Precision measurement of the metastable $^3{\mathrm{P}}_2$ lifetime of neon %using laser-cooled trapped atoms
Martin Zinner
The lifetime of the metastable $^3{\mathrm{P}}_2$ state of neon has been determined to $14.70(13)\, \mathrm{s}$ (decay rate $0.06801(62)\, \mathrm{s}^{-1}$) by measuring the decay in fluorescence of an ensemble of $^{20}$Ne atoms trapped in a magneto-optical trap (MOT). Due to low background gas pressure ($p < 5 \times 10^{-11} {\mbox{mbar}}$) and low relative excitation to the $^3{\mathrm{D}}_3$ state (0.5 \% excitation probability) operation only small corrections have to be included in the lifetime extrapolation. Together with a careful analysis of residual loss mechanisms in the MOT a lifetime determination to high precision is achieved.
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% 
% 3P2  $^3{\mathrm{P}}_2$
% 3D3  $^3{\mathrm{D}}_3$
% (Fig.~\ref{fig:background}(a))
% $\alpha^{-1} = 14.4^{-1}$s 
%
%
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%\preprint{ \today}





\title{Precision measurement of the metastable $^3{\mathrm{P}}_2$ lifetime of neon
%using laser-cooled trapped atoms
}
% Force line breaks with \\

\author{Martin Zinner}
\author{Peter Spoden}
\author{Tobias Kraemer}
\author{Gerhard Birkl\footnote{Author to whom correspondence should be addressed.}}
%\footnote{Author to whom correspondence should be addressed.}
\author{Wolfgang Ertmer}

\affiliation{%
Institut f\"ur Quantenoptik, Universit\"at Hannover, Welfengarten 1, D-30167 Hannover
}%

\date{\today}

\begin{abstract}

The lifetime of the metastable $^3{\mathrm{P}}_2$ state of neon
has been 
determined
to $14.70(13)\, \mathrm{s}$ (decay rate $0.06801(62)\, \mathrm{s}^{-1}$)
by measuring the decay in fluorescence of an ensemble of
$^{20}$Ne atoms trapped in a magneto-optical trap (MOT).
Due to low background gas pressure ($p < 5 \times 10^{-11} {\mbox{mbar}}$)
and low relative excitation to the $^3{\mathrm{D}}_3$ state
(0.5 \%
excitation probability) operation only small corrections
have to be included in the lifetime extrapolation.
Together with a careful analysis of residual loss mechanisms in the MOT
a lifetime determination to high precision is achieved.
\end{abstract}

\pacs{32.70.Cs, 32.70.Fw, 32.80.Pj}
%\keywords{Suggested keywords}%Use showkeys class option if keyword
                              %display desired
\maketitle


In contrast to its importance for various active fields of research
covering such a wide range as
atomic physics, quantum optics, and nuclear physics,
there still has been no
measurement of the natural lifetime of the metastable
$^3{\mathrm{P}}_2$ state \cite{LS} of neon with sufficient precision.
A selection of research activities
profiting from an improved measurement
include the quest for
Bose-Einstein condensation (BEC) of metastable neon,
advanced atomic structure calculations, the investigation
of ultracold collisions, and even precision tests of the
electroweak theory being currently persued by
investigating the nuclear decay of an optically trapped sample of 
$^{19}$Ne in the $^3{\mathrm{P}}_2$ state
\cite{berkeley}.

Ongoing research directed towards
BEC of metastable neon atoms
in the $^3{\mathrm{P}}_2$ state \cite{Beijerinck,uns} clearly will
benefit from an accurate knowledge of the state's lifetime.
This includes the optimization of the production process as well as
the study of collision processes such as Penning-ionization \cite{Penning} 
and elastic s-wave scattering.
Exciting new physics 
complementing the work on metastable helium \cite{HeBEC1,HeBEC2}
can be expected for  
a $^3{\mathrm{P}}_2$ neon condensate.
We would like to point out
the study of higher-order correlations \cite{Correlations} and 
the intriguing possibility of a modification of the
$^3{\mathrm{P}}_2$ decay rate due to the high phase-space
density or the phase coherence in a BEC \cite{Savage}.

Significant advance in atomic structure
calculations has already been and will be
further triggered \cite{Kim} by the work presented here.
To date only a preliminary experimental value of 22 s for the
$^3{\mathrm{P}}_2$ lifetime of neon exists \cite{ShimizuNeon}
with no detailed investigations having been performed \cite{Shimizuprivate}.
A precise determination of the lifetime will
put a more stringent test on theory.
For most rare-gas atoms, the $^3{\mathrm{P}}_2$ lifetime depends sensitively on 
electron correlations and relativistic corrections.
The latter have only a minor effect
for the case of neon, making neon specifically interesting
for a critcal test of  electron correlations.
In addition, our measurement will
close the chain of $^3{\mathrm{P}}_2$ lifetime measurements for rare-gas atoms
beyond helium performed in recent years
for Ar ($38_{-5}^{+8}\, \mathrm{s}$ \cite{ShimizuArKr}),
Kr ($39_{-4}^{+5}\, \mathrm{s}$ \cite{ShimizuArKr} and $28.3_{-1.8}^{+1.8}\, \mathrm{s}$ \cite{Walhout}),
and Xe ($42.9_{-0.9}^{+0.9}\, \mathrm{s}$ and $42.4_{-1.3}^{+1.3}\, \mathrm{s}$ \cite{Rolston}), respectively.
This will deliver the input for
a detailed systematic investigation of the Z-dependence of
metastable lifetimes.

%%%%%%%%%%%%%%%%%%%%%%%Figure 1

\begin{figure}[b]
\centering
%    \includegraphics[width=220pt, keepaspectratio]{Xdecay.eps}
    \includegraphics[width=200pt, keepaspectratio]{LIFEFIG1.eps}
\caption{Decay of the fluorescence for a $^{20}$Ne ensemble in a MOT
for the given excitations to the $^3{\mathrm{D}}_3$ state of
$\pi_3=0.005$ ($\Delta/\Gamma=-5.3$, $I/I_0=1.75$, 
$p = 3.4 \times 10^{-11} {\mbox{mbar}}$) and
$\pi_3 = 0.47$ ($\Delta/\Gamma=-0.16$, $I/I_0=24$,
$p = 4.7 \times 10^{-11} {\mbox{mbar}}$).
Both sets of data 
consist of five curves each, demonstrating
the
high reproducibility of our decay measurements.
}
\label{fig:curves}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%

An accurate experimental determination of
the $^3{\mathrm{P}}_2$ lifetimes of rare-gas atoms
other than helium has only been achievable by the ability
to prepare cold atom samples in a MOT
\cite{ShimizuNeon, ShimizuArKr, Rolston, Walhout}.
For a precision lifetime measurement,
the coupling of the $^3{\mathrm{P}}_2$ to the $^3{\mathrm{D}}_3$ state
by the MOT light 
has to be considered
since it will modify
the observed decay rate.
To date, two different methods making use of a
variable MOT-on/off duty cycle have been employed,
either by directly measuring the rate of UV photons produced 
by the decay during a MOT-off period \cite{Rolston,Walhout}
or by extrapolating the observed decay rate for varying MOT-on/off duty ratios to vanishing
MOT-on periods \cite{ShimizuArKr}.

In our approach, we record the decay of fluorescence 
 in a
MOT (Fig.~\ref{fig:curves}) for different steady-state values of the population $\pi_3$
of the $^3{\mathrm{D}}_3$ state ($0.005 \!<\! \pi_3 \!<\! 0.47$). This
allows the extrapolation of the observed decay rate to $\pi_3 =
0$ thus giving the $^3{\mathrm{P}}_2$ lifetime. 
Extreme care has been taken
to achieve optimized experimental conditions such as low background
gas pressure ($p < 5 \times 10^{-11} {\mbox{mbar}}$) and 
MOT operation for such low values as $\pi_3 = 0.005$. This
minimizes the amount of required corrections dramatically
and the accuracy to which the  $^3{\mathrm{P}}_2$ lifetime of
Ne can be determined is improved considerably.

