††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

Present address: ]Department of Physics, Harvard University, Cambridge, MA
02138, USA

# Observation of the Hanbury Brown and Twiss Effect with Ultracold Molecules

Jason S. Rosenberg  Department of Physics, Princeton University, Princeton,
New Jersey 08544, USA  Lysander Christakis  Department of Physics, Princeton
University, Princeton, New Jersey 08544, USA  Elmer Guardado-Sanchez  [
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Zoe Z. Yan  Department of Physics, Princeton University, Princeton, New Jersey
08544, USA  Waseem S. Bakr  Email<EMAIL_ADDRESS>Department of Physics,
Princeton University, Princeton, New Jersey 08544, USA

###### Abstract

Measuring the statistical correlations of individual quantum objects provides
an excellent way to study complex quantum systems. Ultracold molecules
represent a powerful platform for quantum science due to their rich and
controllable internal degrees of freedom. However, the detection of
correlations between single molecules in an ultracold gas has yet to be
demonstrated. Here we observe the Hanbury Brown and Twiss effect in a gas of
bosonic 23Na87Rb, enabled by the realization of a quantum gas microscope for
molecules. We detect the characteristic bunching correlations in the density
fluctuations of a 2D molecular gas released from and subsequently recaptured
in an optical lattice. The quantum gas microscope allows us to extract the
positions of individual molecules with single-site resolution. As a result, we
obtain a high-contrast two-molecule interference pattern with a visibility of
$54(13)\%$. While these measured correlations arise purely from the quantum
statistics of the molecules, the demonstrated capabilities pave the way toward
site-resolved studies of interacting molecular gases in optical lattices.

In a landmark series of experiments in the 1950s, Hanbury Brown and Twiss
(HBT) demonstrated bunching correlations of photons from chaotic sources of
light arriving at two detectors [1, 2]. Their interferometry technique had
practical applications in measuring the angular diameter of stars and led
Glauber to develop a theory of quantum coherence, laying the foundation for
the field of quantum optics [3]. In contrast to conventional interference
observed, for example, in Young’s double-slit experiment, the HBT effect
results from the interference of two-particle rather than single-particle
amplitudes – regardless of whether those “particles” are photons,
quasiparticles, or matter. HBT interferometry has become a workhorse in high
energy and nuclear physics to probe the space-time geometry of collision
volumes [4]. The effect has also been demonstrated with electrons [5, 6],
neutrons [7], and phonons [8, 9].

In the field of ultracold quantum gases, the exquisite control afforded by
modern experimental techniques has stimulated a wealth of intensity
interferometry measurements in atomic systems, both bosonic [10, 11, 12, 13,
14, 15, 16, 17, 18, 19] and fermionic [20, 14, 21]. Molecular gases have
recently been brought into the ultracold regime where quantum effects play a
significant role [22], often by leveraging the powerful approach of
associating two ultracold atoms into a single molecule [23]. These molecules
represent the most complex objects for which full control over all the
motional and internal degrees of freedom has been demonstrated. Recent
experiments with polar molecules in optical lattices have studied strongly-
correlated many-body systems [24] and laid the groundwork for using molecules
as qubits in quantum computing [25].

Figure 1:  Hanbury Brown and Twiss interference of heteronuclear bosonic
molecules. (a) Molecules released from lattice sites separated by a distance
$\ell$ are detected after a time of flight at a variable separation d. The
amplitudes associated with the shown two-particle trajectories (black and
blue) interfere, leading to a joint detection probability that depends on $d$.
(b) The experiment in (a) is performed with molecules in an optical lattice.
Top: multiple molecules are trapped in a 2D optical lattice. Middle: the 2D
lattice is abruptly shut off, allowing the molecules to expand freely in the
plane. An additional vertical lattice (not pictured) is left on to levitate
the molecules against gravity. Bottom: the molecules are recaptured in the 2D
lattice after time of flight, and their positions are measured with a quantum
gas microscope. Figure 2:  Microscopy of molecules in an optical lattice. (a)
Three-step process for detecting the molecules. Top: an array of NaRb
molecules is prepared in a lattice. Middle: a microwave Landau-Zener sweep
flips the Na atoms from the $|1,1\rangle$ to the $|2,2\rangle$ state, breaking
apart the molecules into unbound Rb (red) and Na (blue) atoms. Bottom: a pulse
of resonant light (blue beam) removes the Na atoms from the lattice. The
remaining Rb atoms are laser cooled (red beam) and the fluorescence photons
are collected with an objective, revealing the original positions of the
molecules with single-site resolution. (b) Left: a sample fluorescence image
of Rb atoms tagging the positions of molecules in the lattice. Right: molecule
occupancy of the lattice for the fluorescence image. (c) Radially-averaged
density profile of the NaRb molecules (blue) as well as the Rb atoms (red)
prior to associating the atoms into molecules. We observe peak fillings of
$0.73(2)$ for the Rb atoms and $0.15(1)$ for the molecules. Both density
profiles are averaged over 30 experimental repetitions. Figure 3:  Molecule
binding energy versus magnetic field $B$. For each field, we drive the Na
$|1,1\rangle{\rightarrow}|2,2\rangle$ transition to break apart the molecules
(upper left inset) and measure the resulting dissociation spectrum. We extract
the bound-free transition frequency from the onset of the spectrum. We also
measure the free atomic transition frequency at each field. The difference
between the atomic and molecular transition frequencies gives the binding
energy (blue circles). The black dashed line is the predicted binding energy
from a coupled-channel calculation. Lower right inset: example molecular
dissociation spectrum at 343.4 G (red circles) fit to an asymmetric Gaussian
(solid black line). The error on the binding energies is taken as the half-
width of the shaded region of the spectrum. Figure 4:  Observation of the HBT
effect with molecules. (a) Typical fluorescence image obtained after molecule
time of flight. (b) Integrated density profile after time of flight, averaged
over 764 images. (c) HBT correlations for Rb atoms, post-selected for images
that yielded between 130 and 300 atoms. (d) 1D cut of the Rb HBT correlations
averaged along the horizontal and vertical axes. Red arrows indicate the
predicted position of the interference peaks. (e) HBT correlations for
molecules averaged across the lattice symmetries. Images with fewer than 20
molecules were excluded from the analysis. (f) 1D cut of the NaRb HBT
correlations averaged along the horizontal and vertical axes. Red arrows
indicate the predicted position of the interference peaks.

As two-particle correlations will form a crucial experimental probe for
interacting many-body systems comprising polar molecules, we first study
whether intensity interferometry can be used to characterize distributions of
non-interacting molecules. We perform an HBT experiment in which we measure
fluctuations in the density distribution of individual clouds of molecules on
a grid of detectors after a time of flight [26]. Particles originating from
sources $A$ and $B$ can arrive at detectors $1$ and $2$ in two different ways
($A\rightarrow 1,B\rightarrow 2$ and $A\rightarrow 2,B\rightarrow 1$), shown
in Fig. 1. If the two paths are indistinguishable then their quantum
mechanical amplitudes interfere, as reflected in the two-point density
correlation function. The correlation relates to the Fourier transform of the
source density distribution in the case of a distribution of sources with no
phase coherence. As a result, if the positions of the molecules are initially
discretized by an optical lattice, the interference pattern exhibits
constructive interference peaks whose separation and width are inversely
related to the lattice spacing and in-situ cloud size, respectively [11, 20].

There are two requirements for measuring a high-visibility HBT interference
pattern: particle shot noise-limited detection and a detector spatial
resolution better than the interference peak width. To fulfill these
conditions, we detect the positions of individual molecules with single-site
resolution using a quantum gas microscope [27, 28]. To date, single neutral
ultracold molecules have only been detected in small tweezer arrays containing
up to four molecules [29, 30, 31]. With the molecular quantum gas microscope
introduced in this work, we prepare and detect over one hundred molecules
simultaneously by dissociating them into atoms and using the atoms as tags for
the molecule positions.

