# Isotope shift measurement of the $423$-$\text{\,}\mathrm{nm}$ transition in
neutral Ca

David Röser<EMAIL_ADDRESS>Physikalisches Institut, Universität
Bonn, Nussallee 12, 53115 Bonn, Germany  Lukas Möller  Physikalisches
Institut, Universität Bonn, Nussallee 12, 53115 Bonn, Germany  Hans Keßler
Physikalisches Institut, Universität Bonn, Nussallee 12, 53115 Bonn, Germany
Simon Stellmer<EMAIL_ADDRESS>Physikalisches Institut, Universität
Bonn, Nussallee 12, 53115 Bonn, Germany

###### Abstract

We report on saturated absorption spectroscopy measurements of the
$(4s^{2})^{1}S_{0}\rightarrow(4s4p)^{1}P_{1}$ transition for the four most
abundant even-mass isotopes in calcium. By referencing the laser locked to an
ultralow expansion cavity and carefully investigating systematic errors,
isotope shifts are determined with a precision below
$100\text{\,}\mathrm{kHz}$, improving previously reported values by a factor
of about five. A King plot analysis employing literature values of the
$729$-$\text{\,}\mathrm{nm}$ transition in $\mathrm{Ca}^{+}$ ions shows
excellent linearity. The field and mass shift parameters are determined from
King plots with other transitions.

††preprint: APS/123-QED

##  I  Introduction

Laser spectroscopy plays a key role in investigating the atomic or molecular
structure [1]. Over the past decades, spectroscopic techniques have been
developed and improved to outstanding precision on the single
$\text{\,}\mathrm{Hz}$ level [2, 3, 4, 5, 6, 7]. These measurements can be
used to define frequency standards or explore physics beyond the standard
model, e.g., to search for temporal variations in fundamental constants [8],
breaking of local Lorentz invariance [9, 10], CPT violation [10], or a new
gauge boson [11, 12, 13, 14, 15, 5, 16, 17].

The term isotope shift refers to the energy difference of electronic states
between different isotopes. The main contributions to the isotope shifts are
the mass and the field shift. The mass shift originates from a change of the
electron’s reduced mass and momentum correlations of electrons and is
proportional to the relative mass difference between isotopes. The field shift
arises from a different nuclear charge distribution due to different neutron
numbers [18].

In a simplified case, the field shift is assumed to be proportional to the
difference of the squared nuclear charge radii. Then, isotope shifts of
different transitions in atoms or ions of the same species are linearly
related to each other, when they undergo a King plot analysis [19]. Deviations
from the linear behavior allow insights into the nuclear potential, as higher-
order corrections or new physics terms might be determined [20, 5].

A possible new boson might be a force carrier for an interaction between
neutrons and electrons. Their coupling would change the nuclear potential by a
Yukawa-type potential, which is dependent on the neutron number [13]. Hence,
the interaction could be visible in isotope shift measurements and appear as a
nonlinearity in a King plot [20, 15, 18].

It has been found that strong nonlinearities can occur due to large nuclear
deformation. Here, the simple field shift approximation fails, as the
quadratic field shift and the next higher order Seltzer moment have to be
taken into account [13, 5]. Still, the combination of various transitions in
the Yb system allowed to constrain the parameter space for the new boson
coupling [6]. An overview of the work on $\mathrm{Yb}$ can be found in [21].

For isotope shift measurements, group II elements are of particular interest.
Their even-mass isotopes feature an even number of protons and an even number
of neutron leading to zero nuclear spin. The absence of hyperfine structure
circumvents center-of-gravity determinations and thus enhances measurement
precision.

Among the group II elements, calcium holds particular significance, as it
offers two ’doubly magic’ nuclei. They feature a closed proton and a closed
neutron shell, resulting in non-deformed nuclei [22]. Isotope shift
measurements on neutral or ionized Ca have been measured on numerous
transitions with different methods, e.g., [3, 12, 23, 24, 25, 26, 15, 27, 4,
7]. Especially the shifts on the $4s^{2}S_{1/2}\rightarrow 3d{}^{2}D_{5/2}$
quadrupole transition at $729\text{\,}\mathrm{nm}$ in $\mathrm{Ca}^{+}$ have
been measured with outstanding precision [3, 7]. Reported King plots show
excellent linearities, further narrowing the bounds on the coupling strength
of a new possible boson [7]. An overview about previous work up to 2020 can be
found in Ref. [28].

