%SourceDoc ./tesis.tex
\chapter{Electromagnetic Field Quantization for Open Optical Cavities}
\label{chap:1} 

A fundamental problem in quantum optics is the description of the
radiation damping due to leakage out of an open cavity. Conventionally
this problem is treated on a phenomenological level by adopting the
system-and-bath model \cite{Seni59,Seni60,Gard00} to describe the
resonator losses: The resonator field is modeled by a discrete set of
independent quantum harmonic oscillators associated with the normal
modes of the closed cavity. Each mode is \emph{weakly} coupled to its
own bath represented by a continuous set of oscillators. The coupling
to the reservoir introduces both losses and noise for the resonator
field dynamics. Baths corresponding to different modes are
statistically independent \cite{Hake84}. In spite of its intuitive 
appeal and it success in applications, this model of damping has been
criticized. The model lacks a microscopic justification as the
coupling amplitudes between the system and bath only enter as
phenomenological parameters. Moreover, the model is only reliable for
almost closed cavities with very high-$Q$ for which the spectrum has
well-isolated resonances.

In the last decade the interest on a quantum description of the
electromagnetic field inside open optical cavities has reemerged as
consequence of the experimental observation of lasing in random media
\cite{Cao03} and the need for a quantum theory of excess
noise \cite{Exte01}. In this chapter we address this problem and
develop a quantization method for open optical cavities which provides
an exact description of the cavity field leakage for
spectrally-overlapping resonances. Our method is of significance for
several reasons: it provides a quantum description of excess noise, it
allows for multi-mode lasing, and upon combination with a description
of wave chaos it can account for lasing in random media.

Our quantization method applies both to open cavities defined by
mirrors of arbitrary shape and quality, and to dielectric media
characterized by a real dielectric constant which might be spatially
nonuniform. The system may be coupled to an arbitrary number of escape
channels. The quantization of the field is performed in terms of the
exact eigenmodes of Maxwell's equation defined in the whole space
\cite{Glau91}. The essence of the proposed quantization method lies in
the use of a technique known in nuclear and condensed-matter physics
as the Feshbach projection formalism \cite{Fesh62}. The projection
formalism separates the field into two contributions corresponding,
respectively, to the cavity region and its exterior. Each region
entails a set of normal modes in terms of which the electromagnetic
field can be represented by associating to each mode standard creation
and annihilation operators. In the resonator region the modes form a
discrete set and vanish outside the cavity, while there is a
continuous set of external modes which vanish inside the resonator and
fulfill scattering-type boundary conditions at infinity. The
separation of the field allows for a rigorous representation of the
system Hamiltonian by a bilinear form in the cavity and exterior
operators, with the same structure as the system-and-bath Hamiltonian
used in phenomenological approaches \cite{Seni59,Seni60,Gard00}. In
contrast to these approaches, our derivation provides explicit
expressions for the coupling amplitudes. The field Hamiltonian
generally includes both resonant and non-resonant terms. When both are
kept, the quantization method accounts for overdamping (when the field
damping rate becomes of the order of the frequency). In the case of
weak damping (when the field damping rate is much smaller that the
mean frequency separation) our approach reduces to the well-known
field quantization of the phenomenological approach.

We note that our system-and-bath representation of the open resonator
dynamics is not unique but depends on the choices made in the
separation of the space into two subregions. This arbitrariness is
usually encountered with the projection technique \cite{Savi03}; its
origin goes back to the freedom to chose the ``inside'' and ``outside''
subregions in which to separate the space. Consider for example a
cavity with an opening, i.e., a hole in the material walls of the
cavity boundary. Any choice of a fictitious surface covering the hole
yields its own inside/outside separation. Moreover, different boundary
conditions may be imposed at the chosen separating surface.
Nevertheless, each such surface and boundary condition entail
eigenmodes allowing to represent the electromagnetic field almost
everywhere. We emphasize that the field expansion cannot be expected
to converge uniformly or even pointwise, and in particular not on an
arbitrarily chosen boundary. Still the system-and-bath representation
is exact in the $L^2$ sense required by quantum mechanics. The freedom
in the choice of the separating surface and the boundary conditions
manifests itself in the Hamiltonian and the field expansions, as the
coupling amplitudes, and the internal and external basis sets, all
depend explicitly on these choices. However, we show below that all
physical observables, in particular the electromagnetic fields and the
scattering amplitudes, do not rely on these choices and only depend on
the physical boundary conditions imposed by Maxwell's equations.

Our quantization procedure can naturally be apply to random media. For
such media the internal system Hamiltonian can be modelled using
random matrix theory. The statistical properties of the system
observables are then obtained by averaging over the pertinent
ensemble. In addition by providing a separation of the inside/outside
field in the case of overlapping resonances our quantization method
provides a solid ground for the construction of a laser theory for
multi-mode fields (see Chapter \ref{ch:LEQ}).

The outline of the chapter is as follow. In
Sec.~\ref{sec:normal_modes} the field quantization is described in
terms of the exact eigenmodes of Maxwell's equations in the whole
space. In Sec.~\ref{sec:cavity_bath} the system-and-bath Hamiltonian
is derived using Feshbach's projector technique. Our derivation is
valid for fields with arbitrary polarization. Finally, in
Sec.~\ref{sec:examples} we demonstrate our method for a number of
models frequently used for optical resonators. We compute their
respective system-and-bath Hamiltonians, and show that the
electromagnetic field and the scattering properties agree exactly with
results obtained by direct solutions of the problem. Performing the
computations for different sets of boundary conditions along the
surface separating the resonator and channel region, we demonstrate
that the physical observables are independent of that choice of
boundary conditions.

The results presented in this chapter have been reported in
Refs.~\cite{Hack02,Vivi03,Vivi04}.

\section{Normal Modes}
\label{sec:normal_modes}

We consider the quantization of the electromagnetic field taking into
account the three spatial dimensions and the complete vector character
of the field. The space might be filled with a dielectric medium and
we allow for the presence of material surfaces, e.g., mirrors, on
which physical boundary conditions are imposed. The medium is
characterized by a scalar dielectric constant
$\epsilon(\vect{r})$\footnote{Generalization of the quantization
  theory for a real tensor dielectric function is straight forward.}
which depends on position and that we assumed to be real and frequency
independent. We are meanly interested in resonator-like configurations
suggested by particular geometrical arrangements of the material
surfaces or by special behaviors of the dielectric constant, e.g., a
piece of dielectric surrounded by free space. We thus assume that
far from the region of space defining the cavity the electromagnetic
field propagates freely, i.e., $\epsilon(\vect{r})$ becomes constant.

The electromagnetic field for the total system can be represented by a
mode expansion involving the exact eigenmodes of Maxwell's equations
which are defined throughout the whole space. This so-called
modes-of-the-universe approach \cite{Lang73,Glau91,Knol91,Voge94}
serves as a starting point for the derivation of the system-and-bath
Hamiltonian in Sec.~\ref{sec:cavity_bath}; we therefore summarize the
main steps of this approach below.

It is convenient to formulate the quantization procedure in terms of
the vector potential $\vect{A}$ and the scalar potential $\phi$. We
work in the Coulomb gauge which, in the absence of sources,
corresponds to the choice $\phi = 0$ and the generalized
transversality condition $\nabla \cdot [\epsilon({\bm r}) {\vect{A}}] =
0$.  The magnetic and electric fields then follow from the potentials
via the familiar relations
\begin{equation} \label{eq:ebfield}
\vect{E} = - \frac{1}{c} \frac{\partial \vect{A}}{\partial t}
\quad \text{and} \quad \vect{B} = \nabla \times \vect{A}.
\end{equation}
The electromagnetic Hamiltonian of the problem is given by
\begin{equation}
\label{eq:huniv}
H = \frac{1}{2} \int d {\bm r} \left[ \frac{c^2 {\vect{\Pi}}({\bm
      r},t)^2}{\epsilon({\bm r})} + \bigl( \nabla \times \vect{A}({\bm 
     r},t) \bigr)^2
\right],
\end{equation}
where $\vect{\Pi}({\bm r},t)= \epsilon({\bm r}) \dot {\bm
  A}(\vect{r},t)/c^2$ is the canonical momentum field.  The
  quantization of the fields may be achieved by imposing a suitable
  commutation relation between $\vect{A}({\bm r},t)$ and
  $\vect{\Pi}({\bm r},t)$. An alternative but equivalent procedure is
  to expand the fields in a complete set of mode functions and to
  impose canonical commutation relations for the expansion
  coefficients. We follow the second procedure here, and expand the
  vector potential in terms of the normal modes $\vect{f}_m
  (\omega,{\bm r})$, defined as solutions of the wave equation
\begin{equation} \label{eq:evequniv}
\nabla \times \bigl( \nabla \times \vect{f}_m(\omega,{\bm r}) \bigr) -
\frac{\epsilon({\bm r})\omega^2}{c^2} \vect{f}_m(\omega,{\bm r})={\bm
  0} .
\end{equation}
The solutions automatically satisfy the transversality condition $
\nabla \cdot \left[\epsilon({\bm r}) \vect{f}_m(\omega, {\bm r})
\right] = 0$.  The eigenmodes are labeled by the continuous
frequency $\omega$ and a discrete index $m$ that specifies the
asymptotic boundary conditions far away from the resonator region
(including the polarization). We consider asymptotic conditions
corresponding to a scattering problem with incoming and outgoing
waves.  Then $\vect{f}_m(\omega,{\bm r})$ represents a solution with
an incoming wave only in the single channel $m$ and with outgoing
waves in all open scattering channels. The definition of the channels
depends on the problem at hand: For a dielectric coupled to free
space, one may expand the asymptotic solutions in terms of angular
momentum states.  Then $m$ corresponds to an angular momentum quantum
number. On the other hand, for a cavity connected to external
waveguides, $m$ may represent a transverse mode index. The field
expansions then take the form
\begin{subequations}
\label{eq:qpexp}
\begin{align}
\vect{A}({\bm r},t) & =  c \sum_{m} \int\limits_{\omega_m}^{\infty} \!
\rmd\omega \;  q_{m}(\omega,t) \vect{f}_m(\omega,\vect{r}), \\ 
\vect{\Pi}({\bm r},t) & = \frac{1}{c} \sum_{m}
\int\limits_{\omega_m}^{\infty} \! \rmd\omega \; p_m (\omega,t) 
\vect{f}_{m}^\dagger (\omega,\vect{r}),
\end{align}
\end{subequations} 
where the expansion coefficients $q_m(\omega, t)$ and $p_m(\omega, t)$
are complex time dependent variables. The frequencies $\omega_m$ are the
threshold frequency above which the $m$-channel becomes open. The
wave functions $\vect{f}_m(\omega,\vect{r})$ have been allowed to be
complex, because it is often convenient to use these rather than real
ones, e.g., in scattering problems.

We follow reference \cite{Glau91} and stick to the usual internal
product with respect to which the differential operator in
Eq.~\eqref{eq:evequniv} is not Hermitian. However the substitution
\begin{equation}
\label{eq:fguniv}
\vect{f}_{m} (\omega,{\bm r}) = \frac{1}{\sqrt{\epsilon({\bm r})}} \bm{\phi}_{m} 
(\omega,{\bm r})
\end{equation}
transforms Eq.~\eqref{eq:evequniv} into the eigenvalue problem
\begin{equation}
\label{eq:hevequniv}
L \bm{\phi}_{m} (\omega, {\bm r}) \equiv
\frac{1} {\sqrt{\epsilon({\bm r})}} \nabla \times 
\biggl( \nabla \times \frac{ \bm{\phi}_{m} (\omega, {\bm r})}
{\sqrt{\epsilon({\bm r})}} \biggr) =
\frac{\omega^2} {c^2} \bm{\phi}_{m} (\omega, {\bm r}) ,
\end{equation}
for the Hermitian differential operator $L$. Choosing an orthonormal
set of basis functions $\bm{\phi}_m(\omega, {\bm r})$, it follows
that the associated mode functions $\vect{f}_m (\omega,\vect{r})$ satisfy the
orthonormality condition
\begin{equation}
\label{eq:forth}
\int \! \rmd{\bm r} \; \epsilon({\bm r}) \vect{f}^*_m(\omega,{\bm r}) 
\cdot \vect{f}_{m^\prime}(\omega^\prime,{\bm r}) = \delta_{mm^\prime} \, 
\delta(\omega - \omega^\prime).
\end{equation}
The functions ${\bm \phi}_m(\omega)$ form a complete set in the
subspace of $L^2$ functions defined by the transversality
condition
\begin{equation}
\label{eq:gauge}
\nabla \cdot \left[\sqrt{\epsilon({\bm r})} {\bm \phi}_m 
(\omega,{\bm r}) \right] = 0.
\end{equation} 
The associated mode functions $\vect{f}_m(\omega, {\bm r})$ form a
complete set in the space of transverse functions
\cite{Glau91,Voge94}, with completeness relation given by
\begin{equation}
\label{eq:fcomplt}
\sum_{m} \int\limits_{\omega_m}^{\infty} \!  \rmd\omega \;
f_{mi}(\omega,\vect{r}) f_{mj}^{*} (\omega,\vect{r}') =
\delta_{ij}^{(\epsilon)}(\vect{r},\vect{r}').
\end{equation}
where $f_{mi}(\omega,\vect{r})$ are the vector components of
$\vect{f}_{m}(\omega,\vect{r})$. In the space of transverse functions
the generalized $\delta$-function
$\delta_{ij}^{(\epsilon)}(\vect{r},\vect{r}')$ acts like the usual
$\delta$-function, $\delta_{ij}^{(\epsilon)}(\vect{r},\vect{r}') =
\delta_{ij}\delta(\vect{r}-\vect{r}')$; that is
\begin{equation}
\int \! \rmd\vect{r}' \; \epsilon(\vect{r}')
\delta_{ij}^{(\epsilon)}(\vect{r},\vect{r}') f_{j}(\vect{r}')= f_{i}(\vect{r}),
\end{equation}
for $\vect{f}(\vect{r})$ belonging to the space of transverse
functions (herein we adopt the convention of summation over repeated
vector component indices). If a function $\vect{h}(\vect{r})$ does not
belongs to this space then the action of
$\delta_{ij}^{(\epsilon)}(\vect{r},\vect{r}')$ on $\vect{h}(\vect{r})$
projects the function onto the space of transverse functions.