With no observable changes in shape and temperature of the atom sample, 
the temporal evolution of the observed fluorescence is proportional to
the temporal evolution of the number of trapped atoms $N\!(t)$.
The decay can be described by the differential equation
%
\begin{equation} \label{decay}
\frac{\mbox{d}}{\mbox{d}t}N\!(t)=-\alpha N\!(t) - \beta\! \int \!\! n^2\!(r,t) \, \mbox{d}^3\!r + \mathcal{O}(n^3).
\end{equation}
%
The experimental fluorescence data is fitted to a solution of this
equation. No absolute calibration of the atom number is required
for the determination of the decay constant $\alpha$. In
Fig.~\ref{fig:curves}, non-exponential two-body decay 
due to intra-trap collisions can be seen during the first 30 seconds.
Two-body losses depend on the integral over the
local number density $n(r,t)$ and are described by the parameter
$\beta$, the absolute determination of which
would require an absolute calibration
of $n(r,t)$, which is not the scope of this work.
% The empirical
An uncalibrated value of $\beta$, however, is determined in our
fitting procedure.
% although no absolute value is accessible since
%we did not pursue an absolute calibration of $n(r,t)$.
% due to
%the lack of an absolute calibration of $N(t)$.
For all data runs
no contributions of order $n^3$ or higher could be observed.



%%%%%%%%%%%%%%%%% Figure 2


\begin{figure}[b]
\centering
    \includegraphics[width=150pt, keepaspectratio]{LIFEFIG2.eps}
\caption{Detail of the level scheme of neon. Atomic levels are labeled according to 
LS- and Racah notation.
%In this notation the $^3{\mathrm{P}}_2$ state reads $3s[3/2]_2$.
Transitions are labeled by their multipolar character.
}
\label{fig:scheme}
\end{figure}


%%%%%%%%%%%%%%%%%



At low atom numbers
the decay is dominated by one-body losses with rate
%This exponential decay is caused by the finite $^3{\mathrm{P}}_2$ decay rate,
%by the finite decay rate from the 3D3 state to other states than the 3P2 state
%of the metastable state
%as well as collisions with background atoms
%and other residual losses from the MOT.
%
%The decreasing number of atoms in the MOT shows initially a non-exponential two-body decay
%which is caused by intra-trap collisions,
%which are described by the parameter $\beta$ and depend on the square of the local number density $n(r,t)$.
% Hence, the whole decay can be described by the differential equation ($N$ being the number of atoms in the trap)
%\begin{equation} \label{decay}
%\frac{\mbox{d}}{\mbox{d}t}N\!(t)=-\alpha N\!(t) - \beta\! \int \!\! n^2\!(r,t) \, \mbox{d}^3\!r + \mathcal{O}(n^3).
%\end{equation}
%
%
%The one-body decay rate may be described by
\begin{equation} \label{alpha}
%\alpha = \frac{1}{\tau_2} \left( 1 - \pi_3 \right)  + \frac{1}{\tau_3} \pi_3  + \gamma p +\Xi(B').
\alpha = \frac{1}{\tau_2} \left( 1 - \pi_3 \right)  + \frac{1}{\tau_3} \pi_3  + \gamma p +\mathcal{ML}.
\end{equation}
%
The parameter $\pi_3$ gives the population in the
$^3{\mathrm{D}}_3$-state. Thus $(1-\pi_3)N$ atoms decay from the
$^3{\mathrm{P}}_2$-state with lifetime $\tau_2$.
%whereas $(1-\pi_3)$ is the population in the
%$^3{\mathrm{P}}_2$-state, respecitvely, $\tau_2$ is the natural
%lifetime of the $^3{\mathrm{P}}_2$-state, and
The combined
rate for all decay channels from
the $^3{\mathrm{D}}_3$-state other than the transition to the
$^3{\mathrm{P}}_2$-state (see Fig.~\ref{fig:scheme})
is given by $\tau_3^{-1}$. Background
collisions contribute with a rate of $\gamma p$ \cite{MOT1} with $p$ being the
pressure in the vacuum chamber and $\gamma$ being a constant
depending on the atomic and molecular species present.
% in thevacuum chamber \cite{MOT1}.
The parameter $\mathcal{ML}$ summarizes possible MOT losses.
% from the MOT which can be attributed to a finite trap
%depth $U_0$ or finite escape velocity $v_e$.
The main objective of this work is to determine the lifetime
$\tau_2$ from the observed decay rates $\alpha$ by extrapolating
to vanishing excitation ($\pi_3=0$), background pressure
($p=0$), and MOT losses ($\mathcal{ML} = 0$).
Achieving optimized starting conditions (small $\pi_3$ and $p$)
and minimizing 
potential contributions of $\mathcal{ML}$ by a systematic analysis
of the MOT characteristics have been essential.

%should be the averaged Clebsch-Gordan coefficient of all transitions occuring in the MOT's light field \cite{MOT2}.
%Empirically it is found for a helium MOT \cite{HeMOT} that $C$ deviates from the theoretical value. We adopt the notion of Kuppens et al.\cite{Beijerinck}
% and take $C=0.7 \pm 0.3$.

%For a given $\pi_3$ the number of atoms in the $3s[3/2]_2$-state is $(1-\pi_3)N$, whereas the number of atoms in the $3p[5/2]_3$-state is $\pi_3 N$.
%The one-body decay rate may then be described by
%\begin{equation} \label{alpha}
%\alpha = \frac{1}{\tau_2} \left( 1 - \pi_3 \right)  + \frac{1}{\tau_3} \pi_3  + \gamma p +\Xi(B').
%\alpha = \frac{1}{\tau_2} \left( 1 - \pi_3 \right)  + \frac{1}{\tau_3} \pi_3  + \gamma p +\it{ML}.
%\end{equation}
%
%$\tau_2$ is the lifetime of the $3s[3/2]_2$-state, $\tau_3$ accounts for the possibility
%of a decay channel of the $3p[5/2]_3$-state to the ground state of the
%atom other than the transition via the $3s[3/2]_2$-level (Fig.~\ref{fig:scheme}).
%$p$ is the pressure in the vacuum chamber,
%$\gamma$ is a constant depending on the atomic and molecular
% species present in the vacuum chamber and their collisional cross section \cite{MOT1}
% with metastable neon atoms, $\mathcal{ML}$ summarizes other possible loss mechanisms,
% which can be attributed to a finite trap depth $U_0$ or escape velocity $v_e$.
%
%We infer from a simple two-level model of the MOT that $U_0$ is a function of the applied magnetic
% field's gradient $B'$, namely that $U_0$ diverges as $B'$ tends to $0$.
%In addition, the distance from the trap centre $d_0$ where $U_0$ is reached is a function of $B'$
%and increases with a decreasing gradient.
%Thus, below a certain value of $B'$ this distance is greater than the radius of
%the MOT's laser beams.
% In this case the effective trap depth is limited by the size of the laser beams applied
% and decreases as $B' \to 0$.
% Moreover, $v_e$ is proportional to $d_0$ (e.g. \cite{Weiner}) and therefore
% depends on $B'$.
%From this, we conclude that $\Xi \to 0$ for $B' \to 0$ if the radius
% of the laser beams would be infinite.
% In the limit of ${\tau_3} \to \infty$ and $p \to 0$ the lifetime $\alpha^{-1}$ of the MOT should double by changing $\pi$ from $0$ to $0.5$.
%
%The lifetime $\tau_2$ can be determined from the observed decay rates $\alpha$ by
%extrapolating
%to a vanishing excitation ($\pi_3=0$), to vanishing
%background pressure ($p=0$), and to an infinite trapping volume ($B'=0$).
%The experimental setup is shown in Fig.\ref{fig:setup}.
%and will be described in detail {\bf ELSEWHERE}.
%A metastable atomic beam is generated
%by a DC-discharge and is collimated using two dimensional laser cooling on the degrees of motion perpendicular %to the beam axis.
 %All laser-cooling schemes for $^{20}$Ne employed  in this experiment
%rely on the $640.2 \mbox{nm}$-transition from the $3s[3/2]_2$-state to the $3p[5/2]_3$-state, with a linewidth of
% $\Gamma = 2 \pi \! \times \! 8.18 \mbox{MHz}$ and
% a saturation intensity of $I_s = 4.08 \mbox{ mW/cm}^2$\cite{database}.
%A dye laser, which is frequency stabilized by Doppler free saturation spectroscopy on a neon RF-discharge,
%is used to generate the necessary light fields for laser cooling.
%  The flux of atoms reaching the MOT through several apertures is on the order of $10^{10} \mbox{s}^{-1}$
%including a factor of 40 enhancement due to the collimation stage.