We form ultracold bosonic 23Na87Rb molecules from degenerate gases of Na and
Rb atoms by magnetoassociation [32]. To start, dual Na and Rb Bose-Einstein
condensates (BECs) are prepared in a single layer of a vertical lattice. Fine
adjustment of atom numbers is accomplished using a bichromatic dimple trap,
allowing the reproducible production of clouds of several hundred atoms per
species. Both species are prepared in their $\left|F=1,m_{F}=1\right\rangle$
states, where $F$ and $m_{F}$ are the total angular momentum quantum number
and its projection along the quantization axis, respectively. Next, we make
the mixture miscible by nulling the interspecies scattering length using an
$s$-wave Feshbach resonance at 347.6 G [33]. The atoms are then loaded into a
2D in-plane optical lattice with spacing $a\,{=}\,752$ nm, and tunneling
between sites is frozen by increasing the lattice depth. We associate the
atoms into molecules by adiabatically ramping the magnetic field across the
resonance at a rate of 2.5 G/ms, changing the scattering length from
attractive to repulsive. For lattice sites containing one atom of each
species, the theoretical conversion efficiency is close to unity
(Supplementary Information). We selectively remove remaining free Na and Rb
atoms by transferring them to the $|2,2\rangle$ hyperfine state and applying
resonant light pulses on the optical cycling transitions. The transfer and
removal is repeated four times for each species to achieve high efficiency.

To realize a quantum gas microscope for molecules, we employ a three-step
process to extract site-resolved positions, as shown in Fig. 2(a). We first
ramp the magnetic field back to the attractive side of the Feshbach resonance,
increasing the Franck-Condon overlap between the bound and free states of
atoms on a site. We note that a confinement-induced weakly-bound state exists
here due to the lattice [34], so an adiabatic ramp across the resonance does
not break apart the molecules. The confinement-induced molecules are then
dissociated by addressing Na with another series of microwave transfers and
optical removal pulses. The remaining Rb atoms serve as markers for the
Feshbach molecule locations. Light-assisted collisions during the optical
removal pulses are estimated to be negligible (Supplementary Information).
After ramping the magnetic field to zero, the in-plane lattice depth is
increased to 6000 $E_{\text{r,Rb}}$ (where $E_{\text{r,Rb}}=h^{2}/8m_{\rm
Rb}a^{2}$ is the recoil energy for Rb) and the atoms are loaded into a light
sheet potential. Optical molasses beams both cool the Rb atoms and scatter
photons, which are collected with a high-numerical-aperture objective and
imaged onto a camera. The full width at half maximum of the point spread
function is $1010(15)$ nm, which is sufficient for single-site resolution
using a reconstruction algorithm [28].

Fig. 2(b) shows an example image of an in-situ cloud of 103 molecules. A
radially-averaged density profile, averaged over 30 experimental repetitions,
is shown in Fig. 2(c). We measure a central molecule filling of $0.15(1)$,
compared to the Rb parity-projected central filling of $0.73(2)$. The molecule
filling fraction can be increased in future work by improving the overlap
between the atomic clouds, lowering their temperature, and increasing the
vertical confinement to obtain stronger interactions at the superfluid to Mott
insulator transition.

A high molecule detection fidelity is essential for obtaining high contrast
HBT interference as well as for future applications of the molecule
microscope. Possible limitations on the molecule detection fidelity include
imperfect atom-tagging of sites that had molecules as well as site-to-site
hopping and loss of the Rb tag atoms during fluorescence imaging. We first
measure the false negative molecule detection rate due to imperfect tagging.
After performing the tagging protocol consisting of dissociating the molecules
and optically removing Na, we ramp the field to the repulsive side of the
Feshbach resonance and remove the Rb tag atoms. The false negative rate of
$1.2(1)$% is obtained by repeating the tagging protocol prior to fluorescence
imaging to identify failures from the first attempt. During our 0.5 s imaging
exposure time, we measure site-to-site hopping of $0.1(1)$% and $1.7(3)$%
loss. The latter is consistent with the measured atom lifetime limited by
background gas collisions. We additionally measure a false positive rate of
$0.34(5)$% due to Rb atoms that did not associate into molecules and were not
removed before imaging (Supplementary Information).

We further confirm that the detected particles are molecules by measuring
their binding energy as a function of magnetic field using dissociation
spectroscopy (Fig. 3). Driving the Na $|1,1\rangle{\rightarrow}|2,2\rangle$
transition yields the molecular dissociation spectrum, showing a sharp onset
at a microwave frequency shifted by the binding energy $E_{\rm b}$ from the
atomic transition. Molecules that are not dissociated are not detected in the
fluorescence imaging. We find close agreement between our measured binding
energies and a coupled-channel calculation using the bound package [35] based
on the parameters in Ref. [36]. Confinement effects are ignored since they are
smaller than the resolution of the measurement.

Having established our molecule detection procedure, we now observe the HBT
effect by measuring the density-density correlation function
$g^{(2)}(\textbf{d})$ after a long time of flight (TOF) (Fig. 4). The
correlation function is defined as:

$g^{(2)}(\textbf{d})=\frac{\int\langle\hat{n}(\textbf{x})\hat{n}(\textbf{x+d})\rangle
d^{2}\textbf{x}}{\int\langle\hat{n}(\textbf{x})\rangle\langle\hat{n}(\textbf{x+d})\rangle
d^{2}\textbf{x}}$ (1)

where $\hat{n}(\textbf{x})$ is the number operator at detection position x,
and d is the displacement between the detection positions. Maximal particle
bunching is indicated by $g^{(2)}\,{=}\,2$ while for uncorrelated particles
$g^{(2)}\,{=}\,1$. For a Gaussian source cloud of half-width $s$ (at
$e^{-1/2}$ of maximum), the half-width of the interference peaks is given by
$\delta\,{=}\,\hbar t/ms$ (at $e^{-1}$ of maximum), where $m$ is the particle
mass and $t$ is the TOF. While particle shot noise-limited imaging in a 2D
plane can theoretically achieve peak correlation amplitudes of
$g^{(2)}\,{=}\,2$, the amplitude will be reduced if the width of the peaks is
narrower than the detector size, in our case one lattice site. Therefore, we
carefully choose our source cloud size and use the largest possible TOF given
the constraints from the size of the lattice beams to maximize the HBT
amplitude.

A larger signal-to-noise ratio of the interference pattern can be achieved for
higher in-situ lattice filling fractions. Since we achieve higher fillings
with atoms than with molecules, we first benchmark the interferometry protocol
with Rb atoms. We prepare a gas with $189(20)$ Rb atoms frozen in a 2D optical
lattice (66 $E_{\text{r,Rb}}$ depth), with a peak filling of $0.86(2)$ and an
average source size $s\,{=}\,7(1)$ sites. We abruptly turn off the 2D lattice
to initiate a $9.4(1)$ ms TOF in the vertical lattice. The vertical lattice
confinement is set to $\omega_{\text{Rb}}\,{=}\,2\pi\times(3,4,1000)$ Hz,
providing negligible radial confinement. The TOF satisfies the far-field
detection condition $t\gg 2\pi/\omega_{r}$, where $\omega_{r}\,{=}\,2\pi\times
14$ kHz is the on-site radial trap frequency of the 2D lattice for Rb.
Following the TOF, we turn on the 2D lattice to pin the distribution for
imaging.

The observed atomic HBT correlations are shown in Fig. 4(c). We observe a high
contrast interference pattern with average correlation peak amplitudes of
$1.80(12)$ and an average background value of $0.999(6)$. The measured peak
separation and width from the 1D cut in Fig. 4(d) is $75.9(4)$ and $2.4(2)$
sites respectively, close to the theoretically expected spacing of $ht/m_{\rm
Rb}a\,{=}\,76.2(8)$ and $\delta\,{=}\,1.7(3)$ sites. The symmetric pattern
verifies that the 2D lattice axes are orthogonal and the lattice spacings
along both axes are identical to better than $0.5$%. This implies that
$g^{(2)}(\textbf{d})$ is invariant for d reflected across the $x\,{=}\,0$,
$y\,{=}\,0$, $x\,{=}\,y$, and $x\,{=}\,-y$ symmetry axes, justifying averaging
of the weaker molecular correlations across the lattice symmetries to reduce
noise.

