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{\Large\bfseries
Symmetrical Clock Synchronization with Time-Correlated Photon Pairs\par}
\vspace{3ex}
{\bfseries
Jianwei~Lee$^1$,
  Lijiong~Shen$^{1, 2}$,
  Alessandro~Cer\`{e}$^1$,
  James~Troupe$^3$,
  Antia~Lamas-Linares$^{4, 1}$,
  Christian~Kurtsiefer$^{1, 2}$\par}
{\footnotesize\itshape
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\par}
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%MAIN TEXT
We present a distance-independent synchronization protocol 
based on a symmetric distribution of single-photons between remote clocks~[1].
% 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~[2],
the determination of the clock offsets is independent of the signal propagation times for a symmetrical communication channel, and is therefore inherently robust against symmetric-channel-delay attacks.
Our protocol, based on existing quantum communication techniques,
provides a natural complement to Global Navigation Satellite Systems (GNSS) for quantum key distribution~[3,4],
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.

\begin{figure}[ht]
\center
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=\textwidth]{clocksyncsetup_CEE2019.eps}
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\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=\textwidth]{offsets_separate_atomic_clocks.eps}
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\begin{quotation}
\vspace{-0.5cm}
\noindent\footnotesize\textbf{Fig. 1}
Left:
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.
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{quotation}

% 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 2\,s, 
tracks the drift between the two frequency references (Fig.~1~(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.~1~(right,~b)),
does not show any significant correlation with the estimated propagation time (Fig.~1~(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 ($<1\times10^{-12}$).

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\textbf{References}

\footnotesize
[1]  J. Lee, L. Shen, A.~Cer{\`e}, J. E. Troupe, A. Lamas-Linares, and C. Kurtsiefer, ``Symmetrical Clock Synchronization with Time-Correlated Photon Pairs,'' arXiv:1812.08450.

[2] L. Narula and T. Humphreys, ``Requirements for secure clock synchronization,'' IEEE J. Sel. Topics Signal Process \textbf{12}, 749 (2018).

[3] I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, ``Free-space quantum key distribution with entangled photons,'' Appl. Phys. Lett. \textbf{89}, 101122 (2006).

[4] C. Ho, A. Lamas-Linares, and C. Kurtsiefer, ``Clock synchronization by remote detection of correlated photon pairs,'' New J. Phys. \textbf{11}, 045007 (2009).

% \footnotesize
% [1] J. Itatani, D. Zeidler, J. Levesque, D. M. Villeneuve, and P. B.
% Corkum, ``Controlling High Harmonic Generation with Molecular Wave
% Packets,'' Phys. Rev. Lett. \textbf{94}, 123902 (2005). For journal
% articles, authors are listed first, followed by the article’s full
% title in quotes, the journal’s title abbreviation, the volume number
% in bold, page number, and the year in parentheses.

% [2] G. P. Agrawal, \textit{Nonlinear Fiber Optics}, 3rd. ed.,
% (Academic Press, Boston, 2001). For citation of a book as a whole:
% authors, followed by title in italics, and publisher, city, and year
% in parenthesis.

% [3] R. Kienberger and F. Krausz in, \textit{Few-cycle laser pulse
% generation and its applications}, F.X. Kärtner ed. (Springer Verlag,
% Berlin, 2004). For citation of a book chapter, authors are listed
% first, followed by book title in italics, editors, and publisher,
% city, and year in parenthesis. Chapter number may be added if
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