Unraveling Mirror Properties in Time-Delayed Quantum Feedback Scenarios
Fabian M. Faulstich\textsuperscript{1
2
Manuel Kraft\textsuperscript{1
Alexander Carmele\textsuperscript{1
% We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects in quantum optical platforms. % Our derivation is based on a microscopic Hamiltonian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quantized electromagnetic field, and a single two-level system in front of the dielectric. % The dielectric is modeled as a a system of identical two-state atoms. % The Heisenberg equation yields a system of describing differential operator equations, which we solve in the Weisskopf-Wigner limit. % Due to a finite round-trip time between emitter and dielectric, we yield delay differential operator equations. % Our derivation motivates and justifies the typical phenomenological assumed coupling element and allows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrors with gain dynamics. %
DraftArX.tex
\documentclass[amsmath,amssymb,aps,pra,preprint,groupedaddress,showpacs]{revtex4-1}

\usepackage{color,graphicx}



\begin{document}

\title{Unraveling Mirror Properties in Time-Delayed Quantum Feedback 
Scenarios}

\author{Fabian M. Faulstich\textsuperscript{1,2}}
\email[E-mail me at: ]{faulstich@math.tu-berlin.de}
\homepage[Visit: ]{http://page.math.tu-berlin.de/~faulstich/}
\author{Manuel Kraft\textsuperscript{1}}
\author{Alexander Carmele\textsuperscript{1}}
\affiliation{\textsuperscript{1}Institut f\"ur Theoretische Physik, Technische Universit\"at
Berlin, Hardenbergstra\ss e 36, 10623 Berlin, Germany\\ \textsuperscript{2}Institut f\"ur Mathematik, Technische 
Universit\"at Berlin, Stra\ss e des 17.\ Juni 136, 10623 Berlin, Germany}


\begin{abstract}
%
We derive in the Heisenberg picture a widely used phenomenological 
coupling element to treat feedback effects in quantum optical 
platforms. 
%
Our derivation is based on a microscopic Hamiltonian, which describes
the mirror-emitter dynamics based on a dielectric, a mediating fully 
quantized electromagnetic field, and a single two-level system 
in front of the dielectric.
%
The dielectric is modeled as a a system of identical two-state atoms.
%
The Heisenberg equation yields a system of describing differential operator 
equations, which we solve in the Weisskopf-Wigner limit.
%
Due to a finite round-trip time between emitter and dielectric, we yield 
delay differential operator equations.
%
Our derivation motivates and justifies the typical phenomenological
assumed coupling element and allows, furthermore, 
a generalization to a variety of mirrors, such as dissipative mirrors 
or mirrors with gain dynamics.
%
\end{abstract}

\maketitle

\section{Introduction} 
%
Feedback protocols are successfully applied to stabilize
periodic processes in classical and quantum mechanical systems \cite{wiseman2009quantum,scholl2008handbook}.
%
In semi-classical systems Pyragas control allows
to suppress relaxation oscillations in the switch-on dynamics of
a semiconductor laser \cite{scholl2016control}.
%
In those systems, feedback control is modelled via a Maxwell-based
treatment of the light-matter interaction.
%
The paradigm for a Maxwell theory based feedback control is the 
Lang-Kobayashi model, where part of the laser output is fed back
into the laser dynamics \cite{lang1980external,bardella2016dynamic}.
%
Instead of self-feedback, in quantum systems measurement-based setups
of feedback control are explored.
%
They allow to stabilize Fock states, theoretically predicted and already
experimentally demonstrated, e.g., in cQED systems \cite{sayrin2011real}.
%
Quantum feedback, however, is not restricted to a read-out and 
open quantum system approach, first experiments study the 
many-photon quantum limit of feedback 
\cite{albert2011observing,hopfmann2013nonlinear,reitzenstein2003pronounced}.
%

%
These successful experimental implementations of coherent 
feedback, or non-invasive self-feedback, increase the interest
for models, which allow predictions and interpretations of the 
observed feedback effects.
%
A variety of models have been proposed in the linear and 
nonlinear regime.
%
For example, a cavity-QED system is driven into the strong coupling
regime \cite{carmele2013single}, or a laser-driven two-level system is 
partially interacting with its own emission statistics, showing modified
Mollow triplet signatures \cite{pichler2016photonic,grimsmo2015time}.
%
Another promising route is feedback-induced parametric squeezing
\cite{kraft2016time,nemet2016enhanced},
and enhancing of network entanglement by phase-selective feedback
based state addressing \cite{hein2015entanglement,hein2014optical}.
%
All these models are based on a phenomenological coupling of the 
emitters to the radiation field, namely a momentum dependent 
coupling strength.
%
%


In this article, we justify the assumed coupling element.
%
In order to do this, we analyze the interaction of an 
initially excited two-state atom with a quantized electromagnetic 
field and a dielectric medium of $N$ two-state atoms. 
%
The discussion is restricted to the one-dimensional case where the electromagnetic field modes wave vectors $k$ are considered to be parallel to the $z$-axis.
%
This system is illustrated in Fig.~\ref{fig:Eins}.
%
As a system, we assume a semi-infinite one-dimensional 
waveguide \cite{hoi2015probing}. 
%
Material platform for such systems are, e.g., superconducting
transmission lines \cite{eichler2011experimental}, diamond nanowires
mediating between nitrogen-vacancy centers \cite{babinec2010diamond}, hollow optical fibers with cold atoms \cite{bajcsy2009efficient},
and photonic crystal waveguides coupling quantum dots \cite{laucht2012waveguide}, or plasmonic nanowires \cite{akimov2007generation}.  
%

%
The article is structured as follows. 
%
First, we describe briefly in the next section, Sec.~\ref{sec:HamEff}, the 
phenomenological model that is widely used in the literature \cite{kabuss2016unraveling,dorner2002laser}.
%
This model is our benchmark, and the following sections fulfill the 
purpose to give a microscopic justification for the applied quantum 
optical, momentum-dependent coupling.
%
In order to do this, we present a microscopical Hamiltonian in
Sec.~\ref{sec:MicHam}.
%
Starting from this Hamiltonian, we employ the Heisenberg equation 
approach and derive operator equations of motions for the dynamics 
of an atom near a plane dielectric interface, mathematically rigorously
in Sec.~\ref{sec:OpEq}.
%
This section ends with an effective operator equations in analogy
to the effective model of Sec. \ref{sec:HamEff}.
%
This is the main result of the paper.
%
In Sec.~\ref{sec:Concl}, we conclude the article and give a short outlook,
about possible extensions of the model.
%
\section{Phenomenological Model} \label{sec:HamEff}
%
In this section, we describe the effective and widely used phenomenological
model.
%
Commonly, the interaction between a dielectric and an electromagnetic field is 
simplified by assuming a hard-wall boundary at the position of the dielectric. 
%
This assumption yields the following Hamiltonian 
\cite{dorner2002laser,kabuss2016unraveling}:
%
\begin{equation}
H_{eff}/\hbar
=
\omega_e 
P^{\dag}P+\sum_k\sqrt{\frac{2cK_{0,0}}{L\pi}}\sin(kl)\left(P^{\dag}r_k+r_k^
{\dag}P\right)
+\sum_k\omega_k r_k^{\dag}r_k~,
\label{eq:HamEff}
\end{equation}
%
where $\omega_e$ is the transition frequency of the atom, $K_{0,0}$ is the coupling constant between atom and field, $L$ is the 
length of the quantized box,
%
$\omega_k$ is the frequency of mode $k$ and $l$ is the distance between the 
atom and the dielectric.
%
The ground and excited state of the initially excited atom at position $z_0$ 
are described by the operators $G=|0\rangle\langle0|$ and 
$E=|1\rangle\langle1|$, respectively.
% 
Its excitation (resp. de-excitation) dynamics is denoted by the raising (resp. lowering) operator $P^{\dag}=|2\rangle\langle1|$ (resp. $P=|1\rangle\langle2|$).
%
The $k$-th mode of the electromagnetic field is described by the creation
(resp. annihilation) operator $r^{\dag}_k$ (resp. $r^{}_k$).
%
Note, the dynamics of the dielectric are here fully represented by the 
momentum-dependent coupling element $ \sin(kl) $ and do not appear
explicitly in this effective Hamiltonian.
%
\begin{figure}[h!]
\centering
\includegraphics{Im1.jpg}
\caption{Excited atom located in a distance $l$ to the dielectric medium.}
\label{fig:Eins}
\end{figure}
 
