%!TEX root = clock_main.tex
\begin{frame}{Distance-Independent Clock Synchronization}\footnotetext[1]{e.g. Network Time Protocol}
\begin{columns}
    \begin{column}{0.75\textwidth}
        \textbf{Forward}
        \begin{center}  
        \includegraphics[width=\textwidth]{images/classical_schematic/symmetric_clock_sync_forward.eps}
        \end{center}
        % \vspace{5cm}
        \textbf{Backward}
        \begin{center}
        \hspace*{-25pt}\includegraphics[width=0.9\textwidth]{images/classical_schematic/symmetric_clock_sync_backward.eps}
        \end{center}
    \end{column}
    \begin{column}{0.28\textwidth}
      \begin{flushright}
        $t'_f - t_f = \Delta t_{AB} + \delta$\\
        \vspace{70pt}
        $t_b - t'_b = \Delta t_{AB} - \delta$
      \end{flushright}
    \end{column}
\end{columns}
\vfill
\begin{flushright}
  % $\rightarrow$ \quad 
  \begin{equation}
    \delta = \frac{(t'_f - t_f)-(t_b - t'_b)}{2}\nonumber \text{\quad independent of } \Delta t_{AB}
  \end{equation}
\end{flushright}
\note{
\small{
Where an independent measurement of the propagation delay is impractical,
synchronisation is typically performed in the forward and backward direction.
\\
For the forward direction, the arrival time is delayed by delta, 
whereas for the backwards direction, the emission time is delayed by delta.
These conditions result in different equations for the time tags t and t', propagation delay and delta.
% 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 routinely over the Internet with macroscopic signals,
it has yet been performed with correlated photon pairs between two independent clocks.

Doing so allows us to enhance protocol security using the same tools used in quantum communication protocols.
% 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 Synchronization}
%   \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}\footnotetext[1]{Spontaneous Parametric Down-Conversion}
% 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 conserving process where a pump photon is down-converted
into two lower energy photons.

The generation time of the pairs are typically within a few hundred femtosecs, 
which results in a tight distribution detection times of the two photons.

THe width of the distribution is limited mainly by the response time of the detector than the generation process itself.

For this distribution, the spread is about 1 ns, due to the use of Si APDs.

This is expected to reduce to about a 100 ps with nanowire detectors.
% 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[2]{New J. Phys. 11, 045011 (2009)}
\end{frame}

\begin{frame}\frametitle{Symmetric Clock Synchronization 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.
We first demonstrate how to synchronize with these sources.

Alice and Bob are each given a source, an APD, and a Rubidium clock.
% 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.

For a photon pair 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, 

then the midppoint correponds to the clock offsets,

while the round trip time the separation between the peaks.

% 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 produces different peak separations, but the same mid point, and does not affect estimating the offset.

% 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 clock separations from 2 to 50 m, represented by peak pairs of different colors, 
the mid points 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 Synchronization Independent of Distance}
\begin{center}
  \includegraphics[width=\textwidth]{images/offset_against_distance/offsets.eps}
  \note{
  Plotting clock offsets, we find that they do not differ from each other by more than 7 ps for the different RTT.
  % Plotting estimated clock errors for different clock separations and find that they do not differ from each other by more than 7 ps.
  
  For 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 (45,50) {$\displaystyle \propto\frac{\sigma_j}{\sqrt{RT_a}}$}
   \put (65,50) {\textcolor{gray}{$\sigma_j$: detector jitter}}
   \put (65,45) {\textcolor{gray}{$R$: pair rate}}
  \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 is governed by 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 20\,ps} & < & \text{\quad 100\,ps} & < &\text{\quad 600\,ps}
    \end{align*}
  \end{center}
\note{
Having characterized the precision of our setup,

we now consider the problem of measuring the clock offset when the clocks have different frequency references.

Since the clocks have different `tick rates', we expect the clock error delta, to change with time, represented by the midpt shifting,

and the measured coincidence peaks, to shift likewise.

To determine the peak positions, 
% we sample the peaks at 2 sec intervals so that the 
we sample the peaks quickly enough so that 
the peak drifts are much smaller than the detector resolution, 

but we also ensure that we have enough time to
accumulate statistics 

to allow locating the peaks with sufficient precision.

We determine that measuring the offset every 2 secs meets this requirement.
}
\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 Synchronization}}
% \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 proceed with the synchronization with independent clocks, while
  % 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{Signal Verification}
\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 Synchronization:}
    % \begin{itemize}
    %   \item Ensures distance-independent clock-offset estimation
    % \end{itemize}
    % \end{block}
  \end{center} 
\note{\small
  Now that we have demonstrated the timing aspect, 
  we would like to investigate 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/counterfeited the photons with her own is to distribute and monitor polarization entanglement through the synchronization channel. 

  For e.g., quantifying polarization correlations between A1 and B2 allows verifying that Bob's photons correlate with Alice's and must therefore originate from her source. 

  This is typically performed by measuring a Bell inequality.
  % We do this with a similar setup as before, but instead of using SPDC, we use it to generate a polarization-engangled state.

  % With it, 
  % 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.

  Due to the no-cloning theorem, 
  an adversary Eve, cannot counterfeit the quantum state of the transmitted photon with 100\% fidelity, and will therefore be detected by the Bell measurement. 

  This technique counters signal spoofing attacks faced by existing protocols.
  As the physical theories underpinning this technique is well documented, 
  we instead looked for ways in which our protocol can be hacked without disturbing the distributed entanglement.
  % Monogamy of entanglement also ensures that no other photon, other than the ones in the channel are correlated with Alice, exists.

  % As these theorems have already been established, 
  % we know the security aspect will work in principle,
  % we asked ourselves what are the remaining 

  % The protocol is thus also robust against attacks that intercepts, copies, and replays the signals with an arbitrary delay.



  
  % 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.

}
  \footnotetext[1]{Monogamy of entanglement: Phys. Lett. A 360, 249 (2006).}
  \footnotetext[2]{No-cloning theorem: Nature 299, 802 (1982).}
\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}