%SourceDoc ./tesis.tex
\section{Formal Solution of the Field Equations of Motion}
\label{app:2A}

In this section we give the exact solutions to the equations of motion
for the electromagnetic field in open optical cavities. Two cases are
considered: The exact dynamics generated by the Hamiltonian
\eqref{eq:hlast} and the dynamics in the \emph{rotating wave
  approximation} generated by the Hamiltonian \eqref{eq:hrwa}. 

\subsubsection{Exact Dynamics}

The solutions of the field equations of motion for the exact dynamics
are obtained in a rather simple way if one uses the position and
momentum operators to describe the field. The modes-of-the-universe
Hamiltonian
\eqref{eq:hqp} corresponds to a set of independent harmonic
oscillators,
\begin{subequations}
\label{eq:qpdyn}
\begin{align}
q_{m}(\omega,t) &= q_{m}(\omega,t_0) \cos(\omega (t-t_0)) +
\frac{1}{\omega}  p_{m}^{\dagger}(\omega,t_0) \sin(\omega (t-t_0)), \\
p_{m}^{\dagger}(\omega,t) &= p_{m}^{\dagger}(\omega,t_0) \cos(\omega
(t-t_0)) - \omega  q_{m}(\omega,t_0) \sin(\omega (t-t_0)),
\end{align}
\end{subequations}
where $q$ and $p^{\dagger}$ are the position and the adjoint momentum
operator for the exact modes of Maxwell's equations. The solutions
for the cavity and channel operators are obtained by substitution of
Eq.~\eqref{eq:qpdyn} into definitions \eqref{eq:qop} and
\eqref{eq:pop}, and using the inverse relations \eqref{eq:qpsplit} for
the operators at the initial time $t_0$. The result for the position
operators is
\begin{subequations}
\label{eq:Qexact}
\begin{multline}
Q_{\lambda}(t) = \sum_{\lambda'} \left[ \dot{X}_{\lambda
    \lambda'}(t-t_0) Q_{\lambda'}(t_0) + X_{\lambda \lambda'}(t-t_0)
  P_{\lambda'}^{\dagger}(t_0) \right] \\ 
+ \sum_{m}
\int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \left[
  \dot{X}_{\lambda m}(\omega,t-t_0) Q_{m}(\omega,t_0) + X_{\lambda
    m}(\omega,t-t_0) P_{m}^{\dagger}(\omega,t_0) \right], \\
\end{multline}
\begin{multline}
Q_{m}(\omega,t) = \sum_{\lambda} \left[ \dot{X}_{\lambda
    m}^{*}(\omega,t-t_0) Q_{\lambda}(t_0) + X_{\lambda m}^{*}(\omega,t-t_0)
  P_{\lambda}^{\dagger}(t_0) \right] \\ 
+ \sum_{m'}
\int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega' \left[ \dot{X}_{m
    m'}(\omega,\omega',t-t_0) Q_{m'}(\omega',t_0) + X_{m
    m'}(\omega,\omega',t-t_0) P_{m'}^{\dagger}(\omega',t_0) \right],
\end{multline}
\end{subequations}
while the momenta follow from 
\begin{equation}
\label{eq:Pexact}
P_{\lambda}^{\dagger}(t) = \dot{Q}_{\lambda}(t), \qquad
P_{m}^{\dagger}(\omega,t) = \dot{Q}_{m}(\omega,t).
\end{equation}
The matrix elements of the time evolution transformation $X$ are defined by
\begin{subequations}
\begin{align}
X_{\lambda \lambda'}(t) &= \sum_{n} \int\limits_{\omega_{n}}^{\infty}
\! \rmd\omega'' \; \alpha_{n\lambda}(\omega'')
  \alpha_{n\lambda'}^{*}(\omega'') \frac{1}{\omega''} \sin(\omega'' t), \\
X_{\lambda m}(\omega,t) &= \sum_{n} \int\limits_{\omega_{n}}^{\infty}
\! \rmd\omega'' \; \alpha_{n\lambda}(\omega'')
  \beta_{n m}^{*}(\omega'',\omega) \frac{1}{\omega''} \sin(\omega'' t),
  \\
X_{m m'}(\omega,\omega',t) &= \sum_{n} \int\limits_{\omega_{n}}^{\infty}
\! \rmd\omega'' \; \beta_{n m}(\omega'', \omega)
  \beta_{n m'}^{*}(\omega'',\omega') \frac{1}{\omega''} \sin(\omega'' t).
\end{align}
\end{subequations}
Using the results given in Appendix \ref{app:A1} for the integrals
involving the expansion coefficients $\alpha$ and $\beta$, one can
easily check that
\begin{align}
X_{\lambda \lambda'}(0) &= 0,  & \dot{X}_{\lambda \lambda'}(0) &=
\delta_{\lambda \lambda'}, & \ddot{X}_{\lambda \lambda'}(0) &= 0, \nonumber\\
X_{\lambda m}(\omega,0) &= 0, & \dot{X}_{\lambda m}(\omega,0) &= 0,  &
\ddot{X}_{\lambda m}(\omega,0) &= 0, \\
X_{m m'}(\omega,\omega',0) &= 0,  & \dot{X}_{m m'}(\omega,\omega',0) &=
\delta_{m m'} \, \delta(\omega - \omega'), & \ddot{X}_{m
  m'}(\omega,\omega',0) &= 0. \nonumber
\end{align}