%
This Hamiltonian introduces an interesting quantum feedback due to the structured continuum approach, namely by introducing a wavelength dependent coupling between the emitter and the photons of the reservoir.
%
Known from perturbation theory, this kind of frequency dependent coupling leads to numerous non-Markovian effects and renders Weisskopf-Wigner approaches impossible to treat quantum feedback.
%
Deriving the dynamics of the excitation operator via the Heisenberg equation of motion $ -i\hbar \partial_tO=[H_{eff},O] $, the feedback mechanism becomes apparent:
%
\begin{equation}
\begin{aligned}
\frac{d}{dt}P^{\dag}(t)
&=
-K_{0,0}P^{\dag}(t)
+K_{0,0}e^{-i\omega_e\tau}P^{\dag}(t-\tau)\Theta(t-\tau)(G(t)-E(t))\\
&\quad+\int_{\mathbb{R}}dk~i\sqrt{\frac{2cK_{0,0}L}{\pi}}\sin(kl)e^{
i(\omega_k-\omega_e)t}r_k^{\dag}(0)(G(t)-E(t))~,
\label{eq:DelayEff}
\end{aligned}
\end{equation}
%
where $\Theta(t)$ is the Heaviside step function and $\tau:=2l/c$ is the 
round-trip time 
\cite{kabuss2016unraveling,kabuss2015analytical,trautmann2016dissipation}.
%
To derive Eq.~\eqref{eq:DelayEff} a 
transformation into the rotating frame with respect to the transition
frequency $\omega_e$ was performed and without loss of generality $l\geq0$ 
was assumed. 
%
The first term in Eq.~\eqref{eq:DelayEff} describes the decay of the 
excited state into the reservoir proportional to the constant $ K_{0,0}$.
%
Additionally to this decay, after a round-trip time of $ \tau $, part of the initial signal is fed back into the dynamics of the atomic operator which is described by the second term in Eq.~\eqref{eq:DelayEff}.
%
Note, evaluating this equation in the many-excitation limit is a tedious
task due to the noise contributions of the third term in Eq.~\eqref{eq:DelayEff} 
and increasing reservoir-system entanglement spread. 
%
Possible strategies have been proposed in the Heisenberg picture 
\cite{kabuss2016unraveling}, in the quantum cascaded approach based on 
Liouvillian \cite{grimsmo2015time} or on the quantum stochastic Schr\"odinger 
equation \cite{pichler2016photonic,lu2017intensified}.\\
%
Subsequently, we are interested in the development of a more detailed picture
of the feedback mechanism than described by the Hamiltonian Eq.~\eqref{eq:HamEff}.
%
Therefore, we explicitly include the atomic dynamics of the dielectric 
induced by the field. 
%
The derivation is in analogy to the calculations of P. W. Milonni and R. J. 
Cook \cite{cook1987quantum}.
%
However, in contrast to Milonni et al., the equations are derived in the 
Heisenberg picture, to simplify the many-excitation limit.
%
This is rendered possible by treating the quantum noise contributions
explicitly, which is beyond the scope of the model from Milonni et al.
%
Our results form the backbone for further and detailed investigations by 
expanding the proposed feedback mechanism to a wider family of mirrors 
(metallic, dielectric, active, passive) and to acknowledge possible
connections to the regime of quantum optomechanics 
\cite{sudhir2017appearance,naumann2014steady}.
%
In particular, in the limit of a continuously distributed dielectric 
we derive the proposed $ \sin(kl) $ coupling from a reflecting medium in
distance $ l $ justifying the phenomenological ansatz.
%

In the following, we will derive the polarization equation of motion
in Eq.~\eqref{eq:DelayEff} with a more microscopic model, and discuss 
thereby the limits of validity for the given implementation.
%

