\begin{frame}{Distance-Independent Clock Synchronisation}
\begin{columns}
    \begin{column}{0.7\textwidth}
    \begin{center}  
        \includegraphics[width=\textwidth]{images/classical_schematic/symmetric_clock_sync_forward.eps}
        \vspace{30pt}
        \hspace*{-25pt}\includegraphics[width=0.9\textwidth]{images/classical_schematic/symmetric_clock_sync_backward.eps}
    \end{center}
    \end{column}
    \begin{column}{0.3\textwidth}
      \begin{flushright}
        $t'_f - t_f = \Delta t_{AB} + \delta$\\
        \vspace{65pt}
        $t_b - t'_b = \Delta t_{AB} - \delta$
      \end{flushright}
    \end{column}
\end{columns}
\vfill
\begin{flushright}
  $\delta = \frac{(t'_f - t_f)-(t_b - t'_b)}{2}$
\end{flushright}
\note{
\small{
Where an independent measurement of the clock separation is impractical,
synchronisation is typically performed in the forward and backward directions.
\\
Suppose Bob's clock is always later by delta.
For a signal travelling forward, Bob records a later arrival time, so the time difference is greater than the physical propagation time.
For a signal travelling backward, since Bob records a later departure time, so the time difference is smaller than the physical propagation time.
\\
Solving the simultaneous equations, the offset delta can be solved without knowledge of the propagation delay.
\\
While this two-way, symmetrical synchronization is performed routintely over the internet,
there has yet been a demonstration at a single photon level between isolated, remote clocks,
In the single photon regime, we can distribute entanglement and by monitoring polarization correlations, enhance the security of the protocol.
}
}
\end{frame}

% \begin{frame}{Distance-Independent Clock Synchronisation}
%   \begin{center}    
%     \includegraphics[width=0.8\textwidth]{images/classical_schematic/symmetric_clock_sync_forward.eps}\\
%     % \fbox{$t'_f - t_f = \Delta t_{AB} + \delta$}
%     \vfill
%     % $\text{Alice}\leftarrow\text{Bob}$
%     \hspace*{-30pt}\includegraphics[width=0.72\textwidth]{images/classical_schematic/symmetric_clock_sync_backward.eps}\\
%     % \fbox{$t_b - t'_b = \Delta t_{AB} - \delta$}
%     \vfill
%   \fbox{Solve for $\delta$ without $\Delta t_{AB}$}
%   \end{center}
% \footnotetext[1]{\tiny Precision Time Protocol (PTP), Network Time Protocol (NTP)}
% \note{
% \small{
% Where an independent measurement of the clock separation is impractical,
% synchronisation is typically performed in the forward and backward directions.
% \\
% Suppose Bob's clock is always later by delta.
% For a signal travelling forward, Bob records a later arrival time, so the time difference is greater than the physical propagation time.
% For a signal travelling backward, since Bob records a later departure time, so the time difference is smaller than the physical propagation time.
% \\
% Solving the simultaneous equations, the offset delta can be solved without knowledge of the propagation delay.
% \\
% While this two-way, symmetrical synchronization is performed routintely over the internet,
% there has yet been a demonstration at a single photon level between isolated, remote clocks,
% In the single photon regime, we can distribute entanglement and by monitoring polarization correlations, enhance the security of the protocol.
% }
% }
% \end{frame}

\begin{frame}\frametitle{Time-Correlated Photon Pairs from SPDC}
% Few Hundred Femtoseconds 
\begin{center}
  \includegraphics[width=0.4\textwidth]{images/spdc/Spontaneous_Parametric_Downconversion.eps}\vfill
  \includegraphics[width=0.9\textwidth]{images/spdc/spdc_g2.eps}
  % \def\svgwidth{0.65\textwidth}
  % \import{images/spdc/}{Spontaneous_Parametric_Downconversion.pdf_tex}
  \vfill
\end{center}
\note{
We derive our photons from a process called SPDC.
Its basically an energy and momentum converving process where a pump photon is down converted
into two lower energy photons.
The generation time of the pairs are typically within a few hundreds of femtosecs.
In the graph below, the detection time differences between the photon pairs is histogramed.
We see that the difference spread is of the order of a nanosecond, and is dominated by the Si Avalanche Photodetectors used.  