%%%%%%%%%%%%% Figure 3


\begin{figure}[b]
\centering
    \includegraphics[width=200pt, keepaspectratio]{LIFEFIG3.eps}
\caption{Experimental setup.}
\label{fig:setup}
\end{figure}


%%%%%%%%%%%%%%%%%%


%Applying standard laser-cooling theory, 
Following \cite{MOT2}, the population $\pi_3$
%of the laser-excited $^3{\mathrm{D}}_3$ state
is determined by:
\begin{equation}
\pi_3 = \frac{1}{2} \,\frac{C \cdot I/I_0}{1 + C \cdot I/I_0 + \left(  2 \Delta / \Gamma \right)^2 }.
\end{equation}
with 
%$I/I_0$ given by 
the total intensity $I$ of the MOT light fields,
the saturation intensity $I_0=\pi h c \Gamma / 3 \lambda^3=4.08\, \mathrm{mW/cm}^2$
(linewidth $\Gamma= 2 \pi \! \times \! 8.18\, \mathrm{MHz}$,
$\lambda = 640.4 \,\mathrm{nm}$) and detuning $\Delta$.
%\cite{database}.
The phenomenological parameter $C$
accounts for the deviation of the effective total intensity in a three-dimensional MOT from a
one-dimensional two-level system \cite{MOT2}.
%It was found (e.g. \cite{HeMOT}) that $C$ deviates from the expected theoretical value \cite{MOT2}.
We adopt the notion of \cite{Beijerinck}
and take $C=0.7$ with an uncertainty of $0.3$.
%The population $\pi_3$ is calculated from measured values of the intensity $I$
%and the detuning $\Delta$.
The intensity $I$ is
%obtained by measuring the total power of light entering the MOT with a
%calibrated photodiode
measured with an uncertainty
%in the absolute value and the temporal variation of $I$
%during one decay cycle
%of two minutes
of 10 \%, 
the detuning $\Delta$
%$= \omega_{\mathrm{MOT}} - \omega_{0}$
%is obtained by determining the
%frequency difference of the MOT light and
% $\omega_{\mathrm{MOT}} $
%the $^3{\mathrm{P}}_2 \to ^3{\mathrm{D}}_3$-transition
%can be determined
with an absolute uncertainty of $1\, \mathrm{MHz}$.
% $\omega_{0}$.
%Observed long time frequency drifts of the dye laser were on the order of $240\, \mathrm{kHz/h}$.
%To account for frequency drifts of the dye laser the resonance frequency
%was checked in between the decay measurements on the fluorescence of an ensemble of trapped atoms.
% The frequency drift observed was on the order of $240 \mathrm{kHz/h}$ and was taken into account
%for the determination of $\Delta$.
Therefore, for all data presented here the uncertainty in $C$ is
the dominant contribution to the uncertainty in $\pi_3$.
%, the
% with
%the absolute uncertainty of which% in $\pi_3$
%is minimized for the smallest and largest
%accessible values of $\pi_3$.
%We would like to stress the fact that for very low (high) population,
%i.e. $\pi_3\!\to\!0\,(0.5)$,
%the uncertainty in $\pi_3$  is minimized.
%Accordingly, the two sets of data in Fig.~\ref{fig:curves} have
%been taken for these values.
%Low population ($\pi_3=0.005$) is
%obtained for $\Delta/\Gamma=-5.3$ and $I/I_0=1.75$, large
%population ($\pi_3 = 0.47$) for
%$\Delta/\Gamma=-0.16$ and $I/I_0=24$.

The experimental setup is shown in Fig.~\ref{fig:setup}.
A collimated beam of me\-ta\-stable atoms is decelerated
%from about $650 \mbox{m/s}$ to a final velocity of circa $50 \mbox{m/s}$
in a Zeeman-slower
with the slowing laser beam passing through the center of the MOT.
%This laser beam is circularly polarized and detuned about 20 linewidths above resonance.
%The decelerated atoms are captured in a magneto-optical trap.
The MOT light field consists of six individual beams (diameter
$22\,\mbox{mm}$) with spatially filtered beam profiles.
%and form three mutually orthogonal
%pairs.
%The laser beam of the MOT is
%frequency shifted by an acousto-optical modulator
%{\bf CRYSTAL TECHNOLOGY}
%and passes a single-mode glass fibre for spatial filtering
%spatially filtered using a single-mode glass fibre
%before being split up into six individual beams which form three pairs of mutually perpendicular $\sigma^+ \!\! %- \!\! \sigma^-$-beams
% which have a diameter of 22mm.
%The MOT quadrupole magnetic field is generated by two field coils.
 %of a  Ioffe-Pritchard type magnetic trap, used for magnetic trapping of metastable neon.
%giving gradients
%between $0.3$ and $95 \,\mbox{G/cm}$ along the field's symmetry axis.
%In order to produce an atomic ensemble of nearly isotropic shape
%the intensity of the two MOT laser beams along the symmetry axis
%of the magnetic field is set a factor of 2 lower relative to the
%four other beams. 
All light fields are derived from a dye laser
which is frequency stabilized
%by Doppler free saturation spectroscopy
to a neon RF-discharge.


%distributed among the different MOT laser beams
%In order to produce a nearly spherical cloud of atoms the light intensitywas a factor of two higher in the
%beams perpendicular to the symmetry axis of the magnetic field coils
%than in those parallel to it.
For each
%experimental
 run, the MOT is loaded for
$400\,\mathrm{ms}$ reaching a number of up to $4 \! \times \!
10^8$ atoms, a maximum collision-limited density on the order of $1 \!
\times \! 10^9 \,\mathrm{cm}^{-3}$
and a typical temperature below 1 mK. Then, the loading process is
terminated by blocking the atomic beam and the slowing laser beam
by mechanical shutters, and by switching off the magnetic field of
the Zeeman slower.
%The ions produced within the cloud of trapped atoms due to Penning collisions or collisions with background particles,
% can be detected by
%a multichannel plate detector (MCP).
%two multichannel plates in Chevron configuration with a bias grid.
%The electronical pulses of the MCP are processed to a gated counter.
In the following measurement period of two minutes, the fluorescence of the trapped atoms is
recorded by a CCD camera
capturing five image frames per second.
%As the initial decay is dominated by two-body losses, the MOT was not optimized for particular high number densities.
%In addition, the settings of the CCD camera (shutter opening time, aperture)
% were chosen such that we were able to measure even a low level of fluorescence.
%Thus, at high atom numbers the CCD chip was eventually locally saturated. Images with a single saturated pixel did not enter the data processing.
%
%\begin{figure}[b]
%\centering
%   \includegraphics[width=215pt, height=145pt]{alpha_delta.eps}
%\caption{Decay rates $\alpha$ of the MOT for different detunings $\Delta$ at a constant calculated relative population $\pi=0.05$.}
%
%\label{fig:alphadelta}
%\end{figure}
%\begin{figure}[b]
%\centering
%   \includegraphics[width=211pt, height=139pt]{decay.eps}
%\caption{Fluorescence of the decaying $^{20}$Ne ensemble in a MOT vs time. Shown are two sets of measurements
%consisting of five curves each. Each set was measured for a given excitation $\pi$ of the upper laser cooling level .}
%\label{fig:curves}
%\end{figure}
Fluorescence decay curves
are obtained by integrating the counts of each frame over the region where atoms are present.
Great care has been taken to eliminate the influence of residual stray light.
%which enhances the level of light power recorded by the CCD camera and thus yields an offset which strongly affects
% the determination of the parameter $\alpha$.
% For this purpose we performed a twofold processing of the CCD images:
%The fluorescence signal of trapped atoms
%$F_{\mbox{at}}(t)$
%was obtained by integrating the CCD counts over the region where atoms were present.
%By integrating over a second region of the image where clearly no atoms were present, the stray light could be subtracted from the
%fluorescence signal for each picture.
The decay curves span up to four orders of magnitude in signal.
Fig.~\ref{fig:curves} shows two sets of data taken for different values of $\pi_3$.
For $\pi_3=0.005$ a decay rate of $\alpha = 0.06929(48) \, \mathrm{s}^{-1}$ and a respective
decay time of $\alpha^{-1} = 14.43 \, \mathrm{s}$ is observed.
%Both sets consist of 5 individual decay curves which are almost not distinguishable in the figure
%giving proof of the high reproducibility of our decay measurements.
%Each curve is fitted
%according to a solution of Eq.~(\ref{decay}). The averaged results of $\alpha$ for
%both values of $\pi_3$ are shown.
%For $\pi_3=0.005$ a decay rate of $\alpha = 0.069$$s (\alpha^{-1} = 14.4^{-1}$s$^{-1}$
%is observed.
 % Another region of the images where clearly no atoms
% were present was defined and subtracted from the first.
%To account for an eventually locally varying background signal, the integral over the second region $F_{\mbox{bg}}(t)$
%was scaled by a factor of $c$ to the fluorescence signal of the first region at times close to $120$ s, when no atoms were present any more:
%$c F_{\mbox{bg}}(t) = F_{\mbox{at}}(t)$ for $t \to 120s$.
% For each picture taken, the corrected fluorescence signal $F_{\mbox{Ne}}= F_{\mbox{at}}(t) -c F_{\mbox{bg}}(t) $ exhibiting a decay spanning up to
%four orders of magnitude was obtained
%and fitted according to a solution of Eq.~(\ref{decay}).
%As expected from Eq.~(\ref{alpha}) the decay rate strongly depends on the population
%$\pi_3$ with larger population causing a decreasing decay rate.