Inasmuch as the fields are real, the vector potential and its
canonical momentum fulfill the relations $\vect{A}=\vect{A}^{\dagger}$
and $\vect{\Pi}=\vect{\Pi}^{\dagger}$. Together with
Eq.~(\ref{eq:qpexp}), this implies
\begin{subequations}
\label{eq:qpreal}
\begin{align}
q_m(\omega) &= \sum_{m^\prime} \int\limits_{\omega_{m'}}^{\infty} \!
\rmd\omega^\prime \;
\mathcal{M}_{mm^\prime}^{\dagger}(\omega,\omega^\prime)
q_{m^\prime}^{\dagger} (\omega^\prime), \\ 
p_m^{\dagger}(\omega) &=
\sum_{m^\prime} \int\limits_{\omega_{m'}}^{\infty} \!
\rmd\omega^\prime \;
\mathcal{M}_{mm^\prime}^{\dagger}(\omega,\omega^\prime)
p_{m^\prime}(\omega^\prime),
\end{align}
\end{subequations}
where $\mathcal{M}$ has the matrix elements
\begin{equation}
\label{eq:uuniv}
\mathcal{M}_{m m^\prime}(\omega,\omega^\prime) = \int \! \rmd{\bm r} 
\; \epsilon ({\bm r}) \vect{f}_{m}(\omega,{\bm r}) \cdot 
\vect{f}_{m^\prime} (\omega^\prime,{\bm r}).
\end{equation}
We note that $\mathcal{M}$ is unitary and symmetric\footnote{We use
  the convention $\mathcal{M}^\dagger_{mn} (\omega,
  \omega^\prime)=[\mathcal{M}_{nm}(\omega^\prime ,\omega)]^*.$}.
Moreover, $\mathcal{M}$ only couples degenerate modes, as modes with
different frequencies are orthogonal, $\mathcal{M}(\omega,
\omega^\prime) \sim {\delta(\omega-\omega^\prime)}$.

Substituting the field expansions (\ref{eq:qpexp}) into the
Hamiltonian (\ref{eq:huniv}), using Eq.~(\ref{eq:qpreal}) and the
properties of $\mathcal{M}$, one obtains the Hamiltonian in terms of
the variables $q$ and $p$,
\begin{equation}
\label{eq:hqp}
H = \frac{1}{2} \sum_m \int\limits_{\omega_m}^{\infty} \! \rmd\omega \;
\left[ p_m^\dagger(\omega) p_m(\omega) + \omega^2
q_m^{\dagger}(\omega) q_m(\omega)
\right]. \quad
\end{equation}
Quantization is now achieved by promoting the variables $q(\omega)$
and $p(\omega)$ to operators.  The Heisenberg equations of motion
for $q(\omega)$ and $p(\omega)$ lead to Maxwell's equations,
provided we impose the equal time commutation relations
\begin{align}
\label{eq:qpcomm}
\ [q_m(\omega),q_{m'}(\omega^\prime)] &=
[q_m(\omega),q_{m'}^{\dagger}(\omega^\prime)] = 0, \nonumber \\ 
\ [p_m(\omega),p_{m'}(\omega^\prime)] &=
[p_m(\omega),p_{m'}^{\dagger}(\omega^\prime)] =0, \\
\ [q_m(\omega),p_{m'}(\omega^\prime)] &= \rmi\hbar \, \delta_{mn} 
\, \delta(\omega - \omega^\prime) \nonumber.
\end{align}
Further use of Eq.~(\ref{eq:qpreal}) gives the remaining commutation
relation
\begin{equation}
\label{eq:qpdcom}
[q_m(\omega),p^{\dagger}_{m'}(\omega^\prime)] = \rmi\hbar
\mathcal{M}^{\dagger}_{mm'}(\omega,\omega^\prime).
\end{equation}
Combining the field expansions (\ref{eq:qpexp}) with the completeness
relation (\ref{eq:fcomplt}), it is easily seen that these commutation
relations imply canonical commutation relations for the vector
potential and the canonical momentum field.

The last step in the quantization procedure is to express the
operators $q(\omega)$ and $p(\omega)$ in terms of creation and
annihilation operators,
\begin{subequations}
\label{eq:qpaexp}
\begin{align}
q_m(\omega) &= \left( \frac{\hbar}{2\omega} \right)^{\frac{1}{2}}
  \bigg[ A_m(\omega) + \sum_{m^\prime}
  \int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^\prime \;
  \mathcal{M}_{mm^\prime}^{\dagger}(\omega,\omega^\prime)
  A_{m^\prime}^{\dagger} (\omega^\prime) \bigg], \\ 
p_m(\omega) &= \rmi
  \left( \frac{\hbar \omega}{2} \right)^{\frac{1}{2}} \bigg[
  A_m^{\dagger}(\omega) - \sum_{m^\prime}
  \int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^\prime \;
  \mathcal{M}_{mm^\prime}(\omega,\omega^\prime)
  A_{m^\prime}(\omega^\prime) \bigg].
\end{align} 
\end{subequations}
The latter obey the commutation relations
\begin{eqnarray}
\ [A_m(\omega), A_{m^\prime}(\omega^\prime)] &=& 0, \nonumber \\
\ [A_m(\omega), A^{\dagger}_{m^\prime}(\omega^\prime)] &=&
\delta_{mm^\prime} \, \delta(\omega - \omega^\prime),
\end{eqnarray}
obtained by inverting the relations \eqref{eq:qpaexp} and making use
of the commutation relations \eqref{eq:qpcomm}. In terms of the
creation and annihilation operators the Hamiltonian takes the familiar
form
\begin{equation}
\label{eq:ha}
H = \frac{1}{2} \sum_m \int\limits_{\omega_m}^{\infty} \! \rmd\omega
\, \hbar\omega \left[ A_m^\dagger(\omega) A_m(\omega) + A_m(\omega)
  A_m^\dagger(\omega) \right],
\end{equation}
describing a set of independent harmonic oscillators. Finally,
substituting the representations (\ref{eq:qpaexp}) into
Eqs.~(\ref{eq:qpexp}), one obtains the field expansions
\begin{subequations}
\label{eq:aexp}
\begin{align}
\vect{A} & = c \sum_m \int\limits_{\omega_m}^{\infty} \! \rmd\omega \left(
  \frac{\hbar}{2\omega}\right)^{\frac{1}{2}} \left[ A_m(\omega)
\rme^{-\rmi\omega t} \vect{f}_m(\omega,\vect{r}) + A_m(\omega)^{\dagger}
\rme^{\rmi\omega t} \vect{f}_m^{*} (\omega,\vect{r}) \right] \\
\vect{\Pi} & = -\frac{\rmi}{c} \sum_m
\int\limits_{\omega_m}^{\infty}\! \rmd\omega \left(
  \frac{\hbar\omega}{2}\right)^{\frac{1}{2}} \epsilon(\vect{r}) \left[
A_m(\omega) \rme^{-\rmi\omega t} \vect{f}_m(\omega,{\bm r}) - A_m^{*}(\omega)
\rme^{\rmi\omega t} \vect{f}_m^{*}(\omega,{\bm r}) \right].
\end{align}
\end{subequations}
In empty space, $\epsilon({\bm r}) \equiv 1$, the fields expansions
reduce to the standard mode expansion of the free electromagnetic
field.


\section{Cavity and Channel Fields}
\label{sec:cavity_bath}

In this section we use the Feshbach projection technique \cite{Fesh62}
to provide an exact decomposition of the electromagnetic field into
two components corresponding to two separated subsystems: the resonator
and its exterior. The latter subsystem is usually referred to as the
channel region in the scattering formalism; a notation that we shall
adopt here. In quantum optics, under very general conditions, this
subsystem plays the role of a reservoir. The main result of the
section is the microscopic derivation of the field Hamiltonian which,
under the proposed field decomposition, has the structure of a
system-and-bath Hamiltonian accounting for arbitrarily large coupling
between the two subsystems. 

The starting point of the projection formalism is the inside/outside
separation of the space into the two subsystems. This is accomplished
by introducing a close surfaced that separates the internal or
resonator region from the external or channel region
(Fig.~(\ref{fig:ocavity01})). As mention before, this procedure
entails certainly a degree of arbitrariness: First, the shape and size
of the interface is merely restricted by the requirement that far from
the resonator the field should propagate freely (this allow us to
define asymptotic boundary conditions below). Second, the type of
boundary conditions along the interface for the internal and external
subsystems are only limited by the condition that the Hermiticity of
the problem should be retained
\cite{Savi03}. This arbitrariness is manifest in the representation
obtained for the field expansions and in the explicit expression for
the Hamiltonian. In spite of this, all physical observables can be
shown to be independent of the choices made for the interface and
boundary conditions. Explicit examples illustrating this fact are
presented in Sec.~\ref{sec:examples}.
\begin{figure}[t]
  \begin{center}
    \includegraphics[scale=0.6]{./Figures/open_cavity_01.eps}
    \caption{\label{fig:ocavity01}Sketch of a resonator-like
    configuration in two dimensions. The continuous lines represent
    mirrors that partially bound a region of space filled with a
    dielectric material (shaded region). $I'$ and $I$ are two possible
    choices of cavities for the problem. In both cases the separating
    surfaces are shown as dashed lines. $\partial I'$ defines a cavity
    with circular shape but with an intricate configuration of mirrors
    in its interior, while $\partial I$ follows the boundary of the
    dielectric material, coinciding in some regions with the mirrors,
    and finally enclosing completely the dielectric material.}
    \end{center}
\end{figure}


\subsection{Feshbach Projection}

Formally, the separation of space in two regions is achieved using the
projection operators \cite{Fesh62}
\begin{equation}
\label{eq:defq}
\mathcal{Q} = \int_{{\bm r} \in I} \! \rmd{\bm r} |{\bm r}
\rangle\langle{\bm r}|, \qquad \mathcal{P} = \int_{{\bm r} \not\in I}
\! \rmd{\bm r} |{\bm r} \rangle\langle{\bm r}|,
\end{equation}
where $|{\bm r} \rangle$ denotes a standard position eigenket and $I$
is the region of space occupied by the cavity (see
Fig.~(\ref{fig:ocavity01})). The operators $\mathcal{Q}$ and
$\mathcal{P}$ depend on the choice of $I$, but all physical
observables turn out to be independent of this choice.  One easily
shows that $\mathcal{P}$ and $\mathcal{Q}$ are projection operators,
\begin{subequations}
\label{eq:prop}
\begin{align}
\mathcal{P} = \mathcal{P}^\dagger,\qquad  \mathcal{P}^2 = \mathcal{P}; \\
\mathcal{Q} = \mathcal{Q}^\dagger;\qquad  \mathcal{Q}^2 = \mathcal{Q} .
\end{align}
\end{subequations}
Moreover, they are orthogonal, $\mathcal{QP} = \mathcal{PQ} = 0$, and
complete, $\mathcal{Q} + \mathcal{P} = 1$. Therefore, an arbitrary
Hilbert space function ${\bm \psi}$ and the associated function
$\vect{h} = {\bm \psi}/\sqrt{\epsilon}$ may be decomposed into their
projections onto the resonator and channel space
\begin{subequations}
\label{eq:uvdef}
\begin{align}
{\bm \psi}(\vect{r}) &= \chi_{-}({\bm r}) {\bm \mu} ({\bm r}) 
+ \chi_{+}({\bm r}) {\bm \nu} ({\bm r}), \\
\vect{h}(\vect{r}) &= \chi_{-}({\bm r}) \vect{u} ({\bm r}) 
+ \chi_{+}({\bm r}) \vect{v} ({\bm r}),
\end{align}
\end{subequations}
where $\chi_{\mp}$ are the characteristic functions of the resonator
and the channel region, respectively,
\begin{subequations}
\begin{align}
\chi_{-}(\vect{r}) &= \int_{{\bm r}^\prime \in I} \! \rmd{\bm r}^\prime 
\delta({\bm r} - {\bm r}^\prime) , \\
\chi_{+}({\bm r}) &=  1-\chi_{-}({\bm r}).
\end{align}
\end{subequations}

Acting on ${\bm \phi}$ with the differential operator $L$ defined in
Eq.~(\ref{eq:hevequniv}), we obtain
\begin{equation}
\label{eq:lqph}
L {\bm \psi} = \chi_{-}({\bm r}) L \, {\bm \mu} ({\bm r}) -
\int_{\partial I} \! \rmd^2{\bm r}^\prime K({\bm r}, {\bm r}^\prime) {\bm
  \mu} ({\bm r}^\prime) + \chi_{+}({\bm r}) L \, {\bm
  \nu}({\bm r}) + \int_{\partial I} \! \rmd^2{\bm r}^\prime K({\bm r},
    {\bm r}^\prime) {\bm \nu} ({\bm r}^\prime) ,
\end{equation}
where $K({\bm r}, {\bm r}^\prime)$ is a singular differential
operator defined at the boundary
\begin{multline}
\label{eq:defk}
K(\vect{r},\vect{r}^\prime) {\bm \mu}(\vect{r}^\prime)
\Bigr\rvert_{\vect{r},\vect{r}'\in \partial I} = \left[
  \frac{\delta({\bm r}-{\bm r}^\prime)} {\sqrt{\epsilon({\bm r})}}
       {\bf n}^\prime \times \biggl( \nabla^\prime \times \frac{{\bm
           \mu}({\bm r}^\prime)}{\sqrt{\epsilon ({\bf r}^\prime)}}
       \biggr) \right.\\
\left.  + \frac{\nabla \delta({\bf r}-{\bm r}^\prime)}
             {\sqrt{\epsilon({\bm r})}} \times \biggl( {\bf n}^\prime
             \times \frac{{\bm \mu}({\bf
                 r}^\prime)}{\sqrt{\epsilon({\bm r}^\prime)}}\biggr)
               \right] \Biggr\rvert_{\vect{r},\vect{r}'\in \partial
               I}.
\end{multline}
Here $\nabla^\prime$ denotes a derivative with respect to ${\bf
  r}^\prime$, and ${\bf n}^\prime$ is a unit vector normal to the
boundary. The action of $K({\bm r}, {\bm r}^\prime)$ on ${\bm \nu}
({\bm r}^\prime)$ is defined in a similar fashion, with ${\bm \mu}
({\bm r}^\prime)$ replaced by ${\bm \nu} ({\bm r}^\prime)$.

