text:
\section{Introduction}

Recently, optical schemes have been developed to selectively
produce cold and dense samples of deeply bound molecules in well
defined quantum states \cite{Dan08, La08, Ni08} (see also
\cite{Vit08, Dei08}). This has opened up new possibilities for
cold collision experiments, ultracold chemistry \cite{Sta06,
Kre05, Kre08}, and for testing fundamental laws via precision
spectroscopy \cite{Ze08, DeM08, Ch09}. For such future experiments
it is mandatory that the location and properties of the available
molecular quantum states are well known and understood.

Until recently, the $a \, ^3\Sigma_u^+$ triplet potential of the Rb$_2$
molecule has remained largely experimentally unexplored. Natural
samples of Rb$_2$ are normally found in their $X \, ^1\Sigma_g^+$ singlet
state
 from which the lowest triplet state is somewhat
difficult to reach in an optical Raman process due to a change in
u/g symmetry. Here we report precision spectroscopy of
$^{87}$Rb$_2$ molecules in this lowest triplet state, $a \, ^3\Sigma_u^+$
(5$S_{1/2}$ + 5$S_{1/2})$, where we resolve vibrational,
rotational, hyperfine and Zeeman structure with an accuracy as
high as 30\,MHz. We use dark-state spectroscopy where a gas of
weakly bound Feshbach molecules is irradiated by two laser beams.
Molecular losses, induced by one of the two lasers, are suppressed
when the second laser is tuned into resonance with a bound state
(see Fig.\,\ref{fig:potential}). Fitting a coupled-channel model
to our experimental data, we have constructed an accurate
Born-Oppenheimer potential for $a \, ^3\Sigma_u^+$. This enables us to
calculate the wave functions of  triplet bound states as well as
their binding energies to 60\,MHz$\times h$ accuracy (where $h$ is
Planck's constant) over the whole manifold of vibrational and low
rotational (N$<$5) levels. Our theory and data agree to the extent
that further refinement of the theoretical model requires a
reduction in the experimental uncertainty.

\begin{figure}[tpb]
\centering
\includegraphics{Fig1.eps}
\caption{Dark-state spectroscopy scheme for the $^{87}$Rb$_2$
$a \, ^3\Sigma_u^+$ potential. Lasers 1 and 2 couple the molecular levels
$|\textrm{i}\rangle$ and $|v\rangle$ to the excited level $|\textrm{e}\rangle$ with Rabi
frequencies $\Omega_{1,2}$, respectively. Laser 1 is kept on
resonance while laser 2 can be tuned to any level of the
$a \, ^3\Sigma_u^+$ potential. Inset: Bound state level of Feshbach
molecules  as a function of magnetic field $B_z$. The dashed line
gives the position of the Feshbach resonance. The dot marks the
Feshbach molecule state used in the experiments.}
\label{fig:potential}
\end{figure}

Molecular spectroscopy with cold atomic gases goes back to the
beginnings of laser cooling \cite{Wei99,Jon06,Koh06}. An
experiment closely related to ours is the one by Araujo {\it et
al.} \cite{deA03} where the Na$_2$ triplet ground state was
explored using a magneto optical trap combined with two-color
photoassociation spectroscopy. Our spectroscopy also makes use of
a two-photon transition, but starts from cold Feshbach molecules
rather than from free atoms. Recent investigations of the
$a \, ^3\Sigma_u^+$ potential of Rb$_2$ include the work of \cite{L06},
\cite{Be09} and \cite{Mu10}. Using one-color photoassociation of
the $a \, ^3\Sigma_u^+$ potential of laser cooled $^{85}$Rb, reference
\cite{L06} put tight constraints on the position of the repulsive
wall of the $a \, ^3\Sigma_u^+$ potential but was not able to resolve the
vibrational structure. The two other groups determined several
ro-vibrational levels using fluorescence spectroscopy \cite{Be09}
or pump-probe photoionization spectroscopy of Rb$_2$ formed on
helium nanodroplets \cite{Mu10}. Our work goes well beyond these
measurements as we fully resolve hyperfine, rotational and Zeeman
structure for almost all vibrational states. Highly precise data
of the asymptotic behavior of the coupled $a \, ^3\Sigma_u^+$- $X \, ^1\Sigma_g^+$
system is contained in the large set of observed Feshbach
resonances \cite{marte:2002, roberts:2001, erhard:2003, papp:2006}
and in the two-photon photoassociation measurements of four weakly
bound levels at zero magnetic field by \cite{wynar}.

This article is organized as follows: Section \ref{exp} presents
the experimental setup and typical dark state spectroscopy scans.
Section \ref{quantum} discusses the relevant quantum numbers of
our studies and the assignment of the observed lines. Section
\ref{ccmodel} is a short summary of the coupled-channel model and
the optimization procedure of the Born-Oppenheimer potentials.
Section \ref{vibmode} discusses the progression of the
substructure of the vibrational manifolds. We conclude the paper
with a summary and an outlook towards further experiments in
section \ref{conclusion}.

\section{Experimental setup and dark state spectroscopy}
\label{exp}

The starting point for our experiments is a 50\,$\mu$m-size pure
ensemble of $3\times10^4$ weakly bound Feshbach molecules which
have been produced from an atomic Bose-Einstein condensate of
$^{87}$Rb by ramping over a Feshbach resonance at a magnetic field
of 1007.4\,G (1\,G = 10$^{-4}$\,T) \cite{Vol03}. They are trapped
in the lowest Bloch band of a cubic 3D optical lattice with no
more than a single molecule per lattice site~\cite{Tha06}. The
lattice depth for the Feshbach molecules is 60\,$E_r$, where
$E_r=\pi^2\hbar^2/2m a^2$ is the recoil energy, with $m$ the mass
of the molecules and $a = 415.22\,$nm the lattice period. Such
deep lattices prevent the molecules from colliding with each
other, which suppresses collisional decay. We observe lifetimes of
a few hundred ms. After producing the Feshbach molecules, the
magnetic field is set to 1005.8\,G where the spectroscopy is
carried out. At this magnetic field, the binding energy of the
Feshbach molecules is 4.4(3)\,MHz$\times h$ (Fig.\,
\ref{fig:potential} inset).

Our dark state spectroscopy works as follows: The Feshbach
molecules in state $|\textrm{i}\rangle$ are irradiated by simultaneous
rectangular pulses from lasers 1 and 2. The pulses typically last
10\,$\mu$s with Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$,
respectively (Fig.\,\ref{fig:potential}). We keep laser 1 resonant
with the $|\textrm{i}\rangle-|\textrm{e}\rangle$ transition and at a power $I_1$ of about
0.1\,mW ($\Omega_{1}= 2 \pi \times $ 0.3\,MHz) such that in the
absence of laser 2 about half of the molecules are lost by
spontaneous emission from $|\textrm{e}\rangle$. Laser 2 with its power $I_2$ (up
to a few hundred mW) is scanned. As long as laser 2 does not hit
an $|\textrm{e}\rangle-|v\rangle$ resonance, laser 1 will continue to induce losses.
However, when an $|\textrm{e}\rangle-|v\rangle$ resonance occurs, the initial state
$|\textrm{i}\rangle$ is projected onto a dark state $|\Psi_{dark}\rangle =
(\Omega_{2}|\textrm{i}\rangle - \Omega_{1}|v\rangle)/ \sqrt{\Omega_{1}^2+
\Omega_{2}^2}$. Molecules in this dark state are shielded from
excitation to the short-lived level $|\textrm{e}\rangle$ \cite{La09}. This leads
to a suppression of molecular losses. After the lasers are
switched off we measure the number of molecules via a reverse
magnetic field sweep through the Feshbach resonance, dissociating
the remaining molecules in $|\textrm{i}\rangle$ into atoms which are detected by
absorption imaging. These measurements are destructive, and for
each point in a scan, a fresh sample of Feshbach molecules has to
be prepared.

\begin{figure}[tbp]
\centering
\includegraphics{Fig2.eps}
\caption{(a) Binding energies $E_B(v)$ for the state $a \, ^3\Sigma_u^+$,
where $v$ is the vibrational quantum number. The line is the
result of a coupled-channel model calculation after optimization
of the Born-Oppenheimer potential.  Five levels were not measured.
(b) Residues \emph{i.e.} difference between experimental data and
theory. Large error bars belong to early measurements without
simultaneous
wavemeter readings of both lasers.}
\label{fig:vibladder}
\end{figure}

The level $|\textrm{e}\rangle$ has an excitation energy of about 295
\,THz$\times h$ with respect to $|\textrm{i}\rangle$ and a width $\Gamma$ =
$2\pi \times$ 8\,MHz. Laser 1, a grating-stabilized diode laser,
is Pound-Drever-Hall locked to a cavity which is in turn locked to
an atomic $^{87}$Rb-line. Laser 2, a Ti:Sapphire laser, is
free-running and typically drifts over a frequency range of a few
MHz within seconds. Both lasers have a short-term laser line width
of several tens of kHz.  Their beams have a $1/e^2$ intensity
waist radius of 130\,$\mu$m at the molecular sample, through which
they propagate nearly collinearly. They are polarized parallel to
the magnetic bias field $B_z$ (pointing in the vertical direction)
and thus can only induce $\pi$ transitions. The frequencies of
both lasers are automatically read out using a commercial
wavemeter (WS7 from HighFinesse). The binding energy $E_B$ of
$|v\rangle$ minus the binding energy of our initial Feshbach state
(4.4\,MHz$\times h$) corresponds to the frequency difference of
the two lasers. The wavemeter has a nominal accuracy of 60\,MHz
after calibration. Over the course of days we have observed drifts
of $\pm200$\,MHz, e.g. by repeatedly addressing the same
spectroscopic line. Over the length of a few experimental cycles
(5\,minutes) the wavemeter is stable to within 10\,MHz, which
represents a random noise floor. Assuming a sufficiently smooth
behavior of the wavemeter, drifts of the wavemeter typically
affect the frequency measurements of laser 1 and laser 2 in a
similar way, especially since the laser frequencies only differ by
3\%. Thus, such common mode drifts cancel out in the binding
energy to first approximation. Indeed, based on our experience
with the wavemeter where we have measured binding energies of a
few sharp lines over an extended period of time and via various
intermediate levels we estimate the accuracy to reach 30\,MHz.
This includes peak position uncertainties due to variations of the
number of molecules produced, as well as a frequency drift of
laser 2 during the time between the laser pulse and wavelength
measurement.

Fig.\,\ref{fig:vibladder} a) shows the measured binding energies
of the triplet potential as a function of the vibrational quantum
number $v$ at a magnetic field of 1005.8\,G. There are 41
vibrational states with binding energies ranging from
5\,MHz$\times h$ to about 7038\,GHz$\times h$.  The vibrational
splitting between the two lowest vibrational levels, $v=0$ and
$v=1$, is about 393\,GHz. Each vibrational state has hyperfine,
rotational, and Zeeman substructure. This structure is spread out
over a range of about 20\,GHz as shown in Fig.\,\ref{fig:v0_v6_1g}
for the states $v=0$ and $6$. These spectra typically consist
of roughly 1000 points corresponding to an average step size of 20
MHz. Each point represents one production and measurement cycle
which takes 28\,s.

