\relax \providecommand\hyper@newdestlabel[2]{} \bibstyle{osajnl} \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined \global\let\oldcontentsline\contentsline \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} \global\let\oldnewlabel\newlabel \gdef\newlabel#1#2{\newlabelxx{#1}#2} \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} \AtEndDocument{\ifx\hyper@anchor\@undefined \let\contentsline\oldcontentsline \let\newlabel\oldnewlabel \fi} \fi} \global\let\hyper@last\relax \gdef\HyperFirstAtBeginDocument#1{#1} \providecommand\HyField@AuxAddToFields[1]{} \providecommand\HyField@AuxAddToCoFields[2]{} \citation{wenjun2014two,mills:1991,PTP,Moreira:2009} \citation{narula:17} \citation{lee2019symmetrical,hou2018fiber,lamas2018secure} \citation{quan2021implementation} \citation{lee2019symmetrical} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}\protected@file@percent } \citation{hong1987measurement} \citation{glauber1963quantum} \citation{shi2020stable,lohrmann2020broadband,grieve2019characterizing} \citation{grieve2019characterizing} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Clock synchronization setup. Alice has a source of time-correlated photon pairs based on spontaneous parametric down-conversion (SPDC) and a single-photon nanowire photodetector (SNSPD). One photon of the pair is detected locally, while the other one is sent through a single mode fiber of length\nobreakspace {}$L$ to be detected on the remote side with Bob's InGaAs avalanche photodiode (APD). Times of arrival for all detected photons are recorded at each side with respect to the local clock, each locked to a rubidium frequency reference (10\tmspace +\thinmuskip {.1667em}MHz). Occasionally, a transmitted photon is reflected at the end face of the fiber back to Alice, allowing her to determine the round-trip time and derive the absolute offset between the clocks. \relax }}{2}{figure.caption.1}\protected@file@percent } \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} \newlabel{fig:setup}{{1}{2}{Clock synchronization setup. Alice has a source of time-correlated photon pairs based on spontaneous parametric down-conversion (SPDC) and a single-photon nanowire photodetector (SNSPD). One photon of the pair is detected locally, while the other one is sent through a single mode fiber of length~$L$ to be detected on the remote side with Bob's InGaAs avalanche photodiode (APD). Times of arrival for all detected photons are recorded at each side with respect to the local clock, each locked to a rubidium frequency reference (10\,MHz). Occasionally, a transmitted photon is reflected at the end face of the fiber back to Alice, allowing her to determine the round-trip time and derive the absolute offset between the clocks. \relax }{figure.caption.1}{}} \@writefile{toc}{\contentsline {section}{\numberline {2}Time synchronization protocol}{2}{section.2}\protected@file@percent } \newlabel{eq:round-trip time }{{2}{2}{Time synchronization protocol}{equation.2.2}{}} \newlabel{eq:offset}{{3}{2}{Time synchronization protocol}{equation.2.3}{}} \@writefile{toc}{\contentsline {section}{\numberline {3}Experiment}{2}{section.3}\protected@file@percent } \citation{ho2009clock} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Clock synchronization scheme. Alice and Bob measure detection times $t$ and $t'$ of photon pairs generated from Alice's source using local clocks. Detection times $t_1$ and $t'_2$ are associated with a time-correlated photon pair where one photon of the pair is transmitted to Bob, while $t_3$ and $t_4$ are associated with a pair where one of the photons is reflected at Bob back to Alice. The single-trip time $\tau _{AB}$ of photons in the synchronization channel, calculated from the time difference $t'_2-t_1$, depends on the signal delay $\Delta t_{AB}$ associated with the length of the channel, and the absolute clock offset $\delta $ between the