\section{Introduction}

It took more than 30 years till
Efimov's prediction for the existence of weakly
bound three-particle states  with asymptotic discrete scaling symmetry
 \cite{efimov:plb} was confirmed in the cold-atom physics
experiments \cite{kraemer:06a}. This discovery raised the interest in
few-body systems with resonant short-range interactions,
both experimentally and theoretically.
After the three-body system where semi-analytical
results have been obtained as summarized in Ref.~\cite{braaten:rev},
the next step in the complexity
to explore the Efimov physics is the system of four identical bosons
where several numerical studies
\cite{lazauskas:he,hammer:07a,stecher:09a,schmidt:10a,yamashita:10a,deltuva:10c,stecher:11a}
are already available.
Although there is no four-boson Efimov effect \cite{amado:73a,wang:12a},
 the properties of the four-boson system are strongly
affected by the three-boson Efimov effect.
The observables of the four-boson collisions
 show the same  discrete scaling symmetry
and are related to the respective trimer binding energies in a
universal way  provided the size of the involved few-body bound states
greatly exceeds the interaction range \cite{stecher:09a,deltuva:10c}.
Furthermore,  below each Efimov trimer in a certain regime
there are two tetramers \cite{hammer:07a,stecher:09a}.
The tetramers associated with the excited trimers are
unstable bound states (UBS) \cite{res_cpl,deltuva:11a,stecher:11a}
and lead to resonant effects in the four-boson reactions.
First numerical calculations of the
four-boson recombination  \cite{stecher:09a}
and dimer-dimer collisions  \cite{dincao:09a}
in the adiabatic hyperspherical representation
were followed by the work of Refs.~\cite{deltuva:10c,deltuva:11b,deltuva:12a}
employing the  integral equations for the transition operators
that were solved in the momentum-space framework.
In the latter case the  universal limits for the scattering observables
and tetramer properties such as widths and intersections with thresholds
were obtained with considerably higher accuracy;
furthermore, remarkable resonant effects were predicted in
the atom-trimer collisions as well \cite{deltuva:11a}.
The theoretical results \cite{stecher:09a,dincao:09a,deltuva:11b,deltuva:12a}
are  roughly consistent with
the existing data from  the dimer-dimer and four-atom recombination
experiments \cite{ferlaino:08a,ferlaino:09a,pollack:09a,zaccanti:09a}
performed in a regime that is not strictly universal.

In the present work we will study a different type of bosonic tetramers,
namely, the ones of the approximate dimer-atom-atom structure.
Their existence was predicted in Refs.~\cite{braaten:rev,dincao:09a} as a
consequence of the three-body Efimov effect in the three-body system made off
a dimer and two atoms.
This is illustrated in Fig.~\ref{fig:ae} where we schematically show
the four-boson energy ($E$) spectrum as function of $1/a$ with $a$ being the
two-boson scattering length.
We consider the regime around the special value $a = a_n^d$ that corresponds
to the $n$th Efimov trimer being at the atom-dimer threshold, i.e.,
$b_d = b_n$ where $b_d$ ($ b_n$) is the dimer ($n$th trimer)
binding energy relative to the free particle threshold.
If the trimer is a true bound state with zero width, i.e.,
if deeper dimers are absent (which is not the case in typical
experiments \cite{ferlaino:08a,ferlaino:09a,pollack:09a,zaccanti:09a}),
the atom-dimer  scattering length $A_d$ is infinite at $a = a_n^d$.
 Sufficiently close to $E=-b_n$ and $a = a_n^d$
where $A_d$ greatly exceeds the dimer size being
of the order of $a$, one may expect
to mimic some properties of the few-boson systems
using a model that considers the dimer as a pointlike particle.
Then in the effective three-body system consisting of a dimer and two atoms
there are two atom-dimer pairs with infinite two-body scattering length $A_d$.
In such a three-body model of the four-boson system
the three-body Efimov effect occurs, however,  with a very large discrete
scaling factor $e^{\pi/s_0} \approx 2.016 \times 10^5$ \cite{braaten:rev}.
The resulting Efimov states accumulate at  $E = -b_d$.
Their dimer-atom-atom structure is only an
approximation but no attempt has been made so far to describe them
rigorously as four-boson states. This will be done in the
present work using exact four-particle equations.
We label the tetramers of this type
by two integers $(n,m)$ where $n$ refers to the
associated Efimov trimer and $m=0,1,2,\ldots$ distinguishes between
different states existing around  $a = a_n^d$ with  fixed $n$.
The $(n,m)$th tetramer intersects the  $n$th atom-trimer threshold
at $a= a_{n,m}^t$ and the dimer-atom-atom threshold at $a= a_{n,m}^d$.
The calculation of these tetramers is technically very difficult task
owing their weak binding and very large $e^{\pi/s_0}$; our results
will be limited to $m=0$ states.
Furthermore, all these tetramers  lie above the dimer-dimer threshold
and above all lower atom-trimer thresholds $n'$ with
$n' < n$. Therefore the considered tetramers are UBS  with finite width.
We will extract their properties using rigorous four-particle scattering
calculations. Performing in addition the three-body calculations
we will establish the limitations of the  dimer-atom-atom model.

In Sec.~\ref{sec:th} we shortly recall the technical framework.
In Sec.~\ref{sec:res} we present results for tetramer
properties and their effect on the atom-trimer and dimer-dimer
 scattering observables.
 We summarize in  Sec.~\ref{sec:sum}.

\begin{figure}[!]
\includegraphics[scale=0.5]{a.eps}
\caption{\label{fig:ae} (Color online)
Schematic representation of the four-boson energy spectrum
in the vicinity of the  dimer-atom-atom (dotted line)
and the $n$th atom-trimer (dashed line) threshold intersection
at $a= a_n^d$. The tetramers (solid lines)
accumulate at  $a= a_n^d$ and $E=-b_d$ but
for simplicity only two states ($m=0,1$) are displayed;
only qualitative relations are preserved. The  $(n,m)$th tetramer
intersects the dimer-atom-atom and the  $n$th atom-trimer thresholds
at $a= a_{n,m}^d$ and $a= a_{n,m}^t$, respectively.
}
\end{figure}