In each spectrum of Fig.\,\ref{fig:v0_v6_1g} we observe some 10
lines which vary markedly in line width.  The width of each line
is determined by the coupling between the levels $|\textrm{e}\rangle$ and
$|v\rangle$, \emph{i.e.} the Rabi frequency $\Omega_2$. Interestingly,
for our measuring scheme this width scales as $\Omega_2^2$
\cite{DissLang09} and not as $\Omega_2$ \cite{width, Sho90}, as
one might expect. We have taken advantage of this enhanced
broadening when searching for lines and vibrational manifolds,
which otherwise can be like looking for a needle in a haystack.
For example, for $v=0$ and an intensity $I_2$ of a few hundred
mW we reached line widths of several GHz. The substructure of the
desired vibrational level then appears essentially as a single
broad line with a width of about 20 GHz. Once this level was found
the power was reduced, in order to resolve its substructure.

\begin{figure}[tbp] \centering \includegraphics{Fig3.eps}
\caption{Scans of two vibrational levels: a)  $v = 6$, b)
$v = 0$. Plotted is the remaining molecular fraction as a
function of the binding energy $E_B$, which basically corresponds
to the laser difference frequency. The scans were recorded using
an excited state $|\textrm{e}\rangle$ with $1_{\textrm{g}}$ character. The labels s and d
indicate rotational states $N=0$ and 2, respectively. The numbers
after the labels s or d indicate the position in the spectrum.}
\label{fig:v0_v6_1g}
\end{figure}

In general, we expect spectra of different vibrational levels
to be very similar. Up to $v=35$ this is indeed the case. For
$v \ge 35$ the vibrational manifolds start to overlap, as the
splitting between them becomes smaller than 20 GHz. Spectra a)
($v=0$) and b) ($v=6$) of Fig.\,\ref{fig:v0_v6_1g} are
clearly similar. Some visible differences are artifacts, e.g. it
seems that in spectrum b) two narrow lines are missing at $E_B / h
\approx$\,7021GHz. This can be explained by the fact that the
lines were narrower (width $\leq10$\,MHz) than the local step size
of that scan and thus escaped observation.

\section{Quantum numbers and assignment}
\label{quantum}
\subsection{Quantum numbers}
\label{numbers}

For a deeper understanding of the structure of the
spectrum and its assignment we now discuss the relevant quantum
numbers and selection rules.

For the weakly bound levels like the Feshbach state or the
vibrational levels close to the atomic asymptote Hund's coupling
case (e) with atomic quantum numbers is most appropriate. Here
none of the angular momenta couple to the molecular axis. The
state vector is described by
\begin{align*}
|(f_A,f_B)f, l, F, M\rangle\,,
\end{align*}
\noindent where $f_A,f_B$ are the total angular momenta for atoms
A and B, $f$ is the sum of both atomic angular momenta, $l$ is the
mechanical rotation of the atomic pair, $F$ is the total angular
momentum of the pair and $M$ is its projection onto a space fixed
axis. At low magnetic fields the prepared Feshbach state $|\textrm{i}\rangle$
can be approximated by the state vector $|f_A=2, f_B=2, f = 2,
l=0, F=2, M=2\rangle$. At the magnetic field used in the
experiment (1005.8\,G), $F$ is no longer a good quantum number
because of only weak coupling between $l$ and $f$. A more
appropriate state vector is
\begin{align*}
|(f_A,f_B)f, m_f, l, m_l, M\rangle.
\end{align*}
$M$ is a good quantum number, while for example, $f_A$ and $f_B$
have expectation values \cite{expectval} of about 1.79 instead of
quantum numbers $f_A=f_B=2$ for 0\,G.

Due to the strong hyperfine coupling of Rb and the large exchange
energy, deeply bound levels of the triplet state can be described
by Hund's coupling case (b$_\beta$) at low magnetic fields
\cite{Tow55}, namely:
\begin{align*}
|N, (I,S)f, F, M\rangle\,,
\end{align*}
\noindent where $I$ and $S$ are the total nuclear and electronic
spin quantum numbers, $N$ the molecular rotation including
electron orbital angular momentum, and $F$ and $M$ have exactly
the same meaning as in Hund's case (e). The index $\beta$
indicates that the nuclear spin I is not coupled to the molecular
axis. Since both atoms are in the 5$S$ configuration of Rb, the
molecular electronic orbital angular momentum is zero. This means
that $N=l$ and that $f$ is the same as in Hund's case (e). Here we
get
\begin{align*}
|N, m_N, (I,S)f, m_f, M\rangle\,,
\end{align*}
as an appropriate state vector for higher magnetic fields, where
$M=m_N+m_f$. One can show that owing to the antisymmetry of the
molecular wave function with respect to nuclear exchange (nuclear
spin of $^{87}$Rb $i=\frac{3}{2}$), molecules with even (odd) $N$
in the $a \, ^3\Sigma_u^+$ state must have either a total nuclear spin
$I=1$ or $3$ ($I=0$ or $2$) \cite{Tow55, He50}. For the $X \, ^1\Sigma_g^+$
singlet ground state this relation is reversed because it has $g$
symmetry in contrast to the $u$ symmetry of the triplet state. For
large magnetic fields, $f$ loses its meaning as $S$ and $I$ start
to decouple.

Expectation values for the total nuclear spin and the electron
spin  for our Feshbach level are  $I = 1.56$ and $S = 0.76$,
respectively. Thus we have significant electronic singlet-triplet
mixing and consequently also mixing of the basis vectors with
different $I$. In contrast, the excited intermediate state $|\textrm{e}\rangle$
has well defined quantum numbers $S$ and $I$. Thus $|\textrm{e}\rangle$ largely
determines which quantum numbers the deeply bound $a \, ^3\Sigma_u^+$
levels will have in the Raman transition.

The intermediate level $|\textrm{e}\rangle$ is located in the $v'=13$
manifold of the $1 \, ^3\Sigma_g^+ (5S_{1/2}+5P_{1/2})$ potential. Due to
significant effective spin-spin interaction, the vibrational
manifold is split into two components, $1_{\textrm{g}}$ and $0_{\textrm{g}}^-$,
separated by 47\,GHz. As intermediate level $|\textrm{e}\rangle$ we either
choose a level with $1_{\textrm{g}}$ character or with $0_{\textrm{g}}^-$ character. Its
rotational energy must be low because the Feshbach level has the
lowest rotational quantum number $N=0$. As rotation is low in
$|\textrm{e}\rangle$, decoupling of $S$ and $I$ from the molecular axis by
rotation is not yet important. We can then approximate the state
$|\textrm{e}\rangle$ by quantum numbers for Hund's case (a$_\alpha$) where the
index $\alpha$ indicates that the nuclear spin $I$ is coupled to
the molecular axis,
\begin{align*}
|S\Sigma,I\Omega_I,F,M \rangle\,.
\end{align*}
\noindent The projections of the electronic and nuclear spin onto
the molecular axis appear as quantum numbers $\Sigma$ and
$\Omega_I$, respectively.

The $|\textrm{e}\rangle$ level  with $1_{\textrm{g}}$ character is energetically the
lowest within the $1_{\textrm{g}}(v = 13)$ manifold. It has the quantum
numbers $S=1$, $\left|\Sigma\right|=1$, $I=3$,
$\left|\Omega_I\right| \approx3, $ $F \approx 2$ and $M=2$ for low
magnetic fields $B_z$. Its excitation energy is 294.6264(2)
THz$\times h$ with respect to $|\textrm{i}\rangle$ at a field of 1005.8G.

The  $|\textrm{e}\rangle$ level with $0_{\textrm{g}}^-$ character has an excitation energy
of 294.6736(2)\,THz. It can be described by the quantum numbers
$S=1$, $\Sigma=0$, $I=3$, $M = 2$  and $F = 3$ at $B_z$ = 0.
Because of its low hyperfine coupling the better choice of basis
vector here is
\begin{align*}
|S\Sigma,(JI)F,M \rangle\,,
\end{align*}
where $J$ results from the coupling of the electronic spin and the
molecular rotation ($\mathbf{J}=\mathbf{N}+\mathbf{S}$). For our
$0_{\textrm{g}}^-$ $|\textrm{e}\rangle$ level J is approximately zero. Our $1_{\textrm{g}}$ $|\textrm{e}\rangle$
state is a superposition of N=1 and 3.

As stated before, due to the laser polarization along the magnetic
field only $\pi$ transitions are allowed, which results in the
selection rule $\Delta M = 0$. Further, in a one-photon transition
parity has to change. The Feshbach state $|\textrm{i}\rangle$ has a total parity
``plus"  because it is a $\Sigma^+$ state and $(-1)^N = 1$. Thus
we can only address $|\textrm{e}\rangle$ levels with ''minus" parity and $|v\rangle$
levels with ''plus" parity. This means that the $|v\rangle$ level must
have an even rotational quantum number $N=0,\,2,\,4,\, ...$. For
the $|\textrm{e}\rangle$ level only quantum numbers $N=1,\,3, ...$ or
superpositions of these are available. In fact, as the
$1 \, ^3\Sigma_g^+$ state is well described in a Hund's case (a) basis,
$N$ is in general not a good quantum number for the $|\textrm{e}\rangle$ level.
The selection rule $\Delta N = \pm1$ for $N$ determines the range
of reachable levels for $|v\rangle$ according to the superposition in
$|\textrm{e}\rangle$. The selection rules $\Delta I = 0$ and $\Delta S = 0$ are
important for the transition $|\textrm{e}\rangle$ to $|v\rangle$. For the transition
$|\textrm{i}\rangle$ to $|\textrm{e}\rangle$ they are, however, nearly irrelevant since $I$
and $S$ are not good quantum numbers for level $|\textrm{i}\rangle$.

\subsection{Assignment of spectral lines}
\label{assign}

Figure \ref{fig:v60g} shows measured and calculated (based on the
coupled-channel model, see section \ref{ccmodel}) lines of the
$v = 6$ spectrum, where the excited state $|\textrm{e}\rangle$ with $0_g^-$
character was selected. The lines form three groups with the
quantum numbers $f= 4, 3, \text{and}~2$,  according to the
hyperfine coupling of $S=1$ and $I=3$. This is a clear indication
that the hyperfine energy is still dominant compared to the Zeeman
energy.

Each group consists of one line with $N = 0$  corresponding to a
non-rotating molecule, and several lines with  $N=2$ which are
shifted to lower binding energy by about 2 GHz$\times h$ due to
rotation. The fact that we do not observe lines with $N > 2$ can
be explained as follows: The excited state $|\textrm{e}\rangle$ has ''minus"
total parity. Since $J = 0$ and $S = 1$, $N$ must be equal to one
and is also a good quantum number for this lowest level in the
$0_g^-$ manifold. Thus when using the state with $0_{\textrm{g}}^-$ character
only final states $|v\rangle$ with $N=0,\,2$ can be addressed in the
$a \, ^3\Sigma_u^+$ state due to the selection rule $\Delta N = \pm1$.
\begin{figure}[tbp] \centering \includegraphics{Fig4.eps}
\caption{Scan of the hyperfine, rotational and Zeeman structure in
the $a \, ^3\Sigma_u^+(v=6)$ manifold using an excited level $|\textrm{e}\rangle$ with
$0_{\textrm{g}}^-$-character. The x-axis shows the binding energy $E_B$. The
quantum number $f$ is shown below each of the three groups of
lines. Thick black lines above the spectrum indicate states with
$N=0$, grey lines correspond to states with $N=2$. The upper row
of numbers indicates the expectation values of the magnetic
quantum number $m_f$ at the magnetic field of 1005.8 G.}
\label{fig:v60g}
\end{figure}

