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\begin{document}
\title{Spectral Compression of Narrowband Single Photons with a Resonant Cavity}
\author{Mathias~A.~Seidler}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Xi~Jie~Yeo}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551}

\author{Alessandro~Cer\`{e}}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Christian~Kurtsiefer}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551}
\email[]{christian.kurtsiefer@gmail.com}

\date{\today}
\begin{abstract}
We experimentally demonstrate a spectral compression scheme for heralded
single photons with narrow spectral bandwidth around 795\,nm,
generated through four-wave mixing in a cloud of cold
$^{87}$Rb atoms.
The scheme is based on an asymmetric cavity as a dispersion medium and a
simple binary phase modulator, and can be, in principle,  without any optical
losses. We observe a compression from 20.6\,MHz to less than 8\,MHz, almost
matching the corresponding atomic transition.
\end{abstract}

% insert suggested PACS numbers in braces on next line
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\maketitle
\textit{Introduction --}
Efficient atom-light interactions at the single quantum level is at 
the core of several proposals for storing, processing, and relaying quantum
information~\cite{Cirac1997,Briegel1998,Waks2009, Kimble2008}. 
Many of these schemes
require single ``flying'' photons to match the spectrum of atomic transitions~\cite{Wilk2007,Hammerer2010,Lukin2003,Steiner:17}.
Single photons can be emitted from trapped ions~\cite{Keller2004,Almendros2009}, atoms~\cite{Kuhn2002,Srivathsan:2013,park2019polarization} or solid-state systems~\cite{Kurtsiefer2000,Michler2000,Moreau2001}.
However, the spectral width of the generated photons may not always match the
spectral width of the receiving systems. Therefore, methods to engineer the
photon spectrum may be required.

The simplest method for this is to passively filter the spectrum of
bright broadband sources \cite{Meyer-Scott2017, Schuck2010},
with a sometimes significant reduction of brightness, making photon-atom
interaction experiments that require a high interaction
rate~\cite{Duan2001b,Blinov2004} difficult.
More advanced methods to manipulate the spectrum of single photon
sources to match that of 
atomic transitions include restricting the spectral mode of emitters with cavities~\cite{Moreau2001,Kuhn2002,McKeever2004,Wolfgramm:08}, or 
using electromagnetically induced transparency in atomic
ensembles~\cite{Eisaman2005,zhu2017bright} in the source mechanism altogether.
As spectral filtering or engineering of the photon generation mechanism may
not always be possible, it would be desirable to modify the spectrum of a given
photon source while maintaining the brightness.
To our knowledge, the only experiments to modify the photon spectrum of narrowband single photons use gradient echo quantum memories~\cite{Buchler2010,Sparkes2012}.
However, this was only demonstrated for photons with spectral bandwidths narrower than atomic absorption linewidths.

Here, we demonstrate an alternative technique that compresses the spectral bandwidth
of single photons with a spectral bandwidth a few times broader than the corresponding atomic absorption linewidth, while in principle, maintaining the photon rates.
The technique is based on the ideas of time-lenses invented for temporal
imaging~\cite{Kolner1989,Kolner1994}, where the temporal and spectral
characteristics of ultrafast electromagnetic
pulses~\cite{Lavoie2013,Karpinski2017,Li2017} are manipulated. It turns out
that single photon states can be manipulated in a similar way, complementing
the techniques for lossless temporal envelope manipulation of narrowband
single photon states demonstrated
in~\cite{Srivathsan2014,PhysRevLett.123.133602}.

Spectral compression of single photon wave packets is achieved in two
steps:  First, the wave
packet is spread out in time such that the width of its envelope is compatible
with a narrow spectrum; this can be done using a dispersive element that
spreads out different frequency components of the wave packet in time,
effectively generating a chirped wave packet. 
In the second step, a time-dependent phase shift is applied. 
This step changes the spectral energy distribution of the wave packet. 