\section{Four-boson scattering equations \label{sec:th}}

To describe the collisions in the four-boson system we
use exact Alt, Grassberger and Sandhas (AGS)
equations \cite{grassberger:67} for the transition operators
$\mcu_{\beta\alpha}$ in the symmetrized form
\begin{subequations} \label{eq:U}
\begin{align}
\mcu_{11}  = {}&  P_{34} (G_0  t  G_0)^{-1}
 + P_{34}  U_1 G_0  t G_0  \mcu_{11} + U_2 G_0  t G_0  \mcu_{21} ,
\label{eq:U11} \\
\mcu_{21}  = {}&  (1 + P_{34}) (G_0  t  G_0)^{-1}
+ (1 + P_{34}) U_1 G_0  t  G_0  \mcu_{11} , \label{eq:U21} \\
\mcu_{12}  = {}&  (G_0  t  G_0)^{-1}
 + P_{34}  U_1 G_0  t G_0  \mcu_{12} + U_2 G_0  t G_0  \mcu_{22} ,
\label{eq:U12} \\
\mcu_{22}  = {}&
(1 + P_{34}) U_1 G_0  t  G_0  \mcu_{12} . \label{eq:U22}
\end{align}
\end{subequations}
Here $G_0$ is the free resolvent,  $P_{34}$ is the
permutation operator of particles 3 and 4,
$t = v+vG_0t$ is the two-particle transition matrix
derived from the potential $v$,
and $U_{\alpha}$ are the symmetrized AGS operators \cite{deltuva:07a}
for  $3+1$ ($\alpha=1$) and $2+2$ ($\alpha=2$) subsystems.
We solve the AGS equations \eqref{eq:U}
in the momentum-space framework using the
partial-wave decomposition. In the present study only the
states with zero total angular momentum and positive parity ($0^+$)
corresponding to the quantum numbers of the tetramers need to be considered;
among them  the states with the
nonconserved angular momentum of the three-boson subsystem
$J \le 3$ have to be included for the convergence.
The atom-atom interaction $v$ is taken over from Ref.~\cite{deltuva:10c};
it is given by a separable $S$-wave potential with
one- or two-term gaussian form factors.
Further technical details can be found in
Refs.~\cite{deltuva:12a,deltuva:07a,deltuva:ef}.

The scattering amplitudes for all elastic and inelastic
two-cluster reactions are obtained as the on-shell matrix elements of
the AGS transition operators  \cite{deltuva:07a};
the amplitudes for breakup and recombination \cite{deltuva:12a}
are given by the integrals involving $\mcu_{\beta\alpha}$.

The unstable tetramers manifest themselves as poles of the
operators $\mcu_{\beta\alpha}$ in the complex energy plane
at $-B_{n,m}^d - i\Gamma_{n,m}^d/2$ with
 $-B_{n,m}^d $ being the energy of the $(n,m)$th tetramer
relative to the four-boson breakup threshold and
$\Gamma_{n,m}^d$ being its width.
The UBS pole in the complex energy plane
is located in one of the unphysical sheets that
is adjacent to the physical sheet \cite{res_cpl} and
affects the physical observables
leading to resonant effects in the four-boson scattering.
Thus, the properties of the
 unstable tetramers can be extracted from the behavior of
$\mcu_{\beta\alpha}$ as described in Ref.~\cite{deltuva:11a}.

In the three-body dimer-atom-atom (3BDAA) model
the corresponding reactions are
described by solving the AGS three-body scattering equations \cite{alt:67a}.
These calculations need two pair potentials.
The atom-atom potential $v_{aa} = v$ is the same as in full four-boson
calculations. The atom-dimer potential  $v_{ad}$ is taken in the momentum-space
representation as $\langle \mathbf{k}' | v_{ad} | \mathbf{k} \rangle
= \exp(-{k'}^2/\Lambda_d^2) \, \lambda_d \exp(-{k}^2/\Lambda_d^2)$.
Its strength $\lambda_d$ and momentum cutoff (inverse range)
parameter $\Lambda_d$ for each $a$
are adjusted to reproduce $A_d$, $(b_n-b_d)$ and atom-dimer effective range
predicted by the original three-boson calculations using $v_{aa}$.
 As a consequence,  $v_{ad}$ yields an accurate description
of the low-energy atom-dimer scattering.
We note that  the range of $v_{ad}$ with
$1/\Lambda_d \approx a$ greatly exceeds the range of $v_{aa}$
that is much smaller than $a$.

\section{Results \label{sec:res}}

Unless stated otherwise, our results refer to rigorous four-boson
calculations.
We present them as dimensionless ratios that are
independent of the short-range interaction details in the universal limit.
For this one needs
to consider reactions involving highly excited Efimov trimers
where the finite-range effects become negligible.
As shown in previous calculations
\cite{schmidt:10a,deltuva:10c,deltuva:11a,deltuva:11b,deltuva:12a},
$n\ge 3$ is sufficient for high accuracy (note that our nomenclature
starts with $n=0$ for the ground state).
This is fully consistent with our present results. For example,
for the $(n,0)$th tetramer intersection with the corresponding
atom-trimer threshold we obtain
$a_{3,0}^t/a_3^d \approx 1.609$ and $a_{4,0}^t/a_4^d \approx 1.608$
using one-term form factor in the potential $v$ and
$a_{3,0}^t/a_3^d \approx 1.607$ and $a_{4,0}^t/a_4^d \approx 1.608$
using two-term form factor, respectively.
Thus, we conclude that for large $n$ the universal ratio is
\begin{equation} \label{eq:at}
a_{n,0}^t/a_n^d = 1.608(1),
\end{equation}
corresponding to $A_d/a_{n,0}^t = 5.491(3)$.
The calculation of  the  tetramer intersection with the
dimer-atom-atom threshold, $a_{n,0}^d$,
 turns out to be numerically more complicated and
less accurate due to its proximity to $a_n^d$. We have
found that $a_{n,0}^d/a_n^d > 0.9999$.
The  higher tetramers $(n,m \ge 1)$ may exist only at extremely large
 atom-dimer  scattering length $A_d$, i.e., extremely close to $a=a_n^d$,
and cannot be resolved in our four-boson calculations. Thus, with increasing $m$
the ratios $a_{n,m\ge 1}^t/a_n^d$ $(a_{n,m}^d/a_n^d)$ very rapidly
approach 1 from above (below). For example,
our estimations using the 3BDAA model are
\begin{equation} \label{eq:ad}
1- a_{n,0}^d/a_n^d = 1.2(1) \times 10^{-6},
\end{equation}
corresponding to $A_d/a_{n,0}^d = -1.8(1)\times 10^6$, and
\begin{equation} \label{eq:at1}
a_{n,1}^t/a_n^d -1 = 1.3(1) \times 10^{-6},
\end{equation}
corresponding to $A_d/a_{n,1}^t = 1.7(1)\times 10^6$ and
$(b_n-b_d)/b_d = 2.6(2) \times 10^{-13}$.

