# Characterising a tunable, pulsed atomic beam using matter-wave
interferometry

J Morley  R. Flack  B. J. Hiley  P. F. Barker<EMAIL_ADDRESS>Department
of Physics and Astronomy,
University College London, London, UK

###### Abstract

We describe the creation and characterisation of a velocity tunable, spin-
polarized beam of slow metastable argon atoms. We show that beam velocity can
be determined with a precision below 1 % using matter-wave interferometry. The
profile of the interference pattern was also used to determine the velocity
spread of the beam, as well as Van der Waals co-efficient for the interaction
between the metastable atoms and the multi-slit silicon nitride grating. The
Van der Waals co-efficient was determined to be $C_{3}$=1.84$\pm$0.17 a.u., in
good agreement with values derived from spectroscopic data. Finally, the spin
polarization of the beam produced during acceleration of the beam was also
measured, demonstrating a spatially uniform spin polarization of 96 % in the
m=+2 state.

††preprint: APS/123-QED

##  I Introduction

Atomic and molecular beams are vitally important tools in physics, chemistry
and materials science. They allow the study of both short and long range
atomic and molecular interactions [1] and, are an enabling technology for
scattering experiments that provide elastic and inelastic cross-sections for
scientific [2], astrophysical [3] and industrial processes [4]. Experiments
often require a well-defined velocity and velocity spread which is
conventionally engineered through control of gas temperature, pressure and
flow geometry[5]. More recently, optical forces have been used to create
tunable, low energy beams[6, 7, 8, 9, 10], allowing much lower velocities with
additional control over the beam velocity and velocity spread. Such beams are
particularly useful for matter-wave interferometry and for exploring low
energy atomic and molecular collisions[11].

An important requirement for quantitative analysis of many experiments with
atomic beams is an accurate knowledge of their velocity/energy and also the
spread in these values. Time-of-flight (TOF) methods, which determine an
average atomic velocity, are commonly used and are simple to implement.
However, for beams with time dependent accelerating forces, such as in optical
acceleration, TOF calculations often do not accurately estimate the atomic
velocity. To address this problem, alternative methods that measure the atomic
velocity at a particular location have been employed. Optical techniques such
as Ramsey interferometry[12] and Doppler shifted absorption spectroscopy [13]
has been demonstrated for measuring ultra low velocity atomic beams.

Matter-wave interferometry has developed considerably since its first
demonstration using electrons by Davison and Germer in 1927[14]. For example,
interferometry with cold atoms and molecules have been used to explore the
mass limits to quantum superposition[15], for measuring the gravitational
constant[16]. It has also been used for metrological applications including
gravimetry[17], the measurement of polarisability [18] and for absolution
absorption measurements[19]. In this paper, we use matter-wave interferometry
through a multi-slit grating to characterise a slow atomic beam of metstable
argon atoms. In particular, we use it determine the velocity of the tunable
atomic beam, with significantly higher precision than time-of-flight
techniques. Additionally, using the same method allows us to determine the
$C_{3}$ coefficient that characterises the Van der Waals interaction between
these atoms and the silicon nitride diffraction grating.

##  II Matter-wave interferometry with a nanomechanical grating

To undertake these experiments, we utilise a multi-slit matter-wave
interferometer, with a detector that allows us to record the interference
pattern, as well as the time-of-flight of a packets of atoms within the pulsed
beam. The shape and fringe spacing of the interferogram is dependent on the de
Broglie wavelength, $\lambda_{DB}=h/mv$, of the atoms in the beam and a
measurement of $\lambda_{DB}$ allows us to determine the velocity, $v$, where
$h$ is Planck’s constant and $m$ the atomic mass.