The solutions for the bosonic operators for the cavity and channel
field are obtained as linear combinations of Eqs.~\eqref{eq:Qexact}
and \eqref{eq:Pexact}, by means of Eq.~\eqref{eq:qpbres}.

\subsubsection{Dynamics in the Rotating Wave Approximation}

The solutions of the Heisenberg equations of motion
\eqref{eq:aeqmrwa} and \eqref{eq:beqmrwa} are
\begin{subequations}
\label{eq:solrwa}
\begin{align}
\label{eq:arwaexact}
a_{\lambda}(t) &= \sum_{\lambda'} Y_{\lambda \lambda'}(t-t_0)
a_{\lambda'}(t_0) + \sum_{m} \int \! \rmd\omega Y_{\lambda
m}(\omega,t-t_0) b_{m}(\omega,t_0) , \\ 
b_{m}(\omega,t) &=
\sum_{\lambda} Y_{\lambda m}^{*}(\omega,t-t_0) a_{\lambda}(t_0) +
\sum_{m'} \int \! \rmd\omega' Y_{m m'}(\omega,\omega',t-t_0)
b_{m'}(\omega',t_0).
\end{align}
\end{subequations}
Here the time evolution operator $Y$ has the matrix elements
\begin{subequations}
\begin{align}
Y_{\lambda \lambda'}(t) &= \sum_{n} \int \! \rmd\omega'' \; \alpha_{n\lambda}(\omega'')
  \alpha_{n\lambda'}^{*}(\omega'') \rme^{-\rmi\omega'' t}, \\
Y_{\lambda m}(\omega,t) &= \sum_{n} \int \! \rmd\omega'' \; \alpha_{n\lambda}(\omega'')
  \beta_{n m}^{*}(\omega'',\omega) \rme^{-\rmi\omega'' t},\\
Y_{m m'}(\omega,\omega',t) &= \sum_{n} \int\! \rmd\omega'' \; \beta_{n
m}(\omega'', \omega) \beta_{n m'}^{*}(\omega'',\omega') \rme^{-\rmi \omega'' t};
\end{align}
\end{subequations}
and the expansion coefficients $\ga$ and $\gb$ are to be taken in
the rotating wave approximation. Using the results in Appendix
\ref{app:A1}, one finds the initial values
\begin{equation}
Y_{\lambda \lambda'}(0)=\delta_{\lambda \lambda'}, \qquad  Y_{\lambda
m}(\omega,0)= 0 \qquad \text{and} \qquad Y_{m m'}(\omega,\omega',0) =
\delta_{mm'}\, \delta(\go - \go').
\end{equation}

Within the rotating wave approximation Eq.~\eqref{eq:abess} reduces to
\begin{subequations}
\label{eq:abessrwa}
\begin{gather}
\label{eq:exp1rwa}
\left( \omega - \omega_{\lambda} \right) \alpha_{m \lambda}(\omega) = \sum_{n}\!
\int \! \rmd\omega^\prime \; \mathcal{W}_{\lambda n}(\omega^\prime)
\beta_{mn} (\omega,\omega'), \\
\label{eq:exp2rwa}
\left( \omega - \omega^{\prime} \right) \beta_{mn}(\omega,\omega') = \sum_{\lambda}
\mathcal{W}^{*}_{\lambda n}(\omega^{\prime}) \alpha_{m \lambda}(\omega),
\end{gather}
\end{subequations}
Direct substitution shows that the expressions \eqref{eq:solrwa} are
solutions of the equations of motion \eqref{eq:eqmrwa}.