Previous time-lens based spectral compression schemes were performed on
ultrashort pulses, using optical fibers or diffraction gratings as dispersive
elements~\cite{Lavoie2013,Karpinski2017,Li2017}. 
The suitability of a dispersive element for a spectral compression scheme is related to the spectral bandwidth of the optical pulse. 
The bandwidths of the ultrashort pulses used in previous time-lens based spectral compression schemes are typically on the order of $0.1\ldots1$\,THz, and the length of optical fibers used to generate a significant temporal broadening of these pulses are on the order of $0.1\ldots10$\,km.
However, photonic wave packets interacting with single emitters like atoms or molecules have spectral bandwidths on the order of MHz, 
which would require optical fibers on the order of $10^{8}$\,m to generate a suitable temporal broadening for spectral compression.
 Transmitting light through such a long fiber would not only
be impractical, but also prohibitively lossy.
Similarly, currently available gratings would not be able to significantly disperse photonic wave packets with bandwidths of a few MHz.
We overcome this problem by
using the dispersive properties of an optical cavity instead. While the
dispersion in optical cavities can be much larger,
the process then requires a different time-dependent phase shift in the second
step to complete the spectral compression process.

\begin{figure}
%    \includegraphics[scale=1]{figures/concept/concept.pdf}
    \includegraphics[scale=1]{figure1.pdf}
    \caption{\label{fig:concept}
      Concept of spectral compression. The top row shows temporal intensity
      profiles $\abs{\psi(t)}^{2}$ in various stages of the spectral
      compression, the bottom row the corresponding power spectra $\abs{\Psi(\omega)}^{2}$.
      The initial pulse is dispersed by a cavity, leading to a new temporal
      shape, but an unchanged spectrum. An electro-optical modulator (EOM) manipulates the phase
      $\phi^{\prime}(t)$ of the pulse which leads to a narrower spectrum.}
\end{figure}
\textit{Theory --} To understand the spectral compression scheme, we start with
an initial single photon wave packet, described by an envelope
$\left|\psi(t)\right|^2$ of its intensity in time, and its corresponding  power spectrum
$\left|\Psi(\omega;\, \omega_0, \Gamma_p)\right|^2$,
connected by the Fourier transform~$F$:
$\Psi(\omega)=F\left[\psi(t)\right]$. The nearly-monochromatic wave packet
shall be characterized by a central frequency~$\omega_0$ and a spectral width~$\Gamma_p$.
The spreading out of the wave packet in time is accomplished by reflection off
an asymmetric cavity, with an input/output coupler with a low transmission,
and a second high-reflective mirror, similar to the setup used in~\cite{Srivathsan:2014}.


If the losses in the cavity are negligible compared to the transmission of the
coupling mirror, and the 
cavity linewidth~$\Gamma_c$ and photon bandwidth~$\Gamma_p$ are much smaller
than free spectral range of the cavity,
the action of the cavity to a wave packet near its resonance~$\omega_c$ can be
described by a transfer function
\begin{equation}\label{eq:cavity_tf}
    C(\omega;\, \omega_c, \Gamma_c) \approx
    -\frac{\Gamma_c + i\: 2 (\omega-\omega_c)}
    {\Gamma_c - i\: 2(\omega-\omega_c)}
    \,,
\end{equation}
which modifies the incoming spectral wave packet $\Psi(\omega;\, \omega_0,
\Gamma_p)$ to a new one,
\begin{equation}\label{eq:Psi_dispersed}
    \Psi^{\prime}(\omega;\, \Delta\omega, \Gamma_c, \Gamma_p) =\:
    \Psi\left(\omega;\, \omega_0, \Gamma_p\right)\:C\left(\omega;\, \omega_c,\Gamma_c\right)\,,
\end{equation}
where~$\Delta\omega=\omega_0 - \omega_c$ is the detuning between the wave packet and the cavity resonance.
For a lossless cavity, this wave packet has the same power spectrum
as~$\Psi(\omega)$ because $\left| C(\omega;\, \omega_c, \Gamma_c)\right|^2 = 1$.
The temporal envelope of the reflected wave packet, obtained through the inverse Fourier transform~$F^{-1}$,
\begin{equation}\label{eq:psi_dispersed}
    \psi^{\prime}(t;\,\Delta\omega, \Gamma_c, \Gamma_p)
    =\:
    F^{-1}\left[\Psi^{\prime}(\omega;\,\Delta\omega, \Gamma_c, \Gamma_p)\right]\,,
\end{equation}
is now broader in time, and has acquired a time-dependent phase~
$\phi^{\prime}(t;\,\Delta\omega, \Gamma_p, \Gamma_c)$\,.