\begin{figure}[!]
\includegraphics[scale=0.64]{BG.eps}
\caption{ (Color online)
Position of the $(n,0)$th tetramer relative to the
$n$th atom-trimer threshold  (top) and its width (bottom)
as functions of the two-boson scattering length $a$ and
 the atom-dimer scattering length $A_d$.
The results of the full four-body (solid curves) and the 3BDAA
(dashed-dotted curves) models are compared.
The thin dotted line represents the zero value.}
\label{fig:BG}
\end{figure}

The universal values \eqref{eq:at} and \eqref{eq:ad} indicate
that the $(n,0)$th tetramer as UBS exists at
$0.6219 < a_n^d/a < 1.0000012$.
We show in Fig.~\ref{fig:BG} the  $a$-evolution of the $(n,0)$th
tetramer properties, i.e., its relative distance to the
$n$th atom-trimer threshold $(b_n-B_{n,0}^d)/b_n$ and its width
$\Gamma_{n,0}^d/2b_n$ as dimensionless universal quantities.
We include also the predictions of the  3BDAA model
and demonstrate that it is reliable only for the tetramer
position at $  a_n^d/a > 0.96$ where $|A_d|/a > 50$.
Although the 3BDAA model describes well the $n$th trimer binding and
the low-energy atom-dimer scattering, it does not support
lower atom-trimer channels and it is obviously not appropriate in
the dimer-dimer channel since the 3BDAA model treats the two
dimers asymmetrically.
However, these channels are decisive for the tetramer width
since in their absence $\Gamma_{n,0}^d$ vanishes.
Thus, the failure of the 3BDAA model for $\Gamma_{n,0}^d$
is not surprising. Furthermore,
the pointlike dimer approximation may only be reasonable when its size,
being of order of $a$, becomes negligible as compared to $|A_d|$
(that determines also the size of the shallowest trimer if $A_d>0$), i.e.,
for $|A_d| >> a$. Under this condition the  3BDAA model
may provide a reasonable approximation of the full four-boson model
for particular observables like
 the atom-trimer elastic scattering cross section and
the tetramer position $(b_n-B_{n,0}^d)$, but  not for
reactions involving the dimer-dimer or  lower atom-trimer channels.
The intersection point estimations in Eqs.~\eqref{eq:ad} and \eqref{eq:at1}
refer to $|A_d|/a > 10^6$ and therefore should be reliable
whereas the 3BDAA model fails for $a_{n,0}^t/a_n^d$
as can be seen in in Fig.~\ref{fig:BG}.

The tetramer lies very close to the  atom-trimer threshold, even the
largest deviation around  $a_n^d/a = 0.8$ is only
$(b_n - B_{n,0}^d)/b_n \approx 2.26 \times 10^{-4}$.
For comparison, the difference between the atom-trimer and
dimer-atom-atom thresholds at this point is
$(b_n - b_d)/b_n \approx 8.86 \times 10^{-3}$.
Note that in a narrow interval around $a=a_{n,0}^t$
the tetramer is slightly above the atom-trimer threshold,  i.e.,
$B_{n,0}^d < b_n$. The width  $\Gamma_{n,0}^d$ vanishes at
$a=a_{n,0}^t$ while at $a > a_{n,0}^t$
the $(n,0)$th tetramer  becomes an inelastic virtual state (IVS)
 \cite{res_cpl}. Negative width $\Gamma_{n,0}^d < 0$
in the IVS case  implies the change of the energy sheet.
 The IVS corresponds to the pole of the transition operators $\mcu_{\beta\alpha}$
in the complex energy plane
on one of the nonphysical sheets that is, unlike the one of UBS,
more distant from the physical sheet \cite{res_cpl}. The
IVS has a visible impact on the physical collision observables
only when it is located very near to the scattering threshold
\cite{res_cpl}; an example referring to the four-boson system
can be found in Ref.~\cite{deltuva:11a}.
In contrast, the UBS always leads to a resonant behavior.
As example we show in Fig.~\ref{fig:dd}
the $S$-wave phase shift $\delta_S$ and inelasticity parameter  $\eta_S$
for the dimer-dimer and atom-trimer  scattering
at the total four-boson energy $E \approx -B_{n,0}^d$ and $a_n^d/a = 0.8$.
It is interesting to note that only the  dimer-dimer phase shift
increases by 180 deg while the inelasticity parameter
exhibits a rapid variation also for the collisions of atoms
and trimers in the $(n-1)$th Efimov state.
Thus, in these two cases the
inelastic reactions are significantly enhanced by the tetramer UBS
whereas a pronounced resonant peak in the elastic cross section
is present in the dimer-dimer channel only.
The $(n,0)$th tetramer has very little impact
for the collisions of atoms
and trimers in the $(n-2)$th (and lower) Efimov state.

\begin{figure}[!]
\includegraphics[scale=0.64]{phaser.eps} \quad\quad\quad
\caption{\label{fig:dd} (Color online)
$S$-wave phase shift and inelasticity parameter for the dimer-dimer
(solid curves) and atom-trimer scattering where the trimer is
in the $(n-1)$th (dashed-dotted curves) or
in the $(n-2)$th (dotted curves) Efimov state.
The energy regime around the $(n,0)$th tetramer state
at $a_n^d/a = 0.8$ is shown. }
\end{figure}

\begin{figure}[!]
\includegraphics[scale=0.64]{A.eps}
\caption{  (Color online)
Real (solid curve) and imaginary (dashed-dotted curve) parts
of the atom-trimer scattering length $A_n$
as functions of the two-boson scattering length $a$ and
the atom-dimer scattering length $A_d$. Results for
$\mathrm{Re}\,A_n$ obtained using the 3BDAA model are shown as dashed curve.
The thin dotted line represents the zero value. }
\label{fig:A}
\end{figure}