%
\section{Microscopic Approach to Model Quantum Feedback}\label{sec:MicHam}
%
To derive a more general formula including the dielectric properties, 
we use a microscopic approach describing quantum feedback, applying the 
calculation of Ref.~\cite{cook1987quantum} to the Heisenberg picture. 
%
The interaction Hamiltonian $H_I$ for the system illustrated in Fig. \ref{fig:Eins} is in rotating wave and dipole approximation given by:
%
\begin{equation}
\begin{aligned}
H_I&=
i\hbar\sum_{k\in\mathbb{N}} C_{k,0}\left(P^{\dag} r_ke^{ik z_0}-Pr_k^{\dag}e^{-ik z_0}\right)\\
&\quad+i\hbar \sum_{j=1}^N\sum_{k\in\mathbb{N}} C_{k,j}\left(\sigma_{2,1}^{(j)}r_ke^{ik z_j}-\sigma_{1,2}^{(j)}r_k^{\dag}e^{-ikz_j}\right)~,
\end{aligned}
\end{equation}
%
where
%
\begin{equation}
C_{k,j}=\frac{\mu_j }{\hbar} \left(   \frac{2\pi \hbar\omega_k} {AL}  \right)^{\frac{1}{2}}
\label{eq:CouplingElement}
\end{equation}
%
is the frequency dependent coupling element in the light-matter interaction. 
%
Here, $A$ is an effective area, $L$ is the length along the $z$-axis of the 
quantized box and $\mu_j$ is the magnitude of the transition dipole moment of 
each two-state atom.
%
The electronic system is described via
$\sigma_{1,1}^{(j)}=|1\rangle_{j \, j} \langle1|$ 
(resp. $\sigma_{2,2}^{(j)}=|2\rangle_{j \, j} \langle2|$) 
denoting the operator of the ground state (resp. excited state)
dynamics of the $j$-th atom in the dielectric. 
%
Its excitation (resp. de-excitation) dynamics is described by the operator 
$\sigma_{2,1}^{(j)}=|2\rangle_{j \, j} \langle1|$ (resp. 
$\sigma_{1,2}^{(j)}=|1\rangle_{j \, j} \langle2|$). 
%
The transition frequency of the atom at $z_0$ is denoted $\omega_e$. 
%
We assume the dielectric to consist of identical atoms with resp. transition 
frequencies $\omega=\omega_1=...=\omega_N$. 
%
Further, we neglect contributions orthogonal to the polarization-density, 
denoted $\mathcal{P}^2_{\perp}$ \cite{richter2009two}.
%
As the dielectric consists of identical atoms the magnitude of the transition
dipole moment in the dielectric can be identically chosen to be $\mu_1$ which 
is not necessarily equal to $\mu_0$ the magnitude of the transition dipole 
moment of the initially excited atom in $z_0$.
%
The system Hamiltonian $H$ is given by the non-interacting Hamiltonian and
the interacting Hamiltonian $H_I$. Performing a unitary transformation of $H$ 
into the rotation frame with respect to $\omega_e$, the Heisenberg equation
yields the following system of differential operator equations
%
\begin{subequations}
\begin{align}
\frac{d}{dt}P^{\dag}
&=
\sum_{k\in\mathbb{N}}C_{k,0}e^{-ikz_0}
r_k^{\dag}
\left(G-E\right)\label{eq:SubeqHeisenberg1}\\
\frac{d}{dt}\sigma_{2,1}^{(j)}
&= i(\omega-\omega_e)\sigma_{2,1}^{(j)}
+\sum_{k\in\mathbb{N}}C_{k,1}e^{-ikz_j}r_k^{\dag}
\left(
\sigma_{1,1}^{(j)}-\sigma_{2,2}^{(j)}
\right)\quad,j\in\{1,...,N\}\label{eq:SubeqHeisenberg2}\\
\frac{d}{dt}r_k^{\dag}
&=
i(\omega_k-\omega_e)r_k^{\dag}
-C_{k,0}e^{ikz_0}P^{\dag}
-\sum_{j=1}^NC_{k,1}
 e^{ik z_j}\sigma_{2,1}^{(j)}\quad, k\in\mathbb{N}~.\label{eq:SubeqHeisenberg3}
 \end{align}
\end{subequations}
%
We emphasize that we have already applied the rotating wave approximation. 
%
Hence, we restrict the dynamics of the electromagnetic field to be 
quasi-resonant with the atomic transition frequency $\omega_e$ and, thus, 
the intensity to be sufficiently low. 
%
In consequence, only certain refraction and reflection coefficients are 
rendered possible, i.e., for other material and included susceptibilities
\cite{theuerholz2013influence}, the 
operator dynamics needs to be generalized to the non-rotating wave 
regime \cite{richter2009two}.
%
Our goal, however, is mainly to derive a $ \sin(kl) $ kind of coupling and 
for this purpose alone, we can keep this set of equations of motion as they 
already include the desired feedback mechanism.
%
\section{Microscopic Theory of an Atom Near a Plane Dielectric Interface in
the Heisenberg Picture} \label{sec:OpEq}
%
Having described the model in the previous section, we aim to deduce a valid 
feedback equation.
%
We start by formally eliminating $r_k^{\dag}(t)$ in Eqs.~\eqref{eq:SubeqHeisenberg1} and \eqref{eq:SubeqHeisenberg2}.
%
This is achieved by applying Duhamel's formula
%
\cite{john1982partial}
%
to Eq.~\eqref{eq:SubeqHeisenberg3} and substitute the obtained solution in Eqs.~\eqref{eq:SubeqHeisenberg1} and \eqref{eq:SubeqHeisenberg2}. 
%
For $r_k^{\dag}(t)$ we obtain:
%
\begin{equation}
\begin{aligned}
r_k^{\dag}(t)
&=
e^{i(\omega_k-\omega_e)t}r_k^{\dag}(0)
-\int_0^tdt'~e^{-i(\omega_k-\omega_e)(t'-t)}C_{k,0}e^{ik z_0}P^{\dag}(t')\\
&\quad-\sum_{j=1}^N
\int_0^tdt'~e^{-i(\omega_k-\omega_e)(t'-t)}C_{k,1}e^{ik 
z_j}\sigma_{2,1}^{(j)}(t')~,
\end{aligned}
\label{eq:HeisenBergBildDuhamel}
\end{equation}
%
which implies the differential operator equations
%
\begin{equation}
\begin{aligned}
\frac{d}{dt}P^{\dag}(t)
&=
-\int_0^tdt'~\sum_{k\in\mathbb{N}}C_{k,0}^2e^{-i(\omega_k-\omega_e)(t'-t)}P^{\dag}(t')\left(
G-E
\right)\\
&\quad-\sum_{j=1}^N
\int_0^tdt'~\sum_{k\in\mathbb{N}}C_{k,0}C_{k,1}e^{-ik(z_0-z_j)}e^{-i(\omega_k-\omega_e)(t'-t)}\sigma_{2,1}^{(j)}(t')\left(
G-E
\right)\\
&\quad+
\sum_{k\in\mathbb{N}}C_{k,0}e^{-ik z_0}
e^{i(\omega_k-\omega_e)t}r_k^{\dag}(0)
\left(
G-E
\right)\\
\end{aligned}
\end{equation}
%
and
%
\begin{equation}
\begin{aligned}
\frac{d}{dt}\sigma_{2,1}^{(j)}(t)
&= 
i(\omega-\omega_e)\sigma_{2,1}^{(j)}(t)
+\sum_{k\in\mathbb{N}}C_{k,1}e^{-ik z_j}
e^{i(\omega_k-\omega_e)t}r_k^{\dag}(0)
\left(
\sigma_{1,1}^{(j)}(t)-\sigma_{2,2}^{(j)}(t)
\right)\\
& -\int_0^tdt'~\sum_{k\in\mathbb{N}}C_{k,0}C_{k,1}e^{ik z_0-ik z_j}e^{-i(\omega_k-\omega_e)(t'-t)}P^{\dag}(t')\left(
\sigma_{1,1}^{(j)}(t)-\sigma_{2,2}^{(j)}(t)
\right)\\
&-\sum_{J=1}^N
\int_0^tdt'~\sum_{k\in\mathbb{N}}C_{k,1}^2e^{ik( z_J- z_j)}e^{-i(\omega_k-\omega_e)(t'-t)}\sigma_{2,1}^{(J)}(t')\left(
\sigma_{1,1}^{(j)}(t)-\sigma_{2,2}^{(j)}(t)
\right)
\end{aligned}
\end{equation}
%
for $j\in\{1,...,N\}$. 
%
We emphasize that up to this point, no further approximation have been made.
%
To solve this system, we use two approximations. 
%
We start with the narrow-band approximation, which states that the emission 
spectrum is centered around the atomic transition frequency $\omega_e$. 
%
Consequently, we can restrict the following analysis on a frequency interval
$[\omega_e-\upsilon,\omega_e+\upsilon]$ on which the variation of the 
coupling constants is chosen to be small. 
%
%
Therefore, the dependency of $C_{k,j}$ on the frequency $\omega_k$ is 
negligible. 
%
We yield the vacuum field amplitude with:
%
\begin{equation}
C_{k,j}\approx \frac{\mu_j }{\hbar} \left(   \frac{2\pi \hbar\omega_e} {AL}  \right)^{\frac{1}{2}}=:C_{0,j}~,
\end{equation}
%
for any $j\in\{0,1\}$. 
%
This approximation is within the range of the previously used rotating wave
approximation and therefore does not contradict previous assumptions to the 
system.
%
Further, we pass to the Weisskopf-Wigner approximation. 
%
Here, two assumptions are made. 
%
First, the modes of the field are closely spaced in frequency. 
%
Hence, we will integrate over the frequencies instead of summing.
%
Second, the expectation value of the integrand oscillates rapidly for very
small times $t'\ll t$. 
%
Therefore, there is no significant contribution to the value of the integral. 
%
We derive:
%
\begin{equation}
\begin{aligned}
\int_0^tdt'~\sum_{k\in\mathbb{N}}C_{0,0}^2 e^{-i(\omega_k-\omega_e)(t'-t)}f_1(t')
\longrightarrow
\frac{2\pi\mu_0^2\omega_e} {A\hbar c}\langle\delta\circ g,f_1\rangle_{[0,t]}
\end{aligned}
\end{equation}
%
and
%
\begin{equation}
\begin{aligned}
&\int_0^tdt'~\sum_{k\in\mathbb{N}}C_{0,j}C_{0,l} e^{-ik Z}e^{-i(\omega_k-\omega_e)(t'-t)}f_2(t')
\\
&\longrightarrow
\frac{\pi\mu_j\mu_l\omega_e} {A\hbar c}\left(
\langle \text{exp}({i\omega_e g})\delta\circ h_1,f_2\rangle_{[0,t]}+\langle \text{exp}({i\omega_e g})\delta\circ h_2,f_2\rangle_{[0,t]}
\right)~.
\end{aligned}
\end{equation}
%
In the above limits $g(t'):=t-t'$, $h_1(t'):=t+Z/c-t'$, $h_2(t'):=t-Z/c-t' $
and $f_{1,2}$ are arbitrary but sufficiently smooth functions. 
%
Here, we have introduced the constant $Z\in\mathbb{R}$ which later will be 
replaced by differences of the particle  positions along the $Z$-axis, i.e.,
$z_i-z_j$ for $i.j\in\{1,...,N\}$. 
%
We further used the standard notation where $\langle\cdot , \cdot
\rangle_{[0,t]}$ denotes the $L^2$-scalar product restricted on the domain 
$[0,t]$. 
%
For the sake of simplicity, we used calculus notation in the dual pairing,
e.g., $\text{exp}({i\omega_e g})\delta\circ h_1$ describes the composition of 
$\delta$ with $h_1$ multiplied by $\text{exp}({i\omega_e g})$.

