresponding mode will be the first to lase. Equation
\eqref{eq:km-th} shows that the laser threshold is fixed by the
interplay between the empty cavity resonances and the gain provided by
the atomic medium. Generally, however, one expects the lasing mode to
be close to the center of the gain profile where the amplifying
spectrum can be approximated by a flat spectrum. Then the second term
in Eq.~\eqref{eq:km-th} can be neglected, and the lasing threshold is
given by the condition that the lasing mode gain should equal its
loss, $V g^2 \gL_{\text{th}}/\gga_{\bot} =
\gk_{l}$. Using this condition Beenakker and coworkers determined the
distribution of the laser threshold for lasers in chaotic open
cavities with overlapping \cite{Scho00} and isolated \cite{Misi97}
resonances. For the latter they found the threshold distribution to
be wide, its mean value being much smaller that the pumping rate necessary
to compensate for the average loss.


\subsection{Emission Power Spectrum}
\label{sec:EPS}

Below threshold the field dynamics inside the resonator is linear and
the emission spectrum is easily recovered from the steady state
solution \eqref{eq:d-lnw} using the input-output formalism for linear
amplifiers developed in Sec.~\ref{sec:linear}. The emission power
spectrum is defined in terms of the Wienier--Khintchine theorem
\cite{Mand95}
\begin{equation}
\label{eq:PowerS}
\mathcal{S}(\go)= \sum_m^{M(\go)} \int\limits_{-\infty}^{\infty}\! \rmd t \,
\rme^{-i\omega t} \langle b_{m}^{\text{out}\dagger}(t)
b_{m}^{\text{out}}(0) \rangle.
\end{equation}
The bosonic operators $b^{\text{out}}$ and $b^{\text{out}\dag}$ are the
creation and annihilation operators of the output field, $m$ indexes
the exterior channels, $M(\go)$ is the number of open channels at
frequency $\go$, and $\langle \hdots \rangle$ denotes the quantum
average respect to the initial state of the output electromagnetic
field. Using the definitions \eqref{eq:aout}, the above expression
reduces to
\begin{equation}
\label{eq:la-spec}
\mathcal{S}(\omega)= \frac{1}{2\pi} \sum_m^{M(\go)} \langle
b_{m}^{\text{out}\dag}(\omega) b_{m}^{\text{out}}(\omega) \rangle.
\end{equation}

The output field operators are related with the cavity field operators
through the input-output relations \eqref{eq:in-outnonw}. Combining
Eqs.~\eqref{eq:in-outnonw}, \eqref{eq:d-lnw}, and \eqref{eq:G-lamp1}
we may express the output field operators in terms of the input
operators and the cavity field noise,
\begin{equation}
\label{eq:bout-la}
b^{\text{out}}(\go) = S(\go)b^{\text{in}}(\go) + V(\go)TF_{\bot}(\go),
\end{equation}
where we combined the input and output operators at frequency $\go$ to
$M$-component vectors and the noise operators to an $N$-component
vector, $V(\go)= \mathcal{W}^{\dag}T D^{-1}(\go)T^{-1}$ is an $M\times
N$ matrix, and the $M\times M$ scattering matrix is given by 
\begin{equation}
\label{eq:Sm-la}
S(\go) = \openone - 2\pi\rmi \mathcal{W}^{\dag} T D^{-1}(\go) T^{-1}
\mathcal{W}.
\end{equation}

The linear amplifier emission spectrum is obtained by substituting
Eq.~\eqref{eq:bout-la} into Eq.~\eqref{eq:la-spec} and performing the
quantum average. In the absence of external illumination ($\la
b^{\text{in}\dag}b^{\text{in}}\ra =0$) the emission spectrum reduces
to
\begin{equation}
\mathcal{S}(\go) = \frac{1}{2\pi} \sum_{kl} [T^{\dag}V^{\dag}(\go)
  V(\go)T]_{kl} \la F_{\bot k}^{\dag}(\go) F_{\bot l}(\go)\ra.
\end{equation}
For an atomic medium uniformly distributed over the cavity volume the
correlation function of the noise operators reduces to,
\begin{equation}
\la F_{\bot k}^{\dag}(\go) F_{\bot l}(\go')\ra = \frac{4\pi V g^2
  \gga_{\bot} \mean{S_{1}}}{\gga_{\bot}^2 + (\nu-\go)^2} \la
\vect{L}_{l} | \vect{L}_{k} \ra \delta(\go - \go'),
\end{equation}
where we used the relations in Appendix \ref{app:3B}. Taking into
account that $\la \vect{L}_{l} | \vect{L}_{k} \ra =
(T^{-1}T^{-1\dag})_{lk}$, the emission spectrum can be written as
\begin{equation}
\label{eq:EES-VV}
\mathcal{S}(\go) = \frac{2V g^2 \gga_{\bot}
  \mean{S_{1}}}{\gga_{\bot}^2 + (\nu-\go)^2} \tr[V^{\dag}(\go)V(\go)].
\end{equation}
Using the definition \eqref{eq:Sm-la} of the scattering matrix, one
easily shows that $S(\go)S^{\dag}(\go)-\openone = \big(4\pi V g^2
\gga_{\bot} S_{0} / (\gga_{\bot}^2 + (\nu-\go)^2)\big)
V(\go)V^{\dag}(\go) $, so that the emission spectrum can be expressed
in terms of the system scattering matrix,
\begin{equation}
\mathcal{S}(\go) = \frac{\mean{S_{1}}}{2\pi (\mean{S_{1}}
  -\mean{S_{0}})} \tr [S(\go)S^{\dag}(\go)-\openone].
\end{equation}
Here $\mean{S_{1}}$ and $\mean{S_{0}}$ are the mean atomic population
operators for the upper and lower atomic levels, and we used
$\gL=\mean{S_{1}} - \mean{S_{0}}$ , which holds below threshold, to
express the spectrum in terms of the incomplete population
inversion. This result for the emission spectrum corresponds to
\emph{Kirchkoff's law}, relating the thermal emission with the
amplification \cite{Been99,Scho00}.