The $(n,0)$th tetramer intersection with the  $n$th atom-trimer threshold
manifests itself most prominently in the
atom collisions with the $n$th  trimer at
vanishing relative kinetic energy.
The corresponding atom-trimer scattering length $A_n$  is
shown in Fig.~\ref{fig:A} as a function of the
two-boson scattering length $a$.
$A_n$ exhibits a resonant behavior
with $|\mathrm{Im}\,A_n|$ having a peak at $a=a_{n,0}^t$.
The increase of $\mathrm{Re}\,A_n$ shown in Fig.~\ref{fig:A}
near  $a_{n}^d/a = 0.99$ is not related to the tetramers; it is due to
the scaling of the elastic cross section
with the spatial size of the trimer that grows
with increasing $A_d$  and decreasing $(b_n - b_d)$, i.e.,
$\mathrm{Re}\,A_n \sim A_d \sim (b_n - b_d)^{-1/2}$.
This is evident in the inset of Fig.~\ref{fig:A}
where the ratio $\mathrm{Re}\,A_n/A_d$ exhibits no
resonant behavior near  $a_{n}^d/a = 0.99$.
The inelastic reactions quantified by $\mathrm{Im}\,A_n$
are not significantly enhanced by the increased size of the trimer.
The failure of the 3BDAA model, especially for $\mathrm{Im}\,A_n$
and the resonant peak position, can be expected based on Fig.~\ref{fig:BG}
and its discussion. We show in Fig.~\ref{fig:A} the 3BDAA
prediction only for  $\mathrm{Re}\,A_n$
at larger $A_d/a$ values. Only at $A_d/a > 50$, i.e.,
$a_{n}^d/a > 0.96$, it agrees with the four-boson result
within 7\% or better. In the same regime the  3BDAA model underpredicts
 $\mathrm{Im}\,A_n$ almost by a factor of 40 as compared to the four-boson
result. As estimated in Eq.~\eqref{eq:at1}
using the 3BDAA model, the $A_n$ resonance due to the
$(n,1)$th tetramer is expected to take place much closer to $a=a_{n}^d$.
Finally we note that  rapid variations of $A_n$ near
 $a_{n}^d/a = 0.15$  are caused by the proximity of
the dimer-dimer threshold and the  Efimov tetramer
(of a different type)
that intersect the  $n$th atom-trimer threshold at
$a_{n}^d/a = 0.14730(1)$ \cite{deltuva:11b}
and $a_{n}^d/a = 0.14706(1)$ \cite{deltuva:11a},
respectively. The $A_n$ evolution at
  $a_{n}^d/a < 0.15$ is presented in Ref.~\cite{deltuva:11a}.

\begin{figure}[t]
\includegraphics[scale=0.64]{rlx.eps}
\caption{ (Color online)
Inelastic reaction rates for vanishing relative energy
atom scattering from the $n$th trimer as functions of the two-boson
scattering length $a$ or the atom-dimer scattering length $A_d$.
Rates for the dimer production (solid curve)
and the trimer relaxation to the
 $(n-1)$th (dashed-dotted curve) and to
the $(n-2)$th (dotted curve) Efimov state are compared.}
\label{fig:rlx}
\end{figure}

The inelastic reaction rate for the atom scattering from the $n$th
trimer in the ultracold limit is given \cite{deltuva:10c} by
$\beta_n = -(16\pi \hbar/3m) \mathrm{Im}\,A_n$ where $m$ is the
boson mass.
 $\beta_n$ has contributions corresponding to the dimer production
$\beta_n^{dd}$ and the trimer relaxation $\beta_n^{n'}$ to more deeply
bound states $n'$, i.e.,
\begin{equation} \label{eq:rlx}
\beta_n = \beta_n^{dd} + \sum_{n'=0}^{n-1} \beta_n^{n'}.
\end{equation}
The respective dimensionless quantities $\beta_n^{dd} m / \hbar a_n^d$
and  $\beta_n^{n'} m / \hbar a_n^d$
are compared in Fig.~\ref{fig:rlx}. Consistently with our previous
findings for two-cluster collisions \cite{deltuva:10c,deltuva:11b}
and four-atom recombination  \cite{deltuva:12a},
in most of the considered
regime the dominating inelastic reaction is the one leading to the
most weakly bound final channel, i.e., the dimer-dimer channel
in the present case.
However, in a narrow interval around   $a_{n}^d/a = 0.38$
an interesting and unusual situation takes place:
the dimer production rate $\beta_n^{dd}$ is strongly suppressed and
 the trimer relaxation process becomes the most important
inelastic reaction. Another exceptional situation can be seen
 around   $a_{n}^d/a = 0.52$  where $\beta_n^{n-1}$ has a minimum
such that $\beta_n^{n-1} < \beta_n^{n-2}$.

The considered regime is not yet explored experimentally.
According to our predictions, the   $(n,0)$th tetramer
most clearly could be seen as a dimer production rate peak
in ultracold collisions of atoms and excited trimers in the $n$th state,
since  $a_{n,0}^t$ is quite well separated from other special
values of $a$. Of course, in real experiments the  resonance position may
deviate from the universal value (\ref{eq:at})
due to finite-range effects; however, even without the dimer-trimer
intersection, i.e., without the  $a_{n}^d$ point in Fig.~\ref{fig:ae},
the tetramer may still exist as it happens in our model with $n=1$.
On the other hand, the observation of the resonant behavior in
elastic or inelastic dimer-dimer collisions requires a very fine
energy resolution since the tetramer lies very close to the
atom-trimer threshold.
The situation  is even more complicated
if  deeply bound dimer states are present such that the Efimov trimers
become UBS with finite width and lifetime.
According to Ref.~\cite{braaten:rev},
the trimer width is of the order $\eta_{*}b_n$ with $\eta_{*}$
being the atom-dimer inelasticity parameter;
for example, $\eta_{*} \approx 0.03$  in the experiment of
Ref.~\cite{zaccanti:09a} performed at $a>0$ and
 $\eta_{*} \approx 0.1$ in the experiment of
Ref.~\cite{ferlaino:09a} performed at $a<0$.
Thus, the width of the trimer greatly exceeds the distance $(B_{n,0}^d-b_n)$
between the tetramer and the atom-trimer threshold,
such that from the experimental point of view
the $n$th trimer and the $(n,0)$th tetramer are on top of each other.
Furthermore, in the presence of deep dimers the atom-dimer scattering length
 $A_d$ is complex with finite real and imaginary parts.
For example, $|\mathrm{Re}\,A_d/a| < 40$ at $\eta_{*} \approx 0.03$
 \cite{braaten:rev}. This should be sufficient for the existence
of the $(n,0)$th tetramer at $a$ slightly above $a_n^d$
but excludes all the others $(n,m\ge 1)$. For larger $\eta_{*}$
values even the  $(n,0)$th tetramer may be absent.
Thus, the experimental observation of the considered tetramers
would be extremely difficult.