The above notation is used to emphasize that the delta-distribution is a 
functional generated by the Dirac measure. 
%
Hence, the evaluation in the point $t'=0$ is only possible for a domain in which zero is an inner point. 
%
As this is not the case for the given domain the dual pairing of the delta
distribution and the respective functions are not well-defined.
%
Expanding the respective functions by the Heaviside step function yields that
the result of the dual pairing can be multiplied by any constant 
$\alpha\in[0,1]$ and is therefore not unique. 
%
However, the only choice ensuring the commutator relation to hold for any $t$
is $\alpha=1/2$ which 
therefore will be used subsequently. 
%
For further discussion we refer the reader to Appendix \ref{App: delta}.
%
Setting $f_1(t')=P^{\dag}(t')(G(t)-E(t))$ and 
$f_2(t')=\sigma_{2,1}^{(j)}(t')(G(t)-E(t))$ (respectively 
$f_2(t')=\sigma_{2,1}^{(j)}(t')(\sigma_{1,1}^{(j)}(t)-\sigma_{2,2}^{(j)}(t))$ 
and $f_2(t')=P^{\dag}(t')(\sigma_{1,1}^{(j)}(t)-\sigma_{2,2}^{(j)}(t))$) for 
$j\in\{1,...,N\}$, we deduce the following system of delay differential 
operator equations:
%
\begin{subequations}
\begin{align}
\begin{split}
\frac{d}{dt}P^{\dag}(t)
&=
-K_{0,0}P^{\dag}(t)\\
&\quad-\sum_{j=1}^NK_{0,1}e^{-ik_0l_j}\sigma_{2,1}^{(j)}(t-l_j/c)\left(
G(t)-E(t)
\right)\Theta(t-l_j/c)\\
&\quad+
K_{0,0}\Delta B ^{\dag}(0,0,t)\left(
G(t)-E(t)
\right)\end{split}\label{eq:HeisenbergbildOperatorDGLAtomX}\\
\begin{split}
\frac{d}{dt}\sigma_{2,1}^{(j)}(t)
&=-\left(i(\omega_e-\omega)+K_{1,1}\right)\sigma_{2,1}^{(j)}(t)\\
&\quad-
K_{0,1}e^{-ik_0l_{j}}P^{\dag}(t-l_{j}/c)
\left(
\sigma_{1,1}^{(j)}(t)-\sigma_{2,2}^{(j)}(t)
\right)\Theta(t-l_{j}/c)\\
&\quad-
\sum_{J\in\{1,...,N\}\setminus\{j\}}K_{1,1}e^{-ik_0l_{j,J}}\sigma_{2,1}^{(J)}(t-l_{j,J}/c)\left(
\sigma_{1,1}^{(j)}(t)-\sigma_{2,2}^{(j)}(t)
\right)~\Theta(t-l_{j,J}/c)\\
&\quad+
K_{1,1} \Delta B^{\dag}(1,j,t)\left(
\sigma_{1,1}^{(j)}(t)-\sigma_{2,2}^{(j)}(t)
\right)~,
\end{split}\label{eq:HeisenbergBildOperatorsigmaJ2}
\end{align}
\label{eq:SystemVirtualPhotonExch}
\end{subequations}
%
where $k_0=\omega_e /c$, $l_{j,J}:=|z_j-z_J|$, with the special case $l_{0,j}=:l_j$, ${K_{i,j}:=\pi\mu_i\mu_j\omega_e /(\hbar Ac)}$ and
%
\begin{equation}
\Delta B^{\dag} (l,j,t)
:=\left(\frac{LA\hbar}{2\pi^3\omega_e\mu_l^2}\right)^{\frac{1}{2}}\int_{-\infty}^{\infty} d\omega'~ e^{i(\omega'z_j/c-(\omega_e-\omega')t)}
r_{k'}^{\dag}(0)~.
\end{equation}
%
Eqs.~\eqref{eq:SystemVirtualPhotonExch} explicitly expose the delay effect 
in the atom-atom coupling via a mediating electromagnetic field.
%
Subsequently, we use the standard notation $\tau_j:=2l_j/c$ and $\tau := 2l/c$.