[It was already shown in our group that by using one source, 
the arrival times at two different clocks can be compared to discover their time differences.
Here, we performed experiment where we used two such sources to find the error between two clocks without knowing their separation.]
% The main aim of introducing SPDC in this project is that it can be upgraded to produce entanglement between the two photons, 
% and that entanglement can be used to introduce a security feature that I'll describe later on.
}
\footnotetext[1]{ \tiny Ho et al. 2009 New J. Phys. 11 045011}
\end{frame}

\begin{frame}\frametitle{Symmetric Clock Synchronisation with SPDC}
\begin{center}
  \includegraphics[width=0.8\textwidth]{images/setup/clocksyncsetup.eps}
  % \includegraphics[width=\textwidth]{images/paper_images/figure1.eps}
\end{center}
\begin{center}
  \vspace{-1cm}
  \vfill
  \def\svgwidth{\textwidth}
  \hspace*{45pt}\import{images/offset_schematic/}{offset_concept.pdf_tex}
  % \fbox{$\delta$ = midpoint between two coincidence peaks}
\end{center}
\note{
% We first present the experiment we performed using the 
% tight time-correlation to discover clock error for various clock distances.
% The proposal to implement this in a quantum-secure way will be presented later.
The sources are used by both Alice and Bob to generate the timing signal.
Each of them also have a Si-APD, and a time-stamp unit referenced to a Rubidium oscillator.

When a photon pair is created by Alice at time t, 
one of the pair will be coupled through a FBS into Alice's Avalanche Photodiode, registering time t 
the other of the pair will be transmitted to Bob's APD, registering time t'.

When the time differences t-t' is histogrammed, Alice's source will produce a peak on the right, while Bob's source produces a peak on the left.
% The time difference for this event is histogrammed below and this results in the right peak. 
% The peak is due to the physical separation L and due to the error delta between the clocks.
% The same analysis of the photons from Bob's side results in a peak on the left.
If we assume that the forward and backward propagation times are the same, the offset can be calculated to be the mid-point of the two peaks.
% It can be shown that the peaks are separated by the round trip time.
% and if we assume that the forward and backward progogation times are the same, 
% then the offset is given by the mid pt of the two peaks. 
The peak separation, instead, represents only the round trip time.
As a result, changing the clock separation L results in different peak separations, but the same mid point.

% To demonstrate the the protocol measures the same offset independently of the clock seperation L, 
% we impose a constant offset between the clocks by referencing both time stamps to a common frequency reference while switching between different fibres of different L.
}
\end{frame}

\begin{frame}\frametitle{Constant $\delta$, Varying $L$\\Common Frequency Reference}
% \frametitle{Experimental Results\\\small{Common Frequency Reference $\rightarrow$ Constant $\delta$}}
\begin{center}
  % \includegraphics[width=0.9\textwidth]{images/g2/g2.eps}
  \includegraphics[width=0.9\textwidth]{images/paper_images/figure3.eps}
\end{center}
\note{
And this is exactly what we observe - 
for various lengths L ranging from 2 to 50 m, represented by peak pairs of different colors,
the mid points of the peaks can be coincide with a precision of a few picoseconds.
% The blue peaks represents the shortest separation of 1.7 m.
% The orange peaks, 51.7 m are the most widely separated.
% We observe that the mid point of the peaks,
% which is the error between the two clocks,
% coincide quite well for different separations L.
% Having an average value of about 3.8 ms.
% Let's have a closer look at the clock error.
}
\end{frame}

\begin{frame}\frametitle{Clock Synchronisation Independent of Distance}
\begin{center}
  \includegraphics[width=\textwidth]{images/offset_against_distance/offsets.eps}
  \note{