For a plane wave of wavenumber $k$, and momentum, $\hbar
k=\frac{2\pi}{\lambda_{DB}}$, incident on a grating with $N$ slits, the signal
intensity $I_{0}$, as a function of velocity and transverse position $x$
observed on the detector, is a product of the incident intensity $I_{inc}$,
the grating function, and a slit function $f_{slit}$ as[20],

$I_{0}(x,v)=I_{inc}\left[\frac{\sin\left(\frac{1}{2}Nkvd\,\sin(\frac{x}{L})\right)}{\sin\left(\frac{1}{2}kvd\,\sin(\frac{x}{L})\right)}\right]^{2}|f_{slit}(x,v)|^{2},$
(1)

where $L$ is the grating-detector distance and $d$ is the grating slit
separation. Both functions are modified by the atomic velocity $v$ and so can
be used to characterise the velocity of the beam. The slit function is given
by

$f_{slit}(x,v)=\frac{2\cos(\frac{x}{L})}{\sqrt{\lambda_{DB}(v)}}\int^{\frac{s_{0}}{2}}_{0}\,d\zeta\cos\left[kv\,\sin\left(\frac{x}{L}\right)\left(\frac{s_{0}}{2}-\zeta\right)\right]\tau(\zeta,v),$
(2)

in which the standard single slit diffraction function is modified by $\tau$,
which accounts for the attractive Van der Waals (VdW) potential,
$V=-C_{3}/l^{3}$, between the atoms and the slit walls separated by $l$. The
actual slit width is $s_{0}$, $\zeta=s_{0}/2-x$ and $\tau$ is given by

$\tau(\zeta,v)=exp\left[i\frac{t\cos\beta}{\hbar
v}\frac{C_{3}}{\zeta^{3}}\frac{1+\frac{t}{2\zeta}\tan\beta}{(1+\frac{t}{\zeta}\tan\beta)^{2}}\right],$
(3)

where $C_{3}$ is the VdW coefficient, $t$ is the grating thickness and $\beta$
the angle of the tapered edge of the slit profile.

The transverse coherence of the atomic beam contributes to the contrast of the
fringes and is included in the model by convolving equation 3 with the atomic
beam profile, $A(x)$. The profile, which can be seen in figure 9, is defined
by the collimation slit configuration, scattering from the slit walls and the
velocity group of the atoms being measured. This is discussed in further
detail below. The beam width also defines the number of grating slits that are
being illuminated, which leads to a new intensity given as

$I_{c}(x,v)=\int_{-\infty}^{\infty}I_{0}(x^{\prime},v)A(x-x^{\prime})dx.$ (4)

Finally, to account for the observed spread of longitudinal velocities,
equation 4 is integrated over the longitudinal velocity spread.

$I(x,v)=\int_{-\Delta v/2}^{\Delta v/2}I_{c}(x,v)dv.$ (5)

This intensity profile can be fitted to the experimental data to measure
$v,\Delta v,$ and also $C_{3}$. An example of how the peak velocity modifies
the pattern is shown in figure 1(a). To provide an initial estimate of atomic
velocity, $v$, the position of each interference order is plotted for the
three atomic velocities shown in figure 1(b). The gradient of the fitted line
is equal to $\frac{h}{mv}\frac{L}{d}$.

Figure 1: (a) Equation 1 across 5 mm showing one half of the symmetric
interference signal. As the atomic velocity changes from 12 ms-1 (black), 14
ms-1 (red) and 16 ms-1 (blue) the peak positions shift. In each case the
velocity spread is 1 ms-1.
(b) The change in peak positions of successive orders for 12 ms-1 (black), 14
ms-1 (red) and 16 ms-1 (blue). This has been calculated for a velocity
distribution of 1 ms-1 and $C_{3}$ value of 2.17 a.u.

Although, the velocity spread affects the peak height and contrast, and to
some extent the fringe spacing shown in figure 2(a), to a good approximation,
a measurement of the fringe spacing of the interferograms gives a very good
estimate of the velocity. The modification due to the velocity spread
described in equation 5 is more pronounced at higher interference orders, and
therefore, the shape of the interferogram can be used to characterise the
beam’s velocity spread and the effective grating slit width.

