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{\Large\bfseries
O-band Polarization-Entanglement\\
Quantum Key Distribution with Co-Propagating Classical Channel
\par}
\vspace{3ex}

% AUTHORS
{\bfseries
Justin~Yu~Xiang~Peh$^{1}$,
Hou~Shun~Poh$^{1}$,
Syed~Abdullah~Aljunid$^{2}$,
Darren~Ming~Zhi~Koh$^{1}$,
Cyrus~Proctor$^{3}$,
Lijiong~Shen$^{1}$,
Christian~Kurtsiefer$^{1,4}$\par}
{\footnotesize\itshape
1. Center for Quantum Technologies, 3 Science Drive 2, Singapore 117543\\
2. S-Fifteen Instruments Pte. Ltd., 79 Ayer Rajah Crescent 139955, Singapore\\
3. Amazon Web Services Inc., 410 Terry Avenue North, Seattle, WA 98109-5210, U.S.A.\\
4. Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542\par}
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\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{Hou~Shun~\surname{Poh}}
\affiliation{Center for Quantum Technologies, National University of Singapore, 3 Science Drive 2 117543, 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}
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% Clock synchronization is necessary for communication and distributed computing tasks, due to network latencies from the physical propagation of signals. The required synchronization accuracy varies across different tasks; in quantum communication systems, photo-detection events need to be discriminated with sub-nanosecond accuracy in order to distill information from correlations at a sufficiently high rate. This typically entail significant resource overheads, such as the use of ultrastable frequency references, that scale with the number of communicating endpoints.
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Clock synchronization is necessary for communication and distributed computing tasks. In quantum communication systems, photo-detection events need to be discriminated with sub-nanosecond accuracy in order to distill information from correlations at a sufficiently high rate. This typically entails significant resource overheads, such as the use of ultra-stable frequency references, that will also scale with the number of communicating endpoints.

Here, we demonstrate an alternative clock synchronization scheme using weak thermal correlations without the need for external frequency standards. The scheme relies on tracking timing correlations in the photon bunching behaviour of thermal light generated using a sub-threshold laser. The 10~MHz quartz oscillators onboard the time-taggers have a typical frequency accuracy of $10^{-4}$; by performing cross-correlation measurements to identify the position and drift rate of the bunching peak, we can continuously correct for the frequency difference to within $10^{-9}$ of each other, comparable to that of a Rubidium frequency standard~[1]. We also identify lower bounds for peak identification using the cross-correlation method.

% Can also shift this to the first paragraph and mention ', but suffers from relatively lower brightness that limit the tolerable losses.' Then add commentary on our measured brightness and tolerable losses using SMF-28 fiber.
A similar line of work relies on strong timing correlations in photon pairs from spontaneous parametric down-conversion sources for synchronization~[2][3]. We believe our scheme using thermal light sources significantly improves clock synchronization over longer distances due to its intrinsically higher brightness and low attenuation within the C-band optical window of G.652D standard-compliant telecommunication fiber, whilst requiring only an easily-acquirable laser diode and simple temperature control for operation.

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    \caption{\noindent\footnotesize\textbf{Fig. 1.} 
        Experimental setup for generating thermal light at 1550~nm wavelength from a laser diode running at sub-threshold. A brightness of $240\times10^{6}$~counts per second and bandwidth of around 1~GHz is observed after successive spectral filtering using etalons and a bandpass filter.
    }
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    \caption{\noindent\footnotesize\textbf{Fig. 2.}
        Second-order correlation measurement of photon bunching from a thermal source, using independently running time tagging units without external frequency references. A frequency deviation of $8,539.5\times{}10^{-9}$ and timing offset of 844,335.0~ns is compensated for between the pair of time taggers.
    }
    \label{fig:1}
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    \caption{\noindent\footnotesize\textbf{Fig. 3.} 
        Reconstructed timing and frequency drift between two typical quartz crystal oscillators. The frequency offset aligns with the instantaneous gradient of the time drift with an observed maximum frequency drift of $3.5\times{}10^{-7}$ over several minutes, which corresponds to a modest tracking window of around 100~ns/s required for continuous timing and frequency drift compensation.
    }
    \label{fig:1}
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\textbf{References}

\footnotesize\

[1] Stanford Research Systems Inc., \textit{FS725 — Benchtop rubidium frequency standard}, (2015).

[2] C. Spiess, S. Töpfer, S. Sharma, A. Kržič, M. Cabrejo-Ponce, U. Chandrashekara, N. L. Döll, D. Rieländer, and F. Steinlechner, ``Clock Synchronization with Correlated Photons,'' Phys. Rev. Applied \textbf{19}, 054082 (2023).

[3] C. Ho, A. Lamas-Linares, and C. Kurtsiefer, ``Clock synchronization by remote detection of correlated photon pairs,'' New J. Phys. \textbf{11}, 045011 (2009).

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