%%%%%%%%%%%%%%%%%% Figure 4


\begin{figure}[b]
%\centering
    \includegraphics[width=100pt, keepaspectratio]{LIFEFIG4.eps}
    \includegraphics[width=100pt, keepaspectratio]{LIFEFIG5.eps}
\caption{(a)~Decay rates $\alpha$ obtained for
$\pi_3<0.1$ after extrapolation to 
$\pi_3 = 0$ as a function of the nominal background pressure $p_{display}$.
(b)~Coefficients $c_1$
proportional to the number of ionizing background collisions as a function of 
$p_{display}$ allowing the 
determination of the vacuum gauge offset. }
%\caption{Determination of the vacuum gauge's offset. Coefficient of ionizing background collisions $c_1$ vs. pressure.}
\label{fig:background}
\end{figure}


%%%%%%%%%%%%%%%%%%%%


%\begin{figure}[b]
%\centering
%   \includegraphics[width=220pt, height=161pt]{c1p5.eps}
%\caption{Determination of the vacuum gauge's offset. Coefficient of ionizing background collisions $c_1$ vs. pressure.}
%
%\label{fig:background}
%\end{figure}
%
%
%A given value of $\pi$ can be realized by different combinations of $s$ and $\Delta$. This offers the opportunity to check the
% dependency of $\alpha$ on $\pi$. By recording the decay  at a given pressure ($6.1\times10^{-10}\,\mathrm{mbar}$) for different pairs of $s$ and $\Delta$,
% yielding the same relative population $\pi$, it was observed that $\alpha$ remained constant in that case (Fig.\ref{alphadelta}).
%As $\pi$ does not depend uniquely on a specific combination of $s$ and $\Delta$, the dependency of $\alpha$ on $\pi$ can be checked by
%measuring $\alpha$ for different pairs of $s$ and $\Delta$, yielding the same relative population $\pi$ at a given pressure. It was observed,
%that $\alpha$ remained constant in that case.
%
%
%
%
In the following sections we describe our procedure to extrapolate the observed decay rate $\alpha$
to the $^3{\mathrm{P}}_2$ decay rate $\tau_2^{-1}$.
For the extrapolation of $\alpha$ to vanishing background pressure
a systematic investigation of the MOT decay as a function of pressure
has been carried out
(Fig.~\ref{fig:background}(a)).
%After %During
%an intermediate increase in pressure following the loading phase of the MOT
% the background pressure rises due to ground state neon atoms present in the atomic beam. However,
%the base pressure of the vacuum chamber is restored after
%approximately $20\, \mbox{s}$,
% This indicates
%indicating that it
%the then measured pressure
%is %not
%no longer dominated by
% the neon partial pressure.
%neon.% entering during the loading phase.
The pressure is given as
the nominal reading of the vacuum gauge (Balzers IKR070).
%Pressures below $1 \times 10^{-10}\, {\mbox{mbar}}$ were obtained
%by filling a reservoir surrounding a titanium sublimation pump with liquid nitrogen. Pressures above this value %were generated
%by heating that part of the vacuum chamber where the pumps are connected to.
The specific composition of
the background gas is not known. However, we could observe a linear dependence of $\alpha$
%extrapolated to $\pi_3 = 0$
on pressure over a range of almost two orders of magnitude,
indicating a pressure-independent composition of
the background gas.
From a linear fit for data taken for $\pi_3 < 0.1$, we can determine the slope
$\gamma = 5.2(1) \!\times\! 10^{7} \,\mathrm{mbar}^{-1} \,\mathrm{s}^{-1}$.
% which is within the same order of magnitude as the values given in \cite{ShimizuArKr} for Argon and Krypton.
%The same procedure applied to decay measurements recorded at $\pi_3 > 0.4$ yielded
%$\gamma=4.7(1) \!\times\! 10^{7} \,\mathrm{mbar}^{-1} \mathrm{s}^{-1}$ which
%confirms the previous findings that the collisional cross section depends on the
%internal state of the involved neon atoms ZITAT.
%The observed linear dependence of $\alpha$ on pressure would be sufficient
%for an extrapolation for a known offset of the pressure reading.
%On the other hand, the $p\!\!\to\!\!0$-extrapolation remains
%inaccurate if the pressure gauge exhibits an offset.
As a special feature of our setup,
we can directly determine a possible offset by simultaneously measuring the
fluorescence and the rate of ions
produced by collisions
% of metastable neon atoms
%with residual gas atoms
using a multichannel plate (MCP) ion detector.
%The MCP was operated at a voltage of $-2 \,\mathrm{kV}$ and was
%biased with an electrostatic grid at $-250\, \mathrm{V}$.
Under typical operating conditions
we detect ions of different origin:
(a) Ne$^+$-ions originating from intra-trap collisions
%show a
(rate
%which scales with
$\propto \int \!\! n^2\!(r,t) \, \mbox{d}^3\!r$).
(b) Ions which originate from collisions of
metastable neon atoms with residual gas constituents
% are produced
%with a rate which scales with
(rate $\propto N(t) \times p$). % and $p$).
%We note that the observed fluorescence $F_{\mathrm{Ne}}(t)$ is proportional to $N(t)$.
%As we did not attempt to determine the specific detection efficiencies,
%and the fluorescence $F_{\mathrm{Ne}}(t)$ is directly proportional to $N(t)$
Thus, the ion count rate $R_{\mathrm{ion}}(t)$ can be modelled by
% a term linear in $F_{\mathrm{Ne}}(t)$ and one quadratic in $F_{\mathrm{Ne}}(t)$:
%
%
\begin{equation}
%R_{\mathrm{ion}}(t) = c_1(p)  F_{\mathrm{Ne}}(t) + c_2 F_{\mathrm{Ne}}^2(t),
R_{\mathrm{ion}}(t) = c_1 N\!(t) + c_2 \! \int \!\! n^2\!(r,t) \, \mbox{d}^3\!r,
\end{equation}
%
%
where $c_1$ is assumed to be proportional to $p$.
% and background signals on $F_{\mathrm{Ne}}(t)$ and  $R_{\mathrm{ion}}(t)$ aving been subtracted.
The values of $c_1$ obtained for different pressures are shown in
Fig.~\ref{fig:background}(b). From a linear fit we can determine
%a value for
the offset of the pressure gauge reading to $(4 \pm 7) \times
10^{-12} \,{\mathrm{mbar}}$ also allowing for a vanishing
offset. By combining the results for $\gamma$ and the
offset, we can extrapolate the observed
decay rates to $p =0$. Due to the already low
background pressure for our decay measurements only small
corrections arise: At $p = 5 \!\times\! 10^{-11}
\,{\mathrm{mbar}}$ the background collision rate of $\gamma p =
1/(420\,{\mathrm{s}})$ is more than a factor of 25 smaller than
the observed decay rate $\alpha$.
% and
%the induced inverse of the lifetime $\tau_2^{-1}$.

