%SourceDoc ./tesis.tex
\section{Atomic Decay Rate}
\label{app:C}

Here we derive the atomic decay rate $\gga_{\text{at}}$ dropping the
rotating wave approximation for the field dynamics. Our starting point
is Eq.~\eqref{eq:clap}, which when written in terms of the exact modes of
the total system takes the form \cite{Glau91}
\begin{equation}
C_{ij}(t-t^{\prime}) = \Theta(t-t^\prime) \sum_m
\int\limits_{\go_{m}}^{\infty} \! \rmd\omega \; \frac{\hbar \omega}{2}
f_{mi}(\omega,\vect{r}_0) f^{*}_{mj}(\omega,{\bf r}_0)
\rme^{-\rmi\omega(t-t^{\prime})}.
\end{equation}
The Fourier transform of this equation is readily evaluated,
\begin{equation}
\label{eq:ctmu}
C_{ij}(\nu + \rmi\epsilon) = \rmi \sum_m
\int\limits_{\go_{m}}^{\infty} \! \rmd\omega \frac{\hbar\omega}{2} \frac{
  f_{mi}(\omega,\vect{r}_0) f^{*}_{mj}(\omega,\vect{r}_0)}{(\nu -
  \omega) + \rmi\epsilon}. 
\end{equation}
Substitution into Eq.~\eqref{eq:gb} yields the golden rule result
\begin{equation}
\label{eq:gexa}
\gamma_{\text{at}} = \frac{\pi}{\hbar^2} \sum_{ij} \sum_m
\int\limits_{\go_{m}}^{\infty} \! \rmd\omega \; \hbar \omega [ d_{i}
  d^{*}_{j} f_{mi}(\omega,\vect{r}_0) f^{*}_{mj}(\omega,\vect{r}_0)
  \delta(\omega - \nu)]. 
\end{equation}
The atom is located inside the cavity. Therefore, we can use the
mode expansion \eqref{eq:fsep} to replace the modes of the total
system by the cavity modes,
\begin{equation}
\label{eq:gexb}
\gamma_{\text{at}} = \frac{\pi}{\hbar^2} \sum_{ij}
\int\limits_{\go_{m}}^{\infty} \! \rmd\omega \; \hbar \omega\, d_{i}
d^{*}_{j} \biggl[ \sum_{\lambda \lambda^{\prime}}
\sum_{m} u_{\lambda i}(\vect{r}_0) u^{*}_{\lambda^{\prime}
j}(\vect{r}_0)  \alpha_{\lambda m}(\omega)
  \alpha^{*}_{\lambda^{\prime}m}(\omega) \biggr] \delta(\omega - \nu).
\end{equation}
The quantity in square brackets is proportional to the local
density of states $\rho (\omega,\vect{r}_{0})$ (see
Eq.~\eqref{eq:rldos1}), which can be written in terms of the
Green's function projected onto the resonator space \eqref{eq:Gqq},
\begin{equation}
\label{eq:rloca}
\gr (\go_{0},\vect{r}_{0}) = -\frac{2\nu}{\pi c^2} \text{Im} \la
\vect{r}_{0} |\mathcal{G}_{\mathcal{QQ}}(\go_{0}) | \vect{r}_{0} \ra
\, .
\end{equation}
Hence, the atomic decay rate has the same form as in Eq.\
(\ref{eq:gfina}) but with a modified local density of modes
\begin{equation}
\rho (\nu,\vect{r}_0) = \frac{2\nu}{\pi c^2} \text{Im}
\left[\sum_k \frac{ l^{*}_{k}(\nu,\vect{r}_0)
    r_{k}(\nu,\vect{r}_{0})}{ \sigma_{k}^{2}(\nu) -
    \frac{\nu^{2}}{c^{2}}}\right], 
\end{equation}
where $l_k$ and $r_k$ are the left and right eigenmodes and
$\sigma_{k}^{2}$ the eigenvalues of the non-Hermitian operator
$L_{\textrm{eff}} (\nu)$. Equation (\ref{eq:gfin}) is recovered
in the rotating wave approximation. Then the eigenmodes of
$L_{\textrm{eff}}$ are simply related to the eigenmodes of the
non--Hermitean matrix $D^{-1}$, $l_k\approx L_k$ and $r_k\approx
R_k$, and the eigenvalues of $L_{\textrm{eff}} (\nu)$ can be
approximated by
\begin{equation}
\sigma_{k}^{2} \approx \left(\frac{\Omega_k}{c}\right)^2 -
\frac{i 2 \nu}{c^2} \kappa_{k},
\end{equation}
where $\Omega_k$ and $\gk_k$ are the mode frequency and the mode
broadening, respectively. 

