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\begin{document}

\title{O-band Polarization-Entanglement \\
Quantum Key Distribution over Long Distance Deployed Fiber}

\author{Justin Yu Xiang \surname{Peh}}
\affiliation{Center for Quantum Technologies, National University of Singapore, 3 Science Drive 2 117543, Singapore}
\author{Syed Abdullah \surname{Aljunid}}
\affiliation{S-Fifteen Instruments Pte. Ltd., 79 Ayer Rajah Crescent 139955, Singapore}
\author{Darren Ming Zhi \surname{Koh}}
\affiliation{Center for Quantum Technologies, National University of Singapore, 3 Science Drive 2 117543, Singapore}
\author{Cyrus \surname{Proctor}}
\affiliation{Amazon Web Services Inc., 410 Terry Avenue North, Seattle, WA 98109-5210, U.S.A.}
\author{Lijiong \surname{Shen}}
\affiliation{Center for Quantum Technologies, National University of Singapore, 3 Science Drive 2 117543, Singapore}
\author{Christian \surname{Kurtsiefer}}
\affiliation{Center for Quantum Technologies, National University of Singapore, 3 Science Drive 2 117543, Singapore}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3 117542, Singapore}

\date{\today}

\begin{abstract}
  XXXX
\end{abstract}

\maketitle

\section{Introduction}
% Introduction:
%   - Why write this paper?
%   - What is new in this paper?
%   - What are you trying to prove with this paper?
% This differs from the discussion where the emphasis is more on what's the way forward.
%
% Hint: To motivate yourself to write, consider writing it like a tutorial. And also write on the level of individual paragraphs instead of single lines.

% QKD is important today
Quantum key distribution (QKD) provides a means for secret sharing between two remote parties, with information-theoretic security bounded only by laws of quantum physics.
It fills the long-term security gap in secured classical communication by provisioning symmetric keys for session encryption, which are typically negotiated using public key cryptography known to be vulnerable to quantum cryptanalysis.

% BBM92 is a good protocol to use
Early QKD proposals like BB84~\cite{bennettQuantumCryptographyPublic1984} have since been extended to other protocols such as E91~\cite{ekertQuantumCryptographyBased1991a} and BBM92~\cite{bennettQuantumCryptographyBell1992}.
BBM92 in particular, which uses entanglement as a resource for information exchange, is a useful protocol due to its good balance between security and usability.
Randomness is instrinsically embedded in the measurement process of the shared entangled state, decoupling the need for a randomness generator.
The detectability of source tampering also mitigates an entire class of source attacks~\cite{waksSecurityQuantumKey2002a} whilst providing key rates comparable to single channel BB84.

% Fiber-based QKD is useful despite the versatility of free-space implementation
There are mainly two different media over which a photonic quantum channel can be established, free-space and in fiber.
The former, on top of the line-of-sight requirement, tends to be strongly impacted by mode distortions from atmospheric turbulence and beam divergence that result in more frequent availability loss.
In metropolitan settings where line-of-sight is not guaranteed, the use of fiber channels are a versatile alternative, and is less likely to be impacted by environmental effects and availability loss.

% 1310~nm is a good wavelength to use in metropolitan settings
Previous QKD systems that employ BBM92 use the 1550~nm wavelength for the transmission signal~\cite{treiberFullyAutomatedEntanglementbased2009,wengerowskyEntanglementDistribution96kmlong2019,neumannContinuousEntanglementDistribution2022a,quantumOpticsJena2023}.
1550~nm is a natural choice for the transmission signal due to the low loss optical window in telecommunication fibers, but suffers from strong dispersion effects which will increase the noise floor in QKD deployments~\cite{treiberFullyAutomatedEntanglementbased2009}, necessitating the laying of special dispersion-shifted fibers.
The advantage of employing 1310~nm instead lies in avoiding dispersion effects in the fiber due to the zero-dispersion window of standard telecommunication fiber  likely already deployed as part of existing telecommunication infrastructure.

% Existing works are comparable, but we demonstrate with a setup that is designed for portability
In this paper, we demonstrate BBM92 QKD with 1310~nm as the transmission wavelength, over standard telecommunication fiber and using commercial off-the-shelf avalanche photodetectors (APDs).
Using a previously constructed highly wavelength-asymmetric pair source~\cite{shenDistributingPolarizationEntangled2022}, we generate intrinsically narrow bandwidth pairs~(\roughly{}120~GHz) that pick up negligible dispersion through the fiber.
We achieve a long-running bit rate of 70~bits/s over a span of three days.


\section{Methods}
% Setup:
%   - What is the basic concept of the experiment, independent of any resource use?
%
% For a first write, just throw in facts first! Sort them out later. A good draft is a draft that has words in them.