Figure 2: (a)The change in the interference pattern for a peak velocity of 14
ms-1, as the velocity distribution changes from 0.5 ms-1 (black), 1 ms-1 (red)
and 2 ms-1 (blue). The $C_{3}$ coefficient is 1.55 a.u..
(b) The relative peak height, measured from the 0th order peak height for the
2nd (black), 3rd (red), 4th (blue) and 5th (pink) interference orders. Plotted
for a peak velocity of 14 ms-1 and $C_{3}$ coefficient of 1.55 a.u.

A measurement of the peak contrast is isolated from the peak height in figure
2(b) since the peak height is also dependent upon the peak velocity and the
VdW interaction of the slit function.

With the average velocity measured directly from the fringe spacing, the upper
envelope of the signal contains the information to measure $\Delta v$ and
$C_{3}$. The VdW potential changes the slit function and hence the upper
envelope of the signal, which simultaneously adjusts the peak height and
contrast as shown in figure 3(a). As a result, $\Delta v$ and $C_{3}$ are
varied when fitting equations 1 to 5 and have co-dependent uncertainties. In
figure 3(b), the height of the 2nd-6th order of interference peaks are plotted
for a changing VdW potential.

Figure 3: (a) Interferograms with a peak atomic velocity of 14 ms-1 and
velocity spread of 1 ms-1 for a VdW coefficient of 1 a.u. (black), 1.5 a.u.
(red), 2 a.u. (blue).
(b) The relative peak height, measured from the 0th order peak height plotted
over a changing of VdW potential, for the 2nd (black), 3rd (red), 4th (blue)
and 5th (pink) interference order fringe. For an atomic velocity of 14 ms-1
and velocity spread of 1 ms-1

##  III Tunable, low-velocity, metastable argon beam

We create the tunable, low-energy, source of metastable argon by accelerating
them out of a magneto optical trap (MOT) using a single laser beam as shown in
figure 5. To load the MOT, a thermal effusive beam of metastable argon atoms
is decelerated in a Zeeman slower. Neutral atoms are introduced into an RF
discharge by the effusive beam of neutral argon atoms[21]. The MOT sits in a
chamber held at 1$\times$10-8 mbar and the atoms are laser cooled using 3
orthogonal, counter propagating beams arranged such that there is no cooling
beam in the vertical axis. We expand upon the work of Tailandire et al.[10] by
selecting the transverse velocity, as well as the longitudinal velocity. This
makes the transverse velocity dependent on fixed geometry rather than only the
temperature of the trapped atoms. Additionally, a larger, cooler MOT can be
obtained, and could in theory further reduce the atomic beam’s lower velocity
limit.

Figure 4: The experimental arrangement of the metastable argon MOT, the pulsed
atomic beam and the inteferometer. Two, 200CF, 6-way cross chambers house
effusive atomic beam, including a skimmer between the chambers to collimate
the beam. After the 2nd chamber, the Zeeman slower coils extend towards the
spherical octagon MOT chamber, while the Zeeman beam counter-propagates to the
atomic beam. The MOT chamber contains the trapping and push lasers. Attached
to the bottom of this chamber is the detector which records the matter-wave
interferograms and the time-of-flight of the atoms. This signal is recorded on
a CMOS camera.