\section{Summary \label{sec:sum}}

We studied bosonic tetramers that have approximate
dimer-atom-atom structure. They are unstable bound states in
the continuum affecting the scattering processes in the four-boson system.
Exact four-particle equations  for the transition operators
were solved in the momentum-space  framework to
describe the four-boson collisions.
We accurately achieved the universal limit by considering reactions
involving highly excited trimers but, for a given trimer, the
results are restricted to the deepest tetramer only.
We calculated the atom-trimer and dimer-dimer scattering observables
to extract the tetramer position and width.
In particular, we determined the tetramer intersection with the
atom-trimer threshold
and demonstrated that in  ultracold atom-trimer collisions
it leads to a resonant enhancement of elastic and inelastic reactions.
We studied also the reliability of the three-body
dimer-atom-atom model for the atom-trimer collisions:
it may be reasonable for the elastic scattering near threshold
and the real part of the tetramer energy at $|A_d| >> a$
but fails for the inelastic reactions and the tetramer width.


\section{Introduction}
Using elementary quantum systems to encode information is the key to several novel approaches to quantum mechanical based computing \cite{DiVincenzo00, Ladd10}, with one successful technique being linear optics quantum computing (LOQC) in photonic circuits \cite{Knill01, Ralph01,Kok07, OBrien07,Pooley12}. Although this method is in principle scalable, in practice it is limited by the stochastic nature of spontaneous parametric down-conversion (SPDC) sources used to seed the circuits. These photon sources produce simultaneous pairs of photons, one is used as a herald, the other is used to initialize a qubit. Typically the photon is split by a $\eta:(1-\eta)$ beam splitter, leaving it in a superposition of two spatial modes, known as dual-rail encoding \cite{Pittman02:2}. This state can be described as
\begin{equation}
	  |\Psi\rangle=\alpha |10\rangle+\beta\,e^{i\phi}|01\rangle,
 \label{signal}
\end{equation}
where $\phi$ in the phase delay in the transmitted mode, $\ket{10}$ and $\ket{01}$ corresponding to the two spatial modes, and $\alpha=\sqrt{\eta}$ and $\beta = \sqrt{1-\eta}$. In the following, we extend this concept to the temporal modes \cite{Marcikic02} with $\ket{10}$ and $\ket{01}$ now representing a photon in the first or second time bin, respectively. Up until now, time-bin encoded photons have been obtained by ``pumping'' a non-linear crystal with a femtosecond pulse \cite{Marcikic02}, which had initially been passed through a unbalanced Mach-Zehnder interferometer. Eventually, time-bin entangled photons are created in, however, a completely probabilistic fashion.

Here, we demonstrate the deterministic initialisation and delivery of arbitrary time-bin encoded qubits, qutrits and even ququads in one single photon with the help of a strongly-coupled atom cavity system \cite{Kuhn10,Vasilev10,Nisbet11}. While qubits -- used as default information carriers in quantum information processing -- are encoded in the binary system, qutrits and ququads are their equivalent in the ternary and quaternary system, respectively. Without any increase in technological resources, we are able to encode information beyond the binary system with the help of our single photon source. Additionally, our photon generation process is inherently non-probabilistic, versatile, reconfigurable and is not subject to systematic photon losses. We verify the fidelity of the quantum state preparation with time-resolved quantum-homodyne measurements, performed by sending single signal and reference photons into an elementary photonic circuit. Photon correlations monitored in a time-resolved manner \cite{Legero06,Legero04} then allow the state to be partially reconstructed.

\section{Photon source and quantum-homodyne measurement}

We achieve the generation of photons with an a priori defined intensity envelope -- which was first demonstrated by Keller et al. \cite{Keller2004} -- with the help of a controlled Raman transitions on the D2-line of a strongly-coupled $^{87}$Rb atom-cavity system. Having refined the method \cite{Kuhn10,Vasilev10,Kuhn02} and experimentally verified that such photons remain identical \cite{Nisbet11}, our photons are produced with an efficiency of $\eta=85\,\%$  and a repetition rate of $1\,\mathrm{MHz}$, see Fig.\,\ref{fig:Scheme}a and Section \ref{methods}. In the following, we go one essential step further and demonstrate the complete control over the photons' phase profile.

 The amplitude of the driving pulse is modulated to produce $d$-peak photons of 230\,ns peak duration, which yields exactly one $\sin^2$-shaped intensity peak per time bin (or temporal mode). The probability density of twin-peak and triple-peak photons of 460\,ns and 690\,ns duration, respectively, are shown in Fig.\,\ref{fig:Scheme}b. The singleness of these photons is checked with the help of a  Hanbury-Brown-Twiss setup and the second order correlation function at zero detection-time difference is $g^2(\tau=0)<0.05$ \cite{Nisbet11}, which is the upper limit imposed by the shot noise of the detector dark counts. Our source also allows for imposing an arbitrary phase change of $\phi$ between neighbouring peaks within one photon. This is accomplished by applying a phase shift to the driving pulse using an acousto-optic modulator, which intrinsically maps $\phi$ onto the respective time bin. We emphasize that the Hamiltonian of the coupled system stays unitary throughout the whole process, and that any phase change is only imposed when the cavity mode is empty. This ensures an adiabatic evolution and prepares the emitted photon in a coherent superposition of temporal modes. The creation of complex states is now straight forward, as we can directly shape the amplitude and phase profile of the emitted single photons.