The overall structure of each vibrational level can be
understood with a relatively simple effective Hamiltonian
\cite{Du72}:
\begin{align*}
H= \frac{A}{2}\mathbf{S}\cdot\mathbf{I}+B_v\mathbf{N}^2 + \mu_B
\,g_S \, S_z \, B_z
\end{align*}
\noindent with the atomic hyperfine structure constant $A = 3.42$
GHz$\times 4\pi^2 / h$ \cite{Biz99}, the total electronic and
nuclear spin operator $\mathbf{S}$ and $\mathbf{I}$, respectively,
and the operator for molecular rotation $\mathbf{N}$. $B_v$ is the
rotational constant of the desired vibrational level $v$. The last
term describes the Zeeman effect of the electronic spin when
exposed to an external magnetic field $B_z$ in z-direction. The
nuclear Zeeman term is neglected. The Zeeman effect can be
evaluated for the case of strong hyperfine coupling such that $f$
(with $\mathbf{f}=\mathbf{S}+\mathbf{I}$) is still a good quantum
number. Using a simple vector model for $\mathbf{f}$
 one can derive a Land\'e
factor $g_f$:
\begin{eqnarray}
H&= &\frac{A}{2}\mathbf{S}\cdot\mathbf{I}+B_v\mathbf{N}^2+\mu_B
\,g_f \, m_f \, B_z\nonumber\\
\text{with}&g_f&=\frac{g_S}{2}\left(1+\frac{S(S+1)-I(I+1)}{f(f+1)}\right)
\label{hfssimple}
\end{eqnarray}

For each $f$, the $N=2$ group is shifted to lower binding energy
by $6B_v$ compared to $N=0$, corresponding to about 2 GHz for the
$v = 6$ manifold (see Figure \ref{fig:v60g}). Each  $N=2$ group is
split by the Zeeman energy according to the $m_f$ quantum number
of each line. Only those lines which have quantum numbers where
$M=2=m_f+m_N$ can be observed. The level $f=3$ has a small $g_f $
factor (1/6 for $I = 3$), which gives rise to a small Zeeman
splitting. Additionally, one sees that the splitting between $f=4$
and $f=2$, which would be about 12 GHz for the pure hyperfine
part, is enlarged by a Zeeman contribution because of the
different signs of $g_f$ for $f=4$ and $f=2$. In Figure
\ref{fig:v60g} the expectation values for $m_f$ resulting from the
coupled channel model are given in the upper row and show that
these lead to good quantum numbers for $f=4$ and $f=2$ but are
less good for $f=3$, where the hyperfine and electronic Zeeman
energies are small. The mixing of states with different $m_f$ for
$f = 3$ is not included in the simple model of Eq.
\ref{hfssimple}.

As discussed before, for the intermediate state with $0_{\textrm{g}}^-$
character we exclusively observed $N = 0$ and $N = 2$ levels in
the $a \, ^3\Sigma_u^+$ potential. This restriction does not necessarily
apply when using the intermediate level $|\textrm{e}\rangle$ with $1_{\textrm{g}}$
character. The corresponding spectrum in \mbox{Figure
\ref{fig:v0_v6_1g} a)} shows the same $v = 6$ manifold as in
Figure \ref{fig:v60g}, only $|\textrm{e}\rangle$ has $1_{\textrm{g}}$ instead of $0_{\textrm{g}}^-$
character. It has additional lines (at $\sim$4838\,GHz and at
$\sim$4832\,GHz) which match the predicted positions of $N = 4$
levels.

As mentioned in Section \ref{numbers} the level $|\textrm{e}\rangle$ with $1_{\textrm{g}}$
character would be described by a superposition of $N=1$ and $N=3$
states in a Hund's case (b) basis. We thus expect to see
transitions to levels with $N=0, 2, 4$ of the $a \, ^3\Sigma_u^+$ state
according to the selection rule $\Delta N=\pm 1$. In order to
avoid confusion, we note that the $N=0$ and $N=2$ lines between
4825 GHz and 4830 GHz which are clearly visible in Figure
\ref{fig:v60g} are weak or are not even observed in Figure
\ref{fig:v0_v6_1g}. For the same reasons as discussed at the end
of section \ref{exp}, the experimental step size of about 10 MHz
might have been too large compared to the narrow line widths for
these transitions to be seen.

It was not always necessary to carry out a complete scan as in
Figure \ref{fig:v60g} in order to assign quantum numbers to
observed lines in arbitrary vibrational levels. Often it was
sufficient to measure a few characteristic lines and splittings
and to compare them to the calculated spectrum. These data were
then used to optimize the coupled channel model along with its
$a \, ^3\Sigma_u^+$ and $X \, ^1\Sigma_g^+$ Born-Oppenheimer potentials.

\section{Coupled-channel model and optimization of the $a \, ^3\Sigma_u^+$ potential}
\label{ccmodel}

In our work we use a coupled-channel model \cite{Du95, Tie98,
Kr90} that can calculate all bound states of the $X \, ^1\Sigma_g^+$ and
$a \, ^3\Sigma_u^+$ states, which correlate with the atomic asymptote
$5^2S_{1/2}+5^2S_{1/2}$. The program has helped us with the search
for lines as well as with their identification. Using our data we
are able to optimize the Born-Oppenheimer potential of the
$a \, ^3\Sigma_u^+$ state as well as to improve the potential of the ground
state $X \, ^1\Sigma_g^+$ given in \cite{seto:2000}.  In the following we
briefly describe the model and explain how the $X \, ^1\Sigma_g^+ $ and
$a \, ^3\Sigma_u^+$ Born-Oppenheimer potentials are optimized.

To cover the full range of experimental data by a single
theoretical model, only a coupled channel analysis is adequate. It
includes the calculation of the molecular bound states as well as
the scattering resonances. It takes into account the $X \, ^1\Sigma_g^+$ and
$a \, ^3\Sigma_u^+$ potential functions, the hyperfine coupling, the Zeeman
interaction, rotation and the effective spin-spin interaction.
Such a theoretical approach is described in several papers, e.g.
\cite{pas07}.

For the present analysis we include our measurements covering 135
lines of the $a \, ^3\Sigma_u^+$ state as well as data from other work. For
the $X \, ^1\Sigma_g^+$ ground state we added data from Fourier transform
spectroscopy by Seto {\it et al.} \cite{seto:2000} with more than
12000 lines. We also include measurements of Feshbach resonances
for the three isotopologues $^{85}$Rb$_2$, $^{87}$Rb$_2$, and
$^{85}$Rb$^{87}$Rb \cite{marte:2002, roberts:2001, erhard:2003,
papp:2006}, the four asymptotic levels from \cite{wynar}, and
measurements from Fourier transform spectroscopy for the
$a \, ^3\Sigma_u^+$ state reported by Beser {\it et al.} \cite{Be09}.

The full Hamiltonian (cf. \cite{mie00,lau02,pas07}) for a pair of
atoms $A$ and $B$ can be written in the form
\begin{align}
\label{eq:ham}
H &= T_n+U_{\rm X}(R)P_{\rm X} + U_{\rm a}(R)(1-P_{\rm X})\nonumber\\
&+a_A(R)\vec{s}_A\cdot\vec{i}_A+a_B(R)\vec{s}_B\cdot\vec{i}_B\nonumber\\
&+\left[(g_{sA}s_{zA}-g_{iA}i_{zA})+(g_{sB}s_{zB}-g_{iB}i_{zB})\right]
\mu_BB_z\nonumber\\ &+\frac{2}{3} \lambda(R)(3S_Z^2-S^2).
\end{align}
In the present case of the homonuclear molecule $^{87}$Rb$_2$ the
parameters with index A are equal to those of index B. The first
term in the first line shows the kinetic energy $T_n$ where we
take the atomic masses from recent tables by G. Audi {\it et al.}
\cite{aud03}. The next two terms describe the potential energies
$U_{\rm X}$ and $U_{\rm a}$ for the motion of the atoms, where
$P_{\rm X}$ and $1-P_{\rm X}$ are projection operators on to the
uncoupled states X and a, respectively. $R$ corresponds to the
internuclear separation of the two atoms. The second line shows
the hyperfine interaction between the atomic electron spins
$\vec{s}_{A,B}$ and the atomic nuclear spins $\vec{i}_{A,B}$. The
main contribution to the functions $a_{A,B}(R)$ is the Fermi
contact term. The $R$ dependence of the hyperfine parameters
accounts for several effects: It takes into account the electronic
distortions of one atom by the other, \emph{i.e.} the binding, and
an effective coupling of the electron spin of one atom with the
nuclear spin of the other atom. We start with $R$-independent
atomic coupling constants taken from a compilation by Arimondo
{\it et al.} \cite{ari77}. These constants are later refined by a
simple Ansatz for the $R$ dependence (discussed later in Eq.
\ref{eq:hfs}). We neglect the nuclear quadrupole moment in the
hyperfine interaction, which might come into play for deeply bound
levels. The third line in Eq. \ref{eq:ham} gives the Zeeman energy
from the coupling of the electron spin and the nuclear spin with
an external homogeneous magnetic field $B_z$ in $z$ direction. The
electronic and nuclear $g$-factors for the atomic ground state of
the Rb isotopes are taken from the report in \cite{ari77}. This
term couples states with different $f$ quantum numbers. The last
line contains the spin-spin interaction which couples different
$N$ states of basis (b). It is formed by the total molecular spin
$S$ and its projection on the molecule fixed axis $Z$. The
parameter $\lambda$ is a function of $R$, of which one part has
$1/R^3$ dependence as a result from the magnetic dipole-dipole
interaction. In addition, $\lambda$ contains contributions from
second order spin-orbit interactions. The final analysis showed
that such a contribution is significant within the achieved
experimental accuracy. For example, this was important for the
precise location of Feshbach resonances involving $l=1$ and $l=2$
bound levels.

The functional form of the two Born-Oppenheimer potentials,
$X \, ^1\Sigma_g^+$ and $a \, ^3\Sigma_u^+$, is split into three regions on the
internuclear separation axis $R$: the short range repulsive wall
($R < R_\mathrm{SR}$), the asymptotic long range region ($R >
R_\mathrm{LR}$), and the intermediate deeply bound region in
between. The analytic form of the potentials in the intermediate
range, $U_{IR}$, is described by a finite power expansion of a
nonlinear function $\xi$ which depends on the internuclear
separation $R$,
\begin{equation}
\label{eq:uanal} U_{\mathrm {IR}}(R)=\sum_{i=0}^{n}a_i\,\xi^i(R),
\end{equation}
\begin{equation}
\label{eq:xv} \xi(R)=\frac{R - R_m}{R + b\,R_m}.
\end{equation}
Here the $a_i$ are fitting parameters (see table \ref{pota}). We
choose $b$ and $R_m$ such that only few parameters $a_i$ are
needed for describing the steep slope at the short internuclear
separation side and the much smaller slope at the large $R$ side.
$R_m$ is chosen close to the value of the equilibrium separation.
The potential is extrapolated for $R < R_{SR}$  by the short range
part $U_{SR}$ with:
\begin{equation}
\label{eq:rep}
  U_{SR}(R)= u_1 + u_2/R^{N_s}.
\end{equation}
We adjust the parameters $u_1$ and $u_2$ to get a continuous
transition at $R_{\rm SR}$. The final fit uses $N_s\approx 4.5$
for both the  $X \, ^1\Sigma_g^+$\ and $a \, ^3\Sigma_u^+$ states.

For large internuclear distances ($R > R_{\rm LR}$) we adopted the
standard long range form of molecular potentials
\begin{equation}
\label{eq:lrexp} U_{\mathrm {LR}}(R)=-C_6/R^6-C_8/R^8
-C_{10}/R^{10}\pm E_{\mathrm{exch}},
\end{equation}
\noindent where the exchange contribution given by \cite{exc}
\begin{equation}
\label{eq:exch} E_{\mathrm{exch}}=A_{\mathrm{ex}} R^\gamma
\exp(-\beta R)
\end{equation}
is negative for the singlet and positive for the triplet
potential. By adjusting the parameter $a_0$ in Eq. \ref{eq:uanal}
we can  assure a continuous transition from $U_{\mathrm {LR}}$ to
$U_{\mathrm {IR}}$. As mentioned, the data on hand include three
different isotopologues, namely $^{85}$Rb$_2$, $^{87}$Rb$_2$, and
$^{85}$Rb$^{87}$Rb. Using a model developed earlier for LiK
\cite{tie:2009} we checked in the final calculations  that the
data are not sufficiently precise to extract deviations from the
Born-Oppenheimer approximation. Thus we derive the potentials
without any mass scaling correction.