The cold atomic cloud within the MOT has a 1/e2 Gaussian radius of 0.5 mm and
contains up to 3x106 atoms in the metastable 4s[3/2]2 state. A maximum density
of approximately 1010 cm-1 is determined by Penning and associative
ionization. To create the low velocity, pulse atomic beam, the atoms are
accumulated in the MOT for 500 ms seconds before a short pulse of between 0.6
and 7 ms, from a near resonant laser, illuminates the atomic cloud. This
pushes the atoms downwards via the radiation pressure force in the direction
of the vertical push beam, due to repeated recoil events following the
absorption and emission of push beam photons. The push beam passes through an
AOM allowing both the fast switching needed for sub millisecond pulse widths
and for the tuning of the push beam frequency. After acceleration from the
optical forces, the atoms pass through collimating slits placed 13 mm and 43
mm below the cold atomic cloud. The accelerated atoms travel downwards to a
microchannel plate detector with phosphor screen stack. This is placed
approximately 160 mm below the slits. When neutral atoms in the metastable
state strike the surface of the detector they are ionized and create a shower
of electrons which are detected on the phosphor screen. The phosphor screen is
imaged through a viewport using a CCD camera. A gated voltage can be applied
to the MCP detector allows imaging of the beam profile and interference
patterns with a temporal resolution down to 0.1 $\mu$s. For characterising the
beam properties, a silicon nitride grating with a slit spacing of 257 nm, slit
width 90 nm and a depth of 100 nm is placed just below the last collimating
slit.

Figure 5: A schematic showing the layout of the experiment, including both the
matter wave interferometer and a Stern-Gerlach wire which provides a B-field
gradient to separate the 5 magnetic sublevels in the beam.

###  III.1 Longitudinal Velocity

The longitudinal velocity of the atomic beam can be varied between 10-60 ms-1
by adjusting the push beam pulse length, the beam intensity or its frequency
detuning with respect to the atomic resonance. Increasing the pulse length
increases the interaction time between the push beam and hence the period of
acceleration. This effect is most clearly seen in figure 6 which shows the TOF
distribution for four different push beam pulse lengths, for a push beam
detuned 15 MHz from resonance.

Figure 6: The distribution of the atom beam’s TOF for push beam pulse lengths
of 7 ms (black), 4 ms (red), 5 ms (blue) and 2 ms (magenta). Such long pulse
lengths and TOF are possible with low push beam power and with no collimation
slits.

The push beam interaction time is ultimately limited by the attenuation from
the 10 $\mu$m collimation slit that is placed 12mm below the source. This
causes the push beam to diverge rapidly after diffracting through the slit.
This causes the push beam intensity to reduce by a factor of approximately
0.5$I_{0}$ for every mm below the slit which quickly reduces it effect. Given
that the atoms are accelerated through a spatially varying magnetic field (MOT
coils shown in figure 5), the atom’s resonant transition frequency is shifted.
The maximum force occurs when the laser is at resonance in 0 T magnetic field.
Maximal force can be applied by using a detuned the push beam to compensate
for the Zeeman shift in the atomic resonance due to the magnetic field. The
effect of the push beam frequency is shown in figure 7 for a pulse length of
0.6 ms.

Figure 7: The effect of the frequency of the push beam pulse on the velocity
of the atomic beam. For a pulse length of 0.6 $\mu$s and detuned by frequency
4.76 MHz (blue), 8.01 MHz (red) and 11.31 MHz (black). The beam intensity
decreases with the velocity, so the signals are amplified as indicated.

###  III.2 Transverse Velocity

With no collimation from the slits, the divergence angle of the atomic beam
would only be dependent on the temperature of the MOT for a given longitudinal
velocity. For an atomic beam with a 4.5 $\pm$0.5 ms-1 TOF, a beam width of 4
mm FWHM at the detector gives the beam’s divergence angle of 19.4 mrad and an
average transverse atomic velocity of 0.45 $\pm$0.05 ms-1. The MOT cooling
beam frequency adjusts the temperature of the atoms in the MOT and changes the
beam divergence, as seen in figure 8. The minimum beam width is limited by the
minimum MOT temperature of approximately 100 $\mu$K.

Figure 8: The divergence of the atomic beam (black squares) and the equivalent
transverse velocity (blue squares) as a function of the MOT cooling beam
detuning frequency, which controls the MOT temperature. Data is for a 4 ms
peak TOF, using a single 10$\mu$ collimation slit 13mm below the MOT.