\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{qutrits-f1.pdf}
\caption{Setup \textbf{(a)}: An atomic fountain injects $^{87}$Rb atoms into a high-finesse optical cavity, where Raman transitions between hyperfine ground states control the production of single photons of arbitrary spatio-temporal envelope. A $200\,\mathrm{m}$ fibre delays photons to arrive at a fiberized beam splitter together with subsequently emitted photons travelling the shorter path. The half-wave plate (HWP) is used to measure the quantum homodyne signal of parallely and perpendicularly polarized photons, respectively. \textbf{(b)} Probability density of twin- and triple-peak photons. The red-filled area indicates the desired $sin^2$-shaped probability density in every time bin.\label{fig:Scheme}}
\end{center}
\end{figure*}

To verify that the quantum state of the photon corresponds to the desired qubit, the relative phase between individual time bins has to be measured. This is achieved with a quantum-homodyne technique,  based on time-resolved two-photon interference measurements \cite{Legero06,Legero04}. In contrast to the continuous variable regime where a `classical' local oscillator (LO) \cite{Andersen2010} is used, the signal photon interferes on a beam splitter with a single LO photon of an intensity envelope identical to the signal photon, but no phase shift between time bins. For instance, a LO in state $|\Psi_{LO}\rangle = \left(\ket{10}+\ket{01}\right)/{\sqrt{2}}$ is used to analyse twin-peak photons encoding qubits. The photons arrive simultaneously at the beam splitter in spatio-temporal modes $A$ and $B$, such that the input state $\hat{a}_{A}^{\dagger}\hat{a}_{B}^{\dagger}|00\rangle$ translates into a superposition of output modes $C$ and $D$ using the operator relation of the beam splitter, $\hat{a}_{A,B}^{\dagger}\propto\hat{a}_{C}^{\dagger}\pm\hat{a}_{D}^{\dagger}$. Applied to twin-peak signal and LO photons, the unnormalised creator of the quantum state reads
\begin{eqnarray}
(\hat{a}_{C1}^{\dagger})^2-(\hat{a}_{D1}^{\dagger})^2 +e^{i\phi}\Big((\hat{a}_{C2}^{\dagger})^2-(\hat{a}_{D2}^{\dagger})^2\Big)+\label{Eqn:Out2}\\
(\hat{a}_{C1}^{\dagger}\hat{a}_{C2}^{\dagger}-\hat{a}_{D1}^{\dagger}\hat{a}_{D2}^{\dagger})(1+e^{i\phi})-(\hat{a}_{C2}^{\dagger}\hat{a}_{D1}^{\dagger}-\hat{a}_{D2}^{\dagger}\hat{a}_{C1}^{\dagger}) (1-e^{i\phi}),\nonumber
\end{eqnarray}
where the subscripts \{$1,\,2$\} refer to the time bin and \{$C,\,D$\} to the output port of the beam splitter. The first line contains the creation operator pairs where both photons are detected in the same time bin. In this case, the photons coalesce and are detected in the same output port.  The other terms describe the situation where the photons are detected in different time bins. Ideally, no cross-correlations are witnessed for identical photons $(\phi=0)$, which corresponds to the well-known Hong-Ou-Mandel effect \cite{Hong1987}. However, if we choose $\phi=\pi$,  the photons are  forced into opposite output ports whenever the detections occur in different time bins. The photon correlations, which normally follow bosonic statistics, then seem quasi-fermionic. This is explained by the first photo detection projecting the one remaining excitation into a mode superposition of well-defined mutual phase, which subsequently changes by $\pi$ until the second photon is detected. By examining the number of cross-correlations between different time bins, the relative phase difference, $\phi$, between any pair of time bins can be determined.

\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{qutrits-f2.pdf}
\caption{Equivalent circuits: \textbf{(a)} Measurements on twin-peak photons arriving simultaneously at a beam splitter are equivalent to experiments in a virtual circuit with spatial modes corresponding to temporal modes. \textbf{(b)} Correlations from their homodyne measurement as a function of detection-time difference (grey: reference of perpendicular polarization; red: quantum-homodyne signal of parallel polarization, red- and grey-shaded areas: Best model from fitting three auto-correlated $sin^2$-pulses to the measured coincidence probabilities). The red and black solid traces result from summing all coincidences found within a 60\,ns wide interval around each point of the trace. For some of these densly spaced data points, the statistical error is shown. The phase between the signal photon peaks varies from 0 to $\pi$. \textbf{(c)} Relative strength of the cross-correlation side peaks, expectations (blue) and the $95\,\%$ confidence bounds (grey). The $0\rightarrow2$ range is not fully spanned due to coherence-time limitations, background counts and polarisation deficiencies. \textbf{(d)} Relative coincidence probabilities (RCP, see Section \ref{methods}) between virtual detectors:  The time resolution allows the mapping of real detector clicks to virtual detectors C1, C2, D1, and D2 firing. We use this to sub-divide the quantum-homodyne signal  into cases where both detectors fire in the same time bin (brown), or in successive time bins (green). The left coincidence matrix is for non-interfering perpendicularly polarized photons, the right matrix for photons of parallel polarisation with a $\pi$ phase shift between the time bins of the signal photon.\label{fig:Twin}}
\end{center}
\end{figure*}

\section{Experimental qu-d-it preparation and delivery}
An outline of the  arrangement is shown in Fig.\,\ref{fig:Scheme}a and the photon probability density for both the qubit and qutrit in Fig.\,\ref{fig:Scheme}b.  The stream of alternating signal and LO photons is emitted from the cavity and split by a polarizing beam splitter (PBS) between long (upper) and short (lower) beam paths, causing successively emitted photons to arrive simultaneously at the non-polarizing 50:50 beam splitter (see Section \ref{methods}). Perpendicularly polarized photons do not interfere, but as their envelopes are identical, their cross-correlation function reflects the auto-correlation of the photon's shape. For twin-peak photons, a triple-peak convolution is witnessed (grey in Fig.\,\ref{fig:Twin}b), with a large central peak caused by detector clicks in identical time bins which are represented by the annihilation operators $\anniop{C1}\anniop{D1}$ and $\anniop{C2}\anniop{D2}$. The two satellite peaks of equal height arise from cross correlations between time bins represented by $\anniop{C1}\anniop{D2}$ and $\anniop{C2}\anniop{D1}$, respectively.
With parallel polarization, the photons do interfere and the cross-correlation function (red) varies with the relative phase between the peaks of the signal photon. As discussed above, no correlations are found within identical time bins so the central peak vanishes. This is not the case for the satellite peaks at $\pm 230\,$ns, where the number of counts depends strongly on $\phi$ and is expected to vary between zero and twice the reference value found with perpendicularly polarized photons. We also note that a residual signal is always found in the coincidence measurements of parallely polarized photons (Fig. \ref{fig:Twin}b for qubits, Fig. \ref{fig:Trip}b for qutrits and Fig. \ref{fig:Quad}b for ququads, respectively). From photon statistics and the dark count rate of the detectors, half the background can be attributed to correlations involving detector dark counts while the remainder can be explained by a polarization or mode mismatch of the interfering photons due to a residual birefringence of the optical fibres. Dephasing effects play a minor role, as the residual signal does not vary significantly with time \cite{Legero06}.