Having discussed the complete physical model, we are now ready to
calculate all the relevant bound state energies and scattering
properties to compare them with the experimental data. We have
decided to evaluate the Hamiltonian in Hund's basis (e) as $f$ is
still a relatively good quantum number. This is due to the fact,
that the hyperfine interaction is larger than the Zeeman
interaction in our experiment. As the total electron spin is not a
good quantum number in Hund's case (e), the choice of this basis
leads to significant non-diagonal matrix elements from the
Born-Oppenheimer potentials, which are given for a pure singlet or
triplet state.

We evaluate the free parameters of the model with a
self-consistent iteration loop and alternate between: (i) coupled
channel calculations of the full Hamiltonian and (ii) solving the
Schr\"odinger equation separately for the states $X \, ^1\Sigma_g^+$ and
$a \, ^3\Sigma_u^+$ using only the first line of Eq.\,\ref{eq:ham} and
applying the Numerov procedure. The coupled channel calculations
in step (i) are used to determine the hyperfine and Zeeman
structure. From this we construct hyperfine free spectroscopic
data. This data is the input for the Born-Oppenheimer potential
fits in step (ii). We incorporate the fitting routine for the
plain Born-Oppenheimer potentials in step (ii). The optimization
of the singlet and triplet potentials is done simultaneously as
both potentials have a common asymptote. These asymptotic
potentials are given in Eq.\,\ref{eq:lrexp} with equal dispersion
coefficients.

Normally, the iteration loop between the potential function fit
and the coupled channel calculation for producing hyperfine
structure free data converges in a few steps, but we observed some
systematic deviations between measurements and calculations.
Specifically the hyperfine splitting showed variations of a few
percent within the vibrational ladder. Even though our
experimental accuracy is not better than 30\,MHz such small
variations are observable since the hyperfine splitting in
$^{87}$Rb$_2$ of 12\,GHz is so large.

In order to theoretically account for the hyperfine variations we
extended the model. We changed the fixed atomic hyperfine
parameters to a function in $R$. We chose a function which
switches at a distance $R_0$ from the atomic value of the
hyperfine constant to another value for a deeply bound dimer,
\begin{equation}
\label{eq:hfs}
a_{A/B}=a_{Rb}\left(1+ \frac{c_f}{e^{(R-R_0)/\Delta R} +1}\right).
\end{equation}
Here $a_{Rb}$ is the atomic hyperfine constant, $c_f$ the
fractional change of the constant and $R_0$ and $\Delta R$
describe the switching distance and its width, respectively. We
also tried several other simple switching functions, which
produced about equal fit quality. The function in
Eq.\,\ref{eq:hfs} is easily applicable for other isotopes by
introducing the proper atomic hyperfine constant, because the
scaling $c_f$ will be independent of the isotope. We chose
$R_0=11.0~a_0$ and $\Delta R=0.5~a_0$ (where $a_0=0.5292 \times
10^{-10}$\,m is the Bohr radius), such that switching takes place
approximately at the minimum of the $a \, ^3\Sigma_u^+$ potential. From our
fits we obtained an amplitude $c_f= -0.0778$, which corresponds to
a variation of the hyperfine coupling across the potential depth
of up to 8\%. The influence of this hyperfine variation within an
individual vibrational manifold will be smaller because of the
averaging of Eq.\,\ref{eq:hfs} over the vibrational wave function.
As mentioned before, also the spin-spin interaction needs
optimization in order to explain systematic shifts of Feshbach
resonances in s-wave and p-wave scattering channels, as measured
in \cite{marte:2002}. The spin-spin interaction couples different
 partial waves $l$ subject to the selection rule $\Delta
l=0,\,\pm2$ such that resonances in a s-wave scattering channel
involve bound states with $l=0$ and $l=2$. The spin-spin
interaction splits the resonances according to $\left|m_l\right|$,
the projection quantum number of rotation on the space fixed axis.
We use a simple functional form for $\lambda(R)$ in Eq.
\ref{eq:ham}
\begin{equation}
\label{lambda} \lambda(R) = -\frac{3}{4}
\alpha^2\left(\frac{1}{R^3}+a_\mathrm{SO}
\exp{\left(-b(R-R_\mathrm{SO})\right)}\right),
\end{equation}
that consists of two terms. The first term represents the magnetic
dipole-dipole interaction and the second term is the second order
spin-orbit contribution. If Eq.\,\ref{lambda} is given in atomic
units, $\alpha$ is the fine structure constant. Since the few data
at hand cannot be highly sensitive to the actual function, we
adopted values for $b$ and $R_\mathrm{SO}$ from a theoretical
approach by Mies {\it et al.} for Rb$_2$ \cite{Mie:96} (b= 0.7196
a$_0^{-1}$ and R$_\mathrm{SO}$= 7.5 a$_0$). We fitted the
parameter $a_\mathrm {SO}$, which yielded $a_\mathrm{SO}$= -0.0416
a$_0^{-3}$. With these parameters the second part in Eq.
\ref{lambda} contributes significantly to the effective spin-spin
interaction in the internuclear separation interval $R < 20 a_0 $.
This affects bound levels of the triplet state that determine the
Feshbach resonances. After optimization we achieve an accuracy of
0.1\,G for the position of the Feshbach resonance. This correction
to the constant $\lambda$ does not influence the description of
the other bound states within their uncertainty.

Finally, we found that we can improve the fit by adding to the
long range formula Eq.\,6 a term of the form $-C_{26}/R^{26}$, and
retaining all constraints such as continuity as before. From our
fits we obtain an amplitude $C_{26}$ which contributes about a
thousandth of the total long range energy at the connection point
$R_{LR}$. The exponent of 26 was chosen so that the term is
negligible outside a small region around the long range connection
point $R=R_{LR}$.

The final parameter sets of the potentials are shown in table
\ref{potX} and \ref{pota} for the singlet and triplet states,
respectively. The derived potentials and the corrections defined
above agree very closely with all observations to within their
experimental uncertainties, and the normalized standard deviation
(\emph{i.e.} standard deviation / experimental uncertainty) is
close to one. In fact, depending on which of our experimental data
sets (hyperfine free spectra, binding energies from dark state
spectroscopy or Feshbach resonances) we compare to the model
calculations, the normalized standard deviations only vary
slightly, ranging from 1.01 to 1.3. The normalized standard
deviation from the joint calculation over the huge body of
ro-vibrational energies (12459 data points from the states
$X \, ^1\Sigma_g^+$ and $a \, ^3\Sigma_u^+$) is about 1.01, which is quite
satisfactory.

As another result of our analysis we were able to eliminate an
ambiguity in  the rotational assignment of the Fourier transform
spectroscopy data reported by Beser {\it et al.} for the state
$a \, ^3\Sigma_u^+$ \cite{Be09}. The authors stated in the
rotational assignment an ambiguity by $\Delta N=\pm1$. We determined
that the shift of the rotational quantum number must be
$\Delta N=+1$. Afterwards we used the data from \cite{Be09} with
that assignment for the further fits. The result in Table
\ref{pota} includes these data and will give a better confidence
in predicting rotational levels with higher rotational quantum
numbers.

\begin{table}
\fontsize{8pt}{13pt}\selectfont \caption{Parameters of the
analytic representation of the potential of state $X \, ^1\Sigma_g^+$. The
energy reference is the dissociation asymptote and the term energy
$T_e^X$ is the depth of the potential. Parameters with $\ast$ are
set for continuous extrapolation of the potential and those with
$\ast\ast$ see text for interpretation.}
\label{potX}
\begin{tabular*}{1.0\columnwidth}{@{\extracolsep{\fill}}|lr|}
\hline
   \multicolumn{2}{|c|}{$R < R_\mathrm{SR}=$ 3.126 \AA}    \\
\hline
   $N_s$      & 4.53389\\
   $u_1^\ast$ & -0.638904880$\times 10^{4}$ $\mathrm{cm}^{-1}$ \\
   $u_2^\ast$ & 0.112005361$\times 10^{7}$  $\mathrm{cm}^{-1}$ \AA $^{N_s}$ \\
\hline
   \multicolumn{2}{|c|}{$R_\mathrm{SR} \leq R \leq R_\mathrm{LR}=$ 11.000 \AA}    \\
\hline
    $b$ &   $-0.13$              \\
    $R_\mathrm{m}$ & 4.209912760 \AA               \\
    $a_{0}$ &  -3993.592873 $\mathrm{cm}^{-1}$\\
    $a_{1}$ &  0.000000000000000000$ $ $\mathrm{cm}^{-1}$\\
    $a_{2}$ &  0.282069372972346137$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
    $a_{3}$ &  0.560425000209256905$\times 10^{ 4}$ $\mathrm{cm}^{-1}$\\
    $a_{4}$ & -0.423962138510562945$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
    $a_{5}$ & -0.598558066508841584$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
    $a_{6}$ & -0.162613532034769596$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
    $a_{7}$ & -0.405142102246254944$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
    $a_{8}$ &  0.195237415352729586$\times 10^{ 6}$ $\mathrm{cm}^{-1}$\\
    $a_{9}$ &  0.413823663033582852$\times 10^{ 6}$ $\mathrm{cm}^{-1}$\\
   $a_{10}$ & -0.425543284828921501$\times 10^{ 7}$ $\mathrm{cm}^{-1}$\\
   $a_{11}$ &  0.546674790157210198$\times 10^{ 6}$ $\mathrm{cm}^{-1}$\\
   $a_{12}$ &  0.663194778861331940$\times 10^{ 8}$ $\mathrm{cm}^{-1}$\\
   $a_{13}$ & -0.558341849704095051$\times 10^{ 8}$ $\mathrm{cm}^{-1}$\\
   $a_{14}$ & -0.573987344918535471$\times 10^{ 9}$ $\mathrm{cm}^{-1}$\\
   $a_{15}$ &  0.102010964189156187$\times 10^{10}$ $\mathrm{cm}^{-1}$\\
   $a_{16}$ &  0.300040150506311035$\times 10^{10}$ $\mathrm{cm}^{-1}$\\
   $a_{17}$ & -0.893187252759830856$\times 10^{10}$ $\mathrm{cm}^{-1}$\\
   $a_{18}$ & -0.736002541483347511$\times 10^{10}$ $\mathrm{cm}^{-1}$\\
   $a_{19}$ &  0.423130460980355225$\times 10^{11}$ $\mathrm{cm}^{-1}$\\
   $a_{20}$ & -0.786351477693491840$\times 10^{10}$ $\mathrm{cm}^{-1}$\\
   $a_{21}$ & -0.102470557344862152$\times 10^{12}$ $\mathrm{cm}^{-1}$\\
   $a_{22}$ &  0.895155811349267578$\times 10^{11}$ $\mathrm{cm}^{-1}$\\
   $a_{23}$ &  0.830355322355692902$\times 10^{11}$ $\mathrm{cm}^{-1}$\\
   $a_{24}$ & -0.150102297761234375$\times 10^{12}$ $\mathrm{cm}^{-1}$\\
   $a_{25}$ &  0.586778574293387070$\times 10^{11}$ $\mathrm{cm}^{-1}$\\
\hline
   \multicolumn{2}{|c|}{$ R > R_\mathrm{LR}$}\\
\hline

