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%       submission to OSA meetings and conferences    %
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\begin{document}

\title{Multi-Pulse Fitting of Transition Edge Sensor Signals from a Near-Infrared Continuous-Wave Source}

\author{Jianwei~Lee$^1$,
  Lijiong~Shen$^{1, 2}$,
  Alessandro~Cer\`{e}$^1$,
  and Christian~Kurtsiefer$^{1, 2}$}
\address{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}
\email{cqtac@nus.edu.sg}

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\copyrightyear{2019}


\begin{abstract}
  The slow recovery time of
  Transition-edge sensors
  limits their photon number
  resolution and timing accuracy in high photon-flux conditions.
  We present a solution to resolve overlapping events to the limit of the sensors jitter time.
\end{abstract}

\ocis{040.5570, 070.6020.}

\section{Introduction}
  Transition-edge sensors (TES) are photon-number resolving
  calorimetric spectrometers with near unit efficiency~\cite{Lita:2010}.
  Their recovery time, which is on the order of microseconds, limits the number
  resolving ability and timing accuracy in high photon-flux conditions
  because of the finite probability of overlap between detection events.
  When detecting near-infrared radiation, the low signal-to-noise
  precludes the use of established techniques based on signal differentiation
  for high energy radiation~\cite{Fowler:2015ef}.
  We approach the problem by separating it into two distinct phases:
  an initial event identification, followed by a more accurate timing discrimination~\cite{Lee:2018wp}.

\section{Detection localization and fitting}
  We identify photodetection events using a two-level discriminator
  % Its resilience to noise allows us
  to
  coarsely locate both isolated and overlapping pulses, as in Fig.~\ref{fig:setup}.
\begin{figure}[htbp]
\centering
% \begin{minipage}[c]{0.48\textwidth}
%     \includegraphics[width=1\linewidth]{figure1}
% \end{minipage}
% %second column
% \begin{minipage}[c]{0.48\textwidth}
    \includegraphics[width=.55\linewidth]{figure2}
% \end{minipage}
  \caption{\label{fig:setup}
  % (Left)
  % Schematic of the TES biasing and readout electronics.
  % (Right)
  Typical TES
  response
  with overlapping pulses (a),
  qualifying interval identified by the Schmitt trigger (b),
  fit initialization region (c),
  and integration window (d).
  }
\end{figure}

  % with a moderate use of filtering,
  % thus retaining some of the high frequency components of the signal, useful to improve the time accuracy of subsequent operations.
  For monochromatic sources
  % , every detection event has the same energy.
  we can then
  estimate the number of photons for every detection region from the total pulse area, identifying the cases of overlapping events.
  From the number of photons, we calculate a heuristic model function and fit it
  to the signal
  to recover the detection-times, Fig.~\ref{fig:fit}(Left).
\begin{figure}[htbp]
  \includegraphics[width=.48\linewidth]{figure6}
  \includegraphics[width=.48\linewidth]{figure8}
  \caption{\label{fig:fit}
  (Left)
  Fit of a two-photon signal (black) and the heuristic fit model (orange)
  with its two single photon components (blue and red); electrical pulse pair driving the laser diode (purple).
  (Right)
  Normalized second order correlation function $g^{(2)}(\Delta t)$ for   events recorded with a single TES from a coherent light field.}
\end{figure}


To examine the accuracy of the fitting technique over a continuous range of time differences~$\Delta t$,
we extract the normalized second order correlation function~$g^{(2)}(\Delta t)$  for
detection events recorded with a single TES from a coherent light field, Fig.~\ref{fig:fit}(Right).

For~$\Delta t>150$\,ns the events statistics is compatible with the expected value of 1, confirming the reliability of the met
For shorter time differences,
the measured correlation
deviates from the expected behavior, including the unphysical value $g^{(2)}(\Delta t=0)>2$.
This instability region ($\Delta t<150$\,ns)
is comparable with the rise time of the average
single-photon pulse.

\section{Conclusion}

The presented technique provides an alternative to photon counting
using edge detection on the differentiated signal
when signal-to-noise ratio is low.
With the available TES, we can distinguish two photodetection events within
 $\approx150$\,ns,
a value compatible with the detector time jitter,
corresponding
to an increase in the acceptable event detection rate
from~$\approx4.0\times10^5$~s$^{-1}$, discarding overlapping pulses, to~$\approx6.7\times10^6$~s$^{-1}$.

% \bibliographystyle{osajnl}
% \bibliography{pulse_fitting}

\begin{thebibliography}{1}
\newcommand{\enquote}[1]{``#1''}

\bibitem{Lita:2010}
A.~E. Lita, B.~Calkins, L.~A. Pellouchoud, A.~J. Miller, and S.~Nam,
  \enquote{Superconducting transition-edge sensors optimized for
  high-efficiency photon-number resolving detectors,}
  {\protect\JournalTitle{SPIE Defense, Security, and Sensing}} \textbf{7681},
  76810D--76810D--10 (2010).

\bibitem{Fowler:2015ef}
J.~W. Fowler, B.~K. Alpert, W.~B. Doriese, D.~A. Fischer, C.~Jaye, Y.~I. Joe,
  G.~C. O{\textquoteright}Neil, D.~S. Swetz, and J.~N. Ullom,
  \enquote{Microcalorimeter spectroscopy at high pulse rates: A multi-pulse
  fitting technique,} {\protect\JournalTitle{ApJS}} \textbf{219}, 35 (2015).

\bibitem{Lee:2018wp}
J.~Lee, L.~Shen, A.~Cer{\`e}, T.~Gerrits, A.~E. Lita, S.~W. Nam, and
  C.~Kurtsiefer, \enquote{Multi-pulse fitting of transition edge sensor signals
  from a near-infrared continuous-wave source,} arXiv:1808.08830, to appear in
  {\protect\JournalTitle{Rev. Sci. Instrum.}}, (2018).

\end{thebibliography}

\end{document}
