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%       submission to OSA meetings and conferences    %
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\begin{document}

\title{Symmetrical Clock Synchronization with Time-Correlated Photon Pairs}

\author{Jianwei~Lee$^1$,
  Lijiong~Shen$^{1, 2}$,
  Alessandro~Cer\`{e}$^1$,
  James~Troupe$^3$,
  Antia~Lamas-Linares$^{4, 1}$,
  and Christian~Kurtsiefer$^{1, 2}$} 

\address{1. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543\\
  2. Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542\\
  3. Applied Research Laboratories, The University of Texas at Austin, Austin, Texas, USA\\
  4. Texas Advanced Computing Center, The University of Texas at Austin, Austin, Texas, USA}

\email{jianwei.lee@u.nus.edu}

%% Uncomment the following line to override copyright year from the default current year.
\copyrightyear{2019}

\begin{abstract}
We demonstrate a distance-independent clock synchronization protocol, 
using counter-propagating photons from spontaneous parametric down-conversion pair sources, secure against symmetric-delay attacks. 
With rates of 200\,coincidences/s, we record a precision of 51\,ps over 100\,s.
\end{abstract}
\ocis{060.4510, 270.5568.}
% \ocis{060.4510 - Optical communications, 270.5568 - Quantum cryptography.}
% https://www.osapublishing.org/submit/ocis/
\section{Introduction}
Clock synchronization
is crucial to the functioning of modern infrastructure.
Despite its fundamental role, communication between clocks remain susceptible to disruption and manipulation~\cite{shepard:12,bhatti:17}.
We present a distance-independent synchronization protocol 
based on a symmetric distribution of single-photons between remote clocks.
% based on
Tight time correlations of the photon pairs generated by spontaneous parametric down-conversion (SPDC) enables precise synchronization.
Similar to existing bi-directional protocols~\cite{narula:17},
the determination of the clock offsets is independent of the signal propagation times for a symmetrical communication channel.
Our protocol, based on existing quantum communication techniques,
provides a natural complement to Global Navigation Satellite Systems (GNSS) for quantum key distribution~\cite{Marcikic:2006,ho:09},
and is suited for synchronization between mobile stations
where pair rates are typically low and propagation times are constantly changing.
The single-photon regime allows, in principle, an additional security layer 
by distributing polarization entanglement and using a Bell measurement to verify the origin of the photons.

\section{Experiment}
\begin{figure}[htpb]
  \centering
  \includegraphics[width=8.5cm]{clocksyncsetup_CLEO2019.eps}
\caption{Clock synchronization setup. Alice and Bob each have a source of time-correlated photon pairs produced from spontaneous parametric down-conversion (SPDC) and an avalanche photodetector (APD). One photon of the pair is detected locally, while the other photon is sent through a single mode fiber of length~$L$ to be detected on the remote side. Times of arrival for all detected photons are recorded at each side with respect to the local clock.
}
\end{figure}
\noindent
The experimental setup is shown in Fig.~1.
Each of two parties, Alice and Bob, 
transmits photons from time-correlated photon pairs produced from SPDC
to the other party through a shared optical channel.
Each party time-stamps detection events according to a local clock and shares the time stamps over a classical channel.
A histogram of the time difference between detection events at Alice and Bob shows two peaks centered at times~$\tau_{AB} $ and~$\tau_{BA}$,
from which it is possible to derive the time offset between the two clocks~$\delta$ and a round-trip time for the photons~$\Delta T$
\begin{equation}
  \delta\:=\:\frac{1}{2}\left (\tau_{AB}+\tau_{BA}\right )\quad \text{and} \quad
  \Delta T\:=\:\tau_{AB}-\tau_{BA}\,.
\end{equation}

\section{Results}
\begin{figure}
  \centering
  \includegraphics[width=0.45\columnwidth]{precision.eps}
  \includegraphics[width=0.45\columnwidth]{offsets_separate_atomic_clocks.eps}
  \caption{\label{fig:precision}
  Left:
  Standard deviation (precision $\delta t$) of the measured offset between two clocks, both locked to the same frequency reference.
  Solid line: Least-squares fit to a model where $\delta t$ follows Poisson statistics and improves with acquisition time $T_a$.
  Right:
  (a) Measured offset $\delta$ between two clocks with independent frequency references,  with respect to the initial measured offset $\delta_0$. Each point corresponds to a 2\,s acquisition time.
  Continuous blue line: fit used to extract the relative frequency accuracy ($\approx4\times10^{-11}$) between the clocks.
  (b) Residual
  % $r(t)$ 
  of the fit fluctuates due to the intrinsic instability of the individual frequency references.
  (c) The round trip time $\Delta T$ was changed using different fiber lengths.
  }
\end{figure}

To demonstrate the independence of the synchronization from the propagation distance, we initially characterize the precision of the offset estimation~$\delta t$ when both parties share a rubidium frequency reference.
The precision~$\delta t$ depends on detector time jitter and, for constant pair rate, scales with the total acquisition time, as in Fig.~2~(left).

For a more realistic synchronization scenario, each party uses an independent frequency reference,
resulting in a clock offset that changes with
time~$\delta\rightarrow\delta(t)$.
We simulate a symmetric-channel-delay attack by changing the propagation time between Alice and Bob every 5\,mins using four fibers of different lengths.
The measured offset~$\delta(t)$,
evaluated from time stamps collected every $T_a = 2$\,s, 
tracks the drift between the two frequency references (Fig.~2~(right, a)).
The clock drift is well approximated by a parabolic model that we estimate and subtract from the measured $\delta(t)$.
The remaining fluctuation (Fig.~2~(right,~b)),
does not show any significant correlation with the estimated propagation time (Fig.~2~(right,~c)), proving the resilience of the protocol against symmetric-channel-delay attacks.
These fluctuations
are mainly due to the intrinsic frequency instabilities of the frequency references, resulting in the slightly reduced synchronization precision of 51\,ps in 100\,s,
comparable with the Allan deviation of our clocks ($<2\times10^{-12}$).

 

% \bibliographystyle{osajnl}
% \bibliography{clock}

\begin{thebibliography}{1}
\newcommand{\enquote}[1]{``#1''}
\newcommand{\JournalTitle}[1]{\textbf{#1}}

\bibitem{shepard:12}
D.~P. Shepard, T.~E. Humphreys, and A.~Fansler, \enquote{Evaluation of the
  vulnerability of phasor measurement units to {GPS} spoofing attacks,}
  {\protect\JournalTitle{International Journal of Critical Infrastructure
  Protection}} \textbf{5}, 146 (2012).

\bibitem{bhatti:17}
J.~Bhatti and T.~E. Humphreys, \enquote{Hostile control of ships via false gps
  signals: Demonstration and detection,} {\protect\JournalTitle{NAVIGATION:
  Journal of the Institute of Navigation}} \textbf{64}, 51 (2017).

\bibitem{narula:17}
L.~Narula and T.~Humphreys, \enquote{Requirements for secure clock
  synchronization,}  (2017).

\bibitem{Marcikic:2006}
I.~Marcikic, A.~Lamas-Linares, and C.~Kurtsiefer, \enquote{{Free-space quantum
  key distribution with entangled photons},} {\protect\JournalTitle{Applied
  Physics Letters}} \textbf{89}, 101122 (2006).

\bibitem{ho:09}
C.~Ho, A.~Lamas-Linares, and C.~Kurtsiefer, \enquote{Clock synchronization by
  remote detection of correlated photon pairs,} {\protect\JournalTitle{New
  Journal of Physics}} \textbf{11}, 045007 (2009).
\end{thebibliography}

\end{document}