We repeat the HBT interferometry with 56(13) molecules and a mean source size
$s\approx\,17$ sites, which is expected to produce interference peaks whose
widths are on the order of the lattice spacing. The protocol is the same as
that used for atoms, with the molecules released from the 2D lattice at a
magnetic field of $335.1$ G (with binding energy $E_{\text{b}}/h\approx 20$
MHz). Fig. 4(e,f) shows the observed molecular correlations averaged across
the lattice symmetries. Since the TOF is the same as that used in the Rb atom
correlation measurement, the smaller correlation peak spacing for the
molecules is a direct result of their increased mass. The measured spacing is
$60.0(5)$ sites, consistent with the theoretical expectation of $60.3(6)$
sites. While the correlation peaks are narrower than for Rb ($<1$ site), the
peak amplitude remains large at 1.58(13). The average baseline is 1.04(1),
with the deviation from unity caused by correlations at all distances due to
shot-to-shot molecule number fluctuations. The interference contrast is
sensitive to the preparation of the molecules in the same internal state and
the same motional state of the vertical lattice, since these quantum numbers
can provide which-path information during the free expansion; therefore, the
measured contrast of $0.54(13)$ indicates a high degree of
indistinguishability of the molecules.

To conclude, we have demonstrated site-resolved measurements of density
correlation functions in a non-interacting molecular quantum gas after time of
flight expansion. The observation of the HBT effect with molecules paves the
way toward realizing other quantum optical phenomena with molecules of
increasing complexity [37]. By transferring the Feshbach molecules to the
rovibrational ground state [38], a molecular lattice gas with strong dipolar
interactions can be prepared [22]. The interplay of quantum statistics and
interactions can give rise to interesting signatures in HBT measurements [18,
19]. In addition, many correlated quantum states predicted to be realizable
with polar molecules also exhibit real-space density correlations that can be
directly measured with a molecule microscope. These include Wigner crystals
[39] and Mott solids with rational lattice fillings [40]. Finally, inelastic
collisions in molecular gases can lead to strong correlations through the
quantum Zeno effect. While previous experiments have observed these
correlations through the inhibition of loss [41, 24], a molecule microscope
would enable their spatially-resolved measurement.

We would like to thank Geoffrey Zheng, Sejal Aggarwal, Alan Morningstar, and
Ravin Raj for experimental assistance. This work was supported by the NSF
(grant no. 1912154) and the David and Lucile Packard Foundation (grant no.
2016-65128). L.C. was supported by the NSF Graduate Research Fellowship
Program.

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##  I Methods

Here we describe the formation of ultracold bosonic 23Na87Rb molecules by
magnetoassociation from degenerate gases of Na and Rb atoms. NaRb molecules
have been produced in bulk mixtures in Ref. [32].

We start by creating dual Na/Rb Bose-Einstein condensates (BECs) with
typically $2\times 10^{5}$ atoms of each species in the
$\left|F=1,m_{F}=-1\right\rangle$ state, using Na as a sympathetic coolant for
Rb during forced evaporative cooling in a quadrupole magnetic trap followed by
evaporation in a crossed 1064 nm optical dipole trap. To prepare a 2D system,
we load the condensates into a 1064 nm light sheet with tight confinement
along the vertical direction ($\omega_{\text{Na}}=2\pi\times(24,117,1900)$ Hz,
$\omega_{\text{Rb}}=0.86\,\omega_{\text{Na}}$). We then transfer both atom
species to the $\left|F=1,m_{F}=1\right\rangle$ state, the entrance channel
for the relevant Feshbach resonance. To further increase the vertical
confinement, the atoms are loaded into a single layer of a 3.8 $\mu$m spacing
vertical lattice created by two 1064 nm beams intersecting at 16∘
($\omega_{\text{Na}}=2\pi\times(14,20,4500)$ Hz). For our optical lattice
approach of molecule formation, we need the central density of the clouds to
be on the order of one atom per site for each species. Given our trap
parameters, this requires the ability to reproducibly generate small
condensates of order one hundred atoms. We achieve this by performing a second
stage of evaporative cooling in a tightly focused bichromatic dimple trap,
which allows for independent adjustment of the atom number in each species
(Supplementary Information).

Near zero magnetic field, the Na and Rb BECs are immiscible. To increase the
overlap of the spatial distributions prior to magnetoassociation, we tune the
interspecies scattering length using an $s$-wave Feshbach resonance at 347.6 G
to bring the clouds into a miscible regime [33]. We quickly ramp the field
above the resonance to 415.9 G, then slowly decrease the field to the zero
crossing of the interspecies scattering length at 351.9 G in 20 ms. We
subsequently load the 2D mixture into an in-plane square lattice with spacing
$a=752$ nm created by the fourfold interference of a single 1064 nm beam (105
$\mu$m waist) in a bowtie configuration. We freeze tunneling for both species
by ramping the in-plane lattice depth to 36 $E_{\text{r,Na}}$, where
$E_{\text{r,Na}}=h^{2}/8m_{\rm Na}a^{2}$ is the recoil energy for Na. Next, we
ramp the magnetic field below the resonance at a rate of 2.5 G/ms to form
Feshbach molecules. For lattice sites containing one atom of each species, the
theoretical conversion efficiency is very close to unity (Supplementary
Information). We selectively remove remaining free atoms by transferring them
to $|2,2\rangle$ with a microwave Landau-Zener sweep at a field of $\sim
346.6$ G (molecule binding energy $E_{\text{b}}/h\approx 0.7$ MHz) and
applying resonant light on the $|2,2\rangle$ to $|3,3\rangle$ optical cycling
transition. The transfer and removal is repeated four times for each species
to achieve high efficiency as described in the main text.

For imaging, the magnetic field is brought to 0 G, the in-plane lattice depth
is increased to 6000 $E_{\text{r,Rb}}$ and the Rb atoms are loaded back into
the light sheet at a depth of 140 $\mu$K. Optical molasses cooling light
scatters photons into a 0.5 numerical aperture objective, with $\sim 10^{4}$
photons collected on the camera per atom per second.

Supplementary Information

Jason S. Rosenberg, Lysander Christakis, Elmer Guardado-Sanchez, Zoe Z. Yan,
and Waseem S. Bakr

Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

##  1 Preparation of a dual 2D BEC

Our experiment begins by loading a dual-species magneto-optical trap (MOT)
with typically $8\times 10^{8}$ Na atoms and $3\times 10^{6}$ Rb atoms from
two separate 2D-MOTs in 8 s. We deliberately load fewer Rb atoms than Na
because we use the Na as a sympathetic coolant for the Rb during forced
evaporative cooling. For Na, we employ a dark-spot MOT to increase its initial
phase space density. We then cool the Na atoms in an optical molasses, at the
end of which the temperature of the Na atoms is $\sim 140$ $\mu$K. The
$\left|F=1,m_{F}=-1\right\rangle$ atoms are loaded into a magnetic quadrupole
trap by ramping a magnetic field gradient to 174 G/cm. The temperature of the
mixture after loading the magnetic trap is $\sim 160\,\mu$K. We perform forced
microwave evaporation of the Na atoms by flipping atoms in
$\left|1,-1\right\rangle$ to $\left|2,-2\right\rangle$. The Rb atoms are
sympathetically cooled. In order to minimize atom loss near the center of the
trap where the magnetic field goes to zero, we plug the trap with a repulsive
laser beam using 10 W of 532 nm light focused to a 45 $\mu$m waist.