%\begin{figure}[b]
%\centering
%   \includegraphics[width=8cm, height=6cm]{MCP.eps}
%\caption{Determination of the vacuum gauge's offset. Coefficient of ionizing background collisions $c_1$ vs. pressure.}
% \label{fig:MCP}
% \end{figure}


%%%%%%%%%%%%%%%%%%% Figure 5


\begin{figure}[b]
\centering
    \includegraphics[width=120pt, keepaspectratio]{LIFEFIG6.eps}
    %0.24\textwidth
    %keepaspectratio
    \includegraphics[width=120pt, keepaspectratio]{LIFEFIG7.eps}
\caption{Systematic investigation of MOT losses. Decay rates are plotted 
as a function of the magnetic field gradient
for two different excitations (a) $\pi_3=0.016$ and (b)
$\pi_3=0.42$. Shaded areas indicate the gradients
of typical operating conditions.}
\label{fig:gradient}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%


We also investigated the
possibility of additional atom losses from the MOT
(contribution $\mathcal{ML}$
in Eq.~(\ref{alpha}))
attributed to a finite trap depth $U_0$ or finite escape velocity $v_e$
\cite{Weiner}.
%These loss mechanisms are combined to the decay rate 
We assume %from a two-level model of
that a lossless MOT would result from an infinite trapping volume
leading to infinite $U_0$ and $v_e$. 
In principle, this could be achieved 
for the applied magnetic field gradient $B'$
approaching zero if the MOT beams could be made infinitely large.
% However, the distance
%from the trap center where a specific $U_0$ is reached diverges as well.
In all practical implementations
% for small field gradients
the trap depth will remain finite due to the
finite size of the laser beams.
%
%
%Thus, for a
% fixed
%given beam geometry, a decrease in $B'$ should first
%increase the MOT stability until the required beam radius
%surpasses the actual one and then should cause a fast reduction in
%stability for smaller $B'$. Extrapolating the MOT decay
%rates observed in the stable operation regime to $B'\to 0$ should allow the
%extrapolation to vanishing MOT losses.
%
%
Thus, for a
% fixed
given beam geometry, a variation in $B'$ should
allow to investigate the MOT stability and 
to quantify the MOT losses $\mathcal{ML}$
which
include diffusive motion out of the trapping volume \cite{Willems}
leading to an exponentially decreasing loss rate with increasing trap depth
and non-ionizing background gas collisions \cite{MOT1} 
 already being accounted for by the $p \to 0$ extrapolation.
The influence of the magnetic field gradient on the measured decay rate $\alpha$ is shown
in Fig.~\ref{fig:gradient}
for two values of $\pi_3=0.016$ and $0.41$, respectively.
%As expected, for both cases too small values of $B'$ lead to large decay rates. 
%The qualitative picture discussed above is confirmed by the data.
%The minimum decay rate is observed approximately for that gradient where the Zeeman shift
%at the edges of the MOT laser beams compensates the laser detuning
%thus maximizing $U_0$ and $v_e$.
%As indicated by the shaded areas in Fig.~\ref{fig:gradient} all decay
%measurements used for the final determination of $\tau_2$
%were performed for values of $B'$ leading to highest MOT stability.
%For smaller gradients the influence of the laser beam diameter on the trap depth is visible.

In the most relevant case of small $\pi_3$ (Fig.~\ref{fig:gradient}(a)), 
for sufficiently large
$B'> 10\,\mathrm{G/cm}$ a variation of $B'$ does affect the measured $\alpha$
not more than
%at most on the same order of magnitude as the
the uncertainties obtained from the fit.
%in the fitted values of $\alpha$ themselves.
Thus we conclude that for our operating regime (shaded region in Fig.~\ref{fig:gradient}(a))
the trap depth $U_0$ and escape velocity $v_e$
are sufficiently large and that
potential MOT loss contributions to the decay rate $\alpha$ 
are not significant.
%lie within the uncertainty in $\alpha$.
%For small $\pi_3$, $U_0$ and $v_e$ are sufficiently large to prevent relevant MOT
%loss.
For large $\pi_3$ (Fig.~\ref{fig:gradient}(b)), 
a variation of the MOT stability with $B'$ is observed.
%the decay rate increases with increasing $B'$.
Unfortunately, no quantitative model of the MOT at high saturation exists,
the application of the loss models discussed above is not straightforward,
and only a minimum and a maximum estimate of $\mathcal{ML}$ can be given:
Since our decay data for large $\pi_3$ have been taken at
$B'=1.5\,\mathrm{G/cm}$ which gives the best MOT stability, it is
not clear whether MOT losses have a significant contribution on
the determination of $\alpha$ at all.
On the other hand, a linear extrapolation of $\alpha$
to $B'=0$
should overestimate the corrections to be made \cite{Willems,MOT1}
giving $\mathcal{ML}=0.005(2)\,\mathrm{s}^{-1}$ as an upper limit.
Losses for large $\pi_3$ only influence the
$\pi_3 \to 0$ extrapolation which
we take into account 
%the effect of these large-$\pi_3$ MOT losses 
by incorporating $0\,\mathrm{s}^{-1}\!<\!\mathcal{ML}\!<\!0.005\,\mathrm{s}^{-1}$ 
into the uncertainties of 
this extrapolation.
%the $\pi_3 \to 0$
% appropriately.

%On the other hand,
% the relevant decay measurements have been carried out in
%the regime of small $\pi_3$ and 
%so we cannot describe this behaviour quantitatively.
%Possible loss mechanisms include diffusive motion out of the trapping volume \cite{Willems}
%which leads to an exponentially decreasing loss rate with increasing trap depth~$U_0$.
%and non-ionizing background
%gas collisions \cite{MOT1}.
%The latter should already be accounted for by the $p \!\to \! 0$ extrapolation.
%
%In addition, since the escape velocity~$v_e$
%changes with the trapping volume \cite{Weiner}, a part of the $B'$-dependant losses
%may be attributed to non-ionizing background
%gas collisions \cite{MOT1}, which are already accounted for by the $p \!\to \! 0$ extrapolation.
%..
%Nevertheless we follow the argument given above and assume that the most
%stable MOT would be achieved at $B' = 0$ assuming infinite size of the 
%MOT beams. 
%
%
%
%For that
%reason, we assume the correction for large $\pi_3$ to be limited by
%$0\,\mathrm{s}^{-1} < \mathcal{ML} < 0.005\,\mathrm{s}^{-1}$.
%We account for these potential corrections 
%on $\alpha$ for 
%large $\pi_3$ 
%by increasing the uncertainty budget for 
%the $\pi_3 \to 0$ extrapolation appropriately.
%We
%include these corrections in the error budget for the
%determination of $\alpha$ for large values of $\pi_3$.
%
%For the investigation on the $\pi_3$-dependance of $\alpha$,
%measurements at a high excitation level were taken at a gradient of
%$B'=1.5\,\mathrm{G/cm}$. For this situation only a minimal possible
%correction of $\Xi(1.5\,\mathrm{G/cm})=0\,\mathrm{s}^{-1}$ and a maximal
%possible correction of $\Xi(1.5\,{\mathrm{G/cm}})=0.005(2)\,\mathrm{s}^{-1}$
%which stems from a linear extrapolation can be specified.
%This leads to $\Xi(1.5{\mathrm{G/cm}})=0.005(2)$.

%A possible loss mechanism is diffusive motion out of the trapping volume \cite{Willems}
%which leads to a exponentially decreasing loss rate ($\propto \sqrt{U_0}  \cdot e^{-U_0}$.
%Due to the lack of an adequate theoretical model of the MOT at high saturation and since
% no experimental study of gradient dependant losses in this regime exists to our knowledge,
% we cannot describe this behaviour quantitatively.
%In addition, since we expect a dependancy of $v_e$ on $B'$
% a part of the gradient dependant losses may be attributed to background gas collisions,
% as $\gamma$ is a function of $v_e$ \cite{MOT1}, which are already
%corrected by the $p \to 0$-extrapolation.
%Therefore, it is not clear if $\Xi$ has a significant contribution on the minimum decay
% rates observed. Contrariwise,
%a linear extrapolation of $\alpha$ to $B'=0$ should overestimate the $\Xi$-correction
%to be made. Especially for measurements described in the following this leads to
% $\Xi(1.5{\mathrm{G/cm}})=-0.005(2)$.