\graphic[squish=0.98\linewidth,wide,label=schematic]{figures/20231123_source_qkd_horizontal_boxed.pdf}{
  Schematic of the QKD setup with the non-degenerate entangled photon pair source scheme per construction in \cite{shenDistributingPolarizationEntangled2022}.
  405\,nm pump light is split into two paths and undergoes Type-0 SPDC in a ppKTP crystal, generating photon pairs of non-degenerate wavelengths at 586\,nm (signal) and 1310\,nm (idler).
  A pair of $\alpha{}$-BBO crystals separately recombine the signal and idler to create the polarization-entangled state $\ket{\Phi{}^{+}} = \frac{1}{\sqrt{2}}\left(\ket{HH} + \ket{VV}\right)$. DM:~Dichroic mirror; LP:~Longpass filter; BP:~Bandpass filter.
  Each photon is sent through a passive beamsplitter array consisting of 50:50 and polarizing beamsplitters (PBS) for measurement along the rectilinear (HV) and diagonal (DA) polarization bases. Detection events from the individual APDs are registered on corresponding time taggers. A classical authenticated communication channel is used for photon pair coincidence tracking and frequency compensation.
}


% Description of QKD scheme and key points that make the protocol work
We perform entanglement distribution as per the BBM92 QKD protocol:
the $\ket{\Phi^+}$ maximally-entangled state in the polarization degree of freedom is distributed between two parties, one situated locally (Alice) and the other situated remotely (Bob).

% Overall description of source
Highly wavelength-asymmetric photon pairs are generated by Type-0 spontaneous parametric downconversion (SPDC) in a periodically-poled potassium titanyl phosphate (ppKTP) crystal, following the scheme
$$
405\,\text{nm} \rightarrow 586\,\text{nm} + 1310\,\text{nm}.
$$
This is illustrated in Fig~\ref{schematic}.
The linear displacement interformetric configuration of the pair source generates the $\ket{\Phi^+}$ state, by spatially overlapping two downconversion beams from the same nonlinear crystal to yield the $(\ket{HH} + e^{i\theta}\ket{VV})/\sqrt{2}$ state, with phase compensation achieved by tilts in the half-wave plate (HWP).
The signal 586~nm and idler 1310~nm photons are wavelength-separated with a dichroic mirror, the former measured locally by Alice and the latter remotely by Bob.
Figure~\ref{schematic} illustrates the aforementioned pair source setup, and a more complete description of the setup can be found in \cite{shenDistributingPolarizationEntangled2022}.

% Source operating parameters and output properties
We measured 150,000~photon pairs/s using a pair of APDs, using an input pump power of 8.1~mW.
A plot of the source visibility is shown in Fig~\ref{visibility}.
The high asymmetry in downconversion wavelengths results in an instrincally narrow bandwidth of 0.7~nm (120~GHz) for the 1310~nm idler light, without the use of additional spectral filtering optics.

% Old data: Pair rates of $>$10,000\,count/s is observed with $>$22\% heralding efficiency using Si-APD and InGaAs-APD, after detector dark count and afterpulsing corrections, using a pump power of 100\,\textmu{}W. At this pump power, we measured an idler rate of 70,000\,count/s using a Si-APD with 60\% detection efficiency, corresponding to $>$$10^{6}$\,pairs/s/mW source brightness.

\graphic[label=visibility]{figures/sourcevisibility.eps}{
  Pair source visibility as a metric of entanglement fidelity.
  Each curve is obtained by projecting the polarization for 586~nm light, and scanning the polarization of 1310~nm light using a rotating HWP.
}


% Overall description of detection kit
Alice and Bob each perform polarization measurements along the rectilinear (HV) and diagonal (DA) basis to obtain pair correlations used for secret key extraction.
A free-space non-polarizing 50:50 beamsplitter functions as the random choice of measurement bases for the received light, and the polarizing beam splitters performs the measurement in the polarization measurement unit shown in Fig~\ref{schematic}.
Detection of 586~nm and 1310~nm photons is made with Si APDs (Excelitas Technologies SPCM-AQRH) and InGaAs APDs (S-Fifteen Instruments IRSPD1), respectively.

% Compensations
Polarization drifts occur in the fiber due to mechanical stress as a result of temperature fluctuations.
We ensure that the polarization is measured in the same frame of reference on both sides by compensating the drift using a set of liquid crystal variable retarders (LCVRs).
It has been shown that polarization compensation needs to be performed only on one side, and a sequence of four LCVRs aligned along alternating optical axes of 0\textdegree/45\textdegree/0\textdegree/45\textdegree{} is sufficient for arbitrary polarization rotation~\cite{shiFibrePolarisationState2021a}.
Active polarization compensation is employed by performing stochastic perturbation of phase delays across the alternating polarization axes, such that the quantum bit error rate (QBER) loss function is minimized.