The first (third) term on the right hand side of Eq.\ 
(\ref{eq:lqph}) contribute only inside (outside) the resonator. The
second and fourth terms are boundary terms. They involve ${\bm \mu}$,
${\bm \nu}$, and their derivatives at the boundary; these functions
must be evaluated in the limit where the boundary is approached from
the cavity and the channel region, respectively. We note that the
boundary terms generally gives rise to singular behavior. As a
result, the action of $L$ usually goes beyond the Hilbert space.
The range of $L$ within the Hilbert space is defined by the
functions for which the singular terms vanish; this happens in
particular for the eigenfunctions of $L$ in Hilbert space.

We now want to decompose the operator $L$ into a resonator, a channel,
and a coupling contribution. However, it is not obvious how the
decomposition can be carried out for the singular boundary terms. We
therefore replace the boundary integrals by integrations along
surfaces arbitrarily close to the boundary but located inside
respectively outside the resonator region. There is still freedom in
selecting which surface integral should belong to each of the
subsystems. Different choices lead, in general, to different set of
boundary conditions along the separating surface, only subject to the
restrictions that the obtained decomposition of the operator $L$ is
self-adjoint. The freedom in choosing the boundary conditions is
characteristic for the projector technique \cite{Fesh62,Savi03}. For
the sake of clarity and definiteness, in what follows we stick to one
of these possible choices leading to a one particular set of boundary
conditions. Other possibilities are study for specific examples in
Sec.~\ref{sec:examples}. Equation (\ref{eq:lqph}) then becomes
\begin{equation}
\label{eq:Lpro}
L {\bm \psi} = L_{\mathcal{QQ}} {\bm \mu} + L_{\mathcal{QP}}{\bm \nu} + 
L_{\mathcal{PQ}} {\bm \mu} + L_{\mathcal{PP}} {\bm \nu} ,
\end{equation}
where $L_{\mathcal{QQ}}$ and $L_{\mathcal{PP}}$ are the projections of
$L$ onto the resonator and channel space,
\begin{subequations}
\label{eq:lblock1}
\begin{align}
\label{eq:lsep1}
L_{\mathcal{QQ}} {\bm \mu} &= \chi_{-}({\bm r}) \, L {\bm \mu} ({\bf
  r}) - \int_{\partial I_{-}} \! \rmd^2{\bm r}^{\prime}_{-} \Bigg[
  \frac{\delta({\bm r} - {\bm r}^\prime_{-})} {\sqrt{\epsilon({\bm
        r})}} {\bf n}^\prime \times \bigg(\nabla^\prime \times
  \frac{{\bm \mu} ({\bf r}^\prime_{-})}{\sqrt{\epsilon({\bm
        r}^\prime_{-})}}\bigg) \Bigg], \\
\label{eq:lsep2}
L_{\mathcal{PP}} {\bm \nu} &= \chi_{+}({\bm r}) \, L {\bm \nu} ({\bf
  r}) + \int_{\partial I_{+}} \! \rmd^2{\bm r}^{\prime}_{+} \Bigg[
  \frac{\nabla \delta({\bm r} - {\bm r}^\prime_{+})}
       {\sqrt{\epsilon({\bm r})}} \times \bigg( {\bf n}^\prime
  \times \frac{{\bm \nu} ({\bm r}^\prime_{+})} {\sqrt{\epsilon({\bf
        r}^\prime_{+})} } \bigg)\Bigg],
\end{align}
\end{subequations}
and $L_{\mathcal{QP}}$ and $L_{\mathcal{PQ}}$ the coupling terms
\begin{subequations}
\label{eq:lblock2}
\begin{align}
\label{eq:lsep3}
L_{\mathcal{QP}} {\bm \nu} &= + \int_{\partial I_{-}} \! \rmd^2{\bf
  r}^{\prime}_{-} \Bigg[ \frac{\delta({\bm r} - {\bm
      r}^\prime_{-})}{\sqrt{\epsilon({\bm r})}} {\bf n}^\prime \times
  \bigg( \nabla^\prime \! \times \frac{{\bm \nu}
    (\vect{r}^\prime_{+})}{\sqrt{\epsilon({\bm r}^\prime_{+})}}
  \bigg)\Bigg],\\
\label{eq:lsep4}
L_{\mathcal{PQ}} {\bm \mu} &= - \int_{\partial I_{+}} \! \rmd^2{\bf
  r}^{\prime}_{+} \Bigg[ \frac{\nabla \delta({\bm r} - {\bf
      r}^\prime_{+})}{\sqrt{\epsilon({\bm r})}} \times \bigg( {\bf
    n}^\prime \!\!\times \frac{{\bm \mu} ({\bf
      r}^\prime_{-})}{\sqrt{\epsilon({\bm r}^\prime_{-})}}
  \bigg) \Bigg].
\end{align} 
\end{subequations}
The shorthands ${\bm r}^\prime_{\mp}$ indicate that the integrals are
to be evaluated in the limit where ${\bm r}^\prime_{-}$ and ${\bm
r}^\prime_{+}$ approach the boundary from inside respectively outside
the resonator. $L_{\mathcal{QQ}}$ and $L_{\mathcal{PP}}$ define
Hermitian operators on the Hilbert space of functions of the resonator
and the channel region, respectively. Moreover, $L_{\mathcal{QP}} =
L_{\mathcal{PQ}}^{\dagger}$ showing that the decomposition
\eqref{eq:Lpro} preserves the Hermiticity of $L$.

In particular, for the eigenfunctions of the operator $L$ with the
decomposition
\begin{equation}
|{\bm \phi}_{m}(\omega)\rangle = \mathcal{Q}|{\bm
\phi}_{m}(\omega)\rangle + \mathcal{P}|{\bm \phi}_{m}(\omega)\rangle
\equiv |\bar{\bm \mu}_{m}(\omega)\rangle + |\bar{\bm
\nu}_{m}(\omega)\rangle\,, 
\end{equation}
substitution of Eq.~(\ref{eq:Lpro}) into the eigenmode equation
(\ref{eq:hevequniv}) yields the two coupled equations:
\begin{equation}
\label{eq:heveqsep}
\left( \begin{array}{cc}
       L_{\mathcal{QQ}} &  L_{\mathcal{QP}} \\
       L_{\mathcal{PQ}} &  L_{\mathcal{PP}}
       \end{array} \right)
\left( \begin{array}{c}
       \bar{\bm \mu}_{m} (\omega)  \\
       \bar{\bm \nu}_{m} (\omega)
       \end{array} \right)
= \frac{\omega^2}{c^2}
\left( \begin{array}{c}
       \bar{\bm \mu}_{m} (\omega) \\
       \bar{\bm \nu}_{m} (\omega)
       \end{array} \right).
\end{equation}
The requirement that the singular terms on the left hand side vanish
yields the two matching conditions
\begin{subequations}
\label{eq:match1}
\begin{align}
{\bf n} \times \bigl( \bar{\vect{u}}_{m}(\omega) - \bar{\vect{v}}_{m}(\omega) \bigr)
&= {\bm 0}, \\ 
{\bf n} \times \bigl( \nabla \times \bar{\vect{u}}_{m}(\omega)-
  \nabla \times \bar{\vect{v}}_{m}(\omega) \bigr) & = {\bm 0},
\end{align}
\end{subequations}
for all points along the boundary of the resonator region. The gauge
condition $\nabla \cdot [\epsilon \vect{f}(\omega)] =0$ and the
requirement $\nabla \cdot \bigl(\nabla \times \vect{f} (\omega)\bigr)=0$ along
the boundary give the further matching conditions
\begin{subequations}
\label{eq:match3}
\begin{align}
{\bf n} \cdot \bigl( \epsilon \bar{\vect{u}}_{m}(\omega)-\epsilon
\bar{\vect{v}}_{m}(\omega)
\bigr) & = 0 , \\ 
{\bf n} \cdot \bigl( \nabla \times \bar{\vect{u}}_{m}(\omega)-
  \nabla \times \bar{\vect{v}}_{m}(\omega)\bigr) & =  0 .
\end{align}
\end{subequations}
The four matching conditions (\ref{eq:match1}), (\ref{eq:match3})
together with Eq.\ (\ref{eq:ebfield}) realize the well-known
\cite{Jack75} boundary conditions for the electromagnetic field at an
interface in the absence of surface currents or surface charges and
entail the continuity of the electromagnetic field on this surface.


\subsection{Cavity and Channel Modes}
\label{sec:eigenmodes}

The operators $L_{\mathcal{QQ}}$ and $L_{\mathcal{PP}}$ are
self-adjoint operators in the Hilbert space of the resonator and
the channel functions, respectively. The modes of the isolated
resonator $|\vect{\mu}_{\lambda}\rangle$ are solutions of the
eigenvalue equation
\begin{equation}
\label{eq:eveqq1}
L_{\mathcal{QQ}}|\vect{\mu}_{\lambda}\rangle = \left(
\frac{\omega_{\lambda}}{c}\right)^2 |\vect{\mu}_{\lambda}\rangle.
\end{equation}
From Eq.~\eqref{eq:lsep1}, they satisfy
\begin{equation}
\label{eq:eveqqq}
\nabla \times \bigl( \nabla \times \vect{u}_{\lambda} ({\bm r}) 
\bigr) = \frac{\epsilon({\bm r}) \omega^2_{\lambda}}{c^2} 
\vect{u}_{\lambda}({\bm r}).
\end{equation}
with the boundary condition
\begin{equation}
\label{eq:bound1}
{\bf n} \times \bigl( \nabla \times \vect{u}_{\lambda}(\vect{r}) \bigr) 
\Big\rvert_{\vect{r} \in \partial I} = {\bm 0},
\end{equation}
obtained by demanding that the singular term in Eq.~\eqref{eq:lsep1}
vanishes. Hence, the tangential component of $\nabla \times
\vect{u}_\lambda$ vanishes at the boundary. No boundary condition for
the normal component of $\nabla \times \vect{u}_\lambda$ is required
as the three components of this vector are connected through the gauge
condition. We note that the eigenmodes of the resonator form a
discrete set.

In a similar fashion the eigenmodes of the channel region are
solutions of the eigenvalue equation 
\begin{equation}
\label{eq:evepp1}
L_{\mathcal{PP}}|\vect{\nu}_{m}(\omega)\rangle = \left(
\frac{\omega}{c}\right)^2 |\vect{\nu}_{m}(\omega)\rangle.
\end{equation}
Thus, from Eq.~\eqref{eq:lsep2} they satisfy the equation
\begin{equation}
\label{eq:eveqpp}
\nabla \times \bigl( \nabla
  \times \vect{v}_m(\omega,{\bm r}) \bigr) = \frac{\epsilon ({\bf
    r}) \omega^2}{c^2} \vect{v}_{m}(\omega,{\bm r}),
\end{equation}
together with the condition that the tangential component must vanish
at the boundary,
\begin{equation}
\label{eq:bound2}
{\bf n} \times \vect{v}_m(\omega,\vect{r}) \big\rvert_{\vect{r} \in
\partial I} = {\bm 0} .
\end{equation}
The channel modes form a continuum labeled by the frequency
$\omega$ and the index $m$ that specifies the asymptotic conditions
at infinity. We note that the resonator modes $\vect{u}_\lambda$ have
support only within the resonator and vanish in the channel region;
vice versa the channel functions $\vect{v}_m (\omega)$ vanish inside
the resonator and take nonzero values only within the channel
region. The resonator and channel modes form complete and
orthonormal basis sets for the resonator and channel region,
respectively. As a result, the projectors $\mathcal{P}$ and
$\mathcal{Q}$ can be represented in terms of these modes,
\begin{subequations}
\label{eq:qpbas}
\begin{align}
\mathcal{Q} &= \sum_{\lambda} |{\bm \mu}_{\lambda}\rangle \langle
{\bm \mu}_{\lambda}|, \\ 
\mathcal{P} &= \sum_m \int\limits_{\omega_{m}}^{\infty} \!  d\omega \; |{\bm
\nu}_{m}(\omega)\rangle \langle {\bm \nu}_{m}(\omega)|.
\end{align}
\end{subequations}
Together with the eigenmode equation (\ref{eq:heveqsep}) this
reduces the eigenmode problem to the well--known problem
\cite{Fano61,Ditt00} of a discrete number of states coupled to a
continuum.