Similar to Fourier-transform limited pulses, where the time-bandwidth product is minimized by a frequency-independent spectral phase,
we can reduce the spectral bandwidth
of the heralded single photon by removing any time-dependent phase.
This is done by applying a time-dependent phase shift
\begin{equation}\label{eq:eom_dispersion}
    \phi_e(t)= -\phi^{\prime}(t;\;\Delta\omega, \Gamma_p, \Gamma_c)\,,
\end{equation}
resulting in the spectrally-compressed wave packet
\begin{equation}\label{eq:compressed_psi}
    \psi^{\prime\prime}(t;\;\Delta\omega, \Gamma_p, \Gamma_c) =
    \psi^{\prime}(t;\;\Delta\omega, \Gamma_p, \Gamma_c) \: e^{i \phi_e(t)}\,.
\end{equation}

To quantify the compression, we compare the spectral widths before and after
the compression obtained from the respective power spectrum $|\psi^{\prime\prime}(\omega;\;\Delta\omega, \Gamma_p, \Gamma_c)|^2$
obtained through a Fourier transform of Eq.~(\ref{eq:compressed_psi}).

\begin{figure}
 \centering
% \includegraphics{figures/experiment_setup/exp_setup.pdf}
 \includegraphics{figure2.pdf}
 \caption{\label{fig:schematic}
 Schematic setup for generation and spectral compression of heralded single photons. 
   D$_{S,I}$: single-photon detectors, EOM: electro-optical modulator, PBS: polarizing beam splitter, QWP: quarter-wave plate, IF: interference filter. Inset: energy level scheme for four-wave mixing in $^{87}$Rb.}
\end{figure}
We now consider the specific case of a heralded single photon emerging from an
atomic cascade decay, where we intend to compress the idler photon (see inset
of Fig.~\ref{fig:schematic}).
Detection of a signal photon projects the field in the idler mode into the heralded state
\begin{equation}\label{eq:heralded_photon}
    \psi(t) = \sqrt{\Gamma_p}\:e^{-\frac{\Gamma_p}{2}(t - t_0) }
    \Theta(t - t_0)
    % \: e^{-i\omega_0 t}
    \,,
\end{equation}
where~$t_0$ and $t$ are the detection times of the signal and idler photons, respectively.
The exponential decay with the constant~$\Gamma_p$ is a characteristic of the spontaneous process,
while
the Heaviside step function~$\Theta$ is a consequence of the well-defined time order of the cascade decay process. For simplicity, we set~$t_0=0$.

This temporal profile corresponds to a Lorentzian power spectrum for the idler
photons, and its bandwidth is described by the full-width half maximum $\Gamma_p$, 
which also corresponds to the spectral window containing 50\% of the total pulse energy.
However, the compressed spectrum
$|F^{-1}\left[\psi^{\prime\prime}(t)\right]|^2$ has multiple maxima, and is
distinctly different from distributions where the full-width half maximum
naturally quantifies the bandwidth.
Hence, we instead define bandwidth as the smallest spectral width containing 50\% of the total pulse energy,
as this definition of bandwidth is compatible for both a Lorentzian and a generic spectrum.