After this systematic study of the influence of background
collisions and MOT losses
the determination of the $^3\mathrm{P}_2$ lifetime $\tau_2$ can be
obtained by 
extrapolating the measured decay rates to vanishing population $\pi_3$.
% of the $^3{\mathrm{D}}_3$ state can be carried out and the final value of
%the variation of $\alpha$ with
%the population $\pi_3$ can be studied. 
Only data for
a pressure below $5 \times 10^{-11} \, {\mbox{mbar}}$ are considered
and an extrapolation to $p=0$ is performed.
The dependence of $\alpha_{p=0}$ on $\pi_3$ is shown in
Fig.~\ref{fig:Alpha} exhibiting a simple linear behavior.
%These corrections being smaller than $0.0026 {\mathrm{s}}^{-1}$.
%The case of $\Xi(1.5\,\mathrm{G/cm})=0\,\mathrm{s}^{-1}$ shall be discussed first.
%The then obtained values on $\alpha_{p=0}$ for each individual
%decay curve are shown as a function of $\pi_3$ in
%Fig.~\ref{fig:Alpha}, as expected the dependancy is linear.
%As expected $\alpha_{p=0}$ shows a linear
%dependence on $\pi_3$ for all data.
Displayed uncertainties in $\alpha_{p=0}$
are
% given by the combination of the statistically independent
given by the combined uncertainties in the fit of $\alpha$ and
uncertainties in the $p\!\to\!0$ extrapolation.
% As discussed
%before, uncertainties in $\pi_3$ are dominated by the uncertainty
%in the factor $C$.
%the calibration of the photodiode used.
%The parameters for which a very low relative population ($\pi_3=0.005$) was obtained were $\Delta/\Gamma=-5.3$ and $s=1.75$,
%the decay rate being $\alpha\approx1/(14.4\mathrm{s})$.
%Values of $\pi_3$ close to $0.5$ were reached by using $\Delta/\Gamma=-0.16$ and $s=24$.
%We would like to stress that in these cases of a very low (high) relative population, i.e. $\pi_3\!\to\!0\,(0.5)$,
%the error in $\pi_3$ due to the uncertainty of the $C$-value is minimized.
%As expected the decay rate $\alpha {p=0}$ shows a linear dependence on $\pi_3$ for all data.
% At the same time the
%By extrapolating to a vanishing population
% of the $^3{\mathrm{D}}_3$ state can be carried out and the final value of
%the $^3\mathrm{P}_2$-lifetime $\tau_2$ can be
To minimize uncertainties 
in the  $\pi_3 \! \to \! 0$ extrapolation,
only data sets for $\pi_3 < 0.05$
and $\pi_3 > 0.4$ showing the smallest absolute uncertainties in $\pi_3$ have been included.
The final value of the $^3\mathrm{P}_2$ lifetime and decay rate are
$\tau_2=14.70(13)\,\mathrm{s}$ and $\tau_2^{-1}=0.06801(62)\,\mathrm{s}$,
respectively .
% With the uncertainty in $\pi_3$ being minimized for
%smallest and largest $\pi_3$ only data sets for $\pi_3 < 0.05$ and
%$\pi_3 > 0.4$ have been used for the $\pi_3 \to 0$ extrapolation.
%From a linear fit according to $\alpha (p=0) = \frac{1}{\tau_2}
%\left( 1 - \pi_3 \right)  + \frac{1}{\tau_3} \pi_3$ a lifetime
%and the inverse decay rate $\tau_3$
%can be extracted.
%\begin{equation}
%\alpha(\pi_3) = \frac{1}{\tau_2} + \left( \frac{1}{\tau_3} - \frac{1}{\tau_2} \right) \pi_3 .
%\
%$\tau_2 = 14.70(13)\, \mathrm{s}$ can be extracted.
Table~\ref{tab:errors} gives a summary of contributions and
uncertainties for these values.
%We also would like to note that
%The uncertainty given for the MOT loss corrections $\mathcal{ML}$
%for large $\pi_3$ only has a minor influence on the $\pi_3\to 0$
%extrapolation and is accounted for in the corresponding values
%given in Table~\ref{tab:errors}.
% slope of the linear fit and can be easily covered in
%a slightly increased uncertainty of the $\pi_3 \to 0$
%extrapolation.
% We
%would like to note that
%due to our advantageous experimental
%conditions, decay curves for $p < 5 \times
%10^{-11}\,{\mbox{mbar}}$, for
Since stable MOT conditions are directly accessible for
$\pi_3 = 0.005$ and $p < 5 \times 10^{-11}\, {\mbox{mbar}}$
% and for (see Fig.~\ref{fig:curves}). Thus
only
unprecedented small corrections
% for the extrapolation procedures
are required. The final value for $\tau_2$ only deviates from the
uncorrected value of $\alpha^{-1} = 14.43 \, \mathrm{s}$ by 2 \%.
% with each
%correction being smaller than 2.5 \% and partially cancelling each
%other in addition.
%%%%%%%%%%%%%%%%%%%5 Figure 6
\begin{figure}[t]
\centering
    \includegraphics[width=200pt, keepaspectratio]{LIFEFIG8.eps}
\caption{Decay rates $\alpha_{p=0}$ 
%extrapolated to a
%vanishing background pressure 
as a function of the excitation
$\pi_3$ allowing the determination of $\tau_2$ in the 
limit of $\pi_3 \to 0$.}
 \label{fig:Alpha}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[t]
\caption{\label{tab:errors}Contributions to the
$\tau_2=14.70(13)\mathrm{s}$ extrapolation.}
\begin{ruledtabular}
\begin{tabular}{lrr}
%Origin&Decay rate&$\Delta \alpha$ \\
%\hline
MOT decay at $\pi_3=0.005$      &$0.06929\,{\mathrm{s}}^{-1}$   &$\pm 0.00048\,{\mathrm{s}}^{-1}$ \\
$\pi_3\to0$-extrapolation       &$+0.00028\,{\mathrm{s}}^{-1}$  & $\pm 0.00013\,{\mathrm{s}}^{-1}$  \\
$\mathrm{p}\to0$-extrapolation      &$-0.00156\,{\mathrm{s}}^{-1}$  &$\pm 0.00037\,{\mathrm{s}}^{-1}$   \\
\hline
$1/\tau_2$      &$0.06801\,{\mathrm{s}}^{-1}$   &$\pm 0.00062\,{\mathrm{s}}^{-1} $\\
\end{tabular}
\end{ruledtabular}
\end{table}

%Furthermore,
%extrapolating to a hypothetical total excitation of $\pi_3=1$ results in a non-vanishing $\alpha$.
%the slope of $\alpha(\pi_3)$ was found to be
% $\left( {\tau_3}^{-1} - {\tau_2}^{-1} \right)= -0.051(4)\mathrm{s}^{-1}$.
%As the absolute value of this slope is different from the obtained value of
%${\tau_2}^{-1}=0.068\mathrm{s}^{-1}$
%We interpret this as a signature of a second
%decay channel from the $3p[5/2]_3$-state to the atomic ground state different
%from the one via the $3s[3/2]_2$-level.
%We suggest this channel to be a sequence of an M2-transition to either the $J=1$-state $3s[3/2]_1$
% or $3s'[3/2]_1$, followed by an E1-transition to the atomic ground state,
%The decay rate of which being $(2\mathrm{ns})^{-1}$.
%the total inverse decay rate to the intermediate $J=1$-states
%being $\tau_3=59\,\mathrm{s}$.
%Applying the maximal correction of $\Xi(1.5{\mathrm{G/cm}})=0.005(2)\,\mathrm{s}^{-1}$ to decay rates
% obtained at high excitation (i.e. $\pi_3=0.47$) does not considerably change
% the slope of the $\alpha(\pi_3)$-curve
%to $-0.061(4)\mathrm{s}^{-1}$. The then obtained values read
%the value of $\tau_2$ which now reads $14.69(13)\,\mathrm{s}$. Thus, for
%$0\,\mathrm{s}^{-1}<\Xi(1.5{\mathrm{G/cm}})<0.005\,\mathrm{s}^{-1}$ we may combine our results to
%$\tau_2=14.70(14)\,\mathrm{s}$.
%Our investigation of $\alpha (p=0)$ as a function of $\pi_3$
%allows us to
In addition to this result, extrapolating $\alpha_{p=0}$ to $\pi_3=1$, that is to a
hypothetical total population transfer to the
$^3{\mathrm{D}}_3$-state, is also possible. 
From this, we gain the first experimentally determined lower limit 
of $\tau_3\geq59\,\mathrm{s}$
for 
the combined rate $\tau_3^{-1}$ of decay of the
$^3{\mathrm{D}}_3$-state via all channels except the direct
decay to the $^3{\mathrm{P}}_2$ state.
%which combine decay channels towards the 1P1, 3P0, 3P1 and 1S0 states.
%Extrapolating $\alpha$ to $\pi_3=1$, that is to a hypothetical total excitation of the $3p[5/2]_3$-state,
% leads to a non-vanishing decay rate.
%We interpret this as a signature of a second
%decay channel from the $3p[5/2]_3$-state to the atomic ground state different
%from the one via the $3s[3/2]_2$-level.
%We suggest this channel to be a sequence of an M2-transition to either the $J=1$-state $3s[3/2]_1$
% or $3s'[3/2]_1$, followed by an E1-transition to the atomic ground state (Fig.~\ref{fig:scheme}),
%the total inverse decay rate to the intermediate $J=1$-states being $\tau_3$.
However, this value depends sensitively
on the $\mathcal{ML}$-correction applied for large $\pi_3$
and only the stated lower limit is accessible.
%and significantly larger values for $\tau_3$ would be consistent
%with our data.
% We obtain values of $\tau_3=59$ and $141\,\mathrm{s}$ for
%$\mathcal{ML}=0$ and $0.005\,\mathrm{s}^{-1}$ respectively.
%Therefore we can only specify a lower limit on this combined ratio
%to 
%Although we cannot give a
%specific value for $\tau_3$ we can give the value for a lower
%limit of $59 s^{-1}$.