We note that the boundary conditions (\ref{eq:bound1}) and
(\ref{eq:bound2}) on the resonator and channel modes are a consequence
of our separation of the singular terms in Eq.~(\ref{eq:lqph}). As
discussed before, this separation is by no means unique. For example,
the substitution of $\delta( {\bm r}-{\bm r}^\prime)$ by $\delta( {\bf
  r}-{\bm r_+}^\prime)$ and of $\nabla \delta( {\bm r}-{\bf
  r}^\prime)$ by $\nabla \delta( {\bm r}-{\bm r_-}^\prime)$ in
Eq.~(\ref{eq:defk}), leads to a new set of boundary conditions for
which the conditions on the internal and external eigenmodes are just
interchanged. Possible more general types of boundary conditions are
discussed in Ref.~\cite{Savi03}.   

It is worth emphasizing that neither the modes ${\bm
\mu}_{\lambda}$ of the closed resonator nor the channel modes ${\bm
\nu}_{m}(\omega)$ represent exact modes of the total system, as both
of them depend on the choice made for the cavity boundary and the
boundary conditions on it. The exact modes satisfy the matching
conditions derived earlier but, in general, neither of the boundary
conditions (\ref{eq:bound1}) or (\ref{eq:bound2}). Still the
eigenmodes of the total system may be expanded in terms of the
resonator and channel modes as these modes form complete basis sets in
the respective regions. This fact is a consequence of the convergence
in Hilbert space which {\em does not} imply point-wise convergence (at
the boundary). Consequently, the matching conditions must not be
imposed directly at the boundary but hold, as usual \cite{Jack75}, in
a limiting sense infinitesimally close to the boundary.

The eigenmode equation (\ref{eq:heveqsep}) in full space may now be
solved by standard methods \cite{Fano61,Ditt00,Cohe92}. For the
projection onto the channel space one has the Lippmann--Schwinger
type solution
\begin{equation}
\label{eq:ppj}
\mathcal{P}|{\bm \phi}_{m}(\omega)\rangle = |{\bm
  \nu}_{m}(\omega)\rangle + \frac{1}{\Bigl(\dfrac{\omega}{c}\Bigr)^2 -
  L_{\mathcal{PP}}+ \rmi\epsilon} L_{\mathcal{PQ}} |{\bm \phi}_{m}
(\omega)\rangle,
\end{equation}
where the limit $\epsilon \to 0^+$ is implied. Substitution into
the equation for the projection onto the resonator space yields
\begin{equation}
\label{eq:qpj}
\mathcal{Q}|{\bm \phi}_{m}(\omega)\rangle = \frac{1}{ 
\Bigl(\dfrac{\omega}{c}\Bigr)^2 - L_{\textrm{eff}} (\omega)}
 L_{\mathcal{QP}} |{\bm \nu}_{m} (\omega)\rangle,
\end{equation}
where $L_{\textrm{eff}}(\omega)$ is the non--Hermitian
operator
\begin{equation}
\label{eq:Leff}
L_{\textrm{eff}} (\omega) \equiv L_{\mathcal{QQ}} + 
L_{\mathcal{QP}} \frac{1}{\Bigl(\dfrac{\omega}{c}\Bigr)^2 
- L_{\mathcal{PP}}+ \rmi\epsilon} L_{\mathcal{PQ}}.
\end{equation}
The operator $L_{\text{eff}}(\omega)$ is closely related to the
effective Hamiltonian in open quantum systems \cite{Savi03}.  To
simplify notation, we introduce the retarded Green function of the
resonator in the presence of the coupling to the channels
\begin{equation}
\label{eq:Gqq}
\mathcal{G}_{\mathcal{QQ}} (\omega) = \frac{1}{\Bigl(\dfrac{\omega}{c}\Bigr)^2 -
  L_{\textrm{eff}}(\omega)}.
\end{equation}
Combining Eqs.~(\ref{eq:ppj}) and (\ref{eq:qpj}) we arrive at an
expression for the eigenstates $|{\bm \phi}_{m}(\omega)\rangle$,
\begin{equation}
\label{eq:scs}
|{\bm \phi}_{m} (\omega)\rangle = \mathcal{G}_{\mathcal{QQ}} (\omega)
L_{\mathcal{QP}} |{\bm \nu}_{m} (\omega)\rangle + \left[ 1 +
\frac{1}{\Bigl(\dfrac{\omega}{c}\Bigr)^2 - L_{\mathcal{PP}}+
\rmi\epsilon} L_{\mathcal{PQ}} \mathcal{G}_{\mathcal{QQ}}(\omega)
L_{\mathcal{QP}} \right] |{\bm \nu}_m(\omega)\rangle.
\end{equation}
Using the expansion (\ref{eq:qpbas}), this yields an exact
representation of the eigenstates in terms of the resonator and
channel modes
\begin{equation}
\label{eq:univsplit}
|{\bm \phi}_{m} (\omega)\rangle = \sum_{\lambda} \alpha_{m \lambda}
(\omega) | {\bm \mu}_{\lambda}\rangle +
\sum_{m'}\int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^\prime \;
\beta_{m m'}
(\omega,\omega^\prime) |{\bm \nu}_{m'} (\omega^\prime) \rangle ,
\end{equation}
where the expansion coefficients are given by
\begin{subequations}
\label{eq:exa}
\begin{align}
\label{eq:expcoefa}
\alpha_{m \lambda}(\omega) &= \langle {\bm \mu}_{\lambda}|
\mathcal{G}_{\mathcal{QQ}}(\omega) L_{\mathcal{QP}} | {\bm \nu}_{m} (\omega)
\rangle, \\
\label{eq:expcoefb}
\beta_{m m'} (\omega,\omega^\prime) &= \langle {\bm
  \nu}_{m'}(\omega^\prime)| \left[ 1+ \frac{1}{\Bigl(\dfrac{\omega}
{c}\Bigr)^2 -L_{\mathcal{PP}} + \rmi \epsilon} L_{\mathcal{PQ}} 
\mathcal{G}_{\mathcal{QQ}}(\omega) L_{\mathcal{QP}} \right] |{\bm \nu}_{m} 
(\omega)\rangle.
\end{align}
\end{subequations}
The normal modes $\vect{f}_{m}(\omega)$ of the electromagnetic field
are recovered from ${\bm \phi}_{m}(\omega)$ using
Eqs.~(\ref{eq:fguniv}) and (\ref{eq:uvdef}): 
\begin{equation}
\label{eq:fsep}
\vect{f}_{m} (\omega,{\bm r}) = \sum_{\lambda} \alpha_{m\lambda}
(\omega) \vect{u}_{\lambda}({\bm r}) +
\sum_{m'}\int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^\prime \; \beta_{mm'}
(\omega,\omega^\prime) \vect{v}_{m'} (\omega^\prime,{\bm r}) .
\end{equation}


\subsection{Hamiltonian and Field Expansions}
\label{sec:expans}

The decomposition of the electromagnetic field modes into a
resonator and a channel contribution suggests a quantization scheme
different from the modes-of-the-universe approach discussed in
Sec.\ \ref{sec:normal_modes}. Here we carry out the field
quantization on the basis of the resonator and channel modes. Our
starting point is the expansion of the vector potential and the
canonical momentum in terms of these modes; combining
Eqs.~(\ref{eq:qpexp}) and (\ref{eq:fsep}) this expansion takes the form
\begin{subequations}
\label{eq:apsplit}
\begin{align}
\vect{A}({\bm r},t) &= c \sum_{\lambda} Q_{\lambda}
\vect{u}_{\lambda}({\bm r}) + c \sum_{m'}
\int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega \; Q_{m'}(\omega)
\vect{v}_{m'} (\omega,\vect{r}) , \\ \vect{\Pi}(\vect{r},t) &=
\frac{\epsilon(\vect{r})}{c} \left[ \sum_{\lambda} P_{\lambda}
  \vect{u}^{*}_{\lambda}({\bm r}) + \sum_{m'}
  \int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega \; P_{m'}(\omega)
  \vect{v}_{m'}^{*}(\omega,{\bm r}) \right] ,
\end{align}
\end{subequations}
where we have defined the position operators 
\begin{subequations}
\label{eq:qop}
\begin{align}
\label{eq:qopcav}
Q_{\lambda} &= \sum_{m} \int\limits_{\omega_{m}}^{\infty} \!
\rmd\omega \; q_{m}(\omega) \alpha_{m\lambda} (\omega) , \\
\label{eq:qopch}
Q_{m} (\omega) &=  \sum_{m'}
  \int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^\prime \;
  q_{m'}(\omega^\prime) \beta_{m' m}
(\omega',\omega) ,
\end{align}
\end{subequations}
and the momentum operators
\begin{subequations}
\label{eq:pop}
\begin{align}
\label{eq:popcav}
P_{\lambda} &= \sum_{m} \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega
\; p_{m}(\omega) \alpha^{*}_{m\lambda} (\omega), \\
\label{eq:popch}
P_{m}(\omega) &= \sum_{m'} \int\limits_{\omega_{m'}}^{\infty} \!
\rmd\omega^\prime \; p_{m'}(\omega^\prime) \beta_{m' m}^{*}
(\omega',\omega).
\end{align}
\end{subequations}
The $Q_\lambda$ and $P_\lambda$ are time--dependent operators that
represent complex amplitudes associated with the resonator field.
Likewise, the operators $Q_m(\omega)$ and $P_m(\omega)$ are amplitudes
describing the channel field. The reality condition on $\vect{A}$ and
${\bm \Pi}$ implies the following relations between the cavity
operators and their adjoints
\begin{subequations}
\begin{align}
\label{eq:qprealres}
Q_{\lambda} &= \sum_{\lambda^\prime}
\mathcal{N}^{\dagger}_{\lambda \lambda^\prime}
Q^{\dagger}_{\lambda^\prime}\\
P^{\dagger}_{\lambda} &= \sum_{\lambda^\prime}
\mathcal{N}^{\dagger}_{\lambda \lambda^\prime}
P_{\lambda^\prime}.
\end{align}
\end{subequations}
Likewise, the channel operators are connected with their adjoints
via the relations
\begin{subequations}
\begin{align}
\label{eq:qprealout}
Q_m(\omega) &= \sum_{m^\prime} \int\limits_{\omega_{m'}}^{\infty} \!
\rmd\omega^\prime \; \mathcal{N}_{m
  m^\prime}^{\dagger}(\omega,\omega^\prime)
Q_{m\prime}^{\dagger}(\omega^\prime), \\ 
P_m^{\dagger}(\omega) &=
\sum_{m^\prime} \int\limits_{\omega_{m'}}^{\infty} \!
\rmd\omega^\prime \; \mathcal{N}_{m
  m^\prime}^{\dagger}(\omega,\omega^\prime)
P_{m^\prime}(\omega^\prime).
\end{align}
\end{subequations}
The matrix elements
 \begin{subequations}
\begin{align}
\label{eq:usplit}
\mathcal{N}_{\lambda \lambda^\prime} &= \int \! \rmd{\bm r} \;
\epsilon(\vect{r}) \vect{u}_{\lambda}(\vect{r}) \cdot
\vect{u}_{\lambda^\prime}(\vect{r}), \\ 
\mathcal{N}_{m
  m^\prime}(\omega,\omega^\prime) &= \int \! \rmd{\bm r} \;
\epsilon(\vect{r}) \vect{v}_{m}(\omega,\vect{r}) \cdot
\vect{v}_{m^\prime}(\omega^\prime,\vect{r}).
\end{align}
\end{subequations}
are the expansion coefficients of the mode functions
$\vect{u}_\lambda$ ($\vect{v}_m(\omega)$) in terms of the complex
conjugate functions $\vect{u}^*_\lambda$ ($\vect{v}^*_m(\omega)$). The
matrix $\mathcal{N}$ is the projection of the matrix $\mathcal{M}$
defined in Eq.~\eqref{eq:uuniv}, onto the cavity and channel space, so
it is unitary, symmetric and only couples degenerate modes. The
(equal--time) commutation relations of the various operators are
discussed in Appendix \ref{app:A1}. The calculation shows that
operators associated with different subsystems commute.  Moreover, for
each subsystem $Q$ and $P$ behave like the fundamental operators for
position and momentum, respectively.

To discuss the dynamical evolution of the resonator and channel
operators, we must express the field Hamiltonian in terms of these
operators. We use the representation (\ref{eq:qop}) and (\ref{eq:pop})
along with the completeness relation $\mathcal{Q}+\mathcal{P} =1$ to
invert the relation between the operators for the total system and the
operator for the two subsystems
\begin{subequations}
\label{eq:qpsplit}
\begin{align}
q_{m}(\omega) &= \sum_{\lambda} \alpha^{*}_{m\lambda}(\omega)
Q_{\lambda} + \sum_{m'} \int\limits_{\omega_{m'}}^{\infty} \!
\rmd\omega^\prime \; \beta_{mm'}^{*} (\omega,\omega^\prime) Q_{m'}
(\omega^\prime), \\ 
p_{m}(\omega) &= \sum_{\lambda}
\alpha_{m\lambda}(\omega) P_{\lambda} + \sum_{m'}
\int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^\prime \;
\beta_{mm'}(\omega,\omega') P_{m'} (\omega^\prime) .
\end{align}
\end{subequations}
Substitution into Eq.\ (\ref{eq:hqp}) yields the desired expression
for the field Hamiltonian. Using relations between the expansion
coefficients $\alpha$ and $\beta$, that follow from the completeness
and orthogonality of the modes functions (see Appendix \ref{app:B1}),
we can write the result in the form
\begin{multline}
\label{eq:aaham}
H = \frac{1}{2}\sum_{\lambda} \big[ P^\dagger_{\lambda} P_{\lambda} + 
\omega^2_{\lambda} Q^{\dagger}_{\lambda} Q_{\lambda} \big]
+ \frac{1}{2} \sum_m \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega
\left[ P_m^{\dagger}(\omega) P_m(\omega) + \omega^2
Q_m^{\dagger}(\omega) Q_m(\omega) \right] \\
+ \sum_{\lambda} \sum_m
\int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \, W_{\lambda m} (\omega)
Q^{\dagger}_{\lambda} Q_m(\omega),
\end{multline}
with the coupling amplitudes
\begin{equation}
\label{eq:Wlm}
W_{\lambda m}(\omega) = c^2 \langle {\bm \mu}_{\lambda}| L
|{\bm \nu}_m({\omega})\rangle.
\end{equation}
The first two terms clearly describe two independent sets of harmonic
oscillators associated with the two subsystems. The third term shows
that the operators of the subsystems do not simply oscillate as they
would if the subsystems were completely isolated from each other,
since this term couple the motion of the resonator and channel
operators. The coupling reflects the fact that the boundary of the
dielectric is not completely reflecting; thus radiation may leak
through the boundary to the external radiation field.