\section{Corrections to the Rotating Wave Approximation} 
\label{app:2B} 

For resonators with overlapping modes the time scale set by the
inverse of the mean frequency spacing $1/\Delta\go$ is comparable to the
damping time $1/\gG$. Therefore the Langevin or master equation must
be used if they provide a \emph{nonperturbative} description of
damping and noise (in the sense that the mode decay rates $\gG$ may
exceed the mode spacing).

In this section we show that the Langevin equations \eqref{eq:aLEQ} and the
master equations \eqref{eq:ME1} and \eqref{eq:ME2} are nonperturbative
in the above mentioned sense. To this end we address again the exact
field equations of motion and illustrate how they can be used to
compute corrections to the dynamics in the Markov approximation of
Sec.~\ref{sec:DLEQM}. We start from Eq.~\eqref{eq:aexact},
which describes the exact time evolution of the cavity modes
operators $a_{\gl}$,
\begin{equation}
\label{eq:aexact2}
\dot{a}_{\lambda}(t) = -\rmi\omega_{\lambda} a_{\lambda}(t) +
\sum_{\lambda'} \int\limits_{t_{0}}^{t} \! \rmd t' \tilde{C}_{\lambda
\lambda'}(t-t') \biggl[ a_{\lambda'}(t') +
\sum_{\lambda'}\mathcal{N}_{\lambda' \lambda''}^{\dagger}
a_{\lambda''}^{\dagger}(t') \biggr] + \tilde{F}_{\lambda}(t),
\end{equation}
and consider it in the rotating wave approximation made in
Sec.~\ref{sec:DLEQM}. That is, we restrict our analysis to a frequency
bandwidth which is small compared to the central frequency
$\bar{\go}$, and consider the time evolution only for times larger
than the bath correlation times $\tau_{\textrm{bath}}$. In addition, we
assume the coupling to be such that the typical damping rate
$\bar{\gG}$ for the cavity modes are much smaller than $\bar{\go}$,
but of the same order than the mean frequency spacing $\Delta\go$ of
the cavity modes in the frequency range considered.

We first move to a rotating frame
\begin{equation}
\label{eq:abar}
a_{\gl}(t) = \rme^{-\rmi \bar{\go} t} \bar{a}_{\lambda}(t),
\end{equation}
where $\bar{a}_{\lambda}(t)$ are slowly varying
variables. Specifically, $\bar{a}_{\lambda}(t)$ does not change much
on the time scale of the order $\tau_{\textrm{bath}}$ in which
$\tilde{C}_{\lambda\lambda'}(t)$ decays to zero. Substituting the
operators \eqref{eq:abar} into Eq.~\eqref{eq:aexact2} and replacing
$\bar{a}_{\lambda}(t')$ with $\bar{a}_{\lambda}(t)$, one obtains the
equation of motion for the slowly varying operators
$\bar{a}_{\lambda}(t)$,
\begin{multline}
\label{eq:atoapp} 
\dot{\bar{a}}_{\lambda}(t) = \\
- \sum_{\gl} \left[
\tilde{C}_{\lambda\lambda'}(\omega_{\gl'}) \rme^{-\rmi (\go_{\gl'} -
\bar{\go})t} \bar{a}_{\lambda'}(t) +
\tilde{C}_{\lambda\lambda'}(-\omega_{\gl'}) \rme^{\rmi (\go_{\gl'} +
\bar{\go})t} \sum_{\gl' \gl''} \mathcal{N}_{\gl' \gl''}
\bar{a}_{\lambda''}^{\dag}(t) \right] + \rme^{\rmi \go_{\gl}t} \tilde{F}_{\gl}(t).
\end{multline}
Here the channel response matrix in the frequency domain is given by
\begin{equation}
\begin{split}
\tilde{C}_{\lambda\lambda'}(\pm\omega) &= \int\limits_{-\infty}^{t} \!
\rmd t'\;  \tilde{C}_{\lambda\lambda'}(t-t') \rme^{\pm\rmi \go (t-t')}
\\
& = -\frac{\rmi}{\pi} \mathcal{P} \int\limits_{\go_{m}}^{\infty} \!
\rmd \go' \; \frac{\gG_{\lambda \lambda'}(\go')}{\go'-\go} \pm
\gG_{\lambda \lambda'}(\go) + \mathcal{O}(\gamma / 2\bar{\go}), 
\end{split} 
\end{equation}
where we set $t_{0} \to -\infty$ in the integration limits because we
are only considering times $t$ such that $t-t_{0} \gg
\tau_{\textrm{bath}}$. The second term inside the brackets in the
Eq.~\eqref{eq:atoapp} oscillates very fast, with a frequency $\sim
2\bar{\go}$, and can be neglected in the rotating wave
approximation. Equation \eqref{eq:atoapp} then reduces to
Eq.~\eqref{eq:arwaf}, from which the Langevin equation follows. The
neglected terms provide corrections of order
$|C(\bar{\go})|/2\bar{\go} \sim \gG/2\bar{\go}$ which are very small
for the systems of interest in quantum optics. We note that there are
no corrections of order $\gG/ \Delta\go$; thus our field dynamics
correctly describes the regime of overlapping resonances.