To observe interference fringes the beam must have sufficient transverse
coherence. This is achieved by limiting the angle that the source subtends
from the grating and limiting the spread of transverse velocity. To select the
beam’s transverse velocity independently of longitudinal velocity or MOT
temperature, two collimation slits are placed between the MOT and the
detector. The beam width is now controlled by the geometry of the collimation
slit setup. With a 10 $\mu$m slit, 13 mm below the MOT and a 50 $\mu$m slit,
43 mm below the MOT, the beam divergence is 2.4 mrad based on a FWHM of
approximately 0.5 mm shown in figure 9. Minimising the transverse velocity
distribution increases the contrast of the matter-wave interference fringe
peaks used to characterise the longitudinal velocity.

###  III.3 Spin Polarisation of the Atomic Beam

The 4s[3/2]2 metastable state (J =2) has five m-states ($m=-2,-1,0,1,2$). For
the push beam (and the cooling beams) the 4s[3/2]2-4p[5/2]3 is closed cycle
transition. If the push beam is right/left circularly polarised with respect
to the atom’s quantisation axis, the atoms can only be pumped to the m=-2 or
m=-3 stretch state.

Figure 9: Measured atomic beam width profile for the 3 velocities that are
used for the matter wave interferometry (approximately 11 ms-1 (blue), 15 ms-1
(red) and 18 ms-1 (black)). This beam width was narrowed using a 10 $\mu$m
slit and a 50 $\mu$m for the initial and second collimating slit respectively.
There is a background signal indicated by the lower, wider peak that arises
due to elastic collisions with the walls of the slit.

To analyse the spin polarisation purity of the atomic beam, a Stern-Gerlach
(SG) wire was placed between the second collimation slit and the detector
shown in figure 5. A current of around 200 A was pulsed for up to 1 ms through
the wire at the moment the atoms pass the wire. The resultant magnetic field
gradient spatially separates the $m$ states on the detector. Figure 10 shows
the change in the atomic beam spin polarisation as the push beam polarisation
is changed using a $\frac{\lambda}{4}$ waveplate. By finding the fractional
area corresponding to m=+2 or m=-2 state, we determine that greater than 96%
of atoms can be placed in either of the stretch states during acceleration of
the atoms with the push beam.

Figure 10: The $m$ state signal intensity as the push beam polarisation is
changed from $\sigma^{-}$ through $\pi$ to $\sigma^{+}$ by adjusting the
rotation of the $\frac{\lambda}{4}$ waveplate in the push beam path. The
atomic beam passes by a Stern Gerlach wire which creates a magnetic field
gradient to spatially separate the $m$ states.

##  IV Matter-Wave Interferometery of a Slow Metastable Argon Atomic Beam

The collimated atomic beam is pulsed every 0.5 s and interference fringes are
observed on the MCP detector placed 156 mm below the grating.

The grating’s slits lie in a 160 nm thick SiN wafer, positioned 10 mm below
the second slit. The grating has a period of 257 nm, with each slit 97 $\mu$m
wide, shown in figure 11. The atoms’ final positions, after undergoing
diffraction, are observed on the MCP detector. The MCPs have an ultimate
spatial resolution of 6 $\mu$m, but the reduced resolution of the phosphor
screen increases this to $\approx$25 $\mu$m. A 1392 x 1040 pixel CCD camera is
focused on the screen, with each pixel covering 11.4 $\mu$ m. Post processing
of the images is used to find the centre of the Gaussian signal, and therefore
the position of the atom, bringing the spatial resolution of the detector
system down to 11.4 $\mu$m.

Figure 11: A SEM image of the 90x990 nm grating slits created in a SiNi
membrane. The dimensions are slightly distorted due to the non-perpendicular
viewing angle.

The push beam is configured to produce the three velocity distributions shown
in figure 7. Within each distribution, the detector is gated to select a
narrow velocity range. For the three interferograms shown, the $v_{TOF}$
calculated to be 11.65 $\pm$0.49 ms-1, 14.04 $\pm$0.71 ms-1 and 16.67
$\pm$0.54 ms-1 as shown in figures 13 (a)-(c) respectively.