It is illustrative to write down the emission spectrum
\eqref{eq:EES-VV} in terms of the complex eigenfrequencies and
non-orthogonal eigenmodes of the open empty cavity. After a straight
forward calculation one obtains
\begin{equation}
\label{eq:EES-la}
\mathcal{S}(\omega) = \frac{2 V g^{2} \gga_{\bot} \mean{S_{1}} }{\pi
  [\gamma_{\bot}^{2} + (\nu -\go)^{2}]} \left[ \sum_{l} \frac{\gk_{l}
    K_{l}}{|D_{ll}(\omega)|^{2}} + \sum_{k}\sum_{l<k} \text{Im}\left[
    \frac{(\go_{l}^{*} - \go_{k}) \langle
      \vect{R}_{l}|\vect{R}_{k}\rangle \langle \vect{L}_{k}|
      \vect{L}_{l}\rangle}{D_{k k}(\omega)
      D_{ll}^{*}(\omega)}\right]\right],
\end{equation}
where $K_{l}$ is the Petermann factor associated with the $l$-mode as
defined in Eq.~\eqref{eq:Pete-fac}. The first sum inside the brackets
is the contribution to the emission spectrum from the individual
resonances, and is the only term one could anticipate from the
standard theory of lasers \cite{Hake84}. Each term in this sum is
approximated by a Lorentzian centered on the resonance frequency
$\text{Re}\bigl(\bar{\go}_{l}(\gL)\bigr)$ and with fullwidth at half
maximum given by $2\text{Im}\bigl(\bar{\go}_{l}(\gL)\bigr)$. Each of
these contributions is proportional to the Petermann factor. This last
feature is solely due to the non-orthogonal character
\cite{Sieg89a,Sieg89b} of the field modes, and is absent in the case
of isolated resonances. The main difference between our result and the
emission spectrum expected from standard laser theory is the presence
of the double sum term inside the brackets. It accounts for the
correlation among the different modes and is proportional to their
overlap $\langle \vect{R}_{l}|\vect{R}_{k}\rangle \langle
\vect{L}_{k}| \vect{L}_{l}\rangle$. In the limit of orthogonal modes,
right and left eigenmodes coincide and form an orthogonal set, the
cross terms contribution vanish, and the Petermann factor is equal
one.  In this limit, Eq.~\eqref{eq:EES-la} reduces to the emission
spectrum for a linear amplifier with isolated resonances.

\subsubsection{Excess Noise and Linewidth}

Close to threshold the gain experienced by the lasing mode, say the
mode with $k=0$, almost completely compensates its losses. The emission
spectrum is then mainly determined by this mode and we need to retain only
its diagonal contribution. For a mode close to the center of the gain profile, the
emission spectrum is then approximated by the Lorentzian 
\begin{eqnarray}
\label{eq:wct}
\mathcal{S}(\omega) \simeq \frac{2 K_{0} \kappa_{0} \mean{S_{1}}}{\pi
  \gL [(\go - \bar{\gO}_{0})^{2} + \frac{1}{4} \delta\go^{2}]},
\end{eqnarray}
with full width at half maximum $\delta\go$. Here we used $V
g^{2}/\gga_{\bot} \sim \gk_{l}/\gL$, which holds close to threshold
for modes in the center of the gain profile.

It is possible to express $\delta \omega$ in terms of the total output
photon current. The later follows from Eq.~\eqref{eq:wct} upon
integrating over frequency,
\begin{equation}
\label{eq:Ibth}
I_{\text{out}} = \frac{4  K_{0} \gk_{0}^{2}\mean{S_{1}}} {\gL \delta\omega}.
\end{equation}
If we now recall the Schawlow--Townes formula for the linewidth
$\delta\go_{\text{ST}} = 2\gk_{0}^2/I_{\text{out}}$, it follows from 
Eq.~\eqref{eq:Ibth} that the linewidth of the lasing mode is given by
\begin{equation}
\delta \omega = 2K_{0} \frac{ \mean{S_{1}}}{\gL} \delta\go_{\text{ST}},
\end{equation}
where an enhancement as compared with the Schawlow--Townes value is
observed. The factor two in the relation is consequence of the below
threshold calculation in which both phase and amplitude fluctuations
are included; above threshold the non-linearities greatly suppress the
amplitude fluctuations reducing the linewidth by a factor two
\cite{Hake84}. The remaining two other factors are generally named
excess noise factors. The factor $\mean{S_1}/
\gL = \mean{S_1}/ (\mean{S_1} - \mean{S_0})$ accounts for an extra
noise due to the incomplete population inversion. The enhancement due
to the Petermann factor, was discussed above, and is a signature of
the nonorthogonal nature of the electromagnetic modes. A similar below
threshold approach to excess noise was followed by Beenakker and
co-workers in their statistical analysis of the Petermann factor in
chaotic cavities \cite{Frah00,Scho00,Patr00a}. A derivation
of the linewidth enhancement for a lasing mode above threshold is
presented in the following chapter.