  We plot the estimated clock errors for different clock separations and find that they do not differ from each other by more than 7 ps.
    In this acquisition, the pair rate was a modest 200 pairs per sec and each data point was measured in 20 seconds.
    }
\end{center}
\end{frame}

\begin{frame}{Synchronization Precision\\over Acquisition Time}
  \begin{overpic}[width=\textwidth,tics=10]{images/paper_images/figure2.eps}
   \put (60,50) {$\displaystyle \propto\frac{1}{\sqrt{T_a}}$}
  \end{overpic}
  % \includegraphics[width=\textwidth]{images/paper_images/figure2.eps}
  \note{We repeated the measurement for various acquisition times for a constant pair rate of 200 pairs per second and observed that the precision improves with Poissonian statistics as expected.}
\end{frame}

\begin{frame}{}
  \begin{center}
    With Independent Frequency References\\
    the Clock offset $\delta$ drifts with time\\
    \vspace{30pt}
    \def\svgwidth{\textwidth}
    \hspace*{40pt}\import{images/frequency_tracking/}{offset_concept.pdf_tex}
    \vspace{-5pt}
    \begin{align*}
      &\text{precision} & < & \text{\quad peak drift} & < &\text{\quad detector resolution} \\[5pt]
      \text{for 2 sec:\quad} &\text{\quad 0.02\,ns} & < & \text{\quad 0.10\,ns} & < &\text{\quad 0.60\,ns}
    \end{align*}
  \end{center}
\note{
Having characterised the precision of our setup,
Next, we consider the problem of measuring the clock offset when the clocks have different frequency references. 
In this case, we expect the error delta, to change with time.
And the measured coincidence peaks, to shift likewise.

To determine the peak positions, we sample the peaks at 2 sec intervals so that the drifts are much smaller than the detector resolution, while sufficient data is collected to achieve a precision better than the drift.
}
\end{frame}

% \begin{frame}{Clock Synchronization with \\Independent Frequency References\\\small{Clock offset $\delta$ drifts}}
% % {Drift of Difference $\delta$ with Time\\\small{Frequency Difference between Clocks\\\vspace*{-10pt}requires Constant Synchronisation}}
% \vspace*{-20pt}
% \begin{center}
%   \def\svgwidth{\textwidth}
%   \hspace*{40pt}\import{images/frequency_tracking/}{offset_concept.pdf_tex}
% \begin{block}{}
%   \vspace*{-20pt}
%   \begin{align*}
%     &\text{precision}^{1} & < & \text{\quad peak drift}^{1,2} & < &\text{\quad detector resolution} \\
%     &\text{\quad 0.02\,ns} & < & \text{\quad 0.10\,ns} & < &\text{\quad 0.60\,ns}
%   \end{align*}
% \end{block}
% \end{center}
% \footnotetext[1]{acquisition time = 2\,sec}
% \footnotetext[2]{frequency inaccuracy < 5e-11}
% % \footnotetext[1]{ \tiny Ho et al. 2009 New J. Phys. 11 045011}
% \note{
% Having characterised the precision of our setup,
% Next, we consider the problem of measuring the clock offset when the clocks have different frequency references. 
% In this case, we expect the error delta, to change with time.
% And the measured coincidence peaks, to shift likewise.