Further, we consider a dielectric in which the scattered field is small
compared to the incident field on the scatterer. 
%
This level of treatment is consistent with the Born approximation, which 
neglects the interaction of the atoms within the dielectric via photon 
exchange.
%
Assuming that the dielectric and the initially excited atom 
are off-resonant yields that all atoms in the dielectric remain in their
ground state.
%
This and $K_{1,1}\ll |\omega-\omega_e|$ implies
%
\begin{equation}
\begin{aligned}
\frac{d}{dt}\sigma_{2,1}^{(j)}(t)
&=-i(\omega_e-\omega)\sigma_{2,1}^{(j)}(t)-
K_{0,1}e^{-i\omega_e\tau_{j}/2}P^{\dag}(t-\tau_j/2)
\Theta(t-\tau_j/2)+
K_{1,1} \Delta B^{\dag}(j,t)
\end{aligned}
\end{equation}
%
where Duhamel's formula yields the solution
%
\begin{equation}
\begin{aligned}
\sigma_{2,1}^{(j)}(t)
&=
e^{-i(\omega_e-\omega)t}\sigma_{2,1}^{(j)}(0)-
K_{0,1}e^{-i\omega_e\tau_{j}/2}\int_0^tdt'e^{i(\omega_e-\omega)(t'-t)}
P^{\dag}(t'-\tau_j/2)~\Theta(t'-\tau_j/2)\\
&\quad+\int_0^tdt'e^{i(\omega_e-\omega)(t'-t)}
K_{1,1} \Delta B^{\dag}(j,t')~.
\end{aligned}
\end{equation}
%
The following adiabatic approximation is obtained by partial integration and
using that the expectation value of the integrand oscillates rapidly, 
%
as assumed in the Weisskopf-Wigner approximation. 
%
We find:
%
\begin{equation}
\begin{aligned}
\sigma_{2,1}^{(j)}(t)
&=
e^{-i(\omega_e-\omega)t}\sigma_{2,1}^{(j)}(0)+
i\frac{K_{0,1}}{(\omega_e-\omega)}
e^{-i\omega_e\tau_{j}/2}P^{\dag}(t-\tau_j/2)\Theta(t-\tau_j/2)\\
&\quad-i\frac{K_{0,1}}{(\omega_e-\omega)}
e^{-i\omega_e\tau_{j}/2}e^{-i(\omega_e-\omega)(t-\tau_j/2)}P^{\dag}(0)\Theta(t-\tau_j/2)\\
&\quad+\int_0^tdt'e^{-i(\omega_e-\omega)(t'-t)}
K_{1,1} \Delta B^{\dag}(j,t')~.
\end{aligned}
\end{equation}
%
Eliminating the dependency of $\sigma_{2,1}^{(j)}$ in 
Eq.~\eqref{eq:HeisenbergbildOperatorDGLAtomX} by substituting the 
above solution we obtain:
%
\begin{subequations}
\begin{align}
&\frac{d}{dt}P^{\dag}(t) =\notag \\ 
&\quad-K_{0,0}P^{\dag}(t)-
i\frac{K_{0,1}^2}{(\omega_e-\omega)}\sum_{j=1}^Ne^{-i\omega_e\tau_{j}}
P^{\dag}(t-\tau_j)
\left(
G(t)-E(t)
\right)\Theta(t-\tau_j)\label{eq:ErsteZeil}\\
&\quad-
K_{0,1}\sum_{j=1}^Ne^{-i\omega_e\tau_{j}/2}e^{-i(\omega_e-\omega)(t-\tau_j/2)}\sigma_{2,1}^{(j)}(0)\left(
G(t)-E(t)
\right)\Theta(t-\tau_j/2)\label{eq:ZweiteZeil}\\
&\quad+
i\frac{K_{0,1}^2}{(\omega_e-\omega)}\sum_{j=1}^Ne^{-i\omega_e\tau_{j}}
e^{-i(\omega_e-\omega)(t-\tau_j)}P^{\dag}(0)
\left(
G(t)-E(t)
\right)\Theta(t-\tau_j)\label{eq:DritteZeil}\\
&\quad-
K_{0,1}K_{1,1}\sum_{j=1}^Ne^{-i\omega_e\tau_{j}/2}\notag\\
&\quad\quad\times\int_0^{t-\tau_j/2}dt'e^{-i(\omega_e-\omega)(t'-t+\tau_j/2)}
\Delta B^{\dag}(j,t')\left(
G(t)-E(t)
\right)\Theta(t-\tau_j/2)\label{eq:VierteZeil}\\
&\quad+
K_{0,0}\Delta B^{\dag} (1,0,t)\left(
G(t)-E(t)\label{eq:ZFuenftZeil}
\right)~.
\end{align}
\label{eq:GLEICHUNG}
\end{subequations}
%
This delay differential operator equation describes the effect of the dielectric on the polarization of the atom outside the dielectric. 
%
It is the central equation used to describe occupation expectation
values of the initially excited atom, i.e., the system of interest. 
%
We emphasize that the influence of quantum noise terms like Eq.~ \eqref{eq:VierteZeil} makes this equation difficult to handle.\\
%
We now deduce a formula that describes an idealized mirror system 
similar to Eq.~\eqref{eq:DelayEff}.
%
Analogously to the calculations in \cite{cook1987quantum},
we pass to the limit of a continuously distributed dielectric. 
%
This will simplify the system by eliminating the sums as we are using the
Weisskopf-Wigner approximation. 
%
Assuming that the dielectric contains $NA~dz$ identical atoms in the slice $[z,z+dz]$ yields
%
\begin{equation}
\sum_{j=1}^Ne^{-i\omega_e\tau_j}P^{\dag}(t-\tau_j)\Theta(t-\tau_j)
\rightarrow
-i\frac{NA}{2k_0}P^{\dag}(t-\tau)\Theta(t-\tau)e^{-i\omega_e\tau}~,
\end{equation}
%
where a coordinate system was chosen such that $z_0=0$. 
%
We now restrict the system Eq.~\eqref{eq:GLEICHUNG} by only taking terms that scale linearly in $K$ into account. 
%
Hence, we neglect \eqref{eq:ZweiteZeil}, \eqref{eq:DritteZeil} and \eqref{eq:VierteZeil}. 
%
The remaining differential equation is given by
%
\begin{equation}
\begin{aligned}
\frac{d}{dt}P^{\dag}(t)
&=
-K_{0,0}P^{\dag}(t)+
K_{0,0}\Delta B^{\dag} (1,0,t)\left(
G(t)-E(t)
\right)\\
&\quad+\frac{K_{0,1}^2}{(\omega-\omega_e)}
\frac{NA}{2k_0}e^{-i\omega_e\tau}P^{\dag}(t-\tau)\Theta(t-\tau)
\left(
G(t)-E(t)
\right)~.\\
\end{aligned}
\label{eq:FinalEQ}
\end{equation}
%
This delay differential operator equation is indeed similar to Eq.~\eqref{eq:DelayEff}. 
%
To give a more explicit similarity, we identify the Fresnel 
reflection coefficient in the rotating wave approximation 
(see Appendix \ref{App: A}) in Eq.~\eqref{eq:FinalEQ}.
%
Since $K_{0,1}^2=K_{1,1}K_{0,0}$, we observe
%
\begin{equation}
\begin{aligned}
\frac{K_{0,1}^2}{(\omega-\omega_e)}\frac{NA}{2k_0}
&=K_{0,0}
\frac{N\pi\mu_1^2}{2\hbar(\omega-\omega_e)}~.
\end{aligned}
\end{equation}
%
The refraction index of a dielectric of $N$ two-state atoms per unit volume, each of transition frequency $\omega$ and transition dipole moment $\mu_1$ without local field effects, in the rotating wave approximation can be characterized by 
%
\begin{equation}
n^2(\omega_e)-1\approx\text{Re}(\chi(\omega_e))
\approx
\frac{N\pi\mu^2_1}{\hbar(\omega-\omega_e)}~.
\end{equation}
%
For a more detailed derivation see Appendix \ref{App: A}. 
%
We emphasize that for the deduction of primary Eq.~\eqref{eq:DelayEff} a 
negligible absorption of the dielectric is assumed. 
%
Therefore $\chi(\omega_e)=n^2(\omega_e)-1$ is taken as real valued.
%
The reflection coefficient according to the Fresnel formula for normal
incidence is given by $R=-(n-1)/(n+1)$. 
%
In the case ${\omega \approx \omega_e}$ and ${n(\omega_e) \approx 1}$ we obtain:
%
\begin{equation}
R=-\frac{N\pi\mu^2_1}{2\hbar(\omega-\omega_e)}~,
\end{equation}
%
which yields
%
\begin{equation}
\frac{K_{0,1}^2}{(\omega-\omega_e)}
\frac{NA}{2k_0}
=
-K_{0,0}R~.
\end{equation}
%
The quantum noise term can be written as
%
\begin{equation}
\begin{aligned}
K_{0,0}\Delta B^{\dag} (1,0,t)
&=
cK_{0,0}\left(\frac{LA\hbar}{2\pi^3\omega_e\mu_1^2}\right)^{\frac{1}{2}}\int_{-\infty}^{\infty} dk~ i\sin(kl)e^{i(\omega_k-\omega_e)t}
r_{k}^{\dag}(0)~.
\end{aligned}
\end{equation}
%
Using the relation
%
\begin{equation}
\begin{aligned}
cK_{0,0}\left(\frac{LA\hbar}{2\pi^3\omega_e\mu^2}\right)^{\frac{1}{2}}
=
\sqrt{\frac{2cK_{0,0}}{\pi}}
\left(\frac{L}{4\pi}\right)^{\frac{1}{2}}
\frac{\mu_0}{\mu_1},
\end{aligned}
\end{equation}
%
we obtain the delay differential operator equation for the polarization
%
\begin{equation}
\begin{aligned}
\frac{d}{dt}P^{\dag}(t)
&=
-K_{0,0} P^{\dag}(t)-K_{0,0}Re^{-i\omega_e \tau}P^{\dag}(t-\tau)\Theta(t-\tau)
\left(
G(t)-E(t)
\right)\\
&\quad+
R_{\mathcal{N}}\int_{-\infty}^{\infty} dk~ i
\sqrt{\frac{2cK_{0,0}L}{\pi}}
\sin(kl)e^{i(\omega_k-\omega_e)t}
r_{k}^{\dag}(0)\left(
G(t)-E(t)
\right)~,
\end{aligned}
\label{eq:FINAL}
\end{equation}
%
where $R_{\mathcal{N}}:=\mu_0/(\mu_1\sqrt{4\pi})$.\\
%
This equation is similar to the Eq.~\eqref{eq:DelayEff} but differs in 
the appearance of the reflection coefficients $R$ and $R_{\mathcal{N}}$. 
%
As for perfect mirrors $R=-1$ holds, equation \eqref{eq:DelayEff} and
\eqref{eq:FINAL} are identical for perfect mirrors.
%
Further, we emphasize that Eq.~\eqref{eq:FINAL} only holds for $|R|\ll 1$. 
%
However, the results calculated using the model described 
in Sec.~\ref{sec:HamEff} can be generalized to higher reflectivities,