 ${C_6}$ &    0.2270032$\times 10^{8}$ $\mathrm{cm}^{-1}$\AA$^6$      \\
 ${C_{8}}$ &  0.7782886$\times 10^{9}$ $\mathrm{cm}^{-1}$\AA$^8$   \\
 ${C_{10}}$ & 0.2868869$\times 10^{11}$ $\mathrm{cm}^{-1}$\AA$^{10}$   \\
 ${C_{26}^{\ast\ast}}$ & 0.2819810$\times 10^{26}$ $\mathrm{cm}^{-1}$\AA$^{26}$   \\
 ${A_{ex}}$ & 0.1317786$\times 10^{5}$ $\mathrm{cm}^{-1}$\AA$^{-\gamma}$   \\
 ${\gamma}$ & 5.317689    \\
 ${\beta}$ & 2.093816 \AA$^{-1}$   \\
\hline
 \multicolumn{2}{|c|}{Derived constants:} \\
\hline
\multicolumn{2}{|l|}{equilibrium distance:\hspace{2.2cm} $R_e^X$= 4.20991(5) \AA} \\
\multicolumn{2}{|l|}{electronic term energy:\hspace{1.6cm}     $T_e^X$= -3993.5928(30) $\mathrm{cm}^{-1}$}\\
\hline
\end{tabular*}
\end{table}

\begin{table}
\fontsize{8pt}{13pt}\selectfont \caption{Parameters of the
analytic representation of the potential of state $a \, ^3\Sigma_u^+$. The
energy reference is the dissociation asymptote and the term energy
$T_e^a$ is the depth of the potential. Parameters with $\ast$ are
set for continuous extrapolation of the potential and those with
$\ast\ast$ see text for interpretation.}
\label{pota}
\begin{tabular*}{1.0\columnwidth}{@{\extracolsep{\fill}}|lr|}
\hline
   \multicolumn{2}{|c|}{$R < R_\mathrm{SR}=$ 5.07 \AA}    \\
\hline
   $N_s$      & 4.5338950\\
   $u_1^\ast$ & -0.619088543$\times 10^{3}$ $\mathrm{cm}^{-1}$ \\
   $u_2^\ast$ & 0.956231677$\times 10^{6}$  $\mathrm{cm}^{-1}$ \AA $^{N_s}$ \\
\hline
   \multicolumn{2}{|c|}{$R_\mathrm{SR} \leq R \leq R_\mathrm{LR}=$ 11.00 \AA}    \\
\hline
    $b$ &   $-0.33$              \\
    $R_\mathrm{m}$ & 6.0933451 \AA               \\
    $a_{0}$ &  -241.503352 $\mathrm{cm}^{-1}$\\
    $a_{1}$ & -0.672503402304666542$ $ $\mathrm{cm}^{-1}$\\
    $a_{2}$ &  0.195494577140503543$\times 10^{ 4}$ $\mathrm{cm}^{-1}$\\
    $a_{3}$ & -0.141544168453406223$\times 10^{ 4}$ $\mathrm{cm}^{-1}$\\
    $a_{4}$ & -0.221166468149940465$\times 10^{ 4}$ $\mathrm{cm}^{-1}$\\
    $a_{5}$ &  0.165443726445793004$\times 10^{ 4}$ $\mathrm{cm}^{-1}$\\
    $a_{6}$ & -0.596412188910614259$\times 10^{ 4}$ $\mathrm{cm}^{-1}$\\
    $a_{7}$ &  0.654481694231538040$\times 10^{ 4}$ $\mathrm{cm}^{-1}$\\
    $a_{8}$ &  0.261413416681972012$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
    $a_{9}$ & -0.349701859112702878$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
   $a_{10}$ & -0.328185277155018630$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
   $a_{11}$ &  0.790208849885562522$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
   $a_{12}$ & -0.398783520249289213$\times 10^{ 5}$ $\mathrm{cm}^{-1}$\\
\hline
   \multicolumn{2}{|c|}{$ R > R_\mathrm{LR}$}\\
\hline

 ${C_6}$ &    0.2270032$\times 10^{8}$ $\mathrm{cm}^{-1}$\AA$^6$      \\
 ${C_{8}}$ &  0.7782886$\times 10^{9}$ $\mathrm{cm}^{-1}$\AA$^8$   \\
 ${C_{10}}$ & 0.2868869$\times 10^{11}$ $\mathrm{cm}^{-1}$\AA$^{10}$   \\
 ${C_{26}^{\ast\ast}}$ & 0.2819810$\times 10^{26}$ $\mathrm{cm}^{-1}$\AA$^{26}$   \\
 ${A_{ex}}$ &0.1317786$\times 10^{5}$ $\mathrm{cm}^{-1}$\AA$^{-\gamma}$   \\
 ${\gamma}$ & 5.317689    \\
 ${\beta}$ & 2.093816 \AA$^{-1}$   \\
\hline
 \multicolumn{2}{|c|}{Derived constants:} \\
\hline
\multicolumn{2}{|l|}{equilibrium distance:\hspace{2.2cm} $R_e^a$= 6.0940(10) \AA} \\
\multicolumn{2}{|l|}{electronic term energy:\hspace{1.6cm}     $T_e^a$= -241.5034(30) $\mathrm{cm}^{-1}$}\\
\hline
\end{tabular*}
\end{table}

\section{Progression of vibrational levels and their substructure}
\label{vibmode}

In the following we will discuss interesting insights which we
have gained from our analysis of the coupled system. In particular
we investigate the progression of several quantities in the
vibrational ladder and the mixing of singlet and triplet states.
We concentrate on the details of the $a \, ^3\Sigma_u^+$ state, because it
was studied in full resolution of hyperfine and Zeeman energy over
the whole vibrational ladder for the first time. Such a body of
data does not exist for any other alkali dimer.

\subsection{Vibrational ladder and rotational progression}
\label{sec:vibladder} We return  to the vibrational states shown
in Fig.\,\ref{fig:vibladder}, which allows us to compare in detail
the optimized coupled-channel model with our experimental findings
for  all vibrational states.

\begin{figure}[htbp]
\centering \includegraphics{Fig5.eps} \caption{Energy splitting
$\Delta E_B$ between the ``s1"-level ($N=0$) and the ``d1"-level
($N=2$) for a given vibrational quantum number $v$. Large error
bars correspond to early measurements without simultaneous
wavemeter readings (see text). The continuous line is a
calculation based on the coupled channel model.}
\label{fig:vibladders1d1}
\end{figure}

The binding energies given in Fig.\,\ref{fig:vibladder} a)
correspond to the most deeply bound level in each vibrational
level, \emph{i.e.} the state with quantum numbers $N = 0$, $f = 2$
(at 0\,G) and $M = 2$. We will refer to this level as ``s1". The
``s1" level of $v=0$ in the $a \, ^3\Sigma_u^+$ potential is also the
lowest bound state in that potential and has an observed binding
energy of $(7038.067\pm0.050)$\,GHz$\times h$ at 1005.8\,G with
respect to the lowest atomic asymptote $f_A\approx1, m_A=1$ and
$f_B\approx1, m_B=1$.  A list of all the other measured and
calculated bound state energies together with their expectation
values of the quantum numbers can be found in the supplementary
material \cite{EPAPS}.

Fig.\,\ref{fig:vibladder} b) shows the residues between our
experimental ``s1" data and the optimized model. In general the
model agrees very well with the measurements to within the error
bars. The data points with larger error bars belong to early
measurements without simultaneous wavemeter reading of both
lasers, which leads to a significant increase in the experimental
uncertainty (see section \ref{exp}).

We now investigate the progression of the rotational splitting in
the vibrational ladder. For this, we consider the ``s1" level
($N=0$) and its nearest neighbor in our spectra with ($N =2$)
which we call ``d1" (see Figs. \ref{fig:v0_v6_1g} and
\ref{fig:v60g}). The ``s1" and ``d1" levels both have $f=2$ and
$m_f = 2$. Thus the splitting between them is to a high degree
rotational energy. The splitting decreases with increasing $v$,
which is due to the fact, that the mean distance between the
Rb$_2$ nuclei (and hence the effective moment of inertia)
increases as $v$ increases. We have directly observed this
behavior in our experiments (see Fig. \,\ref{fig:vibladders1d1}).
There is very good agreement between the experimental data and the
calculation (continuous line) using the optimized potential of the
$a \, ^3\Sigma_u^+$ state.

Besides the ``s1" level ($N=0, f = 2$) and the ``d1" level ($N=2,
f = 2$) we also observe states with $N=4$ and $f= 2$ for several
low lying vibrational levels. The two lines in Fig. 3 a) at 4839
GHz and the two lines in Fig. 3  b) at 7032 GHz have $N=4$.
Observation of these levels improved the precision when fixing the
position of the $a \, ^3\Sigma_u^+$ potential minimum in terms of the
internuclear distance, or  the effective rotational constant
$B_v$. This turned out to be important for the reassignment of the
observations by Beser {\it et al.} \cite{Be09}, as discussed at
the end of section IV.

\subsection{ Hyperfine splitting and singlet-triplet mixing}
\label{hfsandmixing}

\begin{figure}[tbp]
\centering
\includegraphics{Fig6.eps}
\caption{a) Progression of the substructure of the vibrational
manifold at $B$ = 1005.8 G. Shown are calculated $a \, ^3\Sigma_u^+$ levels
with quantum numbers $I \approx 3$, $M = 2$, $N = 0, 2$ and $v =
0...\,34$. Thick black lines correspond to $N=0$, grey ones to
$N=2$. The ``s2" level serves as energy reference ($\Delta E_B = 0
$) in each vibrational substructure. In addition to the triplet
levels nearby singlet levels of the $X \, ^1\Sigma_g^+$ potential are also
shown (Thick black x's: $N=0$, grey x's: $N=2$). \\b) Symbols as
in part a). To discuss the singlet-triplet mixing from a
theoretical point of view we now also include lines with
$I\approx1$ but only $N=0$ for clarity. On the bottom, the
approximate quantum numbers $(I,f)$ are given. See text for
details.} \label{fig:alltheory}
\end{figure}

It is instructive to investigate the progression of the multiplet
structure of the vibrational levels. Figure
\ref{fig:alltheory}\,a) shows calculated levels for $M=2$ stacked
on top of each other for increasing $v$ of the $a \, ^3\Sigma_u^+$
potential. We concentrate on the triplet states ($S = 1$) with
rotation $N=0,\,2$ and we restrict ourselves further to triplet
levels with $I = 3$. Additionally, we show singlet levels ($S=0$)
with $I = 2$ that are located in the vicinity of the triplet
levels. Thus, the typical stick spectrum of each vibrational
manifold in the $a \, ^3\Sigma_u^+$ state looks like the one in Fig.
\ref{fig:v60g}. There are three groups of lines with $f=4,3,2$,
respectively. The vibrational quantum number $v$ runs from 0 to
34, where the multiplets do not yet overlap for different
vibrational levels.

In order to properly stack the data, we have chosen the level with
$N=0$, $f \approx 3$ to be the energy reference ($\Delta E_B = 0
$) for each vibrational level. We call this level ``s2" (see also
Figs. 3 and 4). This level is quite insensitive to mixing with
singlet levels which makes it a good reference because there is no
singlet level with $f=3$ and even parity for direct coupling. It
is clear from Fig. 6 a), that the multiplet structure of different
vibrational manifolds is similar, at least from $v$ = 0 to about
$v = 30$ as expected from the simple model Hamiltonian in
Eq.\,\ref{hfssimple}. The structure changes in a smooth and
monotonical way with $v$. For each of the quantum numbers
$f=4,\,3,\,2$ the splitting between $N=0$ and $N=2$ decreases with
increasing $v$ as discussed previously in
section\,\ref{sec:vibladder}.