To obtain the optimal cavity parameters, we numerically minimize
the bandwidth of the compressed photon spectrum.
We find that the maximal compression is achieved by a resonant cavity $\Delta\omega=0$ with a bandwidth of $\Gamma_c \approx \Gamma_p/4$.
Under these conditions, the compressed single photon time envelope can be written as 
\begin{equation}\label{eq:comp_psi_exp}
    \psi^{\prime\prime}(t) =e^{-i \phi^{\prime}(t)} \:
    \sqrt{\Gamma_p}\,
    \frac{2\Gamma_c \,e^{-\frac{\Gamma_c}{2} t} -
    (\Gamma_p +\Gamma_c)\, e^{-\frac{\Gamma_p}{2} t}}
    {\Gamma_p -\Gamma_c}\,\Theta(t)\,,
\end{equation}
with a phase function
\begin{equation}\label{eq:phase_function}
    \phi^{\prime}(t) = \pi\,
    \Theta\left(t - 2
    \frac{\log \left(\frac{\Gamma_p +\Gamma_c}
        {2 \Gamma_c}\right)}
    {\Gamma_p -\Gamma_c}
    \right)\,.
\end{equation}
This is a step function changing the phase by $\pi$, with the transition occurring at the minimum of the dispersed photon's temporal intensity profile.
The narrowest bandwidth achievable with compression based on an asymmetric
cavity with this strategy is $\sim\!0.3\Gamma_p$; the temporal envelopes and
power spectra shown in  Fig.~\ref{fig:concept} correspond to this choice.

\textit{Experiment --}
Details of the actual experiment are shown in Fig.~\ref{fig:schematic}.
We generate the time-ordered photon pairs by
four-wave mixing in a cold ensemble of $^{87}$Rb atoms in a cascade level scheme~\cite{Srivathsan:2013}.
Pump beams at 780\,nm and 776\,nm excite atoms from the $5S_{1/2},F=2$ ground level to the $5D_{3/2},F=3$ level via a two-photon transition. The 762\,nm
(signal) and 795\,nm (idler) photon pairs emerge from a cascade decay back to
the ground level, and are coupled to single mode fibers.
Phase matching is ensured with
all four modes propagating collinearly in the same direction.
The two pump have a focus in the cloud with a beam waist of about 400\,$\mu$m. The 780\,nm pump is 55\,MHz blue-detuned from the $5S_{1/2},F=2$ to $5P_{3/2},F=3$ transition and has an optical power of 0.25\,mW.
The 776\,nm pump
has an optical power of 11.4\,mW, and is tuned such that the two-photon transition to the $5D_{3/2},F=3$ state is 5\,MHz blue-detuned.
When the excited atoms decay via the $5D_{1/2},F=2$ state back into the
initial ground state, photons with a wavelength of 762\,nm and 795\,nm photon are emitted~\cite{Srivathsan:2013}.

After suppressing residual pump light and separating signal and idler photons
into different modes, we collect them into single mode fibers. 
The 762\,nm signal photons are detected with an avalanche photo diode and
herald the presence of 795\,nm idler photons.
The time correlation between the detection in the signal and idler modes
(open %blue
circles in Fig.~\ref{fig:photong_g2s}) correspond to the
envelope~$|\psi(t)|^2$ of the intensity in time.

We measure the initial power spectrum of the wave packet (open circles in Fig.~\ref{fig:spectrum}) %blue dots,
by correlating it with a the photon rate transmitted through
a Fabry-P\'{e}rot cavity (FP) with linewidth~
$\Gamma_\text{FP}\approx2\pi\times2.6$\,MHz. 
The transmission is recorded at different detunings of the cavity from the
atomic resonance. 
The observed spectrum was observed to have a full width at half maximum of 20(2)\,MHz, wider
than the atomic line width of 6\,MHz due to collective emission effects in the cloud~\cite{Jen:2012, PhysRevA.98.023835}.

The 795\,nm idler photons are then coupled to the dispersion cavity, with a
coupling mirror of nominal reflectivity~
$R_1=0.97$, and a high reflector with~$R_2=0.9995$
separated by 10.1\,cm, corresponding to a free spectral range of
$1.48$\,GHz, and a measured linewidth
$\Gamma_c\approx2\pi\times7.3$\,MHz.
A Pound-Drever-Hall frequency lock keeps the cavity resonant 
to the central frequency of
the photons throughout the experiment.
The measured time envelope of the single photon wave packet after dispersion is
shown as filled %red
dots in Fig.~\ref{fig:photong_g2s}.