The operators $Q$ and $P$ have a standard representation in terms of
creation and annihilation operators,
\begin{subequations}
\label{eq:qpbres}
\begin{align}
\label{eq:qbres}
Q_{\lambda} &= \left(\frac{\hbar}{2 \omega_{\lambda}} 
\right)^{\frac{1}{2}} \bigg[ a_{\lambda} +  \sum_{\lambda^\prime} 
\mathcal{N}_{\lambda \lambda^\prime}^{\dagger} 
a_{\lambda^\prime}^{\dagger}  \bigg] , \\
P_{\lambda} &= \rmi \left( \frac{\hbar \omega_{\lambda}}{2} 
\right)^{\frac{1}{2}} \bigg[ a_{\lambda}^{\dagger} -  
\sum_{\lambda^\prime} \mathcal{N}_{\lambda \lambda^\prime}
 a_{\lambda^\prime} \bigg].
\end{align}
\end{subequations}
The operators $a_\lambda$ and $a_{\lambda}^\dagger$ obey the
canonical commutation relations
\begin{equation}
\label{eq:a-comm}
[ a_{\lambda}, a_{\lambda^\prime}^\dagger ]  =  \delta_{\lambda 
\lambda^\prime}, \qquad [ a_\lambda , a_{\lambda^\prime} ]  =  0.
\end{equation}
In a similar fashion one derives the representation of the channel
operators $Q_m(\omega)$ and $P_m(\omega)$ in terms of a continuous set
of creation and annihilation operators $b_m^\dagger (\omega)$ and $b_m
(\omega)$. Substituting these representations into the Hamiltonian
(\ref{eq:aaham}) and using the symmetry and unitarity of the overlap
matrices, one arrives at the Hamiltonian
\begin{multline}
\label{eq:hlast}
H = \sum_{\lambda} \hbar \omega_{\lambda} \, a^{\dagger}_{\lambda}
a_{\lambda} + \sum_m \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega
\; \hbar\omega \, b_m^{\dagger} (\omega) b_m(\omega) \\
 + \hbar
\sum_{\lambda} \sum_m \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega
\big[ \mathcal{W}_{\lambda m}(\omega) \, a^{\dagger}_{\lambda}
  b_{m}(\omega) + \mathcal{V}_{\lambda m}(\omega) \, a_{\lambda}
  b_{m}(\omega) + \text{H.c.} \big],
\end{multline}
where we have omitted an irrelevant zero point contribution. The
coupling matrix elements are given by
\begin{subequations}
\label{eq:coupwv}
\begin{align}
\label{eq:coupw}
\mathcal{W}_{\lambda m}(\omega) &= \frac{c^2}{2 \sqrt{\omega_\lambda
    \omega}} \langle {\bm \mu}_{\lambda} | L_{\mathcal{QP}} |{\bm
  \nu}_m(\omega) \rangle , \\
\label{eq:coupv}
\mathcal{V}_{\lambda m}(\omega) &= \frac{c^2}{2 \sqrt{\omega_\lambda
    \omega}} \langle {\bm \mu}_{\lambda}^{*} | L_{\mathcal{QP}} |{\bm
  \nu}_m(\omega)\rangle .
\end{align}
\end{subequations}
The notation $\langle {\bm \mu}_{\lambda}^{*} |$ means $\langle {\bm
  \mu}_{\lambda}^{*} |{\bm r} \rangle \equiv {\bm \mu} ({\bm r})$. In
case the system is time reversal invariant the wavefunctions may be
chosen real, the amplitudes $\mathcal{W}$ and $\mathcal{V}$ then
become real and identical, $\mathcal{W}=\mathcal{V}$. Finally,
substituting the representation (\ref{eq:qpbres}) into
Eq.~(\ref{eq:apsplit}) we find the expansion of the intra-cavity field
\begin{subequations}
\label{eq:afin}
\begin{align}
\vect{A}({\bm r},t) & = c \sum_\lambda \left(\frac{\hbar}{2
\omega_\lambda}  \right)^{1/2} [ a_\lambda \vect{u}_\lambda 
({\bm r})   + a_\lambda^\dagger 
 \vect{u}^*_\lambda ({\bm r})   ] ,\\
\label{eq:pfin}
\vect{\Pi} ({\bm r},t) & =  -\frac{\rmi}{c} 
\sum_\lambda \left(\frac{\hbar 
\omega_\lambda}{2}  \right)^{1/2} \epsilon(\vect{r}) [ a_\lambda  
\vect{u}_\lambda ({\bm r})-a_\lambda^\dagger  
\vect{u}^*_\lambda ({\bm r}) ].
\end{align}
\end{subequations}
The field expansions of the open resonator reduce precisely to the
standard expressions known from closed resonators. However, the field
dynamics is {\em fundamentally} different, as shown in the following
chapter.

The Hamiltonian \eqref{eq:hlast} and the field expansions
(\ref{eq:afin}) are the key results of the quantization
procedure. Although system-and-bath Hamiltonias similar to one above,
are frequently encountered in phenomenological approaches to open
quantum systems \cite{Seni59,Seni60,Gard00}, our result differs from
them in a fundamental aspect: we provide explicit expressions for the
coupling amplitudes between the resonator and the bath subsystems. We
notice that in the Hamiltonian \eqref{eq:hlast} the resonator modes
are coupled to the external radiation field via both resonant
($a^\dagger b$, $b^\dagger a$) and non-resonant ($ab$, $a^\dagger
b^\dagger$) terms. In most cases of interest the mode widths are
smaller than the relevant frequencies, then the rotating-wave
approximation can be made, which amounts to keeping only the resonant
terms in the Hamiltonian. In this approximation, the Hamiltonian
reduces to the well-known system-and-bath Hamiltonian
\cite{Seni59,Seni60,Gard00} of quantum optics. The non-resonant terms
become important only in the case of overdamping, which appears of not
much interest in optics. It has been argued before
\cite{Barn88,Dutr00a,Dutr00d} that the system-and-bath Hamiltonian can
provide only an \emph{approximate} description of open optical
cavities provided they are of very high quality, i.e., when the modes
widths are much smaller than both the typical internal oscillation
frequencies and the spacing between internal frequencies
$\Delta\omega$. Our derivation shows that such pessimism is
inappropriate: the system-and-bath Hamiltonian (\ref{eq:hlast})
describes in an {\em exact} manner isolated modes, spectrally
overlapping modes (when the widths are larger than the mode spacing
$\Delta\omega$), as well as the extreme case of overdamped modes (when
the mode frequencies $\omega_{\lambda}$ are overwhelmed by large
escape rates).

When applied to chaotic cavities or in the presence of disordered
dielectrics our quantization method naturally accommodates for an
statistical description of the field properties. In that case the
isolated cavity Hamiltonian may be represented by a proper ensemble of
random matrices. Then the internal modes $\vect{u}_{\gl}$ share the
statistics of the eigenvectors of the random matrices and the
eigenfrequencies $\omega_{\lambda}$ display level repulsion.

To conclude this section we notice that the cavity resonances, i.e.,
the complex frequencies that determine the cavity response to
external excitations in the presence of the coupling to the outside
world, are also found within the projection formalism. Formally the
resonances are defined as the poles of the resolvent operator
$\mathcal{G}(\omega)$ projected onto the cavity space and analytically
continued in the second Reimannian sheet \cite{Cohe92}. Such a projection
is given by Eq.~\eqref{eq:Gqq}, from which the resonances follows as
the solutions of the eigenvalue problem
\begin{equation}
\label{eq:Leffeveq}
L_{\rm eff}(\omega) |\xi_{i}(\omega)\rangle = \sigma^2_{i}(\omega)
|\xi_{i}(\omega)\rangle .
\end{equation}
Since $L_{\rm eff}(\omega)$ depends parametrically on $\omega$, both
its right eigenstates $|\xi_{i}(\omega)\rangle$ and the complex
eigenvalues $\sigma_{i}(\omega)$ generally depend on $\omega$ as well.
The states $|\xi_{i}(\omega)\rangle$ correspond to the Kapur--Peierls
states \cite{Kapu38,Kuku89} of scattering theory. Combining equations
(\ref{eq:Gqq}) and (\ref{eq:Leffeveq}) the cavity resonances are found
as the solutions of the fixed point equation
$c\sigma_{i}(\omega)=\omega$. We emphasize that this resonance
condition is independent of the inside/outside separation and the
choice of boundary condition made within the system-and-bath
description.


\section{Applications to Open Optical Resonators}
\label{sec:examples}

\begin{figure}[t]
  \begin{center}
    \includegraphics[scale=1]{./Figures/cavities01.eps}
    \caption{\label{fig:cavities} Open optical cavities: (a) A
    one--dimensional dielectric slab of length $l$ with (positive)
    refractive index $n$ bounded on one side $(x=-l)$ by a perfectly
    reflecting mirror. (b) A one--dimensional cavity bounded by a
    perfectly reflecting mirror at $(x=-l)$ and a thin
    semi--transparent mirror of width $d \ll l$ at $(x=0)$. (c) A
    two--dimensional dielectric disk with refractive index $n$.}
\end{center}
\end{figure}
In the previous section we focused on the general quantization of the
electromagnetic field using the Feshbach projection formalism. The main
result was the system-and-bath Hamiltonian \eqref{eq:hlast}. In this
section we explicitly demonstrate the method for a number of models
frequently used for optical resonators. Specifically, we consider the
three types of cavities shown in Fig.~\ref{fig:cavities}: (a) A
one-dimensional dielectric slab with (positive) refractive index $n$
bounded on one side by a perfectly reflecting mirror, (b) a
one-dimensional cavity defined by a perfectly reflecting mirror on one
side and a thin semi-transparent mirror on the other side, and (c) a
two-dimensional dielectric disk with refractive index $n$. The
dielectrics are embedded in empty space. The normal modes of all three
systems in question as well as their scattering properties can be
computed exactly. Likewise, exact expressions can be obtained for the
electromagnetic fields both within the cavities and in the external
region. This makes all three models an ideal testing ground for our
system-and-bath description. Such a test is particularly important to
illustrate how in the projection formalism all physical observables do
not relay in the choice of the separating surface and boundary
conditions, but only on the physical boundary conditions imposed by
Maxwell's equations. For each of the examples considered we compute
their respective system-and-bath Hamiltonian, and show that the
electromagnetic fields and the scattering properties agree exactly
with results obtained by direct solutions of the quantization
problem. In addition, the cavity resonances are found for all the
systems and shown to be independent of the choice of boundary
condition made on the fictitious boundary separating system and bath.



\subsection{One Dimensional Dielectric Cavity}
\label{sec:ex01}

Our first example is the one dimensional dielectric cavity depicted
in Fig.~\ref{fig:cavities}(a) \cite{Ujih75}. The dielectric with
refractive index $n$ is nonabsorbing and nondispersive. It is
bounded by a perfectly reflecting mirror at $x=-l$ while there is
no mirror at the other end of the dielectric at $x=0$. The free
space outside the cavity runs from $x=0$ to infinity, and light
propagates freely there, $n=1$. We assume the electromagnetic field
to be linearly polarized with the electric field vector pointing in
the $z$--direction. Cavity field excitations will decay due to
leakage into the empty half-space. The dielectric function of the
total system including the cavity and the attached half-space is
given by
\begin{equation}
\label{eq:eps01}
\epsilon(x) = n^2 \Theta(-x)  + \Theta(x),
\end{equation}
where the Heaviside--function $\Theta(x)$ is equal to one for positive
$x$ and vanishes for negative $x$. The normal modes of Maxwell's
equations for this problem are given in Eq.~\eqref{eq:f01}. To solve
the problem within the projection formalism, we separate system
and bath at $x=0$.  The cavity thus runs from $x=-l$ to $x=0$, and the
channel region from $x=0$ to $\infty$. The boundary conditions at the
interface are only restricted by the requirement that they lead to a
Hermitian eigenvalue problem.  Below we address two different such
boundary conditions: In the first case we set Neumann boundary
conditions for the cavity; Hermiticity \cite{Savi03} then imposes
Dirichlet conditions for the channel problem. In the second case we
consider the inverse situation with Dirichlet boundary condition for
the cavity and Neumann conditions outside.