The evaporation in the magnetic trap lasts for 20 s. Halfway through the
evaporation, a crossed optical dipole trap (XODT) is ramped up to 18 W. The
dipole trap is created using a single beam at 1064 nm folded into a
$90^{\circ}$ bowtie configuration, with the two intersecting arms orthogonally
polarized. The waist at the atoms is 105 $\mu$m. At the end of the evaporation
in the quadrupole trap, the magnetic field is ramped down and the atoms are
transferred to the optical trap. At this point, the atom number is $2.3\times
10^{6}$ ($6\times 10^{5}$) for Na (Rb) and the temperature of the mixture is
10 $\mu$K. We perform an optical evaporation by lowering the depth of the
optical trap exponentially to 340 mW over 4 s. A bias magnetic field of 8 G is
applied during the optical evaporation to suppress spin-changing collisions.
To avoid differential gravitational sag between the Na and Rb atoms, we turn
on an 80 mW 1064 nm light sheet beam (waists $w_{z}=14$ $\mu$m, $w_{r}=235$
$\mu$m) during the XODT evaporation.

At the end of the evaporation, we achieve a dual BEC with $2\times 10^{5}$
atoms of each species. We perform an RF Landau-Zener transfer of the atoms in
both species to the $\left|1,1\right\rangle$ state, the entrance channel for
the Feshbach resonance we use. Next, we increase the depth of the light sheet
to 2 W to reduce the size of the clouds and transfer the atoms into a single
layer of the vertical lattice (Fig. S1).

Figure S1:  Schematic of relevant optical potentials. The yellow beam depicts
the 1064 nm light sheet, and the red beams represent the vertical lattice
formed by a 16∘ intersection of two 1064 nm beams. Two overlapping dimple
beams with wavelengths 840 nm and 640 nm (shown in blue) help load a small,
reproducible number of Rb and Na atoms on every experimental cycle. The 2D
lattice at 1064 nm forms an in-plane sinusoidal potential.

##  2 Bichromatic dimple trap

The need to prepare gases with central densities on the order of one particle
per site, combined with the spacing of our 2D lattice and the transverse
confinement of the vertical lattice, determines the typical atom numbers we
need to work with. These turn out to be on the order of a few hundred atoms of
each species. Preparing such gases by evaporation in the XODT is unrealistic
because of its very weak confinement. Instead, we load a small fraction of the
atoms in the dual BEC into a bichromatic dimple trap, with the vertical
confinement still provided by the vertical lattice. The dimple trap is formed
by two overlapping beams at 840 nm and 640 nm, each focused through the
microscope objective to a waist of 8 $\mu$m. We use a 14 G/cm magnetic field
gradient to spill atoms from the reservoir. At the end of the evaporation we
lower the dimple depths to zero to load the atoms back into the vertical
lattice. By tuning the relative laser powers in the 840 nm versus 640 nm
dimples (both on the order of 100 $\mu$W), we can independently control the
number of Na and Rb atoms that remain after the evaporation. An example
histogram of the molecule number achieved in this way is shown in Fig. S2.

Figure S2:  Example of molecule number distribution achieved through
evaporation in the bichromatic dimple trap. Here, the average number of
molecules is 138 with a standard deviation of 28.

##  3 Molecule creation efficiency

The main limitation on our molecule creation efficiency is the probability to
have exactly one atom of each species on a lattice site. In addition, both
atoms have to be in the relative motional ground state of the site. Near zero
magnetic field, we can create Rb Mott insulators with $\sim 85\%$ filling (in
the absence of Na). For the mixture, the typical filling we achieve is lower
as reported in the main text. It is difficult to characterize the Na filling
in the lattice due to low signal-to-noise in absorption imaging. Paths for
potential improvement include: (1) increasing the confinement of the vertical
lattice to increase the intraspecies Hubbard interaction strengths, (2)
optimizing the magnetic field and 2D lattice ramps to reduce the temperature
of the mixture, and (3) increasing the overlap between the two species.

For sites that do end up with one atom of each species, the probability of
forming a molecule after the ramp through the Feshbach resonance is given by
the Landau-Zener formula $p=1-e^{-2\pi\delta_{\text{LZ}}}$, where
$\delta_{\text{LZ}}=\frac{hn_{2}}{\mu}\left|\frac{a_{\text{bg}}\Delta}{\dot{B}}\right|$
[42, 30]. Here, $\mu$ is the reduced mass of the two atoms,
$a_{\text{bg}}=76.33\,a_{0}$ is the interspecies background scattering length
[36], $\Delta=4.255$ G is the width of the resonance, $\dot{B}=2.5$ G/ms is
the field ramp rate, and $n_{2}=\int\int
n_{\text{Na}}(\textbf{r})n_{\text{Rb}}(\textbf{r})d\textbf{r}$ is the pair
density on a site. The wavefunctions of the non-interacting atoms in the
ground state of a site of a deep lattice can be approximated by the ground
state of an anisotropic harmonic oscillator with trap frequencies
$\omega_{r,\text{Na}}=2\pi\times 70.5$ kHz and
$\omega_{z,\text{Na}}=2\pi\times 2.4$ kHz
($\omega_{\text{Rb}}=0.86\,\omega_{\text{Na}}$), which gives $n_{2}=4.5\times
10^{19}$ m-3. We therefore expect $p\approx 1$ for our experimental
parameters.

Once we convert the atoms to molecules on these sites, we remove free atoms on
the remaining sites using the procedure described in the main text. The four
resonant optical removal pulses for each species are 50 $\mu$s in duration at
1 mW/cm2. The molecule lifetime in the presence of the Na optical removal
light is 1710(120) $\mu$s, and the lifetime is 5720(360) $\mu$s in the
presence of the Rb light. From these lifetimes, we estimate that the free atom
removal pulses lead to a loss of $14$% of the molecules.

##  4 Quantum Gas Microscopy

Here we describe in more detail the process for detecting the Rb atoms that
tag the positions of the molecules in the lattice. After breaking apart the
molecules and removing the residual Na atoms, the magnetic field is ramped to
zero in preparation for laser cooling. The 2D lattice is ramped to a depth of
6000 $E_{\text{r,Rb}}$ to suppress hopping of the Rb atoms during fluorescence
imaging. The light sheet potential is then pinned to a depth of 140 $\mu$K
after which the vertical lattice depth is ramped down to zero.

We image the Rb atoms in the lattice by applying optical molasses cooling
beams and collecting a portion of the scattered photons. We send two beams
each with a 500 $\mu$m waist containing 30 $\mu$W of cooling and 6 $\mu$W of
repump light through the side of the vacuum chamber, intersecting at a
$90^{\circ}$ angle in the horizontal plane to produce the requisite
polarization gradients. We find that by tilting one of the molasses beams at
$8^{\circ}$ with respect to the horizontal we are able to provide sufficient
cooling along the vertical direction. The cooling light is 30 MHz red-detuned
from the $F=2\rightarrow F^{\prime}=3$ free-space transition, and the repump
light is 42 MHz blue-detuned from the $F=1\rightarrow F^{\prime}=2$
transition. The molasses beams are retroreflected with mirrors mounted on
piezo chips oscillating at 300-400 Hz to smooth out interference patterns in
the cooling beams during the exposure time. We use a custom microscope
objective with NA$\,=0.5$ (Special Optics 54-25-25) mounted directly above the
vacuum chamber to collect $\sim 10^{4}$ photons/atom/second on a sCMOS camera
(Andor Zyla 4.2) with 30x magnification. The full-width at half-maximum of the
point spread function is 1010(15) nm (Fig. S3), which allows us to extract the
positions of the Rb atoms with single-site resolution using a reconstruction
algorithm [43].

We characterize the fidelity of our microscopy by measuring the fraction of Rb
atoms that hop between lattice sites or are lost during the 0.5 s exposure
time. We measure a hopping rate per exposure of $0.1\,(1)$% and a loss rate of
$1.7\,(3)$%. The latter is consistent with the background gas-limited lifetime
of Rb atoms in the optical trap.

Figure S3:  Point spread function with radial averaging (blue stars) and a
double Gaussian fit (red line).