Appreciable mixing of a singlet and a triplet level can occur when
the two levels with $f=2$ are located energetically closely
enough, in our case within a few GHz, corresponding to the
strength of the hyperfine interaction. One effect of mixing is a
shift of the level positions due to level repulsion, clearly seen
in Fig. \,\ref{fig:alltheory}. For $v=24$, for example, the
triplet lines of $f = 2$ are pushed to the left by about 0.4\,GHz
by the close singlet levels on the right hand side. For $v < 20$
we also observe narrow coincidences between triplet and singlet
levels, e.g. for $v=15$. Here almost no perturbation appears in
the graph, in contrast to the case $v=24$. We attribute this to a
significantly lower overlap of the vibrational wavefunctions of
the singlet and triplet levels for the case $v=15$ compared to
that of $v=24$. For vibrational quantum numbers $v > 30$ mixing is
very strong and happens for every vibrational level. Because the
long range behaviour of the $a \, ^3\Sigma_u^+$ and $X \, ^1\Sigma_g^+$ states is
similar, the overlap of the wave functions will become large for
high $v$.  The vibrational spacing will become similar to the
hyperfine splitting, and the state vectors here are best described
by Hund's coupling case (e) \emph{i.e.} quantum numbers of atom
pairs.

Singlet-triplet mixing not only occurs for triplet molecules with
$I = 3$ but also for $I = 1$. Figure \ref{fig:alltheory} b) shows
($N = 0$) triplet levels with $I = 3$ and $I = 1$ for $v = 25 $ to
34. The repulsion of the $I = 1$ levels from the singlet levels is
clearly visible. The figure shows an avoided-crossing-like
behavior for the levels on the right as a function of $v$,
indicating the strong mixing between $I=3$ and $I=2$ which also
means that u/g symmetry is broken for these levels. Further, our
calculations show that only levels with the same $f$ and $N$
quantum numbers mix considerably. The singlet lines shown here
have $f = 2$ and $N = 0$. Apparently, despite the relatively
strong magnetic fields of about 1000 G $f$ is still quite a good
quantum number. Indeed, $f$ appears as quantum number in both
state vectors for Hund's case (b) and (e) and loses its meaning
only for much higher magnetic fields. $N$ is good due to the small
effective spin-spin interaction.

The shift in position due to mixing can be traced more clearly
with a simple difference method discussed in the following.
Fig.\,\ref{fig:dev2_s1} shows the ``discrete second derivative"
$\delta^2 E_B(v)$ of the function for the binding energies
$E_B(v)$, \emph{i.e.} $\delta^2 E_B(v) = E_B(v+1) - 2 E_B(v) +
E_B(v-1)$. In other words it is the difference between the
neighboring energy splittings. The curve calculated from binding
energies ``s1" exhibits sudden jumps for particular vibrational
levels. These are due to singlet-triplet mixing and the $v$
positions are consistent with Fig.\,\ref{fig:alltheory}. In
contrast, the $ \delta^2 E_B(v)$ curve for the ``s2"-state is
smooth, and thus does not indicate mixing with the singlet lines,
which justifies its choice as energy reference in
Fig.\,\ref{fig:alltheory}.
\begin{figure}[tbp] \centering \includegraphics{Fig7.eps}
\caption{ ``Discrete second derivative" of $E_B(v)$,
\emph{i.e.} $\delta^2 E_B(v) = E_B(v+1) - 2 E_B(v) + E_B(v-1) $.
The solid curve
 connects the points from the ``s1" levels while the  diamonds
correspond to the ``s2"-state.}
\label{fig:dev2_s1}
\end{figure}

We have confirmed the singlet-triplet mixing experimentally.
Fig.\,\ref{fig:v28to33_1g} shows scans of parts of the vibrational
levels $v = 28, 31, 33$ where the intermediate level $|\textrm{e}\rangle$ with
$1_{\textrm{g}}$ character was used (see also Fig.\,\ref{fig:v0_v6_1g}). The
``s2"-level is chosen to be the energy reference at $\Delta E_B =
0 $ as before. For $v = 28$ we observe a line structure similar to
that of Figs. \ref{fig:v0_v6_1g} and \ref{fig:v60g}. Thus we take
the $v=28$ spectrum as reference of the pure case "triplet". In
fact, from our calculations we see that the next singlet level is
located about 30 GHz away. This detuning gives rise to only a very
small mixing which slightly lowers the triplet character of the
``s1"-level to 0.99. The singlet level obtains a triplet character
of 0.02. However, due to the selection rule $\Delta S = 0$, the
transition to the intermediate state $|\textrm{e}\rangle$ with its very pure
triplet character would simply be too weak to see.
\begin{figure}[tp]
\centering
\includegraphics{Fig8.eps}
\caption{Observation of strongly perturbed singlet levels. We
compare sections of line spectra for the vibrational levels $v =
28, 31$ and $33$. As in Fig. \ref{fig:alltheory} the ``s2"-level
is chosen as energy reference $\Delta E_B = 0 $.  The lines (and
crosses) above the experimentally observed spectrum are from
coupled-channel calculations. Black lines (crosses) represent
$N=0$ levels, while thin grey lines (crosses) correspond to $N=2$
levels for $S=1$ ($S=0$ respectively). The lines that originate
from the singlet states are labeled with `` $^1\Sigma$ ".}
\label{fig:v28to33_1g}
\end{figure}

This situation changes drastically for $v = 31$ and $33$. Two
additional lines are visible in the spectrum which originate from
singlet states with $M = 2$, $N = 0~\text{and}~2 $. Here the
singlet-triplet mixing is close to 40\% which makes the singlet
lines easily detectable. Additionally, for $v=31$ the ``s1" and
''d1" components are shifted to lower values than expected from
the reference spectrum $v=28$ due to repulsion by the singlet
component on the high energy side. In the case of $v=33$ the
singlet component pushes the ``s1" and ''d1" in the  opposite
direction. Our calculations show that even the chosen energy
reference ``s2" starts to show mixing, indicated by its reduced
triplet character of 0.98. In parallel to the singlet-triplet
mixing, the quantum number $I$ loses its meaning as well, e.g. the
level ``s1" of $v=33$ has an expectation value for $I$ of 2.75
instead of 3, due to a significant contribution of $I=2$ from the
singlet state. Comparing the stick spectrum line positions with
observed lines in Fig.\,\ref{fig:v28to33_1g} we note the very nice
agreement between experiment and theory. We want to mention that
we measured the singlet levels at the predicted positions from
first calculations. This emphasizes the predictive power of the
presented model.

The fact that we observe singlet-triplet mixing for relatively
deeply bound levels with binding energies of a few hundred
GHz$\times h$ is already quite interesting. In addition, it
provides valuable information for fixing the energy position of
the triplet levels with respect to the singlets with high
precision. This is especially important for the large body of data
of the singlet system, see section \ref{ccmodel}, which was
obtained with no connection to the triplet state \cite{seto:2000}.

\subsection{Franck-Condon overlap}
\begin{figure}[htbp] \centering
\includegraphics{Fig9.eps}
\caption{Normalized transition matrix element $\Omega_2 /(2\pi
\sqrt{I_2})$ between the excited level $|\textrm{e}\rangle$ ($v'=13, 1_g$
character) in the $1 \, ^3\Sigma_g^+$ potential and the ``s1" level with
vibrational quantum number $v$ in the lowest triplet potential
$a \, ^3\Sigma_u^+$. } \label{fig:FranckCondon}
\end{figure}
When scanning over all vibrational levels, the transition matrix
elements for the transition from $|\textrm{e}\rangle$ to $|v\rangle$ are not constant
but oscillate as a function of $v$. This oscillation is mainly due
to variations in the Franck-Condon overlap between the $|\textrm{e}\rangle, \
v'=13$ vibrational wave function and the vibrational wave
functions $|v\rangle$ of the $a \, ^3\Sigma_u^+$ potential. Figure
\,\ref{fig:FranckCondon} shows the normalized transition matrix
element, $c_2=\Omega_2 (v) / (2\pi\sqrt{I_2})$, where $\Omega_{2}$
is the Rabi frequency and $I_{2}$ is the intensity of laser 2.
$\Omega_{2}$ is determined from the measured width of the dark
resonance \cite{Method,Win07}.  For these measurements we used the
excited level $1_g$ as $|\textrm{e}\rangle$ and the ``s1" levels as $|v\rangle$ of
the $a \, ^3\Sigma_u^+$ state. The transition matrix element $c_2$ varies
from about\, \mbox{($0.2$ to $33$)\,MHz/$\sqrt{\mbox{Wcm}^{-2}}$}.
In terms of a dipole moment, $\langle e r \rangle = \Omega_2 (v) /
\sqrt{I_2} \times \hbar \sqrt{\epsilon_0 c/2}$, this corresponds
to ($0.05$ to $8.0$)$\times10^{-30}$\,Cm.

In comparison, the amplitude for the transition between $|\textrm{i}\rangle$ and
$|\textrm{e}\rangle$ is $\Omega_{1}/\sqrt{I_{1}}= $0.4\,MHz/
$\sqrt{\mbox{Wcm}^{-2}}$. We determine $\Omega_1$ from resonant
excitation, measuring how quickly $|\textrm{i}\rangle$ molecules are lost for a
given laser intensity $I_1$. The data in Fig.
\ref{fig:FranckCondon} can be used to fix the position of the
$1 \, ^3\Sigma_g^+$ potential relative to the $a \, ^3\Sigma_u^+$ potential
applying the Franck-Condon approximation.

\section{Conclusion and outlook}
\label{conclusion}
In this paper we demonstrate high resolution
dark state spectroscopy of ultracold $^{87}$Rb$_2$ molecules. We
are able to resolve the vibrational, rotational, hyperfine and
Zeeman structure of the lowest $a \, ^3\Sigma_u^+$ potential with an
absolute accuracy as high as 30\,MHz. We find that the hyperfine
structure is weakly dependent on the vibrational level, which also
influences predictions of Feshbach resonances if one would like to
reach accuracies on the order of 0.1 G.