The spectral compression is completed by applying the temporal phase of Eq.~(\ref{eq:phase_function}), in the form of a phase switch
synchronized to the photon passage through a fiber connected 
electro-optical modulator (EOM). 
Since the idler photon is heralded, we use the detection of the signal photon
as a reference signal for triggering the phase switch after an appropriate
time delay.
Our dispersed photon is approximately 80\,ns long, which
corresponds to a spatial spread of 16\,m in a fiber with a refractive index of
1.5.
The phase modulator has an active length of 90\,mm, so at any instant only a
small part of the photon resides inside the EOM, and we are able to modulate
the two parts of the photon with different phases. The correct timing of the phase
modulation is ensured by measuring the length of the fibers and the
electric signal lines with a
timing uncertainty $<$0.5\,ns, on par with the electrical rise time of the phase
change signal. This order of timing uncertainty can be tolerated, since the majority
of second photon wave packet part (Fig.~\ref{fig:photong_g2s}, blue line from
15\,ns to 100\,ns) gets a phase shift of $\pi$. 
The phase flip is applied right after the first part of the dispersed photon exits the modulator, and 
the second part starts to propagate through it. This timing is indicated as dashed line in Fig.~\ref{fig:concept}.
We finally measure the compressed photon spectrum by again recording the photon transmission rate through the Fabry-P\'{e}rot cavity, shown as filled dots in Fig.~\ref{fig:spectrum}.

\begin{figure}
 \centering
% \includegraphics[width=1\columnwidth]{figures/photon_shapes_and_phase_correction/photon_shape.pdf}
 \includegraphics[width=\columnwidth]{figure3.pdf}
 \caption{\label{fig:photong_g2s}
   Detection time distribution for the heralded photon
  before (open blue circles) and after
  (filled red dots) the dispersion cavity.
  We fit an exponential decay Eq.~(\ref{eq:heralded_photon}) to the initial time correlation (blue solid line), from which we infer the photon bandwidth $\Gamma_p$. 
  The simulated temporal profile after the photon passed through the
  dispersion cavity, (red line, calculated from Eq.~(\ref{eq:Psi_dispersed}),
  matches our experimental data (filled dots) well. }

\end{figure}

\begin{figure}
 \centering
% \includegraphics[width=1\columnwidth]{figures/spectra/output.pdf}
 \includegraphics[width=1\columnwidth]{figure4.pdf}
 \caption{\label{fig:spectrum}
 Spectral profile of heralded photons before (blue) and after (red) spectral compression, obtained by measuring the photon transmission rate through the Fabry-P\'{e}rot cavity at different cavity detunings. 
The solid lines are calculated from Eq.~(\ref{eq:compressed_psi}), with $\Gamma_p$ inferred from the temporal envelope measurement of the photons from the source, and $\Gamma_c$ by experimentally characterizing the cavity bandwidth. Shaded areas cover 50\% of the total power for each spectrum.}
\end{figure}


To obtain an initial photon bandwidth $\Gamma_p$, 
we fit the decaying exponential term in Eq.~(\ref{eq:heralded_photon}) to the observed coincidence probability (open circles in Fig.~\ref{fig:photong_g2s}).
The solid red line in Fig.~\ref{fig:photong_g2s} corresponds to an expected temporal
profile of the photon after the dispersion cavity, calculated from Eq.~(\ref{eq:comp_psi_exp}), 
with an inferred photon bandwidth $\Gamma_p=2\pi\times20.6(2)$\,MHz obtained from the fit of the initial photon shape, 
and the cavity linewidth $\Gamma_c\approx2\pi\times7.3$\,MHz measured earlier. 
The observed temporal envelope after the dispersion cavity (full dots in Fig.~\ref{fig:photong_g2s}) agrees very well with the expected profile.