The push beam configured for the atomic velocity used in figure 13 (a)
delivers an average of approximately 3 atoms per push cycle. By averaging the
signal from 80,000 cycles, the interference fringes appear as shown in figure
12.

Figure 12: Detector images indicating the difference between the centroided
(right) and raw (left) detector images for the $v_{TOF}$=11.65 ms-1 velocity
atoms. Figure 13: The interferograms for atoms with $v_{TOF}$ of (a) 11.65
$\pm$0.49 ms-1, (b) 14.04 $\pm$0.71 ms-1 and (c) 16.67 $\pm$0.54 ms-1 (black
squares). The red lines are fits to data using 1. The intensities have been
normalised to the central peak.

###  IV.1 Beam velocity from fringe spacing

The interference peak positions are measured for the three data sets and a
linear fit for is shown for each in figure 14. From the resulting gradient the
average velocity was measured with these results as shown in table I. From
this gradient the average velocity is measured to be 13.92 $\pm$ 0.19 ms-1,
16.82 $\pm$0.23 ms-1 and 19.62 $\pm$0.26 ms-1 for the data sets from figures
13(a)-(c) respectively. After the push beam interaction, the atoms travel
towards the detector accelerated only by gravity. The increase in velocity due
to gravity is $<$1% of these values, so after the interaction with the
grating, the atomic velocity is to a very good approximation constant. The
values are calculated using $L$=156$\pm 1$ mm and $d$=257$\pm$3 nm, The
grating spacing, $d$, is best measured using SEM imaging which can achieve an
accuracy of up 0.4 nm. However, typically these are between 1-20 nm depending
on the instrument and its calibration. The varying peak positions have an
uncertainty of 0.3%.

Figure 14: The interference peak positions in mm (with respect to the zeroth
order peak) plotted as a function of the interference order for the three data
sets in figure 13. The fitted line with gradient, $\frac{h}{mv}\frac{L}{d}$,
provides a measurement of the velocity of 13.92$\pm$0.19 ms-1 (black),
16.82$\pm$0.23 ms-1 (red), 19.62$\pm$0.26 ms-1 (blue).

As the velocity spread increases, there is slight change in the higher order
peak positions and there is subsequently a change in the fitted gradient. The
effect causes a 0.1 ms-1 change in the measured velocity for every 1 ms-1 of
spread in the atomic beam.

###  IV.2 Beam velocity from fitting

To include the beam’s velocity spread, $\Delta v$, and measure the velocity
$v$ more precisely, equation 5 is fitted to the interferogram using a standard
least squares method. The free parameters are $v,\Delta v$ and $C_{3}$ which
gives $v_{g}=13.90\pm$0.04ms -1, $\Delta v=2.34\pm$ 0.10 ms -1 and
$C_{3}=1.75\pm 0.03$ a.u. for figure 13 (a). The beam in figure 13 (b) is
measured to have $v_{g}$=16.73$\pm$0.05 ms-1, $\Delta v=2.60\pm$0.16 ms -1 and
$C_{3}=1.80\pm 0.02$ a.u.. The third data set, figure 13 (c), gives an atomic
beam of $v_{g}$=19.36$\pm$0.13 ms-1, $\Delta v=3.46\pm$0.32 ms -1 and
$C_{3}=2.01\pm 0.03$ a.u.. These measurements of $v_{g}$ are compared with
previous measurements in table I. The fixed parameters for fitting the model
are $t=160\pm 3\,$nm$,\beta=7.0\pm 0.5^{\circ},s_{0}=90\pm 3\,$nm$,d=257\pm
3\,$nm$,L=155.9\pm 0.5\,$mm, and $N$=180. The beam profile $A$, from equation
4, were measured from the beam profile recorded by the MCP detector as shown
in figure 9.