\section{Noise Correlation Functions}
\label{app:3B}

The correlation functions for the atomic noise operators $F_{\bot p}$
and $F_{\| p}$ in the laser equations \eqref{eq:leqom} can be found in
\cite{Sarg74,Hake84}. Those different from zero are
\begin{subequations}
\label{eq:atcf1}
\begin{align}
\langle F_{\bot p}^{\dagger}(t) F_{\bot p}(t^\prime) \rangle & = 
\left( \frac{\gamma_{\|}}{2}(\gL_p - \langle \sigma_{zp}(t)\rangle ) +
2 \gamma_{\bot} \langle \sigma_{1p}(t) \rangle \right)
\delta(t-t^\prime), \\ 
\langle F_{\bot p}(t) F_{\bot p}^{\dagger}(t^\prime) \rangle & =  \left(
-\frac{\gamma_{\|}}{2}(\gL_p - \langle \sigma_{zp}(t)\rangle ) + 2
\gamma_{\bot} \langle \sigma_{0p}(t) \rangle \right)
\delta(t-t^\prime),\\ 
\langle F_{\bot p}(t) F_{\| p}(t^\prime) \rangle
& =  \gamma_{\|} (1-\gL_p) \langle \sigma_{-p}(t)\rangle \,
\delta(t-t^\prime),\\ 
\langle F_{\| p}(t) F_{\bot p}(t^\prime) \rangle
& =  -\gamma_{\|} (1+\gL_p) \langle \sigma_{-p}(t)\rangle \,
\delta(t-t^\prime),\\ 
\langle F_{\bot p}^{\dagger}(t) F_{\|
p}(t^\prime) \rangle & =  -\gamma_{\|} (1+\gL_p) \langle
\sigma_{-p}^{\dagger}(t)\rangle \, \delta(t-t^\prime), \\ 
\langle F_{\|
p}(t) F_{\bot p}^{\dagger}(t^\prime) \rangle & =  \gamma_{\|}
(1-\gL_p) \langle \sigma_{-p}^{\dagger}(t)\rangle \,
\delta(t-t^\prime), \\ 
\langle F_{\| p}(t) F_{\|
p}^{\dagger}(t^\prime) \rangle & =  2 \gamma_{\|} (1-\gL_p \langle
\sigma_{zp}(t)\rangle) \, \delta(t-t^\prime),
\end{align}
\end{subequations}
where $\langle \sigma_{1p}(t) \rangle$ and $\langle \sigma_{0p}(t)
\rangle$ are the atomic mean population in the upper and lower atomic state,
respectively. In the case of two--level atoms they are defined through
the relations $\langle \sigma_{1p}(t) \rangle + \langle \sigma_{0p}(t)
\rangle = 1$ and $\langle \sigma_{1p}(t) \rangle - \langle
\sigma_{0p}(t) \rangle = \langle \sigma_{zp}(t) \rangle$.

The correlation functions for the atomic Langevin density forces
\begin{equation}
F_{\|}(\vect{r},t) = \sum_p F_{\|p}(t)\, \delta(\vect{r}-\vect{r}_p) \quad \text{and} \quad F_{\bot}(\vect{r},t) = \sum_p F_{\bot}(t) \, \delta(\vect{r}-\vect{r}_p)
\end{equation}
in the laser equations \eqref{eq:leqnonH}, follow from the correlation functions
\eqref{eq:atcf1},
\begin{subequations}
\label{eq:atcf2}
\begin{align}
\langle F_{\bot}^{\dagger}(\vect{r},t)
F_{\bot}(\vect{r}^{\prime},t^\prime) \rangle & = \left(
\frac{\gamma_{\|}}{2}(\gL(\vect{r}) - \mean{S_{z}(\vect{r},t)}) + 2
\gamma_{\bot} \mean{S_{1}(\vect{r},t)} \right)
\delta(\vect{r}-\vect{r}^{\prime}) \, \delta(t-t^\prime) \\ 
\langle
F_{\bot}(\vect{r},t) F_{\bot}^{\dagger}(\vect{r}^{\prime},t^\prime)
\rangle & = \left( -\frac{\gamma_{\|}}{2}(\gL(\vect{r}) -
\mean{S_{z}(\vect{r},t)}) + 2 \gamma_{\bot} \mean{S_{0}(\vect{r},t)} \right)
\delta(\vect{r}-\vect{r}^{\prime})\,\delta(t-t^\prime),\\ 
\langle
F_{\bot}(\vect{r},t) F_{\|}(\vect{r}^{\prime},t^\prime) \rangle & =
\gamma_{\|} (1-\gL(\vect{r})) \mean{S_{-}(\vect{r},t)}
\,\delta(\vect{r}-\vect{r}^{\prime})\, \delta(t-t^\prime),\\ 
\langle
F_{\|}(\vect{r},t) F_{\bot}(\vect{r}^{\prime},t^\prime) \rangle & =
-\gamma_{\|} (1+\gL(\vect{r})) \mean{S_{-}(\vect{r},t)} \,
\delta(\vect{r}-\vect{r}^{\prime})\,\delta(t-t^\prime),\\ 
\langle
F_{\bot}^{\dagger}(\vect{r},t) F_{\|}(\vect{r}^{\prime},t^\prime)
\rangle & = -\gamma_{\|} (1+\gL(\vect{r})) \mean{S_{+}(\vect{r},t)} \,
\delta(\vect{r}-\vect{r}^{\prime}) \, \delta(t-t^\prime), \\ 
\langle
F_{\|}(\vect{r},t) F_{\bot}^{\dagger}(\vect{r}^{\prime},t^\prime)
\rangle & = \gamma_{\|} (1-\gL(\vect{r})) \mean{S_{+}(\vect{r},t)}
\,\delta(\vect{r}-\vect{r}^{\prime})\, \delta(t-t^\prime), \\
\langle
F_{\|}(\vect{r},t) F_{\|}^{\dagger}(\vect{r}^{\prime},t^\prime)
\rangle & = 2 \gamma_{\|} (1-\gL(\vect{r}) \mean{S_{z}(\vect{r},t)}) \,
\delta(\vect{r}-\vect{r}^{\prime})\,\delta(t-t^\prime),
\end{align}
\end{subequations}
where $\mean{S_{1}(\vect{r})}$ and $\mean{S_{0}(\vect{r})}$ are the
mean atomic population density operators for the upper and lower
atomic levels. All other correlation functions involving one or two noise operators vanish.