By optimizing mainly the triplet Born-Oppenheimer potential we
obtain a model that can quite accurately predict all vibrational
levels in the $a \, ^3\Sigma_u^+$ and $X \, ^1\Sigma_g^+$ potentials for different
ranges of rotational quantum numbers. After reassigning recent
data from Fourier transform spectroscopy \cite{Be09} we extend the
range of applicability to rotational states as high as $N$ = 70
for the $a \, ^3\Sigma_u^+$ potential. Because of this reassignment also
the molecular parameters like the equilibrium internuclear
separation $R_e$ or the dissociation energy $D_e$ have changed
significantly compared to values reported in \cite{Be09}. The new
values are given in Table \ref{pota}. For rotational levels with
$N\leq4$ the model calculations for triplet levels of any
vibrational level should have a precision similar to that of our
measurements, \emph{i.e.} about 30 MHz. For higher $N$ this
precision will increasingly degrade resulting from the reduced
accuracy of the data from \cite{Be09} of about 300 MHz for $N=70$.
Compared to \cite{seto:2000} the potential of the ground state
$X \, ^1\Sigma_g^+$ is significantly improved close to the atomic asymptote
by including data on the mixed singlet-triplet levels of this
study and data on Feshbach resonances from various other sources.
We recommend the derived potential function for further use. It
can predict the deeply bound levels with an accuracy (about 50
MHz), comparable to that of the Fourier transform spectroscopy in
\cite{seto:2000}. The asymptotic levels are accurate on the order
of a few MHz or better as their position is determined by the
precisely measured location of Feshbach resonances.
\begin{table}
\caption{Scattering lengths (in Bohr radius, $a_0=0.5292 \times
10^{-10}$m). $a_{lowest}$ gives the scattering length of the
energetically lowest hyperfine state at $B_z$ = 0.}
\label{tab:scat}
\begin{tabular}{|r|r|r|r r|} \hline
isotope & {$a_{singlet}$} & {$a_{triplet}$} & { $a_{lowest}$}& ${\rm Rb}+{\rm Rb}$\\
    &  &  &  & $(f,m_f) + (f,m_f)$\\
\hline
    $87/87$ &     $90.35$ &     $ 99.04 $ &  $ 100.36 $&$ (1,1)+(1,1)$   \\
    $85/85$ &     $2720$ &   $ -386.9 $ &       $-460.1  $&$  (2,2)+(2,2) $   \\
    $87/85$ &   $ 11.37 $ &    $ 201.0 $ &      $ 229.4 $&$  (1,1)+(2,2) $   \\
\hline
\end{tabular}
\end{table}
We improved the description of the large set of $^{87}$Rb Feshbach
resonances as compared to ref. \cite{marte:2002} where the
original calculation was based only on a few selected resonances
and a derived asymptotic form of the two potentials. In
\cite{marte:2002} deviations as high as 2\,G appeared and for
example the resonances of asymptote $f_A=1,m_A=0 + f_B=1,m_B=1$
were all calculated systematically too high. In the present model
these discrepancies disappear. The average deviation over all 46
resonances, given in \cite{marte:2002, roberts:2001, erhard:2003,
papp:2006}, is about 0.15\,G \cite{reass}. If one removes the
R-dependence of the hyperfine coupling as given in
Eq.\,\ref{eq:hfs}, deviations in the order of 0.5\,G appear
varying according to the different $f$ levels which correlate to
the Feshbach resonances. To improve the overall fit, one probably
needs to include many-body effects of the collision process for
detecting the resonances. The overall improvement of the long
range behavior also allows the whole set of scattering lengths for
the three isotopologues to be calculated. Table \ref{tab:scat}
gives a selection of these calculations. The error of these data
will be in the last digit shown. These values are consistent with
earlier publications but more precise and in general internally
consistent between the isotopologues.

Our model will be valuable for our planned collision experiments
of ultracold molecules which have to be prepared in well defined
quantum states. The singlet-triplet mixing that we observe is of
interest for a proposed precision experiment to measure the time
dependence of the electron to proton mass ratio \cite{DeM08} (see
also \cite{Ch09,Ze08}). The precision of future measurements would
be mainly limited by the accuracy of the wavemeter and could be
improved by one or two orders of magnitude. Such a high precision
could be used, e.g. to test the fundamental limits of the
coupled-channel model, where we assume the validity of (1) the
Born-Oppenheimer approximation, (2) using Zeeman terms with atomic
parameters only, (3) a limited functional dependence in $R$ of the
hyperfine and spin-spin interaction and (4) neglecting quadrupole
hyperfine coupling.

\begin{acknowledgments}
We thank Marius Lysebo and Leif Veseth for model calculations to
characterize the excited states $|\textrm{e}\rangle$. We thank Matthias Gerster
for studying the simple Hamiltonian in Eq. (1). We acknowledge
helpful predictions from Christiane Koch on the Franck-Condon
overlap for the $|\textrm{e}\rangle - |v\rangle$ transition. This work was supported
by the Austrian Science Fund (FWF) within SFB 15 (project part
17). E.T. thanks the cluster of excellence ``QUEST" for support
and the Minister of Science and Culture for providing a Niedersachsenprofessur.
\end{acknowledgments}

\smallskip
$^*$ These authors both contributed equally to this work.

redpajama_set_name: RedPajamaArXiv


Background The deployed Intensive Therapy Unit (ITU) in the British military field hospital in Camp Bastion, Afghanistan, admits both adults and children. The purpose of this paper is to review the paediatric workload in the deployed ITU and to describe how the unit copes with the challenge of looking after critically injured and ill children.
Methods Retrospective review of patients <16 years of age admitted to the ITU in the British military field hospital in Camp Bastion, Afghanistan, over a 1-year period from April 2011 to April 2012.
Results 112/811 (14%) admissions to the ITU were paediatric (median age 8 years, IQR 6–12, range 1–16).80/112 were trauma admissions, 13 were burns, four were non-trauma admissions and 15 were readmissions.
Mechanism of injury in trauma was blunt in 12, blast (improvised explosive device) in 45, blast (indirect fire) in seven and gunshot wound in 16. Median length of stay was 0.92 days (IQR 0.45–2.65). 82/112 admissions (73%) were mechanically ventilated, 16/112 (14%) required inotropic support. 12/112 (11%) died before unit discharge. Trauma scoring was available in 65 of the 80 trauma admissions. Eight had Injury Severity Score or New Injury Severity Score >60, none of whom survived. However, of the 16 patients with predicted mortality >50% by Trauma Injury Severity Score, seven survived. Seven cases required specialist advice and were discussed with the Birmingham Children's Hospital paediatric intensive care retrieval service. The mechanisms by which the Defence Medical Services support children admitted to the deployed adult ITU are described, including staff training in clinical, ethical and child protection issues, equipment, guidelines and clinical governance and rapid access to specialist advice in the UK.
Conclusions With appropriate support, it is possible to provide intensive care to children in a deployed military ITU.


text:
Q: Control deck losing to Sulfuras quest I play a mage control deck in Hearthstone.
Warriors doing the Sulfuras quest always beat me because they don't mind being controlled. They just have to delay with taunts until they get Sulfuras, then they win.
Is there any way to counter Sulfuras, or should I just concede as soon as I see the quest?
1   Mana Wyrm             4  Cone of Cold x2
2   Amani Bezerker x2     4  Fire Plume Phoenix
2   Faerie Dragon         4  Fireball
2   Frostbolt             4  Gnomish Inventor
2   Pyros                 4  Polymorph x2
2   Ravasaur Runt         4  Spellbreaker
2   Shimmering Tempest    4  Stegodon
3   Counterspell          4  Water Elemental
3   Flesheating Ghoul     5  Gurubashi Berzerker
3   Gluttonous Ooze       6  Blizzard x2
3   Shattered Sun Cleric  7  Flamestrike
3   Stonechill Defender   7  Stormwind Champion
3   Twilight Flamecaller  7  Volcanosaur

A: Let me start by stating that the Taunt Warrior usually has the upper hand in control matchups. When finding yourself in unfavorable matchups you usually have to take more risks than you would take in even matchups. Don't be afraid to make all-in plays if you think that there is no other way to win at some point.
Winconditions
To correctly approach a specific matchup you need to identify either side's wincondition. For the Taunt Warrior this is obviously completing the quest and closing the game out with the Ragnaros heropower.
You wincondition is controlling the board and finishing the game with minion attacks. This is actually true for all your matchups, since you have no other tools to win the game.
This already gives you a rough idea of your general gameplan: Aggressively pushing for board control and keeping it by using your powerful AOE-spells until you minions represent lethal damage.
Execution
You absolutely must secure board control in the early game and pressure the Taunt Warrior. Always keep in mind that most of the Taunt Warriors run two Brawls, so make sure to not overextend. However don't tunnelvision on playing around the second brawl, you probably have to take the risk of them having it ready at some point.
In the midgame, focus on maintaining board control by using Blizzard and Flamestrike. The perfect scenario is having a good board by turn 5 and then following this up with Blizzard and Flamestrike in the following turn.
Other tips:

*

*Don't be afraid to use Fireballs and Frostbolts early to make sure that the Warrior can not take over the board early.

*Always keep in mind that board control is your wincondition. If you lose it, you will probably lose the game, unless you manage to finish the Warrior off with your burn spells.

*Don't use your weapon-remove on Fiery War Axes, unless you absolutely have to to maintain your board. Most Taunt Warriors run Gorehowl, removing this weapon is crucial to keeping your board intact through the midgame.

Deck optimization
Your deck has a lot of one-ofs which is not really common outside of Reno decks (RIP). On the one hand this lets you fit more tech-cards but on the other hand this makes your deck very inconsistent.
You definitely should run two copies of:

*

*Fireball - More burst, opens up a fallback wincondition if you fail to control the board.

*Frostbolt - Same as above and additionally serves as a great early game removal tool.

*Mana Wyrm - Together with Frostbolt this card can get insane trades in the early game and secure board control.

*Arcane Intellect - Reliable card draw in the mid- to lategame.

*Water Elemental - Overall very solid minion and great against weapon classes. Any board-focused deck should run two of those.

This adds 6 more cards to your current deck, so I propose to cut:

*

*both Cone of Colds - This card is only really useful for decks that want to stall the game, like Freeze Mage. But since stalling the game is the complete opposite of your goal and actually the Taunt Warrior's wincondtion, this card does not help you in any way.

*Counterspell - While mage secrets have the strongest effects out of all secrets in the game, they are also fairly expensive with 3 mana. Running secrets without cards like Kirin Tor Mage is usually not a good idea, since you lose a lot of tempo when you pay 3 mana without any effect on the board. Decks like Freeze Mage are okay with that, your deck is not.

*Stonehill Defender - While this is a great card in defensive decks, it's board impact is very insignificant for it's mana cost.

*Flesheating Ghoul - This card is only good if it gets the right setup which you should not rely on.

*Shimmering Tempest - Bad stats for a 2 mana minion and it's effect might not come into play later in the game, depending on the manacost of the generated spell.

Other considerations:
Depending on the type of decks you encounter on the ladder, you should merge some of your early game one-ofs into two copies of the same card. This overall makes your deck a lot more consistent and increases the odds of drawing specific cards in the early game.

A: Denker has given some nice advice already, let me add a bit.
If you queue with a deck there are decks that you're going to be good against and decks that you're bad against. This means that on average, you might only win 35% of your matches against a control warrior. But that still leaves you 1/3 of your games where you win, so do not concede. Freeze mage vs control warrior used to be a typical example where the mage only won like 20% of the time.
In the case of a control warrior, you want to get them down before they start using their new hero power. That's not easy to do as they can also armor up. Adding some extra burn to either kill their minions or burn their face is useful for this. Double fireball and adding a Fireland portal will help.
In general, to make a control mage deck you can either build a freeze or grinder deck. For a grinder deck you want to have cards that provide a card or value advantage so that you can outlast your opponent. Typical inclusions can be Antonidas, Pyros, Medivh, the spell that gives five random spells or Elise.

redpajama_set_name: RedPajamaStackExchange


Wishing everyone a joyous time of year, whether that is Channukah, Christmas, Diwali, Yule or whatever is special to you and those whom you love.
You have probably taken the time to put your feet up… but will your customers?
Many people will be bored on Christmas and start playing with their iPads, iPhones or the old desktop computer lurking in the corner. In other words, just because you have a holiday doesn't mean that your customers and potential customers do. So make sure they have something to do!
What better than to have them spending money on your Ecommerce website – even better if they are using gift tokens that a friend or relative has bought for them. They can find out if Apple Pay or Google Pay really works on their new devices.
In the world of websites, fast is king.
If you want the very best in website hosting, come to us.


Incredible, anti-patriotic French document calling for an end to violence and calling for "L'Humanite". France was considered by most Prussians as being very aggressive regarding the loose coalition of states prior to the formation of the German Empire. The conflict led to the end of the Nalpoleonic era and to France becoming the only republican great power. The document is headed - TUER EST TOUJOURS UN CRIME! - To Kill is Always A Crime.