Data |  TOF |  Gradient |  Fit
---|---|---|---
(a) |  11.65$\pm$0.49 |  13.92$\pm$0.19 |  13.90$\pm$0.04
(b) |  14.04$\pm$0.71 |  16.82$\pm$0.23 |  16.73$\pm$0.05
(c) |  16.67$\pm$0.54 |  19.62$\pm$0.26 |  19.36$\pm$0.13
Table 1: Comparison of three velocity measurements for each data set shown in
figure 13. All values are in ms-1. TOF refers to the use of the time-of-
flight, while gradient is detrmined from the slope of the peak positions shown
in figure 14. Fit refers to the fitting of equations 1-5 to the full
interferogram.

The velocity of each beam measured using the gradient and fit are in good
agreement, but are approximately 20% larger than those derived from the TOF.
This significant discrepancy highlights the inaccuracy of using TOF to
determine the velocity of the beam. The TOF calculation assumes the atoms have
an initial velocity close to $v_{TOF}$ and a constant acceleration due to
gravity. However, the atoms initial velocity (due to the temperature of the
MOT) is much lower and the acceleration is dominated by the short interaction
with the push beam laser. After the first collimation slit (or after the push
beam pulse), the actual atomic velocity should be higher than the average TOF
velocity.

###  IV.3 Measuring the van der Waals interaction between the atoms and the
grating

Fitting equations 1-5 also provides a measurement for the VdW coefficient,
$C_{3}$. For the data in figures 13(a)-(c), we derive a mean value of
1.84$\pm$0.17 a.u. from the three fits. There have been no previous attempts
to measure $C_{3}$ for Ar* and Si3N4 as far as we know. This measurement is in
good agreement with values derived from spectroscopic data ($C_{3}$=1.83
a.u.[22]).

As shown in figure 2 and 3, $\Delta v$ and $C_{3}$ each change the peak height
and signal contrast. In figure 13(a), for the interference peaks above the 6th
order, the signal noise is too large to determine if the observed envelope is
created by the slit function (figure 2(a)) or the longitudinal velocity spread
(figure 3(a)). For a more accurate measurement of velocity and spread, a lower
to signal to noise and/or higher contrast peaks would be required. This could
be achieved with longer data acquisitions or a narrower longitudinal velocity
spread.

##  V Conclusion

We have characterised a slow, spin-polarised, atomic beam created by resonant
acceleration of atoms from a metastable argon MOT. We have shown that matter-
wave interferometry is well suited to the measurement of low velocity beams
produced by optical forces. As the beam experiences acceleration along its
path, we demonstrate that this method is capable of providing precise
measurements of the average velocity and velocity distribution, which cannot
be obtained from typical TOF measurements. Such measurements will find
application in the characterisation of other low velocity beam sources and for
future experiments where we aim to measure the weak value of transverse
momentum inside a matter-wave interferometer[23]. Although not compared
directly with measurements of beam velocity by Doppler shift measurements, the
approach demonstrated here is particularly attractive for low velocity and
pulsed beams, where short transit times and Doppler shifts below the laser
line width limit its application. Finally, we demonstrate that the slit
function, modified by the Van der Waals interactions with the wall, can used
to determine the $C_{3}$ Van der Waals co-efficient for Ar* with the walls of
the multi-slit Si3N4 diffraction grating.

###### Acknowledgements.

The authors would like to thank the Franklin Fetzer Foundation for their
financial support and Markus Arndt from The University of Vienna for the use
of the silicon nitride diffraction grating.