including counter-rotating contributions in the Hamiltonian. 
%
The following graph Fig.~\ref{Fig:2} depicts qualitatively the behavior of the expectation value $\langle E\rangle$ for different $R$ on $[0,2\tau)$.
%
Our derivation allows to choose a reflectivity, and we see the influence
of the mirror properties in the degree of feedback that is measurable
on the emitter's dynamics.
%
For $ R=-1$, the dielectric is a perfect mirror, and the usual 
results from \cite{kabuss2016unraveling,dorner2002laser} are rederived,
however, now within a microscopic model.
%
This is the main result of the paper.
%
Since now the mirror dynamics are explicitly included, we can also start to investigate
regimes, where the mirror is not a passive optical element anymore but
is for example driven.
%
This allows to include gain into the system.
%
Just by hand, we choose a $ R<-1 $ to visualize, how the feedback will
be changed due to gain.
%
We stress, that the gain dynamics is just indicated.
%
\begin{figure}[h!]
\centering
\includegraphics{myfile.jpg}
\caption{\label{Fig:2}Expectation value $\langle E \rangle=\langle P^{\dag}P \rangle$ of the atom outside the dielectric at $z_0$ for different $R$.
%
The values of $R$ increase in the direction indicated by the arrow.
%
$K_{0,0}=1.5$, $\tau = 3$ and $\omega_e=\pi/\tau$.}
\label{fig:Zwei}
\end{figure}
%
%
%
\section{Conclusion} \label{sec:Concl}
%
The presented microscopic approach to quantum feedback yields a delay 
differential operator equation for the polarization $P^{\dag}$ Eq.~
\eqref{eq:GLEICHUNG} including dielectric characteristics. 
%
This expands the analysis of quantum coherent time-delayed feedback to a 
wider class of mirrors, e.g., metallic, dielectric, active, passive etc. 
%
Using an equation of motion approach we provide the possibility to 
describe an excited reflecting dielectric under stimulated emission, i.e., a
description of quantum optical gain in mirror-emitter setups. 
%
In addition, the presented analysis gives access to a wider class of active quantum coherent feedback control where the intrinsic mirror properties are externally steered. 
%
This could have significance both in tailored control of external quantum emitters and multi-photon selective reflection properties. 
%
Furthermore, we have shown that the microscopic approach justifies the $\sin(kl)$ like coupling used in the effective Hamiltonian Eq.~\eqref{eq:HamEff}.
%
In conclusion, the microscopic approach is very promising for further investigations as it takes properties of the dielectric into account and does not contradict the qualitatively motivated model. 
%
\bibliographystyle{tfp}
\bibliography{lib}
%
%
\appendix
\section{Refraction Index} \label{App: A}
%
The microscopic approach to quantum feedback presented in this article makes extensive use of the rotating wave approximation.
%
Subsequently, we deduce the Fresnel coefficients in a framework where the approximation is valid.
%
We start with the general equality relating the refraction index $n$ and the susceptibility $\chi$ 
%
\begin{equation}
n^2-1=\chi~.
\end{equation}
%
Characterizing the 
%
susceptibility  
%
for a two-level system by using the dipole density, expressed for localized atomic levels yields
%
\begin{equation}
P(r,t)=\sqrt{\pi}\mu_1\langle \sigma_{1,2}\rangle N+\sqrt{\pi}\mu_1\langle \sigma_{2,1}\rangle N~.
\end{equation}
%
The density matrix element is described by
%
\begin{equation}
\frac{d}{dt}\langle \sigma_{l,m}\rangle
=
i\omega\langle \sigma_{l,m}\rangle-K_{1,1}-i\sum_{n\in\{1,2\}}\left(
\Omega^*_{l,n}\langle \sigma_{n,m}\rangle-\Omega_{m,n}\langle \sigma_{l,n}\rangle
\right)~,
\end{equation}
%
where $\Omega_{m,l}=\sqrt{\pi}\mu_1E(R,\omega_e)/\hbar$ is the corresponding matrix element of the 
%
Rabi frequency.
%
Since in linear optics the polarization dynamics is only linear in the field, we obtain
\begin{equation}
\frac{d}{dt}\langle \sigma_{l,m}\rangle
=
i\omega\langle \sigma_{l,m}\rangle-K_{1,1}-i\left(
\Omega^*_{l,m}\langle \sigma_{m,m}\rangle-\Omega_{m,l}\langle \sigma_{l,l}\rangle
\right)~.
\label{eq:sigmaFourier}
\end{equation}
%
Transforming to Fourier space, equation \eqref{eq:sigmaFourier} can be solved by
%
\begin{equation}
\langle \sigma_{l,m}\rangle
=
\frac{\sqrt{\pi}\mu_1E(R,\omega_e)}{\hbar}\frac{-i(\langle\sigma_{m,m}\rangle-\langle\sigma_{l,l}\rangle)}{-i(\omega+\omega_e)+K_{1,1}}~.
\end{equation}
%
This yields the dipole density
%
\begin{equation}
P(r,\omega_e)
=
\sum_{l,m\in\{1,2\}}
\sqrt{\pi}\mu_1\frac{\sqrt{\pi}\mu_1E(R,\omega_e)}{\hbar}\frac{-i(\langle\sigma_{m,m}\rangle-\langle\sigma_{l,l}\rangle)}{-i(\omega+\omega_e)+K_{1,1}}N~.
\end{equation}
%
We emphasize that for the sake of consistency we set $\varepsilon_0=1$.
%
As $P=\chi E$ we deduce
%
\begin{equation}
\begin{aligned}
\chi(\omega_e)
&=
N
\sum_{l,m\in\{1,2\}}
\frac{\pi\mu^2_1}{\hbar}\frac{(\langle\sigma_{m,m}\rangle-\langle\sigma_{l,l}\rangle)}{(\omega+\omega_e)-iK_{1,1}}\\
&=
\frac{N\pi\mu^2_1(\langle\sigma_{1,1}\rangle-\langle\sigma_{2,2}\rangle)}{\hbar}
\left(
\frac{(\omega+\omega_e)+iK_{1,1}}{(\omega+\omega_e)^2+K^2_{1,1}}
-
\frac{(\omega_e-\omega)-iK_{1,1}}{(\omega_e-\omega)^2+K^2_{1,1}}
\right)~.
\end{aligned}
\end{equation}
%
We neglect the term containing $(\omega+\omega_e)$ as it can not be resonant.
%
This yields
%
\begin{equation}
\chi(\omega_e)
=
\frac{N\pi\mu^2_1(\langle\sigma_{1,1}\rangle-\langle\sigma_{2,2}\rangle)((\omega-\omega_e)-iK_{1,1})}{\hbar((\omega-\omega_e)^2+K^2_{1,1})}~.
\label{eq:Chi}
\end{equation}
%
Considering an off-resonant setting, we can assume that the part of $\chi$ describing the absorption is close to zero.
%
This part is identified with the imaginary part of Eq.~\eqref{eq:Chi}.
%
Assuming $K_{1,1}\ll |\omega-\omega_e|$ we obtain
%
\begin{equation}
n^2(\omega_e)-1\approx\text{Re}(\chi(\omega_e))
\approx
\frac{N\pi\mu^2_1}{\hbar(\omega-\omega_e)}~.
\end{equation}
%
%
%
\section{Dual-Pairing of Delta-Distribution over Domians not Containing Zero} \label{App: delta} 
In this section we show how to compute the dual-pairing of the delta-distribution with functions over domains not containing zero in physical systems. 
%
The idea is to expand the function in the dual pairing with a Heaviside step function.
%
This yields the problem that the step function can be defined in zero with any number $\alpha\in[0,1]$ multiplied by the function in the dual pairing evaluated in zero.
%
We show that $\alpha$ has to be $\alpha=1/2$ for quantum optical systems, otherwise we would contradict well known commutator relations.\\
%
We start with the qualitatively justified Hamiltonian Eq.~\eqref{eq:HamEff} and deduce for $t\in[0,\tau)$ the operator
%
\begin{equation}
\begin{aligned}
P^{\dag}(t)
&=e^{i\omega_e t}e^{-\kappa t}P^{\dag}(0)+ig_0\int_{0}^tdt'e^{ \kappa(t'-t)}\int_{\mathbb{R}}dk~
e^{-i((\omega_e-\omega_k)t'-\omega_e t)}r^{\dag}_{k}(0)\Delta(t')~,
\end{aligned}
\end{equation}
%
where $\kappa=g^2_0\pi\alpha/c$ and $\Delta(t')=G(t')-E(t')$. 
%
We find a similar solution for $P$. 
%
Assuming a system with weak decay yields $\Delta(t')\approx -1$.
%
We then find
%
\begin{equation}
\,[P(t), P^{\dag}(t)]_+
=
e^{-2\kappa t}+\frac{g_0^2\pi }{2c\kappa}
\left(1-e^{ -2\kappa t}\right)~.
\end{equation}
%
As $[P(t), P^{\dag}(t)]_+=G+E=1$ we know that
%
\begin{equation}
\frac{g_0^2\pi }{2c\kappa}\overset{!}=1~.
\end{equation}
%
Inserting the definition of $\kappa$ yields
%
\begin{equation}
\frac{g_0^2\pi }{2c\kappa}
=\frac{cg_0^2\pi }{2cg^2_0\pi\alpha}
=\frac{  1}{2 \alpha}~.
\end{equation}
%
Hence, the only value of $\alpha\in[0,1]$ for which $[P(t), P^{\dag}(t)]_+=G+E=1$ is $\alpha=1/2$.