% Timestamping
Time tagging of detector clicks is performed with a resolution of 125~ps~\cite{qcrypto}. The timestamps are not tied to a Rubidium frequency standard, instead relying on frequency pre-compensation and active software-based frequency correction to correct for clock skew resulting from temperature and voltage fluctuations~\cite{hoClockSynchronizationRemote2009}.
Timing delays between detections of different polarizations are also compensated for to mitigate timing mismatch attacks~\cite{lamas-linaresBreakingQuantumKey2007}.

\section{Deployment}
% Deployment:
%   - What deployment-specific tasks needed to be done?
%   - Any site-specific constraints?
%   - Any proof of deployment? i.e. characterizations

\begin{figure*}
  \centering
  \begin{minipage}[t]{0.5\textwidth}
    \includegraphics[width=\textwidth]{figures/nussutdfiber_reduced.pdf}
  \end{minipage}
  \begin{minipage}[t]{0.45\textwidth}
    \includegraphics{figures/fiber2.eps}
  \end{minipage}
  \caption{\label{otdr}
    \textbf{Left:} QKD node locations in Singapore, with a straight-line distance of 21~km between the two nodes.
    \textbf{Right:} Optical time-domain reflectometry (OTDR) measurement of the deployed fiber link between NUS and SUTD. The measured distance is 46.4~km with 19.20~dB cumulative loss for 1310~nm light. Reflection spikes and dips occur at fiber patching points.
  }
\end{figure*}

% Deployed fiber info and characterization
Telecommunication fiber (G.652-compliant SMF-28e~\cite{CorningSMF28Ultra2014}) is deployed between Center for Quantum Technologies, National University of Singapore (NUS), and Future Communications Connectivity Lab, Singapore University of Technology and Design (SUTD), with a total fiber length of 46.4~km as shown in Fig~\ref{otdr}.
The entanglement pair source is situated in NUS with the 586~nm photons measured locally, while the 1310~nm photons are sent over the fiber to SUTD for remote measurement.

The chromatic dispersion contributed by the fiber is at most 4~ps, by far smaller than the time resolution used for coincidence matching (125~ps).
The expected polarization mode dispersion of 0.14~ps is less than the coherence time of the downconverted 1310~nm light (8.2~ps), i.e. polarization of the entangled photon remains indistinguishable over the length of fiber.

% Detector settings
The total detection rate at the remote site is approximately 12,000~counts/s, with the InGaAs APDs contributing a total dark count rate of 8,000~counts/s.
The nominal detection efficiency is around 15\% with a measured 4\% afterpulsing probability.

% QKD software and configuration
QKD is performed using \texttt{qcrypto}~\cite{qcrypto}, which performs coincidence matching, error correction and privacy amplification over a classical authenticated channel.
Measurements are packed into $2^{29}$~ns ($0.537$~s) timing packets to minimize communication overhead.
We use the coincidence measurement of photon pairs as a resource for timing synchronization.
The width of the coincidence window is chosen to accomodate the jitter of the Si APDs, and thus allows instantaneous drifts in clock skew of up to 1.4~ns/s to be tolerated with no adverse performance impact.
QBER is estimated during every key reconciliation round, and corrected for in block sizes of 10,000 accumulated bits, using a modified CASCADE/BICONF algorithm~\cite{brassardSecretKeyReconciliationPublic1994a,sugimotoStudySecretKey2000}.
Privacy amplification is then finally performed to yield the final secret keys.

% Key management and delivery after QKD
The secret keys are fed into the key management layer \texttt{guardian}~\cite{guardian} which implements a key delivery interface following the ETSI GS QKD 014 specification~\cite{etsi2019014}.
This software layer is integrated with the QKD software for key routing capabilities, i.e. keys can be consumed by downstream clients across different nodes in a multi-node architecture.
Each QKD node is connected to their respective subnets, and establishes a classical authenticated channel using an encrypted Tailscale peering connection over the internet.