\subsubsection{Cavity with von--Neumann Boundary Conditions}
\label{sec:CN-CD}

The inside/outside decomposition of the differential operator $L$
follows from Eq.~\eqref{eq:lqph},
\begin{subequations}
\label{eq:Lproj01}
\begin{align}
\label{eq:Lqq01}
L_{\mathcal{QQ}} \mu(x) &= -\frac{1}{n^2}\frac{\rmd^2}{\rmd x^2}\mu(x) +
\frac{\delta(x-0_{-})}{n^2} \frac{\rmd}{\rmd x^{\prime}}\mu(x^\prime)
\biggr\vert_{x^{\prime}=0_{-}},\\
\label{eq:Lpp01}
L_{\mathcal{PP}} \nu(x) &= -\frac{\rmd^2}{\rmd x^2}\nu(x) -
\delta^{\prime}(x-0_{+}) \nu(0_{+}),\\
\label{eq:Lpq01}
L_{\mathcal{PQ}} \mu(x) &= \frac{\delta^{\prime}(x-0_{+})}{n}
\mu(0_{-}),\\
\label{eq:Lqp01}
L_{\mathcal{QP}} \nu(x) &= -
\frac{\delta(x-0_{-})}{n}\frac{\rmd}{\rmd x^{\prime }}\nu(x^\prime)
\biggr\vert_{x^{\prime}=0_{+}}.
\end{align}
\end{subequations}
The shorthands $0_{\mp}$ indicate the limits where the interface at
$x=0$ is approached from inside respectively outside the
resonator. The singular terms guarantee the matching conditions for
the electromagnetic field at the interface. In addition, these terms
ensure the Hermiticity of $L_{\mathcal{QQ}}$ and
$L_{\mathcal{PP}}$. The range of the operators in
equations (\ref{eq:Lqq01}) and (\ref{eq:Lpp01}) within Hilbert space is
given by the functions for which the singular term vanishes.

The eigenmodes $\mu_{\lambda}(x)$ of the closed cavity are the
solutions of the eigenvalue problem $L_{\mathcal{QQ}}
\mu_{\lambda}(x) = k^2_{\lambda} \mu_{\lambda}(x)$ with
$k_{\lambda}= \omega_{\lambda} /c$. From Eq.~\eqref{eq:Lqq01},
they satisfy the equation
\begin{equation}
\label{eq:helmres01}
\frac{\rmd^2}{\rmd x^2} \mu_{\lambda}(x) + n^2 k^2_{\lambda} \mu_{\lambda}(x)
= 0,
\end{equation}
subject to the boundary conditions
\begin{subequations}
\begin{gather}
\mu_{\lambda}(-l) = 0 , \\
\frac{\rmd}{\rmd x}\mu_{\lambda}(x) \biggr\vert_{x=0_{-}} = 0.
\end{gather}
\end{subequations}
The first condition is imposed by the perfectly reflecting mirror at
$x= -l$, and the second follows from the requirement that the singular
boundary term applied to $\mu_\lambda$ must vanish. The normalized
solutions of the eigenvalue problem form the discrete set
\begin{equation}
\label{eq:ex01mu}
\mu_{\lambda}(x) = \sqrt{\frac{2}{l}} \sin(n k_{\lambda}(x+l)),
\end{equation}
with wave numbers $k_{\lambda} = (2\lambda +1)\pi / 2 n l  \;
(\lambda = 0,1,2,\dots)$. In the channel region, the eigenvalue problem
reads
\begin{equation}
\label{eq:helmbath01}
\frac{\rmd^2}{\rmd x^2} \nu (k,x) + k^2 \nu (k,x) = 0; \quad (k=\omega/c),
\end{equation}
with Dirichlet conditions at the resonator surface,
\begin{equation}
\nu (k,0_{+}) = 0.
\end{equation}
This determines a continuous set of $\delta$--normalized channel
modes,
\begin{equation}
\label{eq:ex01nu}
\nu (k,x) = \sqrt{\frac{2}{\pi}} \sin(k x).
\end{equation}
Since both the cavity and channel modes are real valued functions,
the coupling amplitudes $\mathcal{W}$ and $\mathcal{V}$ become real
and identical, $\mathcal{W}=\mathcal{V}$. Combining equations
(\ref{eq:coupw}), (\ref{eq:Lqp01}) with the mode functions
(\ref{eq:ex01mu}) and (\ref{eq:ex01nu}), we obtain
\begin{equation}
\label{eq:W01}
\mathcal{W}_{\lambda}(k) = \mathcal{V}_{\lambda}(k)
= \frac{(-1)^{\lambda+1}}{n}
\sqrt{\frac{k}{\pi k_{\lambda} l}}.
\end{equation}
The result for the internal frequencies $\omega_{\lambda} = c
k_{\lambda}$, together with the coupling amplitudes (\ref{eq:W01}),
and the mode functions (\ref{eq:ex01mu}) and (\ref{eq:ex01nu}),
completely specify the system-and-bath Hamiltonian and the electric
and magnetic field.

We now turn to an illustration of our results. To compare the exact
scattering states with their representation in terms of cavity and
channel modes, we combine equations \eqref{eq:defal}, \eqref{eq:defbe}
with the results (\ref{eq:f01}), (\ref{eq:ex01mu}), and
(\ref{eq:ex01nu}). This yields the expansion coefficients
\begin{align}
\label{eq:a01}
\alpha_{\lambda}(k) &= \frac{(-1)^{\lambda + 1} I_k k \cos(n k
  l)}{n\sqrt{\pi l} (k^2-k_{\lambda}^2)} , \\
\label{eq:b01}
\beta(k,k') &=
  \frac{1}{2\pi} \left[ \mathcal{P} \left( \frac{2k' (1+S_k)}{k'^2 -
  k^2} \right) - \rmi \pi (1-S_k) \delta(k' - k) \right] .
\end{align}
The symbol $\mathcal{P}$ denotes the principal value. The scattering
matrix $S_k$ and the amplitude $I_k$ are given, respectively, by
equations (\ref{eq:s01}) and (\ref{eq:I01}). Figure~\ref{fig:diel1d01}
shows the real part of the scattering wave function with wavenumber
$kl=18$. We compare the exact result (solid gray line) with the
system-and-bath expansion (dashed line). In the resonator region we
only included 11 cavity modes with wavenumber centered around $kl=18$.
The agreement is very good; deviations are only visible close to
$x=0$, i.e., near the boundary separating system and bath.  It has
been argued before \cite{Barn88} that cavity or channel expansions
must fail close to the boundary; so a remark concerning the status of
such expansions is in order here: The inclusion of all cavity and all
channel modes yields an exact point--to--point representation of the
scattering function and its derivative, everywhere {\em except} for
the point $x=0$.  This representation does not converge uniformly but
it is exact in the $L^2$ sense. Therefore the system-and-bath
expansion is an exact representation of the scattering state in the
underlying Hilbert space.
\begin{figure}[t]
  \begin{center} \includegraphics[scale=1]{./Figures/die1d01.eps}
    \caption{\label{fig:diel1d01} Real part of the scattering wave
    function corresponding to $kl=18$, computed for a one dimensional
    dielectric cavity with refractive index $n=1.5$. The solid line is
    the exact solution, the dashed line the expansion in terms of the
    resonator and channels modes. Only $11$ cavity modes with
    $k_{\lambda}l$ around $kl=18$ were included. The inset shows the
    cavity gain factor as function of $kl$ for a range around
    $kl=18$.}
\end{center}
\end{figure}

To determine the cavity resonances we solve the eigenvalue problem for
the non-Hermitian operator $L_{\text{eff}}(k)$. Explicit calculation
(Appendix \ref{app:appB}) shows that $L_{\text{eff}}(k)$ acts on an
arbitrary resonator state $\mu(x)$ according to
\begin{equation}
\label{eq:Leff01b}
L_{\text{eff}}(k) \mu(x) = -\frac{1}{n^2}\frac{\rmd^2}{\rmd x^2}\mu(x)
  + \frac{\delta(x-0_{-})}{n^2} \left[ \frac{\rmd}{\rmd
  x^{\prime}}\mu(x^\prime) \biggr\vert_{x^{\prime}=0_{-}} - \rmi k
  \mu(0_{-}) \right].
\end{equation}
Due to the singular term, the action of $L_{\text{eff}}(k)$ generally
goes beyond Hilbert space. The range of $L_{\text{eff}}(k)$ within
Hilbert space is defined by the wave functions for which the singular
term vanishes. It follows that the right eigenstates $\xi_{j}(k,x)$
of $L_{\text{eff}}(k)$ are solutions of the Helmholtz equation
\begin{equation}
\label{eq:leff01evq}
\frac{\rmd^2}{\rmd x^2} \xi_{j}(k,x) + n^2 \sigma_{j}^2(k) \xi_{j}(k,x) = 0,
\end{equation}
that obey the boundary conditions
\begin{subequations}
\begin{gather}
\xi_{j}(k,-l) = 0, \\
\label{eq:bcres01}
\frac{\rmd}{\rmd x^{\prime}}\xi_{j}(k,x^\prime)
\biggr\vert_{x^{\prime}=0_{-}} = \rmi k \xi_{j}(k,0_{-}).
\end{gather}
\end{subequations}
The first condition results from the perfect mirror at $x=-l$ while
the second defines the so--called Siegert boundary condition. It
accounts for the leakage out of the cavity: In the channel region that
boundary condition implies a purely outgoing wave. For a fixed value
of $k$, one finds the discrete set of solutions
\begin{equation}
\label{eq:rres01}
\xi_{j}(k,x) = A_{j}(k) \sin(n \sigma_{j}(k)(x+l)) ,
\end{equation}
with some normalization factors $A_{j}(k)$. Substituting the solutions
into Eq.~\eqref{eq:bcres01} we obtain the secular equation for the
eigenvalues,
\begin{equation}
\label{eq:res01}
\sigma_{j}(k) = \frac{\rmi}{n} k \tan(n \sigma_{j}(k) l).
\end{equation}
The fixed point equation $k = \sigma_{j}(k)$ determines the cavity
resonances. Analytical continuation of Eq.~\eqref{eq:res01} into
the complex plane then yields the resonance condition
\begin{equation}
\label{eq:rescond01}
\tan(n k l) + \rmi n = 0,
\end{equation}
which has solutions only for complex $k$. The resonances can be found
analytically and are given by
\begin{equation}
\label{eq:resex01}
k_j = \frac{1}{n l}\begin{cases} \dfrac{(2j+1)\pi}{2} + \dfrac{\rmi}{2}
  \ln\left(|r|\right);\quad j=0,1,\dots & (n > 1),  \vspace{5pt} \\
  j\pi + \dfrac{\rmi}{2} \ln\left(|r|\right);\quad j=1,2,\dots & (n <
1),
\end{cases}
\end{equation}
where $r = (n-1)/(n+1)$ is the reflection amplitude at the dielectric
surface. Comparison with the direct calculation
(cf. Eq.~\eqref{eq:s01}) shows that the resonances coincide with the
poles of the scattering--matrix. All resonances have the same width
and are located along a straight line in the lower half of the complex
plane. The resonance spacing, i.e., the difference in real parts of
two successive resonances, is constant, $\Delta k=\pi/nl$. The
resonances start to overlap when the modulus of the reflection
amplitude becomes smaller than $|r| = \exp (-\pi)$.

We finally evaluate the cavity gain factor. The free--space local
density of states is $\rho_{0}(k,x) = \sqrt{\frac{2}{\pi}} \sin^2
(k(x+l))$. Integration over the cavity volume yields
\begin{equation}
\label{eq:ldos1d}
\int\limits_{-l}^{0} \! \rmd x \, \rho_{0}(k,x) = \frac{l}{\pi}
\left[ 1 - \frac{\sin(2kl)}{2kl} \right].
\end{equation}
The integrated cavity density of states follows from equations
(\ref{eq:rldos}) and (\ref{eq:a01}) by means of the Poisson sum
rule,
\begin{equation}
\begin{split}
\label{eq:ldoscav01}
\int\limits_{-l}^{0} \! \rmd x \, \rho(k,x) &= \frac{|I_{k}|^2
  k^2 \cos^2(nkl)}{\pi l n^2} \sum\limits_{\lambda=0}^{+\infty}
\frac{1}{(k^2 - k_{\lambda}^2)^2} \\
& = \frac{l|I_{k}|^2}{4\pi} \left[1 - \frac{\sin(2nkl)}{2nkl}\right].
\end{split}
\end{equation}
Combining these results with Eq.~\eqref{eq:I01} for $I_k$, we
obtain the cavity gain factor
\begin{equation}
\label{eq:cgf01}
G^{c}(k) = \frac{n^2
\left[1-\frac{\sin(2nkl)}{2nkl}\right]}{\left[n^2 \cos^2(nkl) +
\sin^2(nkl)\right]\left[1-\frac{\sin(2kl)}{2kl} \right]}.
\end{equation}
The inset in Fig.~\ref{fig:diel1d01} shows the cavity gain factor over a
range of $kl$. The peaks are equally spaced and have approximately the
same high and width as expected from Eq.~\eqref{eq:resex01}.