##  5 Molecule Detection Fidelity

An important condition for measuring correlations between molecules is
ensuring that we are detecting all of the molecules (low false negative rate)
and that the detected particles are in fact molecules and not spurious Rb
atoms (low false positive rate). The procedure for measuring the false
negative rate is described in the main text. The false positive rate is
measured by forming a molecular gas, removing stray atoms, and then reducing
the magnetic field to zero without dissociating the molecules. We measure the
number of Rb atoms detected and compare it to an equivalent sequence in which
we dissociate the molecules before imaging. After 100 repetitions of each
sequence, we find that $0.34\,(5)$% of the Rb tag atoms in the molecule
detection sequence are also present in the sequence in which we do not
deliberately detect the molecules. The false positives are most likely due to
imperfect removal of residual Rb atoms following the molecule formation.

##  6 Light-assisted collisions

When multiple atoms are confined to the same site of a lattice, light-assisted
collisions can cause pairs of atoms to be ejected from the trap when
illuminated with resonant light [44, 45]. Since our molecule detection
procedure requires removing Na atoms with resonant $F=2\rightarrow
F^{\prime}=3$ light, it is important to determine if light-assisted collisions
during the optical pulse can also remove the Rb atoms which tag the molecule
positions. We estimate that the energy gained per photon from a light-assisted
collision is about fourteen orders of magnitude smaller for Na-Rb pairs than
it is for Rb-Rb pairs and is therefore negligible in our system.

The large difference in energy is due to the fact that for homonuclear light-
assisted collisions the atoms experience an attractive potential that scales
with the internuclear distance $R$ as $1/{R^{3}}$, while for heteronuclear
collisions the interaction potential scales as $1/{R^{6}}$. This means that a
heteronuclear atom pair gains significantly less energy during each scattering
event than for a homonuclear pair. Using known values for the $C_{3}$ and
$C_{6}$ coefficients [46, 47], we estimate the energy gained by the atoms
using a classical toy model in the center-of-mass frame with reduced mass
$\mu$. We assume that the atoms are initially at rest with a separation of
approximately 100 nm on a lattice site and that the time spent in the excited
state is given by $\tau=1/\Gamma$, where $\Gamma=2\pi\times 10$ MHz
($2\pi\times 6$ MHz) is the decay rate for Na (Rb). We then numerically
integrate Newton’s equation $d^{2}R/dt^{2}=-3C_{3}/\mu R^{4}$ for the case of
two Rb atoms, or $d^{2}R/dt^{2}=-6C_{6}/\mu R^{7}$ for the case of one Rb and
one Na atom, to determine the change in the internuclear separation during the
excited state lifetime and hence the energy gained per scattering event.

##  7 Molecule lifetimes

We measure the lifetime of the NaRb molecules as a function of lattice depth
and magnetic field. As shown in Fig. S4(a), performed at our typical 2D
lattice intensity of 12.5 kW/cm2, the molecule lifetime in the lattice is
sufficiently long at all magnetic fields studied that loss can be ignored for
the experiments presented in this paper. Following the removal of residual
unassociated atoms at 346.6 G, we ramp the magnetic field to the target value
and hold for a variable length of time. We then return the magnetic field to
346.6 G and repeat the removal of free atoms before dissociating and imaging
the surviving molecules (blue circles).

On the repulsive side of the Feshbach resonance, the molecule lifetime
initially decreases with decreasing magnetic field before saturating at $\sim
1.5$ s. This follows the magnetic field dependence of the closed channel
fraction of the Feshbach molecules. As has been previously observed, larger
closed channel fractions increase the wavefunction overlap with electronically
excited molecular states, enhancing molecule loss due to off-resonant
excitation by the lattice beams [48, 49]. Interestingly, we find that the
lifetime of the molecules also decreases above the Feshbach resonance. Since
the molecules are only weakly bound due to confinement effects above the
resonance [50, 34], we explore whether at these fields the molecule lifetime
is limited by a mechanism that breaks apart the molecules but does not eject
the remaining atoms from the trap, such as noise in the magnetic field. We
therefore repeat this measurement without the final removal of stray atoms
(red squares). The molecule lifetimes show the same behavior as before on the
repulsive side of the resonance, but on the attractive side we now see long
lifetimes consistent with background gas-limited atomic lifetimes. This
indicates that dissociation of the molecules does in fact occur in our system
when holding the weakly bound molecules above the Feshbach resonance.

To examine the role of the light scattering on these lifetimes, we also
measure the lifetimes at a fixed field of 335.1 G for five different lattice
intensities (Fig. S4(b)). We see that the lifetime increases for shallower
lattice depths, consistent with reduced off-resonant excitation of the
molecules by the lattice. For much shallower lattice depths (not shown),
molecule tunneling occurs and a single-body loss rate is no longer appropriate
due to inelastic collisions between the molecules.

Figure S4:  Molecule lifetime in an optical lattice. (a) Molecule lifetime
versus magnetic field for a lattice intensity of 12.5 kW/cm2. Blue circles
indicate the pure molecular lifetime where free atoms are removed both before
and after the hold time, while the red squares do not include the second free
atom removal. (b) Lifetime versus 2D lattice depth at 335.1 G.

##  8 Binding energy estimation

To obtain each point in the dissociation spectrum, we fix the microwave
frequency and scan the magnetic field by $\sim$ 130 mG, corresponding to a
$\sim\,300$ kHz Landau-Zener sweep. The molecular binding energies reported in
Fig. 3 of the main text are extracted by fitting the dissociation curves to a
phenomenological function $N(\omega)$, where $N$ is the number of Na atoms
transferred from the molecular state to the free $|2,2\rangle$ state:

$\displaystyle N(\omega)$
$\displaystyle=\frac{p_{1}}{f(\omega)}e^{-8\log(2)((x-p_{2})/f(\omega))^{2}}$
(S1) $\displaystyle f(\omega)$ $\displaystyle=\frac{2p_{3}}{1+e^{p_{4}(\omega-
p_{2})}}$

Here, $p_{i}$ are free fitting parameters. The binding energy is estimated as
the frequency at which the transfer $N(\omega)$ is half the peak of the fitted
lineshape.


# Haagerup property and Kazhdan pairs via ergodic infinite measure preserving
actions

Alexandre I. Danilenko  B. Verkin Institute for Low Temperature Physics &
Engineering of Ukrainian National Academy of Sciences, 47 Nauky Ave., Kharkiv,
61164, UKRAINE  alexandre.danilenkogmail.com

It is shown that a locally compact second countable group $G$ has the Haagerup
property if and only if there exists a sharply weak mixing 0-type measure
preserving free $G$-action $T=(T_{g})_{g\in G}$ on an infinite $\sigma$-finite
standard measure space $(X,\mu)$ admitting a $T$-Følner sequence (i.e. a
sequence $(A_{n})_{n=1}^{\infty}$ of measured subsets of finite measure such
that $A_{1}\subset A_{2}\subset\cdots$, $\bigcup_{n=1}^{\infty}A_{n}=X$ and
$\lim_{n\to\infty}\sup_{g\in K}\frac{\mu(T_{g}A_{n}\triangle
A_{n})}{\mu(A_{n})}=0$ for each compact $K\subset G$). A pair of groups
$H\subset G$ has property (T) if and only if there is a $\mu$-preserving
$G$-action $S$ on $X$ admitting an $S$-Følner sequence and such that
$S\restriction H$ is weakly mixing. These refine some recent results by
Delabie-Jolissaint-Zumbrunnen and Jolissaint.