The measured spectral profiles before and after compression are shown in Fig.~\ref{fig:spectrum}.
The spectrum of the uncompressed photons (open circles) exhibits a dip around the central
frequency, which was also observed without the compression optics. 
We attribute this to reabsorption of the generated photons by the atomic cloud.
We model this spectrum $S(\omega)$ by considering two processes: 
First, we consider the spectrum $P(\omega)$ of the photon emitted by the atomic cloud 
which can be obtained by the product of
the spectrum of the photon produced by our FWM process $L(\omega; \Gamma_p) = \frac{2}{\pi}\,\frac{\Gamma_p}{4\,\omega^2 + \Gamma_p^2}$
and an absorption term describing the attenuation of the photon by our atomic cloud of optical density OD
\begin{equation}\label{eq:reabsorption}
        P(\omega; \text{OD}, \Gamma_p, \Gamma_{a}) =   A\,L(\omega; \Gamma_p)\,e^{-\text{OD}\frac{\Gamma_a^2}{4\omega^2 + \Gamma_a^2}}\,.
\end{equation}
The scaling factor $A$ is used to account for the detected coincidence rate, 
and $\Gamma_{a}$ is the spectral width of the absorption feature.
From our fit, we extract $\Gamma_{a}=2\pi\times 1.38(1.6)$\,MHz, which does not correspond to the absorption linewidth of the corresponding atomic line width ($2\pi\times5.7$\,MHz) of the $5S_{1/2}\rightarrow5P_{1/2}$ transition. 
Further work is necessary to understand this observation.
The inferred photon bandwidth $\Gamma_p =2\pi\times20.6(2)$\,MHz was determined from the photon coincidence time correlation (Fig.~\ref{fig:photong_g2s}). 
Second, we consider the effect of the Fabry-P\'{e}rot cavity used to sample the spectral profile by convolution the above result with $L'(\omega; \Gamma_{\text{FP}}) = \frac{\Gamma_{\text{FP}}^2}{4\,\omega^2 + \Gamma_{\text{FP}}^2}$ to model the observed spectrum:
\begin{equation}\label{eq:convolution}
    \begin{split}
        S(\omega) & = (P * L')(\omega).
    \end{split}
\end{equation}
We fit the model $S(\omega)$ (Fig.~\ref{fig:spectrum}, black line) to the experimental data corresponding to the spectral profile of the photon 
without sending a signal to the EOM used to compress the photon.
The model, without considering the attenuation of the atomic cloud, is given by $(A\,L * L')(\omega)$ (Fig.~\ref{fig:spectrum}, blue line).
This is provided as a reference for a Fourier-transform limited photon with an exponentially decaying envelope, as emitted by a single atom - the scenario examined in the theory section.

To apply the analysis in the theory section for predicting the spectrum of the compressed photon, 
we first rescale the power spectrum $|\Psi^{\prime\prime}(\omega)|^2$, calculated from Eq.~(\ref{eq:comp_psi_exp}), 
with $A$, which was extracted from the previous fit.
Then, we convolve this spectrum with $L'(\omega; \Gamma_{\text{FP}})$ to obtain the expected compressed photon spectrum $(A\,|\Psi^{\prime\prime}|^2 * L')(\omega)$.
Figure ~\ref{fig:spectrum} (red line) shows the modeled power spectrum slightly deviating from the measured result (red dots), 
exhibiting a lower peak coincidence rate where an absorption dip occurs in the uncompressed photon spectrum (blue dots).
We attribute this difference to the fact that our model does not fully account for the effects imposed by the atomic cloud.

\textit{Discussion --}
By definition, spectral compression reduces the width of a spectral
distribution, resulting in an increased photon rate/intensity at the central
frequency.
In our experiment, we observed a bandwidth of 20(2)\,MHz for the initial photon,
and 8(2)\,MHz for the compressed photon from spectrscopy using a 2.6\,MHz cavity.
This almost matches the natural D transition linewidth of 6\,MHz in $^{87}$Rb.
The maximal photon transmission through the spectroscopic cavity is increased
by a factor of 2.39(4), 
indicating a successful spectral compression of narrowband photons.

The compressions mechanism is, in principle, lossless since both cavity and phase modulators can have arbitrarily low losses. 
In our experiment, however, we observe
an overall transmission of 22\% through the compression optics;  
the dispersion optics alone (PBS, QWP, dispersion cavity) has a transmission
of 72\%, and the fiber-based EOM a transmission of 30\% including the fiber coupling. 