COLUMBUS --Brett Connolly had no idea his repeated attempts to give a puck to a young girl would become a social media sensation or that he'd earn a new fan for his efforts.
Now, the Washington Capitals forward is making plans to meet 6-year-old Keelan Moxley when she attends Game 5 of the Eastern Conference First Round between the Capitals and Blue Jackets at Capital One Arena on Saturday. Moxley and her family will be guests of Capitals owner Ted Leonsis.
Connolly said wife, Katrina, has been in touch with Moxley's mother, Lauren, and filling him in on all that Moxley has been through since Connolly tossed her a puck before Game 2 of the best-of-7 series in Washington on Sunday.
"I'm sure I'll get to meet her. I'll sure we'll do something for her," Connolly said before Game 4 at Nationwide Arena on Thursday (7:30 p.m. ET; USA, SN, TVAS2, FS-O, NBCSWA). "It's pretty cool. She seems like a great young kid with a ton of energy and a good attitude and a good little sense of humor. I'm excited to see where that all goes."
It all began during warmups Sunday, when Connolly was waiting his turn to shoot and spotted Moxley watching from behind the glass.
"I saw her, singled her out and then I tried to give her a puck," Connolly said. "Usually, I throw a puck every game [to a fan]. I'm kind of waiting for my shot, so I do it."
Connolly's first attempt to toss the puck over the glass to Moxley was intercepted by a man who gave the puck to a boy standing to her right instead. Seeing what happened, Connolly tapped his stick blade against the glass pointing at Moxley.
"I saw her face and she was not happy, obviously," Connolly said. "So I said, 'I've got to get another one.'"
The man again intercepted Connolly's toss and gave it to the boy standing to Moxley's left. A frustrated Connolly then banged the glass with his hand, making it clear he was trying to give the puck to Moxley, and went to get another one.
"I kind of looked at the [man] and just made a point. It was, 'Just make sure she gets it,'" Connolly said. "It's funny. Whatever happened to ladies first?"
Connolly tossed a third puck over the glass and this time the man gave it to Moxley. She broke into a smile and turned around to show it to her father, who was standing a few rows back.
The whole sequence was captured by NBC Sports Washington, and the video became the most streamed NHL video on Twitter of all time. Moxley has become a mini celebrity at school and her mother shared a video of her talking about what happened with CBC News.
"I didn't know until after the game my phone was blowing up and everybody had saw it," Connolly said. "It's really, really cool just to see something little that you can do for a young kid, a young girl. I saw her interview. She's extremely well spoken for a young kid. I think she's a fan favorite now, so she's getting a little bit of recognition too. So she's having fun with it. That's what her mom was saying. She's just having the best time ever. All her classmates think she's famous."


text:
Cranfield University and EGB Engineering UK secure business investment from Innovate UK
Published 11 Nov 2020
Press release number PR-SWEE-20-145
Cranfield University and EGB Engineering UK have been successful in securing co-funding from the UK's innovation agency, Innovate UK, for industrial research into low-carbon renewable heating and cooling systems to decarbonise buildings.
Innovate UK, as part of UK Research and Innovation, is investing up to £191 million to fund single and collaborative research and development projects as part of the Sustainable Innovation Fund over the next two years. The aim of these competitions is to help all sectors of the UK rebuild after the effects of COVID-19.
The Sustainable Innovation Fund is funding 1,103 projects; 1,189 UK businesses and totalling over £130 million in support across the UK.
Professor Phil Hart, Director of Energy and Power at Cranfield University, said: "At Cranfield, we are committed to playing our part in delivering Net Zero. The economic impact of the COVID-19 pandemic has increased interest in a 'green recovery', and investment in renewables promises employment, economic growth and reduced emissions.
"We are delighted to be working with our partners EGB Engineering UK on this project which will bring together innovation from academia and industry and help drive the UK towards a more sustainable future."
Dr Kumar Patchigolla, Reader in Low Carbon Energy Systems at Cranfield University and Principal Investigator, said: "This is a fantastic opportunity to develop renewable cooling and heating for commercial and public buildings, and to offer industrial research activities for students at Cranfield University".
Dr Arnold Gad-Briggs, EGB Engineering UK Executive Director, said: "The impacts of the current pandemic on society and the economy presents opportunities for innovation. Heat is the largest energy-consuming sector at over 40% and the single largest contributor to UK emissions. The support from Innovate UK demonstrates confidence in our ability to enable and provide solutions to reduce emissions, thereby stimulating the green economy.
We will also up-skill apprentices and early researchers as part of our commitment to science, technology, engineering and maths (STEM), to plug the skills gaps in the UK."
Innovate UK Executive Chair Dr Ian Campbell, said: "In these difficult times we have seen the best of British business innovation. The pandemic is not just a health emergency but one that impacts society and the economy.
Every initiative Innovate UK has supported through this fund, is an important step forward in driving sustainable economic development. Each one is also helping to realise the ambitions of hard-working people."
About EGB Engineering UK:
EGB Engineering is a UK based company with expertise in the field of power and propulsion. We enable, innovate and provide cost-effective quality engineering products and services to government departments, Original Equipment Manufacturers (OEMs) and end-user clients from various industries such as aerospace, defence, energy and nuclear. For more information visit our website.
About Innovate UK:
Innovate UK drives productivity and economic growth by supporting businesses to develop and realise the potential of new ideas. We connect businesses to the partners, customers and investors that can help them turn ideas into commercially successful products and services and business growth. We fund business and research collaborations to accelerate innovation and drive business investment into R&D. Our support is available to businesses across all economic sectors, value chains and UK regions. Innovate UK is part of UK Research and Innovation. For more information visit our website.
Related Academics
Professor Phil Hart
Professor and Director of Energy and Sustainability
Dr Kumar Patchigolla
Reader in Low Carbon Energy Systems
Links to our media coverage, help to find an expert and information to arrange filming on campus
About the University Creating leaders in technology and management
Theme Page Energy and Sustainability Cranfield's research in energy and sustainability will ensure our future supply needs are met
Centre for Energy Engineering Focusing on the design and engineering of advanced thermal energy systems and components
Industry and businesses choose Cranfield Together we can help to develop your people and technology to meet your organisation's needs

redpajama_set_name: RedPajamaCommonCrawl


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Kelly Clarkson reveals the "hardest" part of her divorce
Weiss Eubanks/NBCUniversal
Divorces aren't often easy – just ask Kelly Clarkson.
During Tuesday's episode of The Kelly Clarkson Show, the 38-year-old host opened up about her split from her husband of six years, Brandon Blackstock, 43.
"There are so many hard parts. The hardest for me is the kids," Clarkson revealed. She shares two children with Blackstock, six-year-old River and four-year-old Remington.
"That's the hardest for me," she shared. "I think, as women especially, we're trained… to take it all on, and you can deal with it, and you're fine, but it's your babies that you worry about."
Clarkson also sought advice from Untamed author Glennon Doyle, who appeared on the show along with Alicia Keys. Doyle previously went through a public split in 2016 from Craig Melton and is currently married to women's soccer star Abby Wambach and was able to provide some perspective, which the pop star found helpful.
"I'm like, 'This isn't happiness, for both of us. I feel both of us deserve better, and neither one of us would want this for our children.' Reading that line so hit home for me, and not selfishly, but for the family. It's like, 'I don't want this for anyone in this scenario right now.' It was such a powerful piece of knowledge," Clarkson told Doyle. "Thank you so much."
Clarkson filed for divorce from Blackstock in June. According to court documents obtained by Entertainment Tonight, Clarkson recently won primary custody of the couple's two children despite originally seeking joint custody.
Additionally, the outlet recently confirmed that Blackstock requested $436,000 a month — $301,000 in spousal support and the remaining $135,000 for child support. If his request is approved, he will receive over $5 million a year.
By Danielle Long


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Events after World War 2
Timeline created by chasity.myers1222
End of World War 2
The end of the war happened after Japan surrendered. Germany surrendered first, making the Allied Powers fall apart. It wasn't until the United States sent atomic bombs to the cities Nagasaki and Hiroshima in Japan that the war finally came to an end.
This was a sepreme court case in which a team of NAACP attorneys challenged the courts in the legality of segregation in public education. In Brown case, the NAACP challenged the "separate but equal" principle.
Her bravery was what started the Mongomery Bus Boycott. She boarded a public bus and sat in the "black section" in the first row. Than a white man came on and she was asked to move because the "white section" was full. She refused to move and was arrested because of her refusal. This sparked chaos in both blacks and whites when it came to giving blacks equal civil rights.
Period: Dec 1, 1955 to Dec 21, 1965
Many blacks refused to ride any kind public bus. They wanted to show their disappointment in Rosa Parks's arrest. Not only did this hurt the economy, but it showed how far blacks would go to rid of segregation.
U-2 Incident
President Eisenhower wasn't happy with the progress that the Soviet Union was making, technology wise. So, as a precaution, he sent U-2 spy planes to search the area. The Soviet Union spotted the planes but couldn't tell they were the property of the U.S. and were unable to shoot them down.
Little Rock Nine Incident
Situation in Little Rock, Arkansas, where nine black students volunteered to go to Central High so schools would become desegregated. However, the Arkansas governor, Orval Faubus, insisted to opposition and called the National Guard, who wouldn't let any of them in. President Eisenhower later stepped in and sent federal troops to Little Rock to protect the students and to enforce the Court's decision.
Sputnik launched
The Soviet Union launched a 184-pound steel ball containing a small satellite which took roughly 98 minutes to orbit Earth. The launchings shocked many Americans, since we suffered when it came to technology in outer space.
Bay of Pigs Invasion
This was an attempt, made by th United States, to overthrow the Cuban government manned by dictatorFidel Castro. The attempt was unsuccessful for the invasion was stopped by Castro's army.
Period: Aug 31, 1962 to Apr 24, 1963
Cuban Missile Crisis
1962 conflict between the U.S. and the Soviet Union resulting from the Soviet installation of nuclear missiles in Cuba.
Letter from Birmingham Jail
Hand written and longer than three pages, King wrote this letter to a fellow cell mate, who was white. He writes in this letter that black boycotts are peaceful and that blacks were ready for civil rights.
1963 demonstration in which more than 200,000 people rallied for economic equality and civil rights.
PROYECTO DE FILOSOFIA MEDIEVAL
Revolution/Republic Timeline
История появления и создания принтера
linea del tiempo de la seguridad social de México y el Mundo
История изобретения и развития мониторов
Principles of Social Media Group 1
Marco cronologico, 1800-1950 Colombia
Línea del tiempo: Evolución de la WEB
U.S history: 1877-2008
Mi linea de vida by: Renato Balbuena Rosales 1A T/M
History of Special Education and Inclusive Education Timeline
EL PRINCIPITO: SU VIAJE POR LOS SIETE PLANETAS.
Historia de la pediatría
The History of Pop Music
Acontecimientos importantes en el sistema educativo mexicano
UnifiedOnline UOIP
The history of the Yorkshire textile industry.
LINEA DEL TIEMPO DE LA EVOLUCIÓN DE LA INFORMÁTICA.
Línea de Tiempo - Modelos atómicos
HISTORIA INVESTIGACION DE OPERACIONES
Época Colonial en México (1521-1821)
Línea del tiempo de fisicos mas importantes
Principales Filosofos De La Calidad Y Sus Aportes
Línea del tiempo de México en siglo XIX
HIstoria de la Terapia Familiar
Medios de comunicación (1900-1968)
Enlightenment's Major Events
Linea del tiempo (IMPERIALISMO)
An Inspector Calls: Context