\subsubsection{Cavity with Dirichlet Boundary Conditions}
\label{sec:CD-CN}

It is interesting to carry out the system-and-bath quantization in a
basis other than that considered in the previous section.  To that end
we reconsider the dielectric resonator of Fig.~\ref{fig:cavities}(a)
but perform the system-and-bath quantization with
\textit{interchanged} boundary conditions at the resonator/channel
interface: The resonator modes are now required to satisfy Dirichlet
boundary conditions at $x=0$ while Neumann conditions hold for the
channel modes. The differential operators corresponding to this choice
have the form
\begin{subequations}
\label{eq:Lproj02}
\begin{align}
\label{eq:Lqq02}
L_{\mathcal{QQ}} \mu(x) &= -\frac{1}{n^2}\frac{\rm\rmd^2}{\rmd x^2}\mu(x) +
\frac{\delta^{\prime}(x-0_{-})}{n^2} \mu(0_{-}) \, ,\\
\label{eq:Lpp02}
L_{\mathcal{PP}} \nu(x) &= -\frac{\rm\rmd^2}{\rmd x^2}\nu(x) - \delta(x-0_{+})
\frac{\rmd}{\rmd x^{\prime}}\nu(x^\prime) \biggr\vert_{x^{\prime}=0_{+}} \, ,\\
\label{eq:Lpq02}
L_{\mathcal{PQ}} \mu(x) &= \frac{\delta(x-0_{+})}{n}
\frac{\rmd}{\rmd x^{\prime }}\mu(x^\prime) \biggr\vert_{x^{\prime}=0_{-}}\, ,\\
\label{eq:Lqp02}
L_{\mathcal{QP}} \nu(x) &= -\frac{\delta^{\prime}(x-0_{-})}{n}
\nu(0_{+}).
\end{align}
\end{subequations} 
The closed cavity eigenmodes of $L_{\mathcal{QQ}}$ solve the Helmholtz
equation and satisfy Dirichlet boundary conditions both at $x=l$ and
$x=0_-$. The second of these conditions follows from the requirement
that the application of $L_{\mathcal{QQ}}$ on any eigenmode must yield
a vanishing singular contribution. The eigenmodes form the discrete
set of functions
\begin{equation}
\label{eq:ex02mu}
\mu_{\lambda}(x) = \sqrt{\frac{2}{l}} \sin(n k_{\lambda}(x+l)),
\end{equation}
with eigenvalues $k_{\lambda} = \pi\lambda/n l \; (\lambda =
1,2,\dots)$. In a similar fashion one finds the continuous set of
channel modes
\begin{equation}
\label{eq:ex02nu}
\nu(k,x) = \sqrt{\frac{2}{\pi}} \cos(k x),
\end{equation}
that satisfy Neumann boundary conditions at $x=0_+$. Substituting the
mode functions into the definitions (\ref{eq:coupwv}) we obtain the
coupling amplitudes
\begin{equation}
\label{eq:W02}
\mathcal{W}_{\lambda k} = \mathcal{V}_{\lambda k} =
 (-1)^{\lambda} \sqrt{\frac{k_{\lambda}}{\pi k l}}.
\end{equation}
We note that the cavity eigenfrequencies and the coupling amplitudes
obtained with the present set of boundary conditions differ from the
results obtained in the previous section. Consequently, two different
system-and-bath Hamiltonians are obtained in the two cases. However,
as we show below both Hamiltonians provide an equivalent, and exact,
description of the field dynamics.

Expanding the modes-of-the-universe $f(k,x)$ in terms of resonator
and channel modes, we find the expansions coefficients
\begin{align}
\label{eq:a02}
\alpha_{\lambda}(k) &= \frac{(-1)^{\lambda} I_k k_{\lambda}\sin(n k
l)}{n \sqrt{\pi l} (k^2-k_{\lambda}^2) }, \\
\label{eq:b02}
\beta(k,k') &= \frac{\rmi}{2\pi} \left[ \mathcal{P} \left( \frac{2k
(1-S_k)}{k'^2 - k^2} \right) - \rmi \pi (1+S_k) \delta(k' - k) \right],
\end{align}
with $S_k$ and $I_k$ defined, respectively, by equations
\eqref{eq:s01} and \eqref{eq:I01}. In Fig.~\ref{fig:diel1d02} we
compare the exact scattering wave function (\ref{eq:f01}) (solid gray
line) with the mode expansion using coefficients (\ref{eq:a02}) and
(\ref{eq:b02}). Perfect agreement is found in the channel region $x >
0$. In the cavity region there is slow convergence close to $x=0$, due
to the Dirichlet boundary conditions at $x=0_-$. The slower
convergence visible in Fig.~\ref{fig:diel1d02} must be compared with
the faster convergence found for the other set of boundary conditions
(Fig.~\ref{fig:diel1d01}). It indicates that, in spite of the freedom
inherent in the projection formalism for the choice of boundary
conditions, certain boundary conditions are better suited for the
problem yielding good approximations with less terms in the mode
expansions.
\begin{figure}[t]
  \begin{center} \includegraphics[scale=1]{./Figures/die1d02.eps}
    \caption{\label{fig:diel1d02} Real part of the scattering wave
      function for a one dimensional dielectric cavity with the
      same parameters as in Fig.~\ref{fig:diel1d01}. The solid gray
      curve is the exact solution. The system-and-bath expansion
      (dotted, dashed line) is based on $11$, respectively, $25$
      cavity modes satisfying Dirichlet boundary conditions at
      $x=0$.  The dashed line for $x>0$ is the representation in
      terms of channel modes.}
\end{center}
\end{figure}

We may now evaluate the system resonances and the cavity gain
factor. With the present boundary conditions the operator
$L_{\text{eff}}(k)$ reduces to (Appendix \ref{app:appB})
\begin{equation}
\label{eq:Leff02b}
L_{\text{eff}}(k) \mu(x) = -\frac{1}{n^2}\frac{\rm\rmd^2}{\rmd x^2}\mu(x) +
  \frac{\delta'(x-0_{-})}{n^2} \left[ \mu(0_{-}) + \frac{\rmi}{k}
\frac{d}{\rmd x'} \mu(x') \biggr\vert_{x^{\prime}=0_{-}} \right].
\end{equation}
It is illustrative to compare this with the result (\ref{eq:Leff01b})
that holds for interchanged resonator/channel boundary
conditions. Both results differ in their singular terms.  However,
upon projection onto the Hilbert space the same operator is recovered
as the singular contributions vanish. In both cases the resulting
boundary condition at the resonator/channel--interface is the Siegert
condition (\ref{eq:bcres01}).

The integrated cavity density of states follows upon combination of
equations (\ref{eq:rldos}) and (\ref{eq:a02}), with the result
\begin{equation}
\label{eq:ldoscav02}
\int\limits_{-l}^{0} \! \rmd x \, \rho(k,x) = 
\frac{l|I_{k}|^2}{4\pi} \left[1 - \frac{\sin(2nkl)}{2nkl}\right].
\end{equation}
It agrees with the result (\ref{eq:ldoscav01}) obtained for the
other set of boundary conditions. This demonstrates that the
physical observables are indeed independent of the choice of
boundary conditions.


\subsection{One Dimensional Cavity with a Semitransparent Mirror}
\label{sec:ex02}

The model of Lay and Loudon \cite{Ley87} is a one dimensional cavity
defined by a totally reflecting mirror at one end and a
semitransparent mirror at the other end (Fig.~\ref{fig:cavities}(b)). The
electric field is linearly polarized in the $z$--direction. Radiation
can leak out through the semitransparent mirror modeled by a
dielectric slab of width $d$ and refractive index $n$. The limit $d
\to 0$ and $n \to \infty$ with $n^2 d=\eta$ fixed is taken at the end
of the calculation; here $\eta$ is a factor characterizing the mirror
transparency. In this limit, the frequency dependent mirror reflexion
and transmission amplitudes are given by
\begin{equation}
\label{eq:rt}
r(k) = \frac{\rmi k\eta}{2 - \rmi k\eta}, \qquad t(k) =
\frac{2}{2 - \rmi k\eta}.
\end{equation}
They obey the common relations for symmetric mirrors, $|r|^2 + |t|^2
=1$ and $rt^{*} + r^{*}t = 0$.

The exact eigenmodes of Maxwell's equations for this problem are given
in Eq. \eqref{eq:f03}. Within the projection formalism there are two
natural ways of a resona\-tor/channel separation: Either one assumes the
mirror to be part of the cavity or the mirror is part of the channel
region. Here, we stick to the latter choice. Accordingly, the cavity
runs from $x=-l$ to $x=0$ and the channel region from $x=0$ to
$\infty$. The alternative definition with the mirror being part of the
cavity can easily be shown to lead to the same physical results. We
choose Dirichlet conditions for the resonator boundary at $x=0_{-}$,
which implies von Neumann conditions for the outside problem. The
differential operators corresponding to these definitions are
\begin{subequations}
\label{eq:Lproj03}
\begin{align}
\label{eq:Lqq03}
L_{\mathcal{QQ}} \mu(x) &= -\frac{\rmd^2}{\rmd x^2}\mu(x) +
\delta'(x-0_{-}) \mu(0_{-}),\\
\label{eq:Lpp03}
L_{\mathcal{PP}} \nu(x) &=
-\frac{1}{n(x)}\frac{\rmd^2}{\rmd x^2}
\left(\frac{\nu(x)}{n(x)}\right) - \frac{\delta(x-0_{+})}{n(x)}
\frac{\rmd}{\rmd x'}\left(\frac{\nu(x')}{n(x')}\right) \biggr\vert_{x'=0_{+}},\\
\label{eq:Lpq03}
L_{\mathcal{PQ}} \mu(x) &= \frac{\delta(x-0_{+})}{n(x)}
\frac{\rmd}{\rmd x'}\mu(x') \biggr\vert_{x'=0_{-}},\\
\label{eq:Lqp03}
L_{\mathcal{QP}} \nu(x) &= -\delta'(x-0_{-})
\frac{\nu(0_{+})}{n(0_{+})},
\end{align}
\end{subequations}
where $n(x)$ is the refractive index in the channel region,
\begin{equation}
n(x) = \begin{cases} n & (0_{+} \le x \le d), \\
                     1 & (d < x).
       \end{cases}
\end{equation}
The eigenvalue problem defined by $L_{\mathcal{QQ}}$ reduces to that
of the dielectric resonator of the first example in
Sec.~\ref{sec:CD-CN} in the case when the dielectric function equals
$1$. Adopting our earlier results in that limiting case we find the
closed cavity eigenmodes
\begin{equation}
\label{eq:ex03mu}
\mu_{\lambda}(x) = \sqrt{\frac{2}{l}} \sin(k_{\lambda}(x+l)),
\end{equation}
with the eigenvalues $k_{\lambda}=\pi \lambda /l \; (\lambda
=1,2,\dots)$. The channel modes are the solutions of the Helmholtz
equation
\begin{equation}
\label{eq:helm03}
\frac{\rmd^2}{\rmd x^2} \nu(k,x) + n^2(x) k^2 \nu(k,x) = 0,
\end{equation}
with Neumann boundary conditions at $x=0_+$. In addition, they must
satisfy the two conditions
\begin{gather}
\label{eq:mirrorBC}
\frac{1}{n} \nu(k,d_{-}) = \nu(k,d_{+}), \\
\frac{1}{n}\frac{\rmd}{\rmd x} \nu(k,x) \biggr\vert_{x=d_{-}} = \frac{\rmd}{\rmd x}
\nu(k,x) \biggr\vert_{x=d_{+}},
\end{gather}
imposed by the continuity of the electric and magnetic field at the
right end of the semitransparent mirror. The shorthands $d_{\pm}$
indicate the limit where $d$ is approached from the left ($d_-$) or
from the right ($d_+$). Solving for $\nu(k,x)$ and taking the limit
$d \to 0$, $n\to \infty$ with $n^2d= \eta$, one obtains the
following continuous set of channel modes,
\begin{equation}
\label{eq:ex03nu}
\nu(k,x) = \frac{1}{\sqrt{2\pi}} \bigl( \rme^{-\rmi kx} + S_{c}(k)
\rme^{\rmi kx}
\bigr), \quad S_{c}(k) = \frac{\rmi-\eta k}{\rmi+\eta k}.
\end{equation}
The coupling amplitudes follow upon substituting the wavefunctions
(\ref{eq:ex03mu}) and (\ref{eq:ex03nu}) into the definitions
(\ref{eq:coupw}) and (\ref{eq:coupv}). The result is
\begin{equation}
\mathcal{W}_{\lambda}(k) = \mathcal{V}_{\lambda}(k) =
\frac{(-1)^{\lambda}}{1-\rmi\eta k} \sqrt{\frac{k_{\lambda}}{\pi k l}}.
\end{equation}
Finally, the representation of the exact modes $f(k,x)$ in terms of
the system and bath modes yields the expansion coefficients
\begin{align}
\label{eq:a03}
 \alpha_{\lambda}(k) &= \frac{(-1)^{\lambda} I_k k_{\lambda}\sin(n k
l)}{\sqrt{\pi l} (k^2-k_{\lambda}^2) }, \\
\label{eq:b03}
 \beta(k,k') &= \frac{\rmi}{2\pi} \left[ \mathcal{P} \left( \frac{1 -
S_{c}^{*}(k') S_{k}}{k' - k} + \frac{S_{c}^{*}(k') - S_{k}}{k'+k}
\right) - \rmi \pi (1+S_{c}^{*}(k)S_k) \delta(k' - k) \right],
\end{align}
where $S_{k}$ and $I_{k}$ are given by Eq.~\eqref{eq:s03} and
Eq.~\eqref{eq:I03}, respectively. Figure~\ref{fig:smt1d} shows the real part of
the scattering wave function with $kl=28.9$. The exact solution
(\ref{eq:f03}) (solid gray line) is compared with the mode expansion
(dashed line) using the expansion coefficients (\ref{eq:a03}) and
(\ref{eq:b03}). The first 35 cavity modes were included. The
deviations from the exact scattering wave function visible near $x=0$
can be made arbitrary small by including more terms in the mode
expansion.
\begin{figure}[t]
  \begin{center} \includegraphics[scale=1]{./Figures/stm1d.eps}
    \caption{\label{fig:smt1d} Real part of the scattering wave
      function with $kl=28.9$ for a one dimensional optical cavity
      with a perfectly reflecting mirror at $x=-l$ and a
      semitransparent mirror at $x=0$. The mirror transparency is
      characterized by $\eta=0.0453$ corresponding to the reflexion
      coefficient $|r(kl=28.9)|^2=0.3$. The solid line is the exact
      solution, the dashed line represents the mode expansion
      truncated to the first 35 modes in the cavity region. Inset:
      Cavity gain factor as function of $kl$ for a range around
      $kl=28$.} \end{center}
\end{figure}