\section{Results}
\begin{figure*}
  \centering
  \begin{minipage}[t]{0.6\textwidth}
    \includegraphics{figures/qkd3_qberrate_squished.eps}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.3\textwidth}
    \includegraphics{figures/qkd4_qberscatter_squished.eps}
  \end{minipage}
  \caption{\label{qberrate}
    \textbf{Top:} Measured QBER over time, with a 12\% QBER limit (with generated keys not used for authentication) marked in red. Sharp spikes in QBER are attributed to human activity, while large drifts in QBER are due to overcorrection of polarization rotation by the active compensation algorithm.
    \textbf{Bottom:} A long-term average secret keyrate of 81~bits/s is achieved for QKD performed over the 46.4~km deployed fiber, marked in black. The highest instantaneous bit rate achieved is 120~bits/s.
    \textbf{Right:} Scatter plot of key generation rate against measured QBER. The dotted line marks the 12\% QBER threshold for successful secret key generation. The graph shows almost one-to-one correspondence between QBER and instantaneous bitrate, modulated by drifts in coincidence rate.
  }
\end{figure*}

\graphic[wide,label=localcounts]{figures/qkd1_remote_rates_wcoinc.eps}{
  Plots of measured (local) count, coincidence, and accidental rates during the QKD.
  The average coincidence and accidental rates are 450~counts/s and 25~counts/s, respectively.
  The gap in coincidence measurement at around 4~pm on 14~October is explained by a QBER spike interrupting QKD and the polarization compensation algorithm taking a longer time for correction.
}


% QKD and bitrate achieved
We performed a continuous QKD run over a span of 3~days, measuring an instantaneous bit rate of around \roughly{}120~bits/s as shown in Fig~\ref{qberrate}.
The long-run averaged bit rate is \roughly{}81~bits/s, with an average of 450 coincidences per second.
The average count rate on the local and remote sides are $1.3\times{}10^{6}$~counts/s and $1.2\times{}10^3$~counts/s respectively.


\setlength{\tabcolsep}{0.5em}% for horizontal padding
{\renewcommand{\arraystretch}{1.2}% for vertical padding
\begin{table}
  \begin{tabular}{|l|c|}
    \hline \textbf{Cause} & \textbf{T (dB)} \\
    \hline Coupling into deployed fiber & -0.1 \\
    \hline Deployed fiber & -19.2 \\
    \hline LCVR transmission & -2.2 \\
    \hline Fiber-fiber coupling in free space & -1.0 \\
    \hline Detector efficiency & -9.2 \\
    \hline
  \end{tabular}
  \caption{\label{losses} Transmission loss contributions from different components in the 1310~nm channel. }
\end{table}
}

% QBER contributions
The measured QBER is minimally at 4\%, with fluctuations in QBER directly attributed to imperfect polarization compensation in the LCVRs.
Accidental coincidences with dark counts from the InGaAs APDs contribute 2.2\% QBER, and the rest is due to the fidelity of state preparation as well as measurement.
Losses from transmission, coupling and detection efficiencies contribute to the lower detection rates measured on the remote side, as per Table~\ref{losses}.

% Commentary on timing drift
The active clock skew compensation algorithm is in effect, resulting in minimal timing drifts during the QKD, as per Fig~\ref{freqcorr}.
The fluctuations in clock skew is comparable to that between two Rubidium frequency standards.
The drift in timing due to clock skew is not observed to affect the QBER, suggesting that the resolution of the compensation algorithm is sufficient.

\graphic[wide,label=freqcorr]{figures/qkd2_freqcorr.eps}{
  Measured clock skew between Alice and Bob, obtained by tracking the instantaneous time difference between both parties. The clock skew values are normally distributed, with an effective compensation resolution of 0.233~ns/s.
  \textbf{Top:} Quartz crystals without active frequency compensation.
  \textbf{Middle:} Quartz crystals with active frequency compensation.
  \textbf{Bottom:} Rubidium clock with active frequency compensation. Averaged clock skew is $0.00\pm{}0.75$~ns/s.
}


\section{Discussion}
The system has similar key rates with similar systems running in the C-band that minimizes optical losses in the fiber.

Several avenues exist to improve the bitrate.
In particular, the use of a high-efficiency SNSPD (with near unity quantum efficiency) and inline motorised polarization controllers will increase the loss budget by up to 12~dB.
This will also bring the QKD operation out of the dark-count limited regime, and drastically reduce the overall QBER.
The additional 40\% transmission loss incurred due to fiber patching for the 586~nm light at the local side, can additionally contribute another 40\% extra coincidence counts.
With a QBER of 3\%, the overall expected bitrate is\ldots{}

With an improvement in the source efficiency from an averaged 15\% to 20\% as originally measured in~\cite{shenDistributingPolarizationEntangled2022}, the overall bitrate will be increased as well.


\section{Acknowledgments}
\begin{acknowledgments}
  This research is supported by the Quantum Engineering Programme through Grant NRF2021-QEP2-04-P01. We would like to acknowledge National Quantum Safe Network Singapore for provisioning of the deployed fiber, and NetLink for assistance with fiber patching. We would like to thank S-Fifteen Instruments for the equipment support, and FCCLab, FCP SUTD for hosting the fiber endpoint.
\end{acknowledgments}

\newpage

\bibliography{bibliography.bib}

\end{document}
