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

\title{Upper Bound on the Duration of Quantum Jumps}

\author{Mathias~A.~Seidler$^1$,
  Alessandro~Cer\`{e}$^1$,
  Ricardo~Guti\'errez-J\'auregui$^2$,
  Roc\'io~J\'auregui$^3$,
  and Christian~Kurtsiefer$^{1, 4}$}
\address{1. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543\\
  2. Institute for Quantum Science and Engineering, Texas A\&M University, College Station, TX 77843, USA\\
  3. Instituto de F\'isica, Universidad Nacional Aut\'onoma de M\'exico, M\'exico D.F., M\'exico\\
  4. Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542}
\email{e-mail address}

%% Uncomment the following line to override copyright year from the default current year.
\copyrightyear{2019}


\begin{abstract}
We estimate the time scale of quantum jumps from the
time correlation of photon pairs generated from a cascade decay in a cold cloud of~$^{87}$Rb.
We find an upper bound of~$21\pm11$\,ps.
\end{abstract}

\ocis{270.0270, 000.2659, 270.5290}

\section{Introduction}
  Since the original formulation of the quantum theory~\cite{Bohr:1913gl},
  there is no satisfactory description of
  % the dynamics of
  the transition between discrete energy states of atoms, or quantum jumps.
  The established technique for observing quantum jumps, based on shelving configuration~\cite{Itano:2015ee}, has an intrinsically limited time resolution because it relies on a null measurement~\cite{Pegg:1991hp}.

  In this work~\cite{Seidler:2018uo} 
  we consider an alternative configuration for the observation of quantum jumps in atomic systems: a monitored cascade three-level system.
  The second order time correlation of the generated photon pair (Fig.~1(Left))
  shows an exponential decay, associated with spontaneous decay, and a sharp rise, associated with the quantum jump.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=.25\columnwidth]{levels.pdf}
 \includegraphics[width=.48\columnwidth]{figure3.pdf}
 \caption{\label{fig:diamond}
  (Left)
  Atomic level configuration in a four-wave mixing experiment
  and
  time evolution of the population of level $\vert 3\rangle$, conditioned on the detection of a signal photon.
  (Right)
  Histogram \(G_{\FWM}(\Delta t)\) of detection time differences for photons
   pairs generated by four-wave mixing in the cold cloud of~$^{87}$Rb.
   The continuous line shows the result of the best fit to the heuristic model.
   % Eq.~(\ref{eq:fit_model}).
   Inset: detail of the sharp rise corresponding to a quantum jump.
  }
  \end{figure}

\section{Experiment}
  Pump beams at 780\,nm and 776\,nm excite $^{87}$Rb atoms in a cold cloud from the $5S_{1/2},F=2$
  ground level to the $5D_{3/2},F=3$ level via a two-photon transition.
  The
  signal (762\,nm) and idler (795\,nm) photons emerge from a cascade decay
  back to the ground level through the $5P_{1/2}$ level, and are coupled to single mode fibers.
  Phase matching ensures
  all four modes propagating in the same direction.
  The linearly polarized pump mode at
  780\,nm is red-detuned by 40\,MHz from the $5S_{1/2},F=2$ to $5P_{3/2},F=3$ while the orthogonally polarized pump mode at 776\,nm is tuned such that the two-photon
  excitation is blue-detuned by 4\,MHz from the difference between the ground
  state and the $5D_{3/2}, F=3$ level.
  We record the detection event time differences of the photon pairs with
  an
  effective time resolution
  below 10\,ps (Fig.~1(Right)); the single photon avalanche detectors themselves have a nominal
  timing jitter around 50\,ps FWHM.

  To improve the time resolution for observing any possible jump dynamics, we
  measured the impulse response of the single photon detectors
  using
  photon pairs with a large optical bandwidth generated by
  spontaneous parametric down-conversion in a nonlinear optical crystal.

\section{Results and Conclusions}
  We do not have a model for the transition dynamics,
  but we can establish an upper bound on its duration
  using a smooth heuristic transition that
  admits the step function as a limiting scenario to describe it
  \begin{equation}\label{eq:sigmoid}
      \sigma (x)\:=\:\frac{1}{1+\exp\left(-x\right)}\,
  \end{equation}
  combined with an exponential decay
  \begin{equation}\label{eq:jump_model}
    y(\Delta t; \alpha, \tau) \:=\:
    \sigma\!\left(\frac{\Delta t}{\alpha}\right)
    \,\exp\left( -\frac{\Delta t}{\tau}\right)\,.
  \end{equation}

  From combining Eq.~(\ref{eq:jump_model})  with the measured detectors impulse response and fitting the resulting functions to the measured time correlation in Fig.~1(Right)
  we estimate$\alpha = 4.7 \pm 2.5$\,ps, corresponding to
  a~10\%--90\% rise time associated with the jump of~$21\pm11$\,ps.
\begin{thebibliography}{1}
\newcommand{\enquote}[1]{``#1''}

\bibitem{Bohr:1913gl}
N.~Bohr, \enquote{I. on the constitution of atoms and molecules,}
  {\protect\JournalTitle{The London, Edinburgh, and Dublin Philosophical
  Magazine and Journal of Science}} \textbf{26}, 1--25 (1913).

\bibitem{Itano:2015ee}
W.~M. Itano, J.~C. Bergquist, and D.~J. Wineland, \enquote{Early observations
  of macroscopic quantum jumps in single atoms,}
  {\protect\JournalTitle{International Journal of Mass Spectrometry}}
  \textbf{377}, 403--409 (2015).

\bibitem{Pegg:1991hp}
D.~T. Pegg, \enquote{Wavefunction collapse time,}
  {\protect\JournalTitle{Physics Letters A}} \textbf{153}, 263--264 (1991).

\bibitem{Seidler:2018uo}
M.~A. Seidler, R.~Guti{\'e}rrez-J{\'a}uregui, A.~Cer{\`e}, R.~J{\'a}uregui, and
  C.~Kurtsiefer, \enquote{{Upper bound on the duration of quantum jumps},}
  {\protect\JournalTitle{arXiv:1812.00129}}  (2018).

\end{thebibliography}

\end{document}