Fig.\,\ref{fig:Twin}c shows the strength of these satellite peaks relative to the reference signal as a function of $\phi$; the expected sinusoidal behaviour is exhibited, with minima and maxima witnessed at $\phi=0$ and $\phi=\pi$, respectively. Taking the 500\,ns coherence time into account, the expected change in peak strength has been calculated (solid trace) and is found to agree well also with the data (dashed) within the $95\,\%$ confidence bounds of a $\chi^2$-test. The limitation of the coherence time of the photons to 500\,ns is mainly due to the used laser system and the drift in the cavity resonance frequency, as the cavity is not actively locked while the atom cloud is flying through its centre. Furthermore, changes in the magnetic environment beyond our control also reduce the coherence time.

Greater insight can be obtained from the equivalent `virtual' photonic circuit in Fig.\,\ref{fig:Twin}a where the various time bins (or temporal modes) are represented as if they were spatial modes. Two photons are generated, each existing in a superposition of two time bins with a possible phase shift,  interfering on a single beam splitter with detectors $C$ and $D$ in the output ports. This beam splitter is used twice -- once per time bin -- and so the equivalent virtual circuit consists of  four beam splitters, where two mimic the state preparation. Due to the photons being much longer than the detectors' time resolution, we are able to distinguish whether individual photons are detected in the first or second time bin, respectively. We therefore associate the two physical detectors $C$ and $D$ with four virtual detectors, $C1, C2, D1,$ and $D2$, where for example the virtual detector $C1$ stands for the physical detector $C$ measuring events only in the first time bin. We then retrieve the four cross-correlation signals between all virtual detectors,  $\anniop{C1}\anniop{D1}$ and $\anniop{C2}\anniop{D2}$ (same time bin), or $\anniop{C1}\anniop{D2}$ and $\anniop{C2}\anniop{D1}$ (different time bins) using the absolute time of each photodetection along with the information \emph{which} virtual detector fired.
We also use the virtual-detector concept for detecting qutrits (six virtual detectors) and ququads (eight virtual detectors). The respective data can be found in Fig.\,\ref{fig:Trip}c and Fig.\,\ref{fig:Quad}c.
Fig.\,\ref{fig:Twin}d shows these correlation matrices for twin-peak photons of perpendicular and parallel polarization,  with a $\phi=\pi$ phase shift in the signal photon. By comparing the correlations to  expectations, a qubit-preparation fidelity of $F=0.96\pm 0.01$ is found, which is deduced in Section \ref{methods}.

 \begin{figure*}[!t]
\begin{center}
\includegraphics[width=\textwidth]{qutrits-f3.pdf}
\caption{Qutrits: \textbf{(a)} For triple-peak photons, the  virtual circuit shows 7 beam splitters and 6 detectors, whilst the physical apparatus is unchanged.  \textbf{(b)} Time-resolved homodyne signal  for photons of perpendicular (black) and parallel (blue) polarization, with the signal photon having a phase shift in the central time bin of $\phi_1=\pi$ (red) and with phase shifts of  $\phi_1=\pi/2$ in the central and $\phi_2=\pi$ in the final time bins (green). As before, the solid traces result from summing all coincidences found within a 60\,ns wide interval around each point of the trace. For some of these data points, the statistical error is shown. The shaded areas under the plotted lines correspond to the fit of five auto-correlated $sin^2$-pulses to the measured coincidence probabilities.
\textbf{(c)} Relative coincidence probabilities (RCP, see Section \ref{methods}) between virtual detectors (blue: detections within the same time bin; green: detections in successive time bins; red: detections two time-bins apart). \label{fig:Trip}}
\end{center}
\end{figure*}

The control of the photon-generation process goes far beyond simple qubits in time-bin encoding. We successfully added more time bins of arbitrary amplitudes and relative phases, thus producing a large variety of arbitrary qudits.
In Fig.\,\ref{fig:Trip} and \ref{fig:Quad} the correlations of triple- and quad-peak photons, representing the states
\begin{eqnarray}
\ket{\Psi_3} \propto \ket{100}+e^{i\phi_1}\ket{010}+e^{i\phi_2}\ket{001},\nonumber\\
\ket{\Psi_4} \propto\ket{1000}+e^{i\phi_1}\ket{0100}+e^{i\phi_2}\ket{0010}+e^{i\phi_3}\ket{0001},
\end{eqnarray}
are shown. The virtual circuit in Fig.\,\ref{fig:Trip}a illustrates the preparation and analysis of a qutrit, $\ket{\Psi_3}$. Although the physical apparatus is unchanged, it now mimics six detectors C1-3 and D1-3. Fig.\,\ref{fig:Trip}b and \ref{fig:Quad}b show the time-resolved correlation signal for qutrits and ququads encoded in triple- and quad-peak photons, respectively, which we also present as correlations between virtual detectors in Fig.\,\ref{fig:Trip}c and \ref{fig:Quad}c. For the 920\,ns long quad-peak photons, the probabilities for $\anniop{C1}\anniop{D4}$ and $\anniop{C4}\anniop{D1}$ correlations are small and others fluctuate due to exceeding the 500\,ns coherence time. Also the state-preparation fidelity drops from 0.94 for qutrits to 0.89 for ququads (see Section 5). Hence only within the coherence time,  the probabilities for observing correlations between time bins depend on the relative phases as predicted. Further increasing the state-vector dimension seems nonetheless feasible. About eight well-separable time bins could be produced within the coherence span, as the dynamics of the atom-cavity system is only restricted by the atom-cavity coupling strength,  $g_0 = 2\pi\cdot 15\,$MHz \cite{Nisbet11}.

 \begin{figure*}[!t]
\begin{center}
\includegraphics[width=\textwidth]{qutrits-f4.pdf}
\caption{Ququads: \textbf{(a)} encoded in quad-peak photons and their \textbf{(b)} resolved homodyne signal of perpendicular (black), parallel (blue) polarization, and for phase shifts of $\phi_1=\phi_3=\pi$ in the signal photon (red). As before, the solid traces are obtained from summing all coincidences found within an 60\,ns wide interval around each point of the trace and for some of these data points, we show the statistical error. The shaded areas in the plot correspond to the fit of seven auto-correlated $sin^2$-pulses to the measured coincidence probabilities. \textbf{(c)} RCP between the eight virtual detectors in this latter case.\label{fig:Quad}}
\end{center}
\end{figure*}