clocks. The round-trip time $\tau _{AA}$ of the channel is estimated using $t_4-t_3$. Assuming a symmetric delay channel, $\delta $ can be derived from $\tau _{AB}$ and $\tau _{AA}$ without \textit {a priori} knowing $\Delta t_{AB}$. \relax }}{3}{figure.caption.2}\protected@file@percent } \newlabel{fig:syncscheme}{{2}{3}{Clock synchronization scheme. Alice and Bob measure detection times $t$ and $t'$ of photon pairs generated from Alice's source using local clocks. Detection times $t_1$ and $t'_2$ are associated with a time-correlated photon pair where one photon of the pair is transmitted to Bob, while $t_3$ and $t_4$ are associated with a pair where one of the photons is reflected at Bob back to Alice. The single-trip time $\tau _{AB}$ of photons in the synchronization channel, calculated from the time difference $t'_2-t_1$, depends on the signal delay $\Delta t_{AB}$ associated with the length of the channel, and the absolute clock offset $\delta $ between the clocks. The round-trip time $\tau _{AA}$ of the channel is estimated using $t_4-t_3$. Assuming a symmetric delay channel, $\delta $ can be derived from $\tau _{AB}$ and $\tau _{AA}$ without \textit {a priori} knowing $\Delta t_{AB}$. \relax }{figure.caption.2}{}} \citation{bousonville2009velocity} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Timing correlations showing coincidence peaks due to (a) round-trip and (b) single-trip propagation of photons in the synchronization channel. (a) $r\left (\tau \right )$: auto-correlation function $R\left (\tau \right )$ normalized to background coincidences extracted from Alice's timestamps acquired over 90\tmspace +\thinmuskip {.1667em}s. (b) $g^{(2)}\left (\tau \right )$: cross-correlation function $G^{(2)}\left (\tau \right )$ normalized to background coincidences extracted from Alice and Bob's timestamps acquired over 3\tmspace +\thinmuskip {.1667em}s. Solid lines: fits to heuristic model. $\tau _{AA}$ and $\tau _{AB}$: peak positions of respective distributions. Error bars: propagated Poissonian counting statistics. \relax }}{4}{figure.caption.3}\protected@file@percent } \newlabel{fig:fits}{{3}{4}{Timing correlations showing coincidence peaks due to (a) round-trip and (b) single-trip propagation of photons in the synchronization channel. (a) $r\left (\tau \right )$: auto-correlation function $R\left (\tau \right )$ normalized to background coincidences extracted from Alice's timestamps acquired over 90\,s. (b) $g^{(2)}\left (\tau \right )$: cross-correlation function $G^{(2)}\left (\tau \right )$ normalized to background coincidences extracted from Alice and Bob's timestamps acquired over 3\,s. Solid lines: fits to heuristic model. $\tau _{AA}$ and $\tau _{AB}$: peak positions of respective distributions. Error bars: propagated Poissonian counting statistics. \relax }{figure.caption.3}{}} \citation{lee2019symmetrical} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Precision of the round-trip (red) and single-trip (blue) times, and the clock offset (black) between two clocks. Both clocks are locked to the same frequency reference. Error bars: precision uncertainty due to errors in determining the positions, $\tau _{AB}$ and $\tau _{AA}$, of the coincidence peaks. \relax }}{5}{figure.caption.4}\protected@file@percent } \newlabel{fig:precision}{{4}{5}{Precision of the round-trip (red) and single-trip (blue) times, and the clock offset (black) between two clocks. Both clocks are locked to the same frequency reference. Error bars: precision uncertainty due to errors in determining the positions, $\tau _{AB}$ and $\tau _{AA}$, of the coincidence peaks. \relax }{figure.caption.4}{}} \@writefile{toc}{\contentsline {section}{\numberline {4}Distance-independent clock synchronization with the same reference clock}{5}{section.4}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {5}Distance-independent clock synchronization with independent