In order to find the resonances one must solve the eigenvalue equation
for $L_{\text{eff}}(k)$. The real space representation of $L_{\rm
eff}(k)$ for our choice of boundary conditions follows upon
combination of equations \eqref{eq:Leff} and (\ref{eq:Lqq03}) (see
Appendix \ref{app:appB}), with the result
\begin{equation}
\label{eq:Leff03b}
 L_{\text{eff}}(k) \mu(x) = -\frac{\rmd^2}{\rmd x^2}\mu(x) +
\delta'(x-0_{-})
\left[ \mu(0_{-}) - \frac{1}{k (\rmi +\eta k)} \frac{\rmd}{\rmd x'} \mu(x')
  \biggr\vert_{x^{\prime}=0_{-}} \right],
\end{equation}
where $\mu(x)$ is an arbitrary resonator function. The range of $L_{\rm
eff}(k)$ within Hilbert space is defined by the resonator functions
for which the singular term on the right hand side vanishes. In
particular, this holds for the right eigenstates $\xi_{j} (k,x)$. It
follows that these states satisfy the boundary condition
\begin{equation}
\label{eq:bcres03}
\xi_{j}(k,x) = \frac{1}{k(\rmi +\eta k)}
\frac{\rmd}{\rmd x}\xi_{j}(k,0_{-}) \biggr\vert_{x=0_{-}} \, ,
\end{equation}
at the semitransparent mirror. There is a discrete set of solutions,
\begin{equation}
\label{eq:rres03}
\xi_{j}(k,x) = A_{j}(k) \sin(\sigma_{j}(k)(x+l)),
\end{equation}
where $A_{j}(k)$ is a normalization constant. Substitution into
Eq.~\eqref{eq:bcres03} yields the equation for the eigenvalues
$\sigma_{j}(k)$,
\begin{equation}
\label{eq:res03}
\sigma_{j}(k) \cot(\sigma_{j}(k) l) = k(\rmi + k\eta).
\end{equation}
After analytical continuation of the fixed point equation $k =
\sigma_{j} (k)$ into the complex plane we obtain the resonance
condition
\begin{equation}
\label{eq:rescond03}
\rmi + \eta k - \cot(k l) = 0 \, , 
\end{equation} 
that coincides with the equation for the poles of the $S$-matrix (cf.\
Eq.~\eqref{eq:s03}).

To quantify the resonant response of the cavity to external
excitations we compute the integrated local density of states, again
using the Poisson sum rule,
\begin{equation}
\begin{split}
\label{eq:ldoscav03}
\int\limits_{-l}^{0} \! \rmd x \, \rho(k,x) &= \frac{l|I_{k}|^2
 \sin^2(kl)}{l^2\pi} \sum\limits_{\lambda=1}^{+\infty}
\frac{k_{\lambda}^2}{(k^2 - k_{\lambda}^2)^2},\\
& = \frac{l|I_{k}|^2}{4\pi} \left[1 - \frac{\sin(2kl)}{2kl}\right].
\end{split}
\end{equation}
Combination with the free--space density of states (\ref{eq:ldos1d})
gives the cavity gain factor
\begin{equation}
\label{eq:cgf03}
G^{c}(k) =
\frac{1}{\left[1- \eta k \sin(2kl) + (\eta k)^2 \sin^2(nkl)\right]}.
\end{equation}
With increasing $kl$ sharper resonances are found in the cavity
gain factor (see Fig.~\ref{fig:smt1d} (inset)). The reason is the
reduction of the mirror transmission for large $kl$ that, in turn,
enhances the lifetime of the cavity resonances.


\subsection{Dielectric Disk}
\label{sec:ex04}

In this section we demonstrate our quantization technique for
resonators of spatial dimension larger than one. Specifically, we
consider a two--dimensional circular dielectric of radius $R$ and
refractive index $n$ (Fig.~\ref{fig:cavities}(c)). The resonator
is embedded in free space.  We restrict ourselves to TM modes with the
electric field polarized in the $z$--direction. It is convenient to
use polar coordinates ${\bm r} = (r,\phi)$ below. The dielectric
function then reads
\begin{equation}
\epsilon (r) = n^2 \Theta (R-r) + \Theta (r - R).
\end{equation}
The scattering problem at the resonator can be solved exactly. The
normal modes of Maxwell's equations are summarized in Eq.~\eqref{eq:f04}.

To apply our quantization technique we separate system and bath along
the boundary of the dielectric: The dielectric disk ($r \le R_{-}$) is
taken as the cavity, while the free space ($r \ge R_{+}$) becomes the
channel region.  For the cavity we assume Dirichlet boundary
conditions at $r=R_{-}$, which implies Neumann conditions at $r=R_{+}$
for the channel problem. The differential operator resulting from this
choice reads
\begin{subequations}
\label{eq:Lproj04}
\begin{align}
\label{eq:Lqq04}
L_{\mathcal{QQ}} \mu(r,\phi) &= -\frac{1}{n^2} \nabla^2 \mu(r,\phi) +
\frac{\partial}{\partial r} \left( \frac{1}{r} \delta (r-R_{-})
\right) \frac{R_{-}}{n^2} \mu(R_{-},\phi),\\
\label{eq:Lpp04}
L_{\mathcal{PP}} \nu(r,\phi) &= -\nabla^2 \nu(r,\phi) -
\delta(r-R_{+}) \frac{\partial}{\partial r'} \nu(r',\phi) \biggr\vert_{r'=R_{+}},\\
\label{eq:Lpq04}
L_{\mathcal{PQ}} \mu(r,\phi) &= \frac{\delta(r-R_{+})}{n}
\frac{\partial}{\partial r'} \mu(r',\phi) \biggr\vert_{r'=R_{-}},\\
\label{eq:Lqp04}
L_{\mathcal{QP}} \nu(r,\phi) &= - \frac{\partial}{\partial r} \left(
\frac{1}{r} \delta (r-R_{-}) \right) \frac{R_{-}}{n} \nu(R_{+},\phi).
\end{align}
\end{subequations} Due to the rotational symmetry we can choose the
eigenstates to be angular momentum eigenstates.

The eigenmodes $\mu_{m\lambda}$ of the closed cavity are labeled by
the angular momentum number $m$ and the radial quantum number
$\lambda$. They solve the Helmholtz equation
\begin{equation}
\label{eq:cav04}
\nabla^2 \mu_{m\lambda}(r,\phi) = - n^2k_{m\lambda}^2
\mu_{m\lambda}(r,\phi),
\end{equation}
and satisfy the Dirichlet condition $\mu_{m\lambda}(R_{-},\phi) =
0$. The normalized eigenstates are given in terms of Bessel functions
of the first kind,
\begin{equation}
\label{eq:mu04}
\mu_{m\lambda}(r,\phi) = \frac{\rme^{\rmi m\phi}
J_{m}(nk_{m\lambda}r)}{\sqrt{\pi}R J_{m+1}(x_{m\lambda})} ,
\end{equation}
with $m = 0,\pm 1, \pm2, \dots$ and $\lambda = 0,1,2,\dots$. The
eigenvalues are $k_{m\lambda} = x_{m\lambda}/ nR$ where
$x_{m\lambda}$ denotes the $\lambda$-th zero of $J_{m}(r)$. In a
similar fashion, one determines the eigenstates in the channel
region. They can be written in terms of Hankel functions,
\begin{equation}
\label{eq:nu04}
\nu_{m}(k,r,\phi) = \sqrt{\frac{k}{8\pi}} \rme^{\rmi m\phi} (H_{m}^{(2)}(kr) +
S_{m}(k) H_{m}^{(1)}(kr)),
\end{equation}
with the diagonal element of the scattering matrix
\begin{equation}
S_{m}(k) = - \frac{H_{m}^{\prime (2)}(kR)}{H_{m}^{\prime (1)}(kR)}.
\end{equation}
The channel states obey the Neumann condition
$\frac{\partial}{\partial r}\nu_{m} (k,r,\phi) \bigr\vert_{r=R_{+}} =
0$.

In addition to the internal frequencies $\omega_{m\lambda} =
ck_{m\lambda}$ we need the coupling amplitudes $\mathcal{W}$ and
$\mathcal{V}$ to fully determine the system-and-bath Hamiltonian.
Combination of Eq.~\eqref{eq:Lqp04} with the mode functions
(\ref{eq:mu04}), (\ref{eq:nu04}) and the definitions
(\ref{eq:coupw}) and (\ref{eq:coupv}) yields after a short
calculation
\begin{subequations}
\begin{align}
\mathcal{W}_{m\lambda,n}(k) &= \frac{-\rmi \sqrt{2 k_{m\lambda}}}{\pi kR 
H_{m}^{\prime (1)}(kR)}\, \delta_{m,n}, \\
\mathcal{V}_{m\lambda,n}(k) &= \frac{-\rmi \sqrt{2 k_{m\lambda}}}{\pi kR
H_{m}^{\prime (1)}(kR)}\, \delta_{m,-n}.
\end{align}
\end{subequations}
We note that the resonant amplitude $\mathcal{W}$ couples only cavity
and channel modes with the same angular momentum, while the
antiresonant amplitude $\mathcal{V}$ couples modes with opposite
angular momentum. This feature guarantees angular momentum
conservation: The resonant terms $\mathcal{W}_{\lambda m}
a_{m}^\dagger b_{m}$ and $\mathcal{W}_{\lambda m}^* a_{m}
b_{m}^\dagger $ account for the creation of a photon with angular
momentum $m$ and the simultaneous annihilation of a second photon with
the same angular momentum. By contrast, the antiresonant terms,
$\mathcal{V}_{\lambda m} a_{m\lambda}b_{-m}$ and $\mathcal{V}_{\lambda
m}^* a_{m\lambda}^{\dagger} b_{-m}^{\dagger}$ describe the
simultaneous annihilation or creation of two photons with opposite
value of angular momentum. In both cases, the total angular momentum
is conserved.

\begin{figure}[t]
  \begin{center} \includegraphics[scale=1]{./Figures/ddisk2d.eps}
    \caption{\label{fig:ddk2d} Real part of the radial component of
      the scattering wave function with angular momentum $m=13$ and
      $kR=10.5$ inside a dielectric disk with radius $R$ and index
      of refraction $n=3.3$. The solid gray curve is the exact
      result. The dotted (dashed) line follows from the mode
      expansion taking into account the first 11 (25) cavity
      modes.}
\end{center}
\end{figure}
We now turn to the electromagnetic field and the cavity resonances.
In the cavity region the exact scattering states $f_{m}(k)$ can be
represented in terms of the cavity modes $u_{m\lambda}$, with the
expansion coefficients
\begin{equation}
\label{eq:a04}
\alpha_{m\lambda,n}(k) = \sqrt{\frac{k}{2}} \frac{ k_{m\lambda} I_{mk}
J_{m}(nkR)}{ n (k^2 - k_{m\lambda}^2)} \delta_{m,n},
\end{equation}
where $I_{mk}$ is given by Eq.~\eqref{eq:I04}. It suffices to
compare the radial component of the scattering wave functions. In
Fig.~\ref{fig:ddk2d} we show the real part of that component for
angular momentum $m=13$ and $kR=10.5$. The solid gray line is the
exact result (\ref{eq:f04}) while the dotted and dashed lines
represent the system-and-bath expansion using the first 11 and 25
cavity modes, respectively.

The cavity resonances are obtained by solving the eigenvalue problem
for $L_{\text{eff}}(k)$. The calculation is presented in Appendix
\ref{app:appB} and yields the resonance condition
\begin{equation}
\label{eq:rescircle}
J_{m}(n k R) H_{m}^{\prime (1)}(kR) - n J_{m}^{\prime}(n k R)
H_{m}^{(1)}(kR) = 0,
\end{equation}
which is equivalent to the equation that determines the poles of
the $S$-matrix (cf.\ Eq.~\eqref{eq:s04}).

\begin{figure}[b]
  \begin{center} \includegraphics[scale=1]{./Figures/ddiskcgf.eps}
    \caption{\label{fig:ddcgf} Cavity gain factor as function of $kR$
      for a dielectric disk with radius $R$ and index of refraction
      $n=3.3$. The sharp resonances correspond to states with
      angular momentum $10< m <18$, very sharp resonances with $m
      \gg 18$ are not resolved.}
  \end{center}
\end{figure}
Substitution of Eq.~\eqref{eq:a04} into Eq.~\eqref{eq:rldos} yields
the integrated local density of states inside the dielectric disk
(Appendix \ref{app:appC}),
\begin{equation}
\label{eq:ldos04}
\int\limits_{\rm disk} d\bm{r} \rho(k,\bm{r}) =
\sum_{m=-\infty}^{+\infty} \frac{k R^2 |I_{mk}|^2}{8 n^2} \left(
J_{m}^{2}(nkR) - J_{m+1}(nkR)J_{m-1}(nkR) \right).
\end{equation}
Together with the free--space local density of states $\rho_{0}(k)
= k/2\pi$, we obtain the cavity gain factor
\begin{equation}
G^{c}(k) = \sum_{m}\frac{4 \left(J_{m}^{2}(nkR) - J_{m+1}(nkR)J_{m-1}(nkR)
\right)}{(\pi n k R)^2 \big| J_{m}(nkR) H_{m}^{\prime(1)} (kR) - n
  J_{m}^{\prime}(nkR) H_{m}^{(1)} (kR) \big|^2},
\end{equation}
where we used the explicit expression (\ref{eq:I04}) for the mode
amplitude $I_{mk}$. The cavity gain factor displays a set of very
sharp resonances (see Fig.~\ref{fig:ddcgf}), corresponding to states
with angular momentum $kR < m <nkR$, superimposed over a smooth
background due to broad resonances with $m < kR$ \cite{Hent02}.