%DAS FOLGENDE VERSTEHE ICH NOCH NICHT !!!
%The latter correction allowing for the possibility of a vanishing $\alpha$ at $\pi_3=1$.
%Therefore, our data does not clearly proof the existence of a second decay channel.

%However, the value of $\tau_3$ depends quite sensitively on the $\Xi$-corrections.
%Extrapolating to $\pi_3=1$ we arrive at $\tau_3=141\,\mathrm{s}$, but cannot rule out the possibility of
%a vanishing $\alpha$.
% and
%$\tau_3=141(97)\mathrm{s}$. Taking into account the extreme of the errors of this
%the $\Xi(1.5{\mathrm{G/cm}})=-0.005(2)$-
%correction leads to a situation which is compatible with
% $\tau_3^{-1}=0\mathrm{s}$ and $\tau_2=14.75\mathrm{s}$.
% Therefore, our data does not clearly proof the existence of a second decay channel.
%
%In conclusion, by varying the relative population of the atomic levels involved in
% laser-cooling we could demonstrate
%that the decay rate of the MOT can be changed continuously if the lower atomic level has a
% finite lifetime
%much shorter than the inverse background collision rate.
%Utilizing this feature this lifetime can be determined if the observed decay rates of the MOT are carefully extrapolated to
% vanishing systematic losses.

To conclude we would like to compare our result
of $\tau_2=14.70(13)\,\mathrm{s}$ to
theoretical values obtained by different atomic structure calculations.
Prior to our work, published theoretical values  
were given in  \cite{Small} to 24.4 s, which can be
rescaled to 20.0 s using more accurate
input parameters,
and in \cite{Kim2} to 22 s.
Significant remaining discrepancies 
%to our experimental value, in part 
initiated a reexamination of atomic structure calculations for the whole set 
of $^3{\mathrm{P}}_2$ lifetimes of rare-gas atoms.
A preliminary result based on a multiconfiguration Dirac-Fock calculation
gives a value of 18.9 s for neon \cite{Kim}. A high degree of sensitivity
to the included electron correlations has been found.
Theoretical values are still expected to be larger than the
experimental value, since the inclusion of additional electron correlations
should further reduce the calculated lifetime.
Refined calculations are forthcoming \cite{Kim}.
% and further
%work including more advanced correlation schemes is in progress \cite{Kim}.
Most recent ab-initio calculations using MCHF methods
give a lifetime of $16.9\,\mathrm{s}$ \cite{FroeseFischer}.
Considering the combined progress in experiment and theory, we anticipate
that the determination of the $^3{\mathrm{P}}_2$ lifetime of neon may continue to serve as 
an important test for precision atomic structure calculations in the future. 
%
%strongly deviates from the theoretical
% and experimental values known so far \cite{ShimizuNeon, Small, Kim2}.
%However, most recent ab-initio calculations \cite{FroeseFischer}
%predict a lifetime of $16.9\,\mathrm{s}$. In addition our result
% initialized a reexamination of atomic structure calculations for the whole set of $^3{\mathrm{P}}_2$-lifetimes
%  \cite{Kim}.
%These results ($18.9$, $34.2$, $39.2$, and $50.9\,\mathrm{s}$ for Ne, Ar, Kr, and Xe)
%are in better agreement with the experimentally determined ones.

We thank Y.-K. Kim for valuable discussions and are grateful for financial support by the DFG within the
{\it Schwerpunktprogramm SPP 1116}.

%Up to now the predicted theoretical $^3{\mathrm{P}}_2$-lifetimes of rare gas atoms
% predicted by Small-Warren and Chow-Chiu
% \cite{Small, Kim2}
%as well as those by Inedelicato, Desclaux and Kim
% are significantly longer than the experimentally determined ones,
% except for metastable Neon, where the
%measured lifetime of $22\, \mathrm{s}$ \cite{ShimizuNeon} is in good agreement with the
%predicted $24\, \mathrm{s}$  \cite{Small}
%and $22\, \mathrm{s}$ \cite{Kim2} respectively.







%Using relativistic wave functions
%Desclaux, Indelicato and Kim obtain preliminary multiconfiguration Dirac-Fock (MCDF)
%results \cite{Kim} for the $^3{\mathrm{P}}_2$-lifetimes
%of metastable rare gases of $18.9\mathrm{s},34.2\mathrm{s},39.2\mathrm{s},50.9\mathrm{s}$.
%which are shown in Table \ref{tab:table4} and compared with the experimental values konwn so far.
%These new results are in better agreement with the experimental values to date.



%\begin{table}
%\caption{\label{tab:table4}Comparison of experimental and theoretical values for the metastable
%states of Ne, Ar, Kr and Xe. }
%\begin{ruledtabular}
%\begin{tabular}{cccc}
%%Element&Experiment [s]&Theory [s]\cite{Kim}&Ratio\\
%\hline
%$^{20}$Ne      &$14.7(2)$              &$18.86$    &$1.28$ \\
%Ar     &$38^{+8}_{-5}$ \cite{ShimizuArKr}      & $34.20$   & $0.9$ \\
%Kr     &$39^{+5}_{-4}$ \cite{ShimizuArKr}      & $39.20$   & $1.0$ \\
%$^{132}$Xe &$42.9(9)$ \cite{Rolston}       & $50.94$   & $1.19$    \\
%$^{136}$Xe &$42.4(13)$ \cite{Rolston}      & ..    & ..    \\
%\end{tabular}
%\end{ruledtabular}
%\end{table}


%begin{figure}[h]

%   \includegraphics[width=5cm, height=3cm, angle=90]{alpha.eps}
%   \caption{Result.}
%   \label{fig:figth}
% \end{figure}


\bibliography{apssamp}% Produces the bibliography via BibTeX.

\begin{references}


\bibitem{LS} The notation of states throughout this paper is based on
LS-coupling. Racah notation
is included in Fig.~\ref{fig:scheme}.

\bibitem{berkeley} see e.g. \texttt{http://weak0.physics.berkeley.edu/weakint/}\\
\texttt{research/neon/}.

\bibitem{Beijerinck} S.J.M.~Kuppens et al., 
% J.G.C. Tempelaars, V.P. Mogendorff, B.J. Claessens, H.C.W. Beijerinck, and E.J.D. Vredenbregt,
Phys. Rev. A {\bf65}, 023410 (2002).

\bibitem{uns} M.~Zinner, Ph.D. thesis, unpublished (2002);
\texttt{http://www.iqo.uni-hannover.de/html/ertmer/}\\
\texttt{atom\_optics/nebec/nebec.html}.

\bibitem{Penning} F.M.~Penning, Naturwissenschaften {\bf15}, 818 (1927);
%F. M. Penning 'Electrische gasontladingen', Philips' Gloeilampenfabrieken, Eindhoven (1957).

\bibitem{HeBEC1} A.~Robert et al.,
%O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C.I. Westbrook, and A. Aspect, 
Science {\bf292}, 461 (2001).

\bibitem{HeBEC2} F.~Pereira Dos Santos et al.,
%J. Leonard, J. Wang, C.J. Barrelet, F. Perales, E. Rasel, C.S. Unnikrishnan, M. Leduc, and C. Cohen-Tannoudji, 
Phys. Rev. Lett. {\bf86}, 3459 (2001).

\bibitem{Correlations} M.~Yasuda, F.~Shimizu,
Phys. Rev. Lett. {\bf77}, 3090 (1996);
E.A.~Burt et al.,
Phys. Rev. Lett. {\bf79}, 337 (1997);
W.~Ketterle, H.J.~Miesner,
Phys. Rev. A {\bf56}, 3291 (1997).