To compare the compression method with simple passive filtering,
we calculate the transmission $T$ of the 20\,MHz bandwidth photons produced by
our source through both bandwidth-limiting schemes, and an analyser cavity
with a bandwidth of 6\,MHz bandwidth and resonant with the central frequency
of the input photons to model an atomic absorption process corresponding to the
$5S_{1/2} \rightarrow$ $5P_{1/2}$ transition in $^{87}$Rb. With a lossless
compression system, we find $T=44\%$, while a resonant filter cavity
of the same bandwidth of 6\,MHz leads to $T=14\%$, illustrating the advantage
of the compression method.
By replacing the fiber-based EOM with a free-space EOM with an optical optical
transmission $>$95\%, the compression method would significantly surpass the
transmission of a passive filter.

Optimal spectral compression of a photon with bandwidth $\Gamma_{p}$ in
the cavity-based scheme is achieved if the dispersion cavity has a bandwidth of $0.25\,\Gamma_p$.
Since the amount of spectral compression is limited by the dispersion
mechanism, dispersion engineering of structured dielectric media~\cite{RevModPhys.78.455,Li_2018,Joannopoulos_2008} or multiple combined optical cavities may allow to further increase the spectral compression.
This method is not limited to the atomic system in our experiment -- it can be
adapted to a wide range of wavelengths and spectral widths, and therefore even
allow to 
match the spectral properties to different types of quantum systems e.g. in a hybrid quantum network~\cite{Kurizki3866}.


We thank Adrian Nugraha Utama for useful discussions on the theoretical
modeling and Jianwei Lee for his valuable input while writing this script.
This work was supported by the Ministry of Education in Singapore. 

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\bibitem [{\citenamefont {Briegel}\ \emph {et~al.}(1998)\citenamefont
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  \BibitemOpen
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\bibitem [{\citenamefont {Wilk}\ \emph {et~al.}(2007)\citenamefont {Wilk},
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  \BibitemOpen
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\bibitem [{\citenamefont {Hammerer}\ \emph {et~al.}(2010)\citenamefont
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\bibitem [{\citenamefont {Srivathsan}\ \emph {et~al.}(2013)\citenamefont
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  \BibitemOpen
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\bibitem [{\citenamefont {Kurtsiefer}\ \emph {et~al.}(2000)\citenamefont
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  \BibitemOpen
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\bibitem [{\citenamefont {Michler}\ \emph {et~al.}(2000)\citenamefont
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\bibitem [{\citenamefont {Moreau}\ \emph {et~al.}(2001)\citenamefont {Moreau},
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\bibitem [{\citenamefont {Meyer-Scott}\ \emph {et~al.}(2017)\citenamefont
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  \BibitemOpen
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  \citenamefont {Huwer}, \citenamefont {Mitchell}, \citenamefont {Hennrich},
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  \BibitemOpen
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  \BibitemOpen
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\bibitem [{\citenamefont {Blinov}\ \emph {et~al.}(2004)\citenamefont {Blinov},
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  \BibitemOpen
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\bibitem [{\citenamefont {McKeever}\ \emph {et~al.}(2004)\citenamefont
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  \BibitemOpen
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\bibitem [{\citenamefont {Wolfgramm}\ \emph {et~al.}(2008)\citenamefont
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  \BibitemOpen
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\bibitem [{\citenamefont {Eisaman}\ \emph {et~al.}(2005)\citenamefont
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  \BibitemOpen
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  \BibitemOpen
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  {\bibfield  {journal} {\bibinfo  {journal} {Applied Physics Letters}\
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\bibitem [{\citenamefont {Buchler}\ \emph {et~al.}(2010)\citenamefont
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  \BibitemOpen
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\bibitem [{\citenamefont {Sparkes}\ \emph {et~al.}(2012)\citenamefont
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  \BibitemOpen
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  \BibitemOpen
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\bibitem [{\citenamefont {Lavoie}\ \emph {et~al.}(2013)\citenamefont {Lavoie},
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\bibitem [{\citenamefont {Karpi{\'{n}}ski}\ \emph {et~al.}(2016)\citenamefont
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\bibitem [{\citenamefont {Li}\ \emph {et~al.}(2017)\citenamefont {Li},
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  {et~al.}(2014{\natexlab{a}})\citenamefont {Srivathsan}, \citenamefont
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  {et~al.}(2014{\natexlab{b}})\citenamefont {Srivathsan}, \citenamefont
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  {Sargent}(2006)}]{RevModPhys.78.455}%
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