\end{document}


DraftArX.bbl
\begin{thebibliography}{10}
\providecommand{\url}[1]{\normalfont{#1}}
\providecommand{\urlprefix}{}

\bibitem{wiseman2009quantum}
Wiseman, H.M.; Milburn, G.J. \emph{Quantum measurement and control}; Cambridge
  university press, 2009.

\bibitem{scholl2008handbook}
Sch{\"o}ll, E.; Schuster, H.G. \emph{Handbook of chaos control}; John Wiley \&
  Sons, 2008.

\bibitem{scholl2016control}
Sch{\"o}ll, E.; Klapp, S.H.; H{\"o}vel, P. Control of self-organizing nonlinear
  systems, \emph{Understanding complex systems (}  \textbf{2016}.

\bibitem{lang1980external}
Lang, R.; Kobayashi, K. External optical feedback effects on semiconductor
  injection laser properties, \emph{IEEE journal of Quantum Electronics}
  \textbf{1980}, \emph{16}~(3), 347--355.

\bibitem{bardella2016dynamic}
Bardella, P.; Chow, W.; Montrosset, I. Dynamic simulations of integrated couped
  cavity lasers, In \emph{Numerical Simulation of Optoelectronic Devices
  (NUSOD), 2016 International Conference on}, IEEE, 2016; pp 11--12.

\bibitem{sayrin2011real}
Sayrin, C.; Dotsenko, I.; Zhou, X.; Peaudecerf, B.; Rybarczyk, T.; Gleyzes, S.;
  Rouchon, P.; Mirrahimi, M.; Amini, H.; Brune, M.; et~al. Real-time quantum
  feedback prepares and stabilizes photon number states, \emph{Nature}
  \textbf{2011}, \emph{477}~(7362), 73--77.

\bibitem{albert2011observing}
Albert, F.; Hopfmann, C.; Reitzenstein, S.; Schneider, C.; H{\"o}fling, S.;
  Worschech, L.; Kamp, M.; Kinzel, W.; Forchel, A.; Kanter, I. Observing chaos
  for quantum-dot microlasers with external feedback, \emph{Nature
  communications}  \textbf{2011}, \emph{2}, 366.

\bibitem{hopfmann2013nonlinear}
Hopfmann, C.; Albert, F.; Schneider, C.; H{\"o}fling, S.; Kamp, M.; Forchel,
  A.; Kanter, I.; Reitzenstein, S. Nonlinear emission characteristics of
  quantum dot--micropillar lasers in the presence of polarized optical
  feedback, \emph{New Journal of Physics}  \textbf{2013}, \emph{15}~(2),
  025030.

\bibitem{reitzenstein2003pronounced}
Reitzenstein, S.; Worschech, L.; Hartmann, P.; Forchel, A. Pronounced switching
  bistability in a feedback coupled nanoelectronic Y-branch switch,
  \emph{Applied physics letters}  \textbf{2003}, \emph{82}~(12), 1980--1982.