## 0\. Introduction

Throughout this paper $G$ is a non-compact locally compact second countable
group. It has the Haagerup property if there is a weakly continuous unitary
representation $V$ of $G$ in a separable Hilbert space $\Cal{H}$ such that
$\lim_{g\to\infty}V(g)=0$ in the weak operator topology and

Of course, the amenable groups have the Haagerup property. The class of
discrete countable Haagerup groups contains the free groups and is closed
under free products and wreath products [CoStVa]. For more information about
the Haagerup property we refer to [Ch–Va]. There is a purely dynamical
description of this property: $G$ is Haagerup if and only if there exists a
mixing non-strongly ergodic probability preserving free $G$-action [Ch–Va,
Theorem~2.2.2] (see §1 for the definitions). Recently, an infinite measure
preserving counterpart of this result was discovered in [DeJoZu]:

###### Theorem A

$G$ has the Haagerup property if and only if there is a 0-type measure
preserving $G$-action $T=(T_{g})_{g\in G}$ on an infinite $\sigma$-finite
measure space $(X,\goth B,\mu)$ admitting a sequence of non-negative unit
vectors $(\xi_{n})_{n=1}^{\infty}$ in $L^{2}(X,\mu)$ such that
$\lim_{n\to\infty}\sup_{g\in K}\langle\xi_{n}\circ T_{g},\xi_{n}\rangle=1$ for
each compact $K\subset G$.

We recall that $T$ is called of 0-type if $\lim_{g\to\infty}\mu(T_{g}A\cap
B)=0$ for all subsets $A,B\in\goth B$ of finite measure. In this paper we
provide a much shorter alternative proof of Theorem A which is grounded on the
Moore-Hill concept of restricted infinite products of probability measures
[Hi].

We note that the 0-type for infinite measure preserving systems is a natural
counterpart of the mixing for probability preserving systems. However unlike
mixing, the 0-type is not a “strong” asymptotic property. It implies neither
weak mixing nor ergodicity. Moreover, the totally dissipative actions are all
of 0-type. In view of that the description in Theorem A does not look sharp
from the ergodic theory point of view. Our first main result in this work is
the following finer ergodic criterion of the Haagerup
property.111Conservativeness, ergodicity, weak mixing and sharp weak mixing
are not spectral invariants of the underlying dynamical systems. Hence the
principal difference of Theorem B from Theorem A is that it provides non-
spectral ergodic characterization of the Haagerup property.

###### Theorem B

The following are equivalent.

We say that $(A_{n})_{n=1}^{\infty}$ is $T$-Følner if $\mu(A_{n})<\infty$ and
$\sup_{g\in K}\frac{\mu(A_{n}\triangle T_{g}A_{n})}{\mu(A_{n})}\to 0$ as
$n\to\infty$ for each compact subset $K\subset G$. If $A_{1}\subset
A_{2}\subset\cdots$ and $\bigcup_{n=1}^{\infty}A_{n}=X$, we say that
$(A_{n})_{n=1}^{\infty}$ is exhausting. We note that sharp mixing (see §1 for
the definition) implies ergodicity and weak mixing. To prove (the non-trivial
part of) Theorem B, we apply the Moore-Hill construction [Hi] to the mixing
non-strongly ergodic $G$-action from [Ch–Va,  Theorem~2.2.2] (cf. the
construction of $II_{\infty}$ ergodic Poisson suspensions of countable
amenable groups from [DaKo]). Then we observe that the action $T$ that we
obtain is IDPFT (see §1 and [DaLe], where such actions were introduced).
Hence, by the properties of IDPFT systems, $T$ is sharply weak mixing whenever
we show that it is conservative. To show the conservativeness of $T$ is
remains to choose the parameters of the Moore-Hill construction in an a
appropriate way.

As a corollary from Theorem B, we obtain one more dynamical characterization
of the Haagerup property in terms of Poisson actions.

###### Corollary C

$G$ has the Haagerup property if and only if there exists a mixing
(probability preserving) Poisson $G$-action that is not strongly ergodic.

Our next purpose is to obtain a “parallel” characterization of property (T)
which is a reciprocal to the Haagerup property. We recall [Jo1,  Definition
1.1] that given a non-compact closed subgroup $H$ of $G$, the pair $H\subset
G$ has property (T) if for each unitary representation $V$ of $G$ satisfying
$(\ast)$, there is a unit vector which is invariant under $V(h)$ for every
$h\in H$. Using the techniques developed for proving Theorem B we obtain an
ergodic (non-spectral) characterization of Kazhdan pairs that refines a
spectral characterization from [Jo2].

###### Theorem D

Let us say that $S\restriction H$ is of weak 0-type if there is a subsequence
$h_{n}\to\infty$ in $H$ such that $\lim_{n\to\infty}\nu(S_{h_{n}}A\cap B)=0$
for all subsets $A,B\in\goth C$ of finite measure. Then replacing the “has no
invariant subsets of positive finite measure” in (i) with a stronger “is of
weak 0-type”, and the “weakly mixing” in (ii) with a weaker “of weak 0-type”
we obtain exactly [Jo2,  Theorem~1.5].

###### Corollary E

A pair $H\subset G$ has property (T) if and only if every (probability
preserving) Poisson $G$-action with weakly mixing $H$-subaction is strongly
ergodic. The same is also true with “ergodic” in place of “weakly mixing”.

The outline of the paper is as follows. In Section 1 we state all necessary
definitions related to the basic dynamical concepts of group actions both in
the nonsingular and and finite measure preserving cases, restricted infinite
powers of probability measures, IDPFT actions and Poisson actions. In Section
2 we prove Theorems B and Corollary C. Section 3 is devoted to the proof of
Theorems D and Corollary E.

## 1\. Definitions and preliminaries

### Nonsingular and measure preserving $G$-actions

Nonsingular actions appear in the proof of Theorem B. We remind several basic
concepts related to them.

###### Definition Definition 1.1

Let $S=(S_{g})_{g\in G}$ be a nonsingular $G$-action on a standard probability
space $(Z,\goth F,\kappa)$.

We also remind some concepts related to finite measure preserving actions.

###### Definition Definition 1.2

Suppose that $\kappa(Z)=1$ and $\kappa\circ S_{g}=\kappa$ for all $g\in G$.

We now state a corollary from the Schmidt-Walters theorem [ScWa,
Theorem~2.3].

###### Lemma 1.3

Let $S=(S_{g})_{g\in G}$ be a mixing measure preserving action on a standard
probability space $(Y,\goth C,\nu)$. Then for each ergodic non-totally
dissipative nonsingular $G$-action $R=(R_{g})_{g\in G}$, the product
$G$-action $S\times R:=(S_{g}\times R_{g})_{g\in G}$ is ergodic.

###### Demonstration Proof

We first note that a mixing action is properly ergodic. Hence if
$R=(R_{g})_{g\in G}$ is properly ergodic then the claim of the proposition
follows immediately from [ScWa,  Theorem~2.3]. If $R$ is not properly ergodic
then there is a noncompact subgroup $H$ in $G$ such that $R$ is isomorphic to
the $G$-action by left translations on the coset space $G/H$ endowed with a
Haar measure. Hence $S\times R$ is ergodic if and only if the $H$-action
$(S(h))_{h\in H}$ on $(Y,\goth C,\nu)$ is ergodic. The later holds because $S$
is mixing. ∎

###### Corollary 1.4

Let $S=(S_{g})_{g\in G}$ be a mixing measure preserving action on a standard
probability space $(Y,\goth C,\nu)$ and let $R=(R_{g})_{g\in G}$ be a
nonsingular $G$-action on a standard probability space $(Z,\goth D,\kappa)$.
The following holds.