\section{Complete Master Equation}
\label{app:2C}

The master equation \eqref{eq:ME1} was obtained upon neglecting both
the frequency dependence of the coupling amplitudes as well as
possible frequency shift terms. Dropping these last approximations,
the more general master equation is given by
\begin{equation}\begin{split}
\dot{\gr} &= -\rmi \sum_{\gl} \go_{\gl} [a_{\gl}^{\dag} a_{\gl}, \gr]
\\
&\quad + \sum_{\gl \gl'}(1+n_{\text{th}}(\go_{\gl})) \bigl( \gG_{\gl
  \gl'}(\go_{\gl}) [a_{\gl'},\gr a_{\gl}^{\dag}] + \gG_{\gl
  \gl'}^{*}(\go_{\gl}) [a_{\gl}\gr,a_{\gl'}^{\dag}] \big) \\
&\quad + \sum_{\gl \gl'} n_{\text{th}}(\go_{\gl}) \big(\gG_{\gl
  \gl'}(\go_{\gl}) [a_{\gl}^{\dag} \gr, a_{\gl'}] + \gG_{\gl
  \gl'}^{*}(\go_{\gl}) [a_{\gl'}^{\dag}, \gr a_{\gl}]  \big) \\
&\quad + \frac{\rmi}{\pi} \sum_{\gl \gl'} \left( \mathcal{P} \int \! \rmd\go
\; \frac{\gG_{\gl\gl'}(\go) n_{\text{th}}(\go)}{\go - \go_{\gl}}
   [a_{\gl'},[\gr, a_{\gl}^{\dag}]] - \mathcal{P} \int \! \rmd\go
\; \frac{\gG_{\gl\gl'}^{*}(\go) n_{\text{th}}(\go)}{\go - \go_{\gl}}
   [a_{\gl'}^{\dag},[\gr, a_{\gl}]] \right) \\
&\quad + \frac{\rmi}{\pi} \sum_{\gl \gl'} \left( \mathcal{P} \int \! \rmd\go
\; \frac{\gG_{\gl\gl'}(\go)}{\go - \go_{\gl}} [a_{\gl'},\gr
  a_{\gl}^{\dag}] - \mathcal{P} \int \! \rmd\go
\; \frac{\gG_{\gl\gl'}^{*}(\go)}{\go - \go_{\gl}} [a_{\gl}\gr,
  a_{\gl'}^{\dag}] \right).
\end{split}\end{equation}
The most relevant terms in this master equation are contained in the
first three lines, accounting for the oscillations, the damping and
the noise in the field dynamics. The last two lines contain the
so-called frequency shift terms which are, due to the principal part
integrals, small corrections to the oscillating part of the master
equation. The terms in the fourth line proportional to $n_{\text{th}}$
constitute the \emph{Stark} shift which has its origin in thermal
fluctuations. The terms on the last line are the \emph{Lamb} shift
which is due to vacuum fluctuations. Contrary to the \emph{Stark}
shift, the \emph{Lamb} shift is present even at zero temperature.