\bibitem{carmele2013single}
Carmele, A.; Kabuss, J.; Schulze, F.; Reitzenstein, S.; Knorr, A. Single photon
  delayed feedback: a way to stabilize intrinsic quantum cavity
  electrodynamics, \emph{Physical review letters}  \textbf{2013},
  \emph{110}~(1), 013601.

\bibitem{pichler2016photonic}
Pichler, H.; Zoller, P. Photonic Circuits with Time Delays and Quantum
  Feedback, \emph{Physical review letters}  \textbf{2016}, \emph{116}~(9),
  093601.

\bibitem{grimsmo2015time}
Grimsmo, A.L. Time-delayed quantum feedback control, \emph{Physical review
  letters}  \textbf{2015}, \emph{115}~(6), 060402.

\bibitem{kraft2016time}
Kraft, M.; Hein, S.M.; Lehnert, J.; Sch{\"o}ll, E.; Hughes, S.; Knorr, A.
  Time-delayed quantum coherent Pyragas feedback control of photon squeezing in
  a degenerate parametric oscillator, \emph{Physical Review A}  \textbf{2016},
  \emph{94}~(2), 023806.

\bibitem{nemet2016enhanced}
N{\'e}met, N.; Parkins, S. Enhanced optical squeezing from a degenerate
  parametric amplifier via time-delayed coherent feedback, \emph{Physical
  Review A}  \textbf{2016}, \emph{94}~(2), 023809.

\bibitem{hein2015entanglement}
Hein, S.M.; Schulze, F.; Carmele, A.; Knorr, A. Entanglement control in quantum
  networks by quantum-coherent time-delayed feedback, \emph{Physical Review A}
  \textbf{2015}, \emph{91}~(5), 052321.

\bibitem{hein2014optical}
Hein, S.M.; Schulze, F.; Carmele, A.; Knorr, A. Optical feedback-enhanced
  photon entanglement from a biexciton cascade, \emph{Physical review letters}
  \textbf{2014}, \emph{113}~(2), 027401.

\bibitem{hoi2015probing}
Hoi, I.C.; Kockum, A.; Tornberg, L.; Pourkabirian, A.; Johansson, G.; Delsing,
  P.; Wilson, C. Probing the quantum vacuum with an artificial atom in front of
  a mirror, \emph{Nature Physics}  \textbf{2015}, \emph{11}~(12), 1045--1049.

\bibitem{eichler2011experimental}
Eichler, C.; Bozyigit, D.; Lang, C.; Steffen, L.; Fink, J.; Wallraff, A.
  Experimental state tomography of itinerant single microwave photons,
  \emph{Physical Review Letters}  \textbf{2011}, \emph{106}~(22), 220503.

\bibitem{babinec2010diamond}
Babinec, T.M.; Hausmann, B.J.; Khan, M.; Zhang, Y.; Maze, J.R.; Hemmer, P.R.;
  Lon{\v{c}}ar, M. A diamond nanowire single-photon source, \emph{Nature
  nanotechnology}  \textbf{2010}, \emph{5}~(3), 195--199.

\bibitem{bajcsy2009efficient}
Bajcsy, M.; Hofferberth, S.; Balic, V.; Peyronel, T.; Hafezi, M.; Zibrov, A.S.;
  Vuletic, V.; Lukin, M.D. Efficient all-optical switching using slow light
  within a hollow fiber, \emph{Physical review letters}  \textbf{2009},
  \emph{102}~(20), 203902.

\bibitem{laucht2012waveguide}
Laucht, A.; P{\"u}tz, S.; G{\"u}nthner, T.; Hauke, N.; Saive, R.;
  Fr{\'e}d{\'e}rick, S.; Bichler, M.; Amann, M.C.; Holleitner, A.; Kaniber, M.;
  et~al. A waveguide-coupled on-chip single-photon source, \emph{Physical
  Review X}  \textbf{2012}, \emph{2}~(1), 011014.

\bibitem{akimov2007generation}
Akimov, A.; Mukherjee, A.; Yu, C.; Chang, D.; Zibrov, A.; Hemmer, P.; Park, H.;
  Lukin, M. Generation of single optical plasmons in metallic nanowires coupled
  to quantum dots, \emph{Nature}  \textbf{2007}, \emph{450}~(7168), 402--406.

\bibitem{kabuss2016unraveling}
Kabuss, J.; Katsch, F.; Knorr, A.; Carmele, A. Unraveling coherent quantum
  feedback for Pyragas control [Invited], \emph{JOSA B}  \textbf{2016},
  \emph{33}~(7), C10--C16.

\bibitem{dorner2002laser}
Dorner, U.; Zoller, P. Laser-driven atoms in half-cavities, \emph{Physical
  Review A}  \textbf{2002}, \emph{66}~(2), 023816.

\bibitem{kabuss2015analytical}
Kabuss, J.; Krimer, D.O.; Rotter, S.; Stannigel, K.; Knorr, A.; Carmele, A.
  Analytical study of quantum-feedback-enhanced Rabi oscillations,
  \emph{Physical Review A}  \textbf{2015}, \emph{92}~(5), 053801.

\bibitem{trautmann2016dissipation}
Trautmann, N.; Alber, G. Dissipation-enabled efficient excitation transfer from
  a single photon to a single quantum emitter, \emph{Physical Review A}
  \textbf{2016}, \emph{93}~(5), 053807.

\bibitem{lu2017intensified}
Lu, Y.; Naumann, N.L.; Cerrillo, J.; Zhao, Q.; Knorr, A.; Carmele, A.
  Intensified antibunching via feedback-induced quantum interference,
  \emph{arXiv preprint arXiv:1703.10028}  \textbf{2017}.

\bibitem{cook1987quantum}
Cook, R.; Milonni, P. Quantum theory of an atom near partially reflecting
  walls, \emph{Physical Review A}  \textbf{1987}, \emph{35}~(12), 5081.

\bibitem{sudhir2017appearance}
Sudhir, V.; Wilson, D.J.; Schilling, R.; Sch{\"u}tz, H.; Fedorov, S.A.;
  Ghadimi, A.H.; Nunnenkamp, A.; Kippenberg, T.J. Appearance and disappearance
  of quantum correlations in measurement-based feedback control of a mechanical
  oscillator, \emph{Physical Review X}  \textbf{2017}, \emph{7}~(1), 011001.

\bibitem{naumann2014steady}
Naumann, N.L.; Hein, S.M.; Knorr, A.; Kabuss, J. Steady-state control in an
  unstable optomechanical system, \emph{Physical Review A}  \textbf{2014},
  \emph{90}~(4), 043835.

\bibitem{richter2009two}
Richter, M.; Carmele, A.; Butscher, S.; B{\"u}cking, N.; Milde, F.; Kratzer,
  P.; Scheffler, M.; Knorr, A. Two-dimensional electron gases: Theory of
  ultrafast dynamics of electron-phonon interactions in graphene, surfaces, and
  quantum wells, \emph{Journal of Applied Physics}  \textbf{2009},
  \emph{105}~(12), 122409.

\bibitem{theuerholz2013influence}
Theuerholz, T.S.; Carmele, A.; Richter, M.; Knorr, A. Influence of F{\"o}rster
  interaction on light emission statistics in hybrid systems, \emph{Physical
  Review B}  \textbf{2013}, \emph{87}~(24), 245313.

\bibitem{john1982partial}
John, F. Partial differential equations, volume 1 of Applied Mathematical
  Sciences, 1982.

\end{thebibliography}