The $(4s^{2}){}^{1}S_{0}\rightarrow(4s4p){}^{1}P_{1}$ transition in Ca I is at
$423\text{\,}\mathrm{nm}$ with a natural linewidth of
$34.5(4)\text{\,}\mathrm{MHz}$ and saturation intensity of
$54(5)\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$.
Isotope shifts on this line have been measured with different methods, e.g.,
ionization spectroscopy [24] and fluorescence detection of an atomic beam [25,
26]. The values reported have typical $1\sigma$ uncertainties of several
$100\text{\,}\mathrm{kHz}$. While the reported shifts for ${}^{44}\mathrm{Ca}$
and ${}^{48}\mathrm{Ca}$ agree very well, there is a
$400\text{\,}\mathrm{kHz}$ discrepancy for ${}^{42}\mathrm{Ca}$ [24, 26].

In this article, we report on a saturated absorption spectroscopy measurement
in a gas cell, narrowing the uncertainty on the isotope shifts to the
sub-$100\text{\,}\mathrm{kHz}$ range. A King plot analysis proves linearity
between our measurement and isotope shifts on the $729$-$\text{\,}\mathrm{nm}$
line in $\mathrm{Ca}^{+}$ [3]. The field and mass shift coefficients for the
$423$-$\text{\,}\mathrm{nm}$ lines are determined from isotope shift data on
dipole transitions in $\mathrm{Ca}^{+}$ [27].

##  II  Experimental Apparatus

###  II.1 Optical setup

The laser light used in the experiment is derived from an infrared diode laser
system equipped a with tapered amplifier operating at
$846\text{\,}\mathrm{nm}$ (Toptica TA pro). A vibration isolated and
temperature stabilized ultra-low expansion (ULE) cavity (Menlo Systems ORC-
Cubic) serves as a frequency reference. The laser undergoes a carrier shift
modulation and the sideband is frequency-stabilized with the Pound-Drever-Hall
method to the ULE cavity. The carrier shift frequency can be tuned with a
direct digital synthesis generator with a $\text{\,}\mathrm{Hz}$ level
precision. The performance of the lock has been evaluated by locking two
identical lasers with different carrier shifts to the same cavity mode. Their
beat signal linewidth is smaller than $2\text{\,}\mathrm{Hz}$ on a timescale
of $1$ second.

The light is converted to $423\text{\,}\mathrm{nm}$ with second harmonic
generation (SHG) in a nonlinear crystal within an optical cavity, which is
locked to the laser using the Hänsch-Couillaud method [29]. A polarization
maintaining fiber is used to clean the mode profile and guide the linearly
polarized light to the experiment table.

To suppress intensity noise in the experiment, the light power is measured on
the experiment table. A PID controller is used to stabilize it by applying
feedback on the tapered amplifier current. The locking bandwidth is measured
to approximately $80\text{\,}\mathrm{kHz}$, which is mainly limited by the
slow response of the SHG cavity.

The laser light is split into two beams of equal intensities
($1.9\text{\,}\mathrm{mW}$ on a Gaussian beam diameter of
$4\text{\,}\mathrm{mm}$), with one serving as pump and the other as the probe
beam. Saturated absorption spectroscopy is performed by sending both beams
counter-propagating through a $1.5\text{\,}\mathrm{m}$ long stainless steel
vacuum system acting as a gas cell [1]. Lenses are slightly tilted from normal
incidence and all other optics, including the vacuum viewports, are wedged to
avoid ethaloning effects due to reflections or stray light impinging on the
photodiodes or interacting with the probed velocity class of atoms.

A lock-in technique is employed by sinusoidal amplitude modulation of the pump
beam with an acousto-optic modulator (AOM). The lock-in frequency is set to
$11\text{\,}\mathrm{kHz}$, as the the spectral noise amplitude is minimal here
and all electronic components reach the white noise floor.

The AOM is driven at a frequency of $200\text{\,}\mathrm{MHz}$. Thus,
interference effects between pump and probe beam are modulated much faster
than the atomic linewidth. To reduce residual Doppler shifts, proper overlap
of pump and probe beam is ensured by coupling the probe beam into the AOM and
observing the diffraction orders [30].