clocks}{5}{section.5}\protected@file@percent } \citation{Xu:2016ji} \citation{riley2008handbook} \citation{lee2019symmetrical,lamas2018secure} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces (a) Measured offset $\delta $ between two clocks, both locked on the same frequency reference. The continuous line indicates the average offset $\overline {\delta }$. Error bars: precision uncertainty due to errors in determining the positions, $\tau _{AB}$ and $\tau _{AA}$, of the coincidence peaks. Dashed lines: one standard deviation. (b) The round-trip time $\Delta T$ was changed using fiber lengths $L=L_0=10\tmspace +\thinmuskip {.1667em}$km, $L_0+1\tmspace +\thinmuskip {.1667em}$m, and $L_0+10\tmspace +\thinmuskip {.1667em}$m. $\Delta T_0 = 103.3\tmspace +\thinmuskip {.1667em}\mu $s. \relax }}{6}{figure.caption.5}\protected@file@percent } \newlabel{fig:offsetvsdist}{{5}{6}{(a) Measured offset $\delta $ between two clocks, both locked on the same frequency reference. The continuous line indicates the average offset $\overline {\delta }$. Error bars: precision uncertainty due to errors in determining the positions, $\tau _{AB}$ and $\tau _{AA}$, of the coincidence peaks. Dashed lines: one standard deviation. (b) The round-trip time $\Delta T$ was changed using fiber lengths $L=L_0=10\,$km, $L_0+1\,$m, and $L_0+10\,$m. $\Delta T_0 = 103.3\,\mu $s. \relax }{figure.caption.5}{}} \@writefile{toc}{\contentsline {section}{\numberline {6}Protocol Security}{6}{section.6}\protected@file@percent } \citation{lee2019asymmetric} \citation{lee2019symmetrical,hou2018fiber} \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces (a) Measured offset $\delta $ between two clocks with different frequency references. Each value of $\delta $ was evaluated from measuring photon pair timing correlations for 3\tmspace +\thinmuskip {.1667em}s. The offset measured at the beginning is $\delta _0$. Continuous blue line: fit used to extract the relative frequency accuracy ($\approx 5.16\times 10^{-11}$) between the clocks. (b) Residual of the fit fluctuates due to the intrinsic instability of the individual frequency references. (c) The round-trip time $\Delta T$ was changed using three different fiber lengths. \relax }}{7}{figure.caption.6}\protected@file@percent } \newlabel{fig:offsetvsdistsepclocks}{{6}{7}{(a) Measured offset $\delta $ between two clocks with different frequency references. Each value of $\delta $ was evaluated from measuring photon pair timing correlations for 3\,s. The offset measured at the beginning is $\delta _0$. Continuous blue line: fit used to extract the relative frequency accuracy ($\approx 5.16\times 10^{-11}$) between the clocks. (b) Residual of the fit fluctuates due to the intrinsic instability of the individual frequency references. (c) The round-trip time $\Delta T$ was changed using three different fiber lengths. \relax }{figure.caption.6}{}} \citation{lamas2018secure} \citation{lee2019symmetrical,hou2018fiber} \citation{hou2018fiber} \citation{hou2018fiber} \bibstyle{osajnl} \bibdata{clockserver} \bibcite{wenjun2014two}{1} \bibcite{mills:1991}{2} \bibcite{PTP}{3} \bibcite{Moreira:2009}{4} \bibcite{narula:17}{5} \bibcite{lee2019symmetrical}{6} \bibcite{hou2018fiber}{7} \bibcite{lamas2018secure}{8} \bibcite{quan2021implementation}{9} \bibcite{hong1987measurement}{10} \bibcite{glauber1963quantum}{11} \bibcite{shi2020stable}{12} \@writefile{toc}{\contentsline {section}{\numberline {7}Conclusion}{8}{section.7}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {8}Backmatter}{8}{section.8}\protected@file@percent } \bibcite{lohrmann2020broadband}{13} \bibcite{grieve2019characterizing}{14} \bibcite{ho2009clock}{15} \bibcite{bousonville2009velocity}{16} \bibcite{Xu:2016ji}{17} \bibcite{riley2008handbook}{18} \bibcite{lee2019asymmetric}{19} \newlabel{LastPage}{{}{9}{}{page.9}{}} \xdef\lastpage@lastpage{9} \xdef\lastpage@lastpageHy{9}