## References

  * Ramsey [1985] N. Ramsey, Molecular beams, Oxford University Press (1985).
  * Zhelyazkova  _et al._ [2020] V. Zhelyazkova, F. B. V. Martins, J. A. Agner, H. Schmutz, and F. Merkt, Phys. Rev. Lett. 125, 263401 (2020).
  * O’Connor  _et al._ [2015] A. P. O’Connor, X. Urbain, J. Stützel, K. A. Miller, N. de Ruette, M. Garrido, and D. W. Savin, American Astronomical Society 219, 6 (2015).
  * Cho and Arthur [1975] A. Cho and J. Arthur, Molecular beam epitaxy, Progress in Solid State Chemistry 10, 157 (1975).
  * Jankunas and Osterwalder [2015] J. Jankunas and A. Osterwalder, Cold and controlled molecular beams: Production and applications, Annual Review of Physical Chemistry 66, 241 (2015).
  * Cacciapuoti  _et al._ [2001] L. Cacciapuoti, A. Castrillo, M. De Angelis, and G. M. Tino, European Physical Journal D 15, 245 (2001).
  * Wang and Buell [2003] H. Wang and W. F. Buell, Journal of the Optical Society of America B 20, 2025 (2003).
  * Maher-McWilliams  _et al._ [2012] C. Maher-McWilliams, P. Douglas, and P. F. Barker, Laser-driven acceleration of neutral particles, Nature 6, 386 (2012).
  * Lu  _et al._ [1996] Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
  * Taillandier-Loize  _et al._ [2016] T. Taillandier-Loize, S. A. Aljunid, F. Correia, N. Fabre, F. Perales, J. M. Tualle, J. Baudon, M. Ducloy, and G. Dutier, Journal of Physics D: Applied Physics 49 (2016).
  * Rakonjac  _et al._ [2012] A. Rakonjac, A. B. Deb, S. Hoinka, D. Hudson, B. J. Sawyer, and N. Kjærgaard, Opt. Lett. 37, 1085 (2012).
  * Carey  _et al._ [2018] M. Carey, M. Belal, M. Himsworth, J. Bateman, and T. Freegarde, Journal of Modern Optics 65, 657 (2018).
  * Slowe  _et al._ [2005] C. Slowe, L. Vernac, and L. V. Hau, Review of Scientific Instruments 76, 1 (2005).
  * Davisson and Germer [1927] C. Davisson and L. H. Germer, The Scattering of Electrons by a Single Crystal of Nickel, Nature (London) 119, 558 (1927).
  * Fein  _et al._ [2019] Y. Y. Fein, P. Geyer, P. Zwick, F. Kiałka, S. Pedalino, M. Mayor, S. Gerlich, and M. Arndt, Quantum superposition of molecules beyond 25 kda, Nature Physics 15, 1242 (2019).
  * Rosi  _et al._ [2014] G. Rosi, F. Sorrentino, L. Cacciapuoti, and et. al, Precision measurement of the newtonian gravitational constant using cold atoms, Nature 510, 518 (2014).
  * Peters  _et al._ [2001] A. Peters, K. Y. Chung, and S. Chu, High-precision gravity measurements using atom interferometry, Metrologia 38, 25 (2001).
  * Berninger  _et al._ [2007] M. Berninger, A. Stefanov, S. Deachapunya, and M. Arndt, Polarizability measurements of a molecule via a near-field matter-wave interferometer, Phys. Rev. A 76, 013607 (2007).
  * Nimmrichter  _et al._ [2008] S. Nimmrichter, K. Hornberger, H. Ulbricht, and M. Arndt, Absolute absorption spectroscopy based on molecule interferometry, Phys. Rev. A 78, 063607 (2008).
  * Grisenti  _et al._ [1999] R. E. Grisenti, W. Schöllkopf, J. P. Toennies, G. C. Hegerfeldt, and T. Köhler, Phys. Rev. Lett. 83, 1755 (1999).
  * Edmunds and Barker [2014] P. D. Edmunds and P. F. Barker, Trapping cold ground state argon atoms, Phys. Rev. Lett. 113, 183001 (2014).
  * Karam  _et al._ [2005] J. C. Karam, N. Wipf, J. Grucker, F. Perales, M. Boustimi, G. Vassilev, V. Bocvarski, C. Mainos, J. Baudon, and J. Robert, Journal of Physics B: Atomic, Molecular and Optical Physics 38, 2691 (2005).
  * Morley  _et al._ [2016] J. Morley, P. D. Edmunds, and P. F. Barker, Journal of Physics: Conference Series 701, 3 (2016).