The Lamb dip is measured with a photodiode followed by a low noise
transimpedance amplifier (FEMTO DHPCA-100), which is then connected to a lock-
in amplifier (Zurich Instruments HF2LI). The demodulated signal is low pass
filtered, typically with a time constant of $5$ seconds. Measurements are
performed by shifting the laser frequency and a subsequent settling time of
twice this time constant. Afterwards, the lock-in signal is recorded and used
for evaluation. This procedure leads to a gain in signal-to-noise ratio (SNR)
of four orders of magnitude compared to unmodulated spectroscopy. A sketch of
the optical setup is shown in Fig. 1.

Figure 1: Sketch of our saturated absorption spectroscopy setup.

###  II.2 Systematic effects

Five grams of naturally abundant calcium are placed into a
$1.1\text{\,}\mathrm{m}$ long reservoir region, which is kept at
$340\text{\,}\mathrm{\SIUnitSymbolCelsius}$ by a magnetic field compensating
heating tape. The temperature inhomogeneity is monitored with sensors and
measured to be smaller than $10\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

The vapor pressure of Ca is estimated to be ${10}^{-7}\text{\,}\mathrm{mbar}$,
leading to approximately $20\text{\,}\mathrm{\char 37\relax}$ absorption of
each beam [31]. At this level, Beer-Lambert’s law can be utilized to show that
the combined light power of probe and pump beam remains constant within the
uncertainty margin of our power measurement. This allows the determination for
AC Stark shifts: Because of the Doppler broadened absorption in the gas cell,
the different isotopes are measured with different power and thus have to be
corrected.

To further minimize systematic errors, any magnetic fields including the
Earth’s magnetic field are compensated with coils in all directions. The
remaining field is significantly smaller than $5\text{\,}\mu\mathrm{T}$ and
reduces Zeeman splitting and thus polarization dependent effects.

In test measurements, pump and probe beam have been deliberately set to a
significant power imbalance to enhance light pressure-induced modifications of
the atomic velocity distribution. This results in line shape asymmetry, which
can be characterized by incorporating a dispersive function into the
Lorentzian profile [32]. The asymmetry was minimized by filling the cell with
$1\text{\,}\mathrm{mbar}$ argon buffer gas to restore the Maxwell Boltzmann
distribution by ’strong’ collisions. In ’strong’ collisions, the change of
velocity is approximately on the order of $v_{0}$, where $v_{0}$ represents
the velocity of the class of atoms interacting with the pump beam. In
precision measurements, it is essential to exercise caution in selecting the
type of buffer gas, as a too high number of ’weak’ collisions may cause a
Gaussian lineshape of the Lamb dip [33]. By ensuring equal intensity between
the probe and pump beams, we can effectively avoid this asymmetry.

Ca-Ar collisions cause a line broadening $\gamma_{\mathrm{c}}$ and a shift
$\delta_{\mathrm{c}}$ [34, 35]. While the broadening insignificantly
deteriorates the line center determination, the shift has to be corrected for.
It is calculated as

$\delta_{\mathrm{c}}=\sigma_{\mathrm{Ca-Ar}}\cdot\overline{v}\cdot
n_{\mathrm{Ar}}\,,$ (1)

where $\sigma_{\mathrm{Ca-Ar}}$ denotes the collisional cross section, which
is precisely measured in Ref. [36], $\overline{v}$ the mean relative collision
velocity, and $n_{\mathrm{Ar}}$ the argon density. For each Ca isotope, we
calculate

$\overline{v}=\sqrt{8k_{\mathrm{B}}T/(3\pi m_{\mathrm{red}})}$ (2)

with

$m_{\mathrm{red}}=M_{\mathrm{A}}m_{\mathrm{Ar}}\,/\,(M_{\mathrm{A}}+m_{\mathrm{Ar}})$
(3)

to accommodate the specific velocity distribution of each isotope $A$ with its
mass $M_{\mathrm{A}}$ [34, 35].

Recoil effects are not measurable, as the atomic linewidth is much larger than
the recoil splitting of the line. With Ref. [37], it can be shown that the
recoil splitting is symmetrical around the resonance up to
$800\text{\,}\mathrm{Hz}$. We expect this to be common mode for all
resonances, but nevertheless an error of $1\text{\,}\mathrm{kHz}$ is added to
the uncertainty margin.