\bibitem{Savage} J.J.~Hope, C.M.~Savage,
Phys. Rev. A {\bf 54}, 3177 (1996).

\bibitem{Kim} J.P.~Desclaux, P.~Indelicato, and Y.-K.~Kim, private communication (2002).

%\bibitem{Kim3} Y.-K. Kim, private communication

\bibitem{ShimizuNeon} F.~Shimizu, Laser Spectroscopy IX, Academic Press, p. 444 ff, (1989).

\bibitem{Shimizuprivate} F.~Shimizu, private communication.

\bibitem{ShimizuArKr} H.~Katori, F.~Shimizu, Phys. Rev. Lett. {\bf70}, 3545 (1993).

\bibitem{Walhout} J.~Lefers et al.,
% N. Miller, D. Rupke, D. Tong, and M. Walhout, 
Phys. Rev. A {\bf 66}, 012507 (2002).

\bibitem{Rolston} M.~Walhout, A.~Witte, and S.L.~Rolston, Phys. Rev. Lett. {\bf72}, 2843 (1994).

\bibitem{MOT1} A.M.~Steane, M.~Chowdhury, and C.J.~Foot, J. Opt. Soc. Am. B {\bf6}, 2142 (1992).

\bibitem{MOT2} C.G.~Townsend et al.,
% N.H. Edwards, C.J. Cooper, K.P. Zetie, C.J. Foot, A.M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard,
Phys. Rev. A {\bf 52}, 1423 (1995).

\bibitem{Weiner} J.~Weiner et al.,
% V.S. Bagnato, S. Zilio, P.S. Julienne,
 Rev. Mod. Phys. {\bf71}, 1 (1999).


\bibitem{Willems} P.A.~Willems et al.,
% R.A. Boyd, J.L. Bliss, K.G. Libbrecht,
 Phys. Rev. Lett. {\bf78}, 1660 (1997).

\bibitem{Small} N.E.~Small-Warren, L.-Y.~Chow-Chiu, Phys. Rev. A {\bf11}, 1777 (1975).

\bibitem{Kim2} J.P.~Desclaux, P.~Indelicato, Y.-K.~Kim, cited in \cite{Rolston}.




%\bibitem{database} $\lambda_{\mathrm{air}}=640.225\,\mathrm{nm}$ and $\Gamma=2\pi \times 8.18 \,\mathrm{MHz}$ according to the Atomic Spectra Database maintained at the National Institute of Standards and Technologies.



%\bibitem{HeMOT} A. Browaeys, J. Poupard, A. Robert, S. Nowak, W. Rooijakkers, E. Arimondo, L. Marcassa, D. Boiron, C. I. Westbrook,
%and A. Aspect, Eur. Phys. J. D {\bf 8}, 199(2000).

%\bibitem{MOT1} A.~M.~Steane, M.~Chowdhury, and C.~J.~Foot, J. Opt. Soc. Am. B {\bf6}, 2142 (1992).
%\bibitem{Beam} R.S. Van Dyck, C.E. Johnson, H.A. Shugart, Phys. Rev. A {\bf5}, 991 (1972)



\bibitem{FroeseFischer} G.~Tachiev,~C.~Froese~Fischer (2002), \texttt{http://atoms.vuse.vanderbilt.edu}.


\end{references}


\end{document}
%End of file apssamp.tex ******

physics0208088.extracted_bib
\bibitem{LS} The notation of states throughout this paper is based on
LS-coupling. Racah notation
is included in Fig.~\ref{fig:scheme}.


\bibitem{berkeley} see e.g. \texttt{http://weak0.physics.berkeley.edu/weakint/}\\
\texttt{research/neon/}.


\bibitem{Beijerinck} S.J.M.~Kuppens et al., 
% J.G.C. Tempelaars, V.P. Mogendorff, B.J. Claessens, H.C.W. Beijerinck, and E.J.D. Vredenbregt,
Phys. Rev. A {\bf65}, 023410 (2002).


\bibitem{uns} M.~Zinner, Ph.D. thesis, unpublished (2002);
\texttt{http://www.iqo.uni-hannover.de/html/ertmer/}\\
\texttt{atom\_optics/nebec/nebec.html}.


\bibitem{Penning} F.M.~Penning, Naturwissenschaften {\bf15}, 818 (1927);
%F. M. Penning 'Electrische gasontladingen', Philips' Gloeilampenfabrieken, Eindhoven (1957).


\bibitem{HeBEC1} A.~Robert et al.,
%O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C.I. Westbrook, and A. Aspect, 
Science {\bf292}, 461 (2001).


\bibitem{HeBEC2} F.~Pereira Dos Santos et al.,
%J. Leonard, J. Wang, C.J. Barrelet, F. Perales, E. Rasel, C.S. Unnikrishnan, M. Leduc, and C. Cohen-Tannoudji, 
Phys. Rev. Lett. {\bf86}, 3459 (2001).


\bibitem{Correlations} M.~Yasuda, F.~Shimizu,
Phys. Rev. Lett. {\bf77}, 3090 (1996);
E.A.~Burt et al.,
Phys. Rev. Lett. {\bf79}, 337 (1997);
W.~Ketterle, H.J.~Miesner,
Phys. Rev. A {\bf56}, 3291 (1997).


\bibitem{Savage} J.J.~Hope, C.M.~Savage,
Phys. Rev. A {\bf 54}, 3177 (1996).


\bibitem{Kim} J.P.~Desclaux, P.~Indelicato, and Y.-K.~Kim, private communication (2002).

%
\bibitem{Kim3} Y.-K. Kim, private communication


\bibitem{ShimizuNeon} F.~Shimizu, Laser Spectroscopy IX, Academic Press, p. 444 ff, (1989).


\bibitem{Shimizuprivate} F.~Shimizu, private communication.


\bibitem{ShimizuArKr} H.~Katori, F.~Shimizu, Phys. Rev. Lett. {\bf70}, 3545 (1993).


\bibitem{Walhout} J.~Lefers et al.,
% N. Miller, D. Rupke, D. Tong, and M. Walhout, 
Phys. Rev. A {\bf 66}, 012507 (2002).


\bibitem{Rolston} M.~Walhout, A.~Witte, and S.L.~Rolston, Phys. Rev. Lett. {\bf72}, 2843 (1994).


\bibitem{MOT1} A.M.~Steane, M.~Chowdhury, and C.J.~Foot, J. Opt. Soc. Am. B {\bf6}, 2142 (1992).


\bibitem{MOT2} C.G.~Townsend et al.,
% N.H. Edwards, C.J. Cooper, K.P. Zetie, C.J. Foot, A.M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard,
Phys. Rev. A {\bf 52}, 1423 (1995).


\bibitem{Weiner} J.~Weiner et al.,
% V.S. Bagnato, S. Zilio, P.S. Julienne,
 Rev. Mod. Phys. {\bf71}, 1 (1999).



\bibitem{Willems} P.A.~Willems et al.,
% R.A. Boyd, J.L. Bliss, K.G. Libbrecht,
 Phys. Rev. Lett. {\bf78}, 1660 (1997).


\bibitem{Small} N.E.~Small-Warren, L.-Y.~Chow-Chiu, Phys. Rev. A {\bf11}, 1777 (1975).


\bibitem{Kim2} J.P.~Desclaux, P.~Indelicato, Y.-K.~Kim, cited in \cite{Rolston}.




%
\bibitem{database} $\lambda_{\mathrm{air}}=640.225\,\mathrm{nm}$ and $\Gamma=2\pi \times 8.18 \,\mathrm{MHz}$ according to the Atomic Spectra Database maintained at the National Institute of Standards and Technologies.



%
\bibitem{HeMOT} A. Browaeys, J. Poupard, A. Robert, S. Nowak, W. Rooijakkers, E. Arimondo, L. Marcassa, D. Boiron, C. I. Westbrook,
%and A. Aspect, Eur. Phys. J. D {\bf 8}, 199(2000).

%
\bibitem{MOT1} A.~M.~Steane, M.~Chowdhury, and C.~J.~Foot, J. Opt. Soc. Am. B {\bf6}, 2142 (1992).
%
\bibitem{Beam} R.S. Van Dyck, C.E. Johnson, H.A. Shugart, Phys. Rev. A {\bf5}, 991 (1972)




\bibitem{FroeseFischer} G.~Tachiev,~C.~Froese~Fischer (2002), \texttt{http://atoms.vuse.vanderbilt.edu}.


