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{\Large\bfseries
Multi-Pulse Fitting of Transition Edge Sensor Signals from a Near-Infrared Continuous-Wave Source\par}
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{\bfseries
Jianwei~Lee$^1$,
  Lijiong~Shen$^{1, 2}$,
  Alessandro~Cer\`{e}$^1$,
  and Christian~Kurtsiefer$^{1, 2}$\par}
{\footnotesize\itshape
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\par}
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%MAIN TEXT
  Transition-edge sensors (TES) are photon-number resolving
  calorimetric spectrometers with near unit efficiency~[1].
  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~[2].
  We approach the problem by separating it into two distinct phases:
  an initial event identification, followed by a more accurate timing discrimination~[3].
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\noindent\footnotesize\textbf{Fig. 1}
Left: Typical TES
response
with overlapping pulses (a),
qualifying interval identified by a two-level discriminator (b),
fit initialization region (c),
and integration window (d).
Right: Normalized second order correlation function $g^{(2)}(\Delta t)$ for events recorded with a single TES from a coherent light field.
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We identify photodetection events using a two-level discriminator
to
coarsely locate both isolated and overlapping pulses, as in Fig.~1(Left).
For monochromatic sources
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.

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.~1(Right).

For~$\Delta t>150$\,ns the events statistics is compatible with the expected value of 1, confirming the reliability of the method.
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.

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\textbf{References}
\newcommand{\JournalTitle}[1]{#1}
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\footnotesize

% \bibitem{Lita:2010}
[1] 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}
[2] 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}
[3] 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,} 
  {\protect\JournalTitle{Rev. Sci. Instrum.}} \textbf{89}, 123108 (2018).
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