##  III Measurements

###  III.1 Modified Lorentzian lineshape

It has been shown in previous theoretical and experimental work, that the Lamp
dip profile is modified in the presence of collisions and thus exhibits
asymmetry [38, 39, 40]. This originates from a finite collision time resulting
in failure of the impact approximation, which requires
$T_{\mathrm{d}}\cdot\Delta\ll 1\,$. Here, $T_{\mathrm{d}}$ denotes the finite
collision time and $\Delta=\nu-\nu_{0}$ the laser detuning from the atomic
resonance $\nu_{0}$ [40]. The actual lineshape that is used to evaluate the
data from this experiment thus reads

$I\left(\Delta\right)\propto\frac{\gamma_{\mathrm{N}}+\gamma_{\mathrm{c}}\left(\Delta\right)}{\Delta^{2}+\left[\,\gamma_{\mathrm{N}}+\gamma_{\mathrm{c}}\left(0\right)\,\right]^{2}}$
(4)

with

$\gamma_{\mathrm{c}}\left(\Delta\right)=\gamma_{\mathrm{c}}\left(0\right)\cdot\left(1+a_{1}\Delta
T_{\mathrm{d}}\right)\,,$ (5)

whereas $\gamma_{\mathrm{N}}$ is the natural linewidth, $a_{1}$ is a
potential-dependent parameter, and $\gamma_{\mathrm{c}}\left(0\right)$ the
collisional broadening from the impact approximation [38, 39]. To fit the
parameters, the background is approximated by a second-order polynomial, as
the resonances of all isotopes lie on the wing of the ${}^{40}\mathrm{Ca}$
resonance. It turns out that the resonances of ${}^{43}\mathrm{Ca}$ and
${}^{44}\mathrm{Ca}$ contribute significantly to the background seen at the
position of the ${}^{42}\mathrm{Ca}$ resonance. To address this, a
comprehensive scan over the resonances of ${}^{42}\mathrm{Ca}$,
${}^{43}\mathrm{Ca}$, and ${}^{44}\mathrm{Ca}$ has been performed and is
fitted in a heuristic way to determine a more precise background
approximation.

###  III.2 Lamb dip measurement

In this work, the Lamb dips of ${}^{40}\mathrm{Ca}$, ${}^{42}\mathrm{Ca}$,
${}^{44}\mathrm{Ca}$, and ${}^{48}\mathrm{Ca}$ are measured. Their abundances
are $97.0\text{\,}\mathrm{\char 37\relax}$, $0.6\text{\,}\mathrm{\char
37\relax}$, $2.1\text{\,}\mathrm{\char 37\relax}$ and
$0.2\text{\,}\mathrm{\char 37\relax}$, respectively. The hyperfine components
of ${}^{43}\mathrm{Ca}$ are not determined, as the lines are not well
separated from the crossover resonances. The abundance of ${}^{46}\mathrm{Ca}$
is $4\cdot${10}^{-3}\text{\,}\mathrm{\char 37\relax}$$, which is too low to
determine its line center with competitive precision [41].

Each isotope undergoes multiple scans in opposite scan directions. For each
scan, Eq. 4 is fitted more than 100 times to the data, each time with a
different fit range. The extracted line centers are averaged and the standard
deviation of their distribution is considered as the statistical error of this
measurement.

The experimental stability is confirmed, as the extracted line centers for
each scan are equal within their uncertainty interval. In between the
measurements for ${}^{42}\mathrm{Ca}$, ${}^{44}\mathrm{Ca}$, and
${}^{48}\mathrm{Ca}$, the resonance frequency of ${}^{40}\mathrm{Ca}$ is
determined to ensure that the measurement precision does not suffer from
possible drifts of the ULE cavity resonance.

Figure 2: The figure shows the demodulated and low pass filtered lock-in
signal exhibiting the measured Lamb dips of ${}^{40}\mathrm{Ca}$,
${}^{42}\mathrm{Ca}$, ${}^{44}\mathrm{Ca}$, and ${}^{48}\mathrm{Ca}$. $\,\nu$
denotes the frequency offset from ${}^{40}\mathrm{Ca}$ in natural frequencies,
while the lock-in signal is depicted in arbitrary units. The resonances are
scaled by a factor of $100$, $50$, and $300$ for ${}^{42}\mathrm{Ca}$,
${}^{44}\mathrm{Ca}$, and ${}^{48}\mathrm{Ca}$, respectively, and the offset
is adjusted for better visibility.