###### Demonstration Proof

It suffices to consider the ergodic decomposition for $R$ and apply Lemma 1.3
to the product of $S$ with each ergodic component of $R$. ∎

### Restricted infinite powers of probability measures

Let $(Y,\goth C,\gamma)$ be a standard non-atomic probability space. Fix a
sequence $\boldkey{B}:=(B_{n})_{n=1}^{\infty}$ of subsets from $\goth C$ of
strictly positive measure. Let $(X,\goth B):=(Y,\goth C)^{\otimes\mathbb{N}}$.
For each $n\in\mathbb{N}$, we set $\boldkey{B}^{n}:=Y^{n}\times B_{n+1}\times
B_{n+2}\times\cdots\in\goth B.$ Then
$\boldkey{B}^{1}\subset\boldkey{B}^{2}\subset\cdots$. We define a measure
$\gamma^{\boldkey{B}}$ on $(X,\goth B)$ by the following sequence of
restrictions (see [Hi] for details):

$\gamma^{\boldkey{B}}\restriction\boldkey{B}^{n}:=\frac{\gamma}{\gamma(B_{1})}\otimes\cdots\otimes\frac{\gamma}{\gamma(B_{n})}\otimes\frac{\gamma\restriction
B_{n+1}}{\gamma(B_{n+1})}\otimes\frac{\gamma\restriction
B_{n+2}}{\gamma(B_{n+2})}\otimes\cdots.$

Since the restrictions are compatible, $\gamma^{\boldkey{B}}$ is well defined.
We note that $\gamma^{\boldkey{B}}$ is supported on the subset
$\bigcup_{n=1}^{\infty}\boldkey{B}^{n}\subset Y$ and
$\gamma^{\boldkey{B}}(\boldkey{B}^{n})=\prod_{j=1}^{n}\gamma(B_{j})^{-1}$ for
each $n$. Hence, $\gamma^{\boldkey{B}}$ is $\sigma$-finite. It is infinite if
and only if $\prod_{n=1}^{\infty}\gamma(B_{n})=0$.

###### Definition Definition 1.5

We call $\gamma^{\boldkey{B}}$ the restricted infinite power of $\gamma$ with
respect to $\boldkey{B}$.

Given a $\gamma$-preserving Borel bijection $T$ of $Y$, we let
$\boldkey{T}:=\bigotimes_{n=1}^{\infty}T$ and
$\boldkey{T}\boldkey{B}:=(TB_{n})_{n=1}^{\infty}$. A straightforward
verification shows that $\gamma^{\boldkey{B}}\circ
T^{-1}=\gamma^{\boldkey{T}\boldkey{B}}$.

###### Proposition 1.6

If $\sum_{n=1}^{\infty}\frac{\gamma(B_{n}\triangle
TB_{n})}{\gamma(B_{n})}<\infty$ then $\boldkey{T}$ preserves
$\gamma^{\boldkey{B}}$.

###### Demonstration Proof

For each $n\in\mathbb{N}$ and an arbitrary Borel subset $A\subset Y^{n}$, we
let $A^{\prime}:=A\times B_{n+1}\times B_{n+2}\times\cdots\in\goth B$. Then
for every $m>n$,

$(\gamma^{\boldkey{T}\boldkey{B}}\restriction\boldkey{B}^{m})(A^{\prime})=\frac{\gamma^{\otimes
n}(A)}{\gamma(B_{1})\cdots\gamma(B_{n})}\prod_{j>m}\frac{\gamma(B_{j}\cap
TB_{j})}{\gamma(B_{j})}.$

Passing to the limit as $m\to\infty$, we obtain that

$\gamma^{\boldkey{T}\boldkey{B}}(A^{\prime})=\lim_{m\to\infty}(\gamma^{\boldkey{B}}\restriction\boldkey{B}^{m})(TA^{\prime})=\frac{\gamma^{\otimes
n}(A)}{\gamma(B_{1})\cdots\gamma(B_{n})}=\gamma^{\boldkey{B}}(A^{\prime}).$

Hence $\gamma^{\boldkey{B}}\circ
T^{-1}=\gamma^{\boldkey{T}\boldkey{B}}=\gamma^{\boldkey{B}}$, as desired. ∎

We note that under the condition of Proposition 1.6,

$\gamma^{\boldkey{B}}(\boldkey{T}\boldkey{B}^{n}\cap\boldkey{B}^{n})={\prod_{j=1}^{n}\gamma(B_{j})^{-1}}\prod_{j>n}\frac{\gamma(B_{j}\cap
TB_{j})}{\gamma(B_{j})}.$

Hence

$\lim_{n\to\infty}\frac{\gamma^{\boldkey{B}}(\boldkey{T}\boldkey{B}^{n}\cap\boldkey{B}^{n})}{\gamma^{\boldkey{B}}(\boldkey{B}^{n})}=1.$
$None$

We note that the Hilbert space $L^{2}(X,\gamma^{\boldkey{B}})$ is the infinite
tensor product of the sequence of Hilbert spaces
$(L^{2}(Y,\gamma(B_{n})^{-1}\gamma))_{n=1}^{\infty}$ along the stabilizing
sequence $(1_{B_{n}})_{n=1}^{\infty}$ of unit vectors (see [Gu] for the
definition). In particular, we see that the linear subspace

$\bigcup_{n=1}^{\infty}\Bigg{\\{}f\otimes\bigotimes_{j>n}1_{B_{j}}\,\bigg{|}\,f\in
L^{2}(Y^{n},\gamma^{\otimes n})\Bigg{\\}}$

is dense in $L^{2}(X,\gamma^{\boldkey{B}})$. Let $U_{T}$ and $U_{\boldkey{T}}$
stand for the Koopman operators associated to $T$ and $\boldkey{T}$ in
$L^{2}(Y,\gamma)$ and $L^{2}(X,\gamma^{\boldkey{B}})$ respectively. The
following proposition is verified straightforwardly.

###### Lemma 1.7

For each $n\in\mathbb{N}$ and arbitrary functions $f,g\in
L^{2}(Y^{n},\gamma^{\otimes n})$, we have that

$\begin{gathered}U_{\boldkey{T}}\bigg{(}f\otimes\bigotimes_{j>n}1_{B_{j}}\bigg{)}=\lim_{m\to\infty}(U_{T})^{\otimes
n}f\otimes\bigg{(}\bigotimes_{j=n+1}^{m}1_{TB_{j}}\bigg{)}\otimes\bigotimes_{j>m}1_{B_{j}}\qquad\text{and}\\\
\bigg{\langle}U_{\boldkey{T}}\bigg{(}f\otimes\bigotimes_{j>n}1_{B_{j}}\bigg{)},g\otimes\bigotimes_{j>n}1_{B_{j}}\bigg{\rangle}=\frac{\langle(U_{T})^{\otimes
n}f,g\rangle}{\prod_{j=1}^{n}\gamma(B_{j})}\prod_{j>n}\frac{\gamma(TB_{j}\cap
B_{j})}{\gamma(B_{j})}.\end{gathered}$

### IDPFT-actions

IDPFT-actions were introduced in [DaLe] in the case, where $G=\mathbb{Z}$. In
[DaKo], IDPFT actions of arbitrary discrete countable groups were studied. In
this paper we consider IDPFT-actions for arbitrary locally compact second
countable groups.

###### Definition Definition 1.8

Let $T_{n}=(T_{n}(g))_{g\in G}$ be an ergodic measure preserving $G$-action on
a standard probability space $(Y_{n},\goth C_{n},\nu_{n})$, let $\mu_{n}$ be a
probability measure on $\goth C_{n}$ and let $\mu_{n}\sim\nu_{n}$ for each
$n\in\mathbb{N}$. We put $(X,\goth
B,\mu):=\bigotimes_{n=1}^{\infty}(Y_{n},\goth C_{n},\mu_{n})$,
$T(g):=\bigotimes_{n=1}^{\infty}T_{n}(g)$ and $T:=(T(g))_{g\in G}$. If
$\mu\circ T(g)\sim\mu$ for each $g\in G$ then the nonsingular dynamical system
$(X,\goth B,\mu,T)$ is called an infinite direct product of finite types
(IDPFT).

We will need the following fact, extending [DaLe,  Proposition~2.3] from
$\mathbb{Z}$-actions to arbitrary $G$-actions.

###### Proposition 1.9

Let $(X,\goth B,\mu,T)$ be an IDPFT system as in Definition 1.8. If $T_{n}$ is
mixing for each $n\in\mathbb{N}$ then $T$ is either totally dissipative or
conservative. If $T$ is conservative then $T$ is sharply weak mixing.