\section{Discussion and Outlook}

The novel approach we have implemented allows for the preparation and deterministic delivery of single photons in arbitrary quantum superpositions of various temporal modes. We made use of this sub-division into time bins to reliably encode a selection of qudits in single photons. Photon-generation efficiencies up to $\eta=85\,\%$ and state-preparation fidelities of $F=96\%$ have been found by quantum homodyning (see Section \ref{methods}). Time-bin encoding takes place upon photon generation which is inherently non-probabilistic, versatile, reconfigurable and not subject to systematic photon losses. These are substantial advantages if compared to other approaches that are probabilistic if based on SPDC and also subject to intrinsic photon loss if they rely on variable amplitude attenuation \cite{Kolchin08}, phase modulation \cite{Specht09} or interferometric techniques for photon shaping \cite{Marcikic02}. The availability of time bins as an additional degree of freedom to LOQC in an essentially deterministic photon-generation scheme is a big step towards large-scale quantum computing in photonic networks \cite{Bennett08}.
Arbitrary single-qubit operations on time-bin encoded qubits seem straightforward to implement with phase-coherent optical delay lines and active optical routing to either switch between temporal and spatial modes, or to swap the two time bins.
Moreover, the probability of having $N$ photons simultaneously available, either from many atom-cavity sources or transmitted through a photon-delay network, scales like $\eta^N$, with $\eta \gg g^{(2)}(0)$. Restrictions as for non-heralded SPDC sources (with $\eta\simeq g^{(2)}(0)$) do  not apply.   Controlling the atom-photon coupling might also allow the mapping of atomic superposition states to time-binned photons \cite{Dilley12,Ritter12}; and the long coherence time, combined with fast detectors, makes  real-time feedback possible during photon generation.

\section{Methods}
\label{methods}

\textbf{Photon source and efficiency:} The method we use for generating and shaping individual photons is theoretically discussed and introduced in \cite{Vasilev10}, and a detailed description of our photon source is to be found in \cite{Nisbet11}. It is based on a single $^{87}$Rb atom coupled to a high-finesse cavity undergoing a Raman transition between two of its hyperfine ground states, labeled $|e\rangle$ and $|g\rangle$ in Fig.\,1a, via an excited state $|x\rangle$. This transition -- driven by a laser pulse with Rabi frequency $\Omega(t)$, and the vacuum field of the cavity mode with coupling strength $g_0$  --  results in the emission of a single photon through the output coupling mirror of the cavity. Atoms are injected into the cavity using an atomic fountain, and couple strongly to the cavity mode for a maximum duration of 200\,$\mu$s. This is long enough to trigger up to 200 subsequent photon emissions. For the most strongly coupled atoms, photons are detected for $20\%$ of the driving pulses. Taking the detector efficiencies, fibre-coupling losses and cavity losses into account, this corresponds to an intra-cavity photon generation efficiency of $85\%$, of which $60\%$ leave our cavity through its output coupling mirror. Increasing the latter to $90\%$ is straightforward if one can afford better mirrors \cite{Ritter12}.

\textbf{Quantum homodyne setup:} The photon source produces an alternating sequence of unpolarized LO and signal photons. A polarizing beam splitter randomly directs these into either a $1\,\mu$s-long delay line (200\,m optical fibre), or directly to the final beam splitter. The fibre-imposed delay matches the time distance between photon productions, such that any pair of successively generated photons arrives simultaneously at the beam splitter with a probability of $25\,\%$. If polarized photons were used in combination with fast switching optical elements, this could be increased to $100\,\%$.
All photo-detection events in the output ports of the beam splitter get recorded with sub-ns time resolution. We use the recorded data stream to determine photon-photon correlations either as a function of the detection-time difference, or within (or between) selected time bins. This allows for monitoring the continuous phase evolution within a photon and for the detection of discrete phase jumps between the time bins of interest. For simultaneously arriving photon pairs,
\textbf{the coincidence probability density} $\times dt$ gives the probability of detecting a coincidence count in an interval of width $dt$ around the specified detection-time difference. Any trace shown here is normalized to the corresponding mean number of photon pairs that we retrieve during the very same experimental run from the amount of coincidences between non-interfering photons of detection-time delay $>1\,\mu$s (the periodicity of the source).

\textbf{The relative coincidence probability (RCP)} is defined by the ratio of quantum-homodyne correlations to the number of reference correlations with perpendicularly polarized and therefore non-interfering photons. In the latter case, the photons randomly split. This implies that the RCP ranges at most from zero (photon coalescence, no correlations) to two (photon anti-coalescence, correlations in any case).

\textbf{Fidelity of the qubit preparation:}  To determine the fidelity of the state preparation, the density matrix $\hat{\sigma}$ has to be reconstructed. For a qubit of $\phi=\pi$, one gets
\begin{equation}
\hat{\sigma} = \frac{1}{2} \left(\begin{array}{ccc}1.01 & -0.85 \\ -0.85 & 0.99\end{array}\right).
\end{equation}
The values on the diagonal are obtained using non-interfering photons of perpendicular polarization in the two-photon coincidence measurement (Fig. 2d). The numbers are proportional to the square root of the number of correlations detected in time bin $1$ and $2$, respectively, and then normalised to one. We further assume that signal and LO photons only differ in phase, such that their off-diagonal elements only differ in sign. Their magnitude is given by the ratio of the square root of the maximum observed to the maximum possible side peak visibility.
Having obtained the density matrix of the single photon state, the fidelity \cite{NielsenChuang} itself is is calculated with $F(\ket{\psi},\hat{\sigma})=\sqrt{\bra{\psi} \hat{\sigma} \ket{\psi}}=0.96\pm 0.01$, where $\ket{\psi} = (\ket{10}-\ket{01})/\sqrt{2}$ is the reference state,  $\hat{\sigma}$ is the reconstructed density matrix, and the error results from propagating the standard deviation associated with the absolute number of $N=418$ correlations forming these side peaks. An equivalent procedure yields fidelities of $0.94\pm 0.01$ and $0.89\pm 0.02$ for qutrits and ququads, respectively.

\section*{Acknowledgements}
This work was supported by the Engineering and Physical Sciences Research Council (EP/E023568/1 and QIPIRC) and the Deutsche Forschungsgemeinschaft (DFG, RU 635).

\section*{References}
\bibliographystyle{unsrt.bst}