To correct the measured line centers for AC Stark shifts, the resonance
frequency of ${}^{40}\mathrm{Ca}$ is measured for different light powers. The
data is shown in Fig. 3. By fitting a linear function, the slope is determined
to be
$-10.0(1.5)\,$\text{\,}\mathrm{k}\mathrm{Hz}\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}\mathrm{)}$$.
The corrections on the isotope shifts are smaller than
$6\text{\,}\mathrm{kHz}$ and thus are one order of magnitude lower than the
statistical errors. Compared to the AC Stark shift parameter of Ref. [26], our
value is approximately a factor of $5$ lower and has the opposite sign. This
indicates that other effects related to power significantly disturb the
determination of the shift parameter. To improve the precision of our
measurements, a more precise work on power dependent shifts would be necessary
to rule out all possible sources of shifts, which goes beyond the scope of
this work.

Figure 3: The resonance of ${}^{40}\mathrm{Ca}$ is measured for different
light powers and is shown as deviation from the extrapolated linecenter at
zero intensity. An error of $2\text{\,}\mathrm{\char 37\relax}$ is estimated
for the power measurement, while the displayed uncertainty of $\Delta\nu_{40}$
originates from the fit. The slope of the fitted linear function serves as AC
Stark shift parameter.

Our measured isotope shifts are presented in Tab. 1. The results agree well
with Ref. [26], but the uncertainty ranges are smaller by a factor of about
$5$. The corresponding corrections and uncertainty budgets are displayed in
table 2. The Lamb dips are shown for each isotopes in Fig. 2.

Table 1: Measured isotopes shifts relative to ${}^{40}\mathrm{Ca}$. and comparison to previously reported values. All values are in MHz.  Isotope | This work | [26] | [24]
---|---|---|---
${}^{42}\mathrm{Ca}$ | $393.090(88)$ | $393.1(4)$ | $393.50(10)$
${}^{44}\mathrm{Ca}$ | $773.814(48)$ | $773.8(2)$ | $773.80(15)$
${}^{48}\mathrm{Ca}$ | $1512.989(73)$ | $1513.1(4)$ | $1513.00(20)$
Table 2: Corrections and uncertainties of the isotope shift measurement. All values are in $$\text{\,}\mathrm{kHz}$.$ Source | Correction | Uncertainty
---|---|---
residual Doppler | $0$ | $10$
second-order Doppler | $0$ | $<0.5$
recoil | $0$ | $1$
AC Stark | $<5$ | $<1$
Zeeman effect | $0$ | $0$
collisions | $<400$ | $<10$
cavity drift | $0$ | $0$
statistical error | $0$ | $<85$

###  III.3 King plot analysis

To further investigate our data, a King plot analysis is performed, as shown
in Fig. 4. We use isotope shifts of the $4s\,^{2}S_{1/2}\rightarrow
3d\,^{2}D_{5/2}$ quadrupole transition at $729\text{\,}\mathrm{nm}$ in trapped
$\mathrm{Ca}^{+}$ with uncertainties below $10\text{\,}\mathrm{H}\mathrm{z}$
[3].

In our King plots, modified isotope shifts
$m\delta_{\mathrm{729\,\mathrm{nm}}}$ are plotted against our measurement
$m\delta_{\mathrm{423\,\mathrm{nm}}}$, with the actual shift being scaled by
$g^{A,A^{\prime}}=\left(1/m_{\mathrm{A^{\prime}}}-1/m_{\mathrm{A}}\right)^{-1}$.
Here, $m_{\mathrm{A}}$ and $m_{\mathrm{A^{\prime}}}$ denote the nuclear masses
of the respective isotopes $\mathrm{A}$ and $\mathrm{A^{\prime}}$ [20]. For
each isotope, the masses are calculated from the respective atomic mass [42]
as

$m_{\mathrm{nucleus}}=m_{\mathrm{atom}}-20\,m_{\mathrm{e}}+\sum_{i=1}^{20}E_{i}^{\mathrm{bind}}\,.$
(6)

Here, $m_{\mathrm{e}}$ denotes the electron mass and
$E_{\mathrm{i}}^{\mathrm{bind}}$ the binding energy of each electron $i$,
which is derived from the NIST database [12, 43].