###### Demonstration Proof

It follows from Corollary 1.4(i) that

$\Cal{D}(T)=Y_{1}\times\cdots\times
Y_{n}\times\Cal{D}\bigg{(}\bigotimes_{j>n}T_{j}\bigg{)}\quad\mod 0$

for each $n>0$. By Kolmogorov’s 0-1 law that either $\mu(\Cal{D}(T))=0$ and
hence $T$ is conservative or $\mu(\Cal{D}(T))=1$ and hence $T$ is totally
dissipative. Thus, the first claim is proved. We do not provide a proof for
the second claim because it is an almost verbal repetition of the proof of
[DaLe,  Proposition~2.3]: just replace the reference to [DaLe,  Theorem~B]
there with a reference to Corollary 1.4(ii). Of course, $\mu$ is not
concentrated on a single orbit. ∎

### Poisson suspensions rm (see [CoFoSi] and [Ro] for details)

Let $(X,\goth B)$ be a standard Borel space and let $\mu$ be an infinite
$\sigma$-finite non-atomic measure on $X$. Let $X^{*}$ be the set of purely
atomic ($\sigma$-finite) measures on $X$. For each subset $A\in\goth B$ with
$0<\mu(A)<\infty$, we define a mapping $N_{A}:X^{*}\to\mathbb{R}$ by setting
$N_{A}(\omega):=\omega(A)$. Let $\goth B^{*}$ stand for the smallest
$\sigma$-algebra on $X^{*}$ such that the mappings $N_{A}$ are all $\goth
B^{*}$-measurable. There is a unique probability measure $\mu^{*}$ on
$(X^{*},\goth B^{*})$ satisfying the following two conditions:

Then $(X^{*},\goth B^{*},\mu^{*})$ is a Lebesgue space. For each
$\mu$-preserving $G$-action $T=(T_{g})_{g\in G}$, we define a $G$-action
$T^{*}=(T_{g}^{*})_{g\in G}$ on $(X^{*},\goth B^{*},\mu^{*})$ by setting

$T_{g}^{*}\omega:=\omega\circ T_{g}^{-1}\qquad\text{for all $\omega\in X^{*}$
and $g\in G$.}$

Then $T^{*}$ preserves $\mu^{*}$.

###### Definition Definition 1.10

The dynamical system $(X^{*},\goth B^{*},\mu^{*},T^{*})$ is called the Poisson
suspension of $(X,\goth B,\mu,T)$. A probability preserving $G$-action is
called Poisson if it is isomorphic to a Poisson suspension of some infinite
$\sigma$-finite measure preserving $G$-action.

### Unitary representations of $G$ and Koopman representations of measure
preserving actions

Let $V=(V(g))_{g\in G}$ be a weakly continuous unitary representation of $G$
in a separable Hilbert space $\Cal{H}$. We will always assume that $V$ is a
complexification of an orthogonal representation of $G$ in a real Hilbert
space.

###### Definition Definition 1.11

$V$ is called weakly mixing if $V$ has no nontrivial finite dimensional
invariant subspaces.

The Fock space $\Cal{F}(\Cal{H})$ over $\Cal{H}$ is the orthogonal sum
$\bigoplus_{n=0}^{\infty}\Cal{H}^{\odot n}$, where $\Cal{H}^{\odot n}$ is the
$n$-th symmetric tensor power of $\Cal{H}$ when $n>0$ and $\Cal{H}^{\odot
0}:=\mathbb{C}$. By $\exp V=(\exp V(g))_{g\in G}$ we denote the corresponding
unitary representation of $G$ in $\Cal{F}(\Cal{H})$, i.e. $\exp
V(g):=\bigoplus_{n=0}^{\infty}V(g)^{\odot n}$ for each $g\in G$.

Let $T=(T_{g})_{g\in G}$ be a measure preserving $G$-action on a
$\sigma$-finite nonatomic standard measure space $(X,\goth B,\mu)$. Denote by
$U_{T}=(U_{T}(g))_{g\in G}$ the associated (weakly continuous) unitary Koopman
representation of $G$ in $L^{2}(X,\mu)$:

$U_{T}(g)f:=f\circ T_{g}^{-1},\quad\text{for all }g\in G.$

Let $L^{2}_{0}(X,\mu):=L^{2}(X,\mu)\ominus\mathbb{C}=\\{f\in
L^{2}(X,\mu)\mid\int_{X}fd\mu=0\\}$. We will need the following fact.

###### Fact 1.12

Let $V$ and $(X,\goth B,\mu,T)$ be as above in this subsesction.

## 2\. The Haagerup property

In this section we prove Theorem B (and hence Theorem A) and Corollary C.
Prior to this we state without proof a folklore proposition.

###### Proposition 2.1

Let $S=(S_{g})_{g\in G}$ be a measure preserving $G$-action on an infinite
$\sigma$-finite standard measure space $(Z,\goth Z,\kappa)$. Let $\goth
Z_{0}\subset\goth Z$ stand for the ring of subsets of finite measure. Let
$Z_{1}\subset Z_{2}\subset\cdots$ be a sequence of subsets from $\goth Z_{0}$
such that $\bigcup_{n=1}^{\infty}Z_{n}=Z$ and for each $n>0$, there is a
finite partition $\Cal{P}_{n}$ of $Z_{n}$ into subsets of equal measure
satisfying the following:

###### Demonstration Proof of Theorem B

The implications (ii)$\Rightarrow$(i) and (iii)$\Rightarrow$(i) are trivial.

We now prove (i)$\Rightarrow$(ii). Let $G$ have the Haagerup property. By
[Ch–Va,  Theorem~2.2.2], there is a mixing measure preserving free $G$-action
$T=(T_{g})_{g\in G}$ on a standard probability space $(Y,\goth C,\gamma)$ and
a $T$-asymptotically invariant sequence $\boldkey{B}:=(B_{n})_{n=1}^{\infty}$
such that $\gamma(B_{n})=0.5$ for each $n\in\mathbb{N}$. Passing to a
subsequence, if necessary, we may (and will) assume that for each compact
subset $K\subset G$,

$\sum_{n=1}^{\infty}\sup_{g\in K}\gamma(T_{g}B_{n}\triangle B_{n})<+\infty.$
$None$

We now let $(X,\goth B):=(Y,\goth C)^{\otimes\mathbb{N}}$. Endow this standard
Borel space with the ($\sigma$-finite) restricted infinite power
$\gamma^{\boldkey{B}}$ of $\gamma$ with respect to $\boldkey{B}$. Since
$\prod_{n\in\mathbb{N}}\gamma(B_{n})=0$, it follows that
$\gamma^{\boldkey{B}}(X)=\infty$. For $g\in G$, let
$\boldkey{T}_{g}:=T_{g}^{\otimes\mathbb{N}}$. In view of 2-1, it follows from
Proposition 1.6 that
$\gamma^{\boldkey{B}}\circ\boldkey{T}_{g}=\gamma^{\boldkey{B}}$. Thus,
$\boldkey{T}:=(\boldkey{T}_{g})_{g\in G}$ is a measure preserving $G$-action
on $(X,\goth B,\gamma^{\boldkey{B}})$. Since the map $X\ni
x=(y_{n})_{n=1}^{\infty}\to y_{1}\in Y$ intertwines $\boldkey{T}$ with $T$ and
$T$ is free, $\boldkey{T}$ is free too. For each $n\in\mathbb{N}$, define a
subset $\boldkey{B}^{n}\subset X$ is the same way as in §1. It follows from
1-1 that the sequence $(\boldkey{B}^{n})_{n\in\mathbb{N}}$ is
$\boldkey{T}$-Følner. Moreover, it is exhausting and
$\gamma^{\boldkey{B}}(\boldkey{B}^{n})=2^{n}$ for each $n$. To show that
$\boldkey{T}$ is of 0-type, we first note that since $T$ is mixing then for
each pair of integers $n<m$,

$\lim_{g\to\infty}\prod_{j=n}^{m}\frac{\gamma(T_{g}B_{j}\cap
B_{j})}{\gamma(B_{j})}=\prod_{j=n}^{m}\gamma(B_{j})=2^{-m+n-1}.$

Hence given two functions $f,r\in L^{2}(Y^{n},\gamma^{\otimes n})$, we deduce
from Lemma 1