% To determine the peak positions, we sample the peaks at 2 sec intervals so that the drifts are much smaller than the detector resolution, while sufficient data is collected to achieve a precision better than the drift.
% }
% \end{frame}

\begin{frame}\frametitle{Drifting $\delta$, Varying $L$\\Independent Frequency References}
% \frametitle{Experimental Results\\\small{Independent Frequency Reference $\rightarrow$ Changing $\delta$}}
% {Clock Synchronization with \\Independent Frequency References}
  \includegraphics[width=\textwidth]{images/offset_against_distance_individual_atomic_clocks/offsets_separate_atomic_clocks.eps}
  \note{
  We then measured the clock offset while changing the clock separation,
  changing the round-trip time by more than an order of magnitude.
  We observed that 
  - the offset changes approximately linearly at a rate consistent with the frequency inaccuracy expected between two Rb clocks
  - the offset fluctuates about this line with a allan deviation consistent with frequency instability of each clock.
  This demonstrates that we can recover meaningful clock parameters even while a symmetric delay attack is under way.
  }
\end{frame}

% \begin{frame}
% Outlook
% \end{frame}

\begin{frame}\frametitle{Secure Clock Synchronisation}
\vfill
  \begin{center}    
    \begin{flushleft}
        $\ket{\psi^-} = \frac{1}{\sqrt{2}} \left( \ket{HV} - \ket{VH}\right)$
    \end{flushleft}
    % \includegraphics[width=\textwidth]{images/secure_setup/setup_secure.eps}
    \def\svgwidth{0.9\textwidth}
    \import{images/secure_setup/}{setup_secure.pdf_tex}
    \vfill
    \begin{block}{Violation of Bell's Inequality:}
    \begin{itemize}
      \item Ensures photons detected are part of the same entangled~pair
    \end{itemize}
    \end{block}
    \begin{block}{Symmetric Clock Synchronisation:}
    \begin{itemize}
      \item Ensures distance-independence of clock-differences
    \end{itemize}
    \end{block}
  \end{center} 
\note{
  Now that we have demonstrated the timing aspect of the protocol, we would like to look further into the security aspect. 

  So far we have assumed that the photons received by Alice and Bob do originate from their intended sources.
  One way to check that an adversary has not somehow replaced the photons with her own photons is to perform a Bell inequality.

  We do this with a similar setup as before, but instead of using SPDC, we use a psi - state.

  By using a polarisation-entangled photon pair source, 
  Alice can perform polarisation correlation measurements with detectors at A1 and B2 to check if the photons were indeed part of the same pair originating from Alice.
  And Bob can do likewise.
  % The violation is then compared against channel noise, source and detector non-idealities.
  % This analysis is still work in progress by my colleagues.
  % We have yet to analyse the secure precision of the protocol, i.e. the trustworthy precision of the clock error given the violation. 
  % The clock error estimation is the same as before - by comparing time differences between the two clocks, and retains the same advantages, that it is independent of the distance between Alice and Bob.

}
\end{frame}

% \begin{frame}{Asymmetric Delay Attacks}
%   \begin{block}{We have assumed that delays were symmetric:}
%   \begin{center}
%   % \vfill
%   \vspace*{-15pt}
%   \def\svgwidth{\textwidth}
%   \hspace*{45pt}\import{images/offset_schematic/}{offset_concept.pdf_tex}
%   \fbox{$\delta$ = midpoint between two coincidence peaks}
% \end{center}
%   \end{block}
%   \begin{block}{To avoid asymmetric delay attacks:}
%     \begin{itemize}
%       \item  Alice and Bob's photons must have the same spectra~\&~spatial mode
%       \end{itemize}
%   \end{block}
%   \note{
%     So far, we have assumed that the delay between Alice and Bob is the same in both directions.
%     We histogram t and t' and assume that the midpoint between the peaks is the clock error.
%     However, this may not be necessarily true and
%     Violating Bell's inequality in the polarization degree of freedom will not by itself guarantee that the timing information has not been manipulated.

%     A simple way to avoid such an attack, 
%     is to make sure that the direction of the photons is uncorrelated with any other degree of freedom such as the frequency spectra or spatial modes.
%     For example, if the photons have different frequencies, 
%     Eve can filter out Bob's photons from Alice's photons and add a delay to their path.
%   }
% \end{frame}