A linear function

$m\delta_{\mathrm{729\,\mathrm{nm}}}=K_{729\,\mathrm{nm}}-\frac{F_{729\,\mathrm{nm}}}{F_{423\,\mathrm{nm}}}K_{423\,\mathrm{nm}}+\frac{F_{729\,\mathrm{nm}}}{F_{423\,\mathrm{nm}}}\,m\delta_{423\,\mathrm{nm}}\,$
(7)

is fitted to the data, with $F_{i}$ and $K_{i}$ the field and mass shift
parameters of the transition $i$, respectively [20].

Our King plot reveals an excellent linearity between our measurement and Ref.
[3], as the line intersects all uncertainty intervals of our data points. From
the fit, we find
$F_{729\,\mathrm{nm}}/F_{423\,\mathrm{nm}}=$2.07336(0.00085)\text{\,}$$ and
$K_{729\,\mathrm{nm}}-({F_{729\,\mathrm{nm}}}/{F_{423\,\mathrm{nm}}})K_{423\,\mathrm{nm}}=16.4290(29)\cdot${10}^{5}\text{\,}\mathrm{MHz}\text{\,}\mathrm{u}$\,$.

From isotope shift measurements of the ${}^{2}S_{1/2}\rightarrow{}^{2}P_{1/2}$
and ${}^{2}D_{3/2}\rightarrow{}^{2}P_{1/2}$ transitions in trapped
$\mathrm{Ca}^{+}$, Ref. [27] determines the field and mass shift coefficients
for the respective transitions. Using their data to perform a King plot
analysis, we can deduce the field and mass shift coefficients
$F_{423\,\mathrm{nm}}=-178.4(3.2)\,$\text{\,}\mathrm{M}\mathrm{Hz}\mathrm{/}\mathrm{f}\mathrm{m}^{2}$$
and
$K_{423\,\mathrm{nm}}=363.6(1.4)\,$\text{\,}\mathrm{GHz}\text{\,}\mathrm{u}$$.

Figure 4: King plot generated with our measurements and isotopes shifts of the
$4s\,^{2}S_{1/2}\rightarrow 3d\,^{2}D_{5/2}$ transition at
$729\text{\,}\mathrm{nm}$ in trapped $\mathrm{Ca}^{+}$ [3]. The errorbars of
$\delta m_{729\,\mathrm{nm}}$ are too small to be visible.

##  IV Conclusions and Outlook

We have reported isotope shifts for the most abundant isotopes,
${}^{42}\mathrm{Ca}$, ${}^{44}\mathrm{Ca}$, and ${}^{48}\mathrm{Ca}$, relative
to ${}^{40}\mathrm{Ca}$. Our measurement agrees very well with Ref. [26] and
reduces the uncertainty by almost a factor of five to the permille scale of
the linewidth.

By performing a King plot analysis with data from Ref. [3], we find excellent
linearity. Furthermore, by comparison to the field- and mass shift parameters
determined in [27], the field- and mass shift constants are calculated for the
$423$-$\text{\,}\mathrm{nm}$ line.

In future measurements, the SNR can be improved by several orders of
magnitude. As demonstrated in Ref. [44], double modulation schemes with an
amplitude-modulated pump beam in modulation transfer spectroscopy can be
applied. This is advantageous, as the frequency can be modulated with several
$100\text{\,}\mathrm{MHz}$. Typical residual amplitude modulation noise from
the frequency modulation occurs at the modulation frequency itself and thus
can be mitigated by double modulation. With successful implementation,
uncertainties of approximately $50\text{\,}\mathrm{kHz}$ might be possible for
all even-mass isotopes, including ${}^{46}\mathrm{Ca}$.

Further, changing to copropagating pump and probe beams eliminates the
crossover resonances and allows for precise determination of the hyperfine
coupling constants in ${}^{43}\mathrm{Ca}$ [45].

##  V Acknowledgments

We acknowledge fruitful discussions with W. Alt and F. Vewinger and thank F.
Wenger and A. Hamer for assistance in the setup and operation of the
experiment. Financial Support from DFG through the Cluster of Excellence ML4Q
(EXC2004/1 – 390534769) and projects 496941189 and INST 217/978-1 FUGG, as
well as from BMBF (project 13N16371 QuantumGuide), is gratefully acknowledged.
Further, H. K. and S. St. acknowledge funding by the state of North Rhine-
Westphalia through the EIN Quantum NRW program.

Experimental data will be made available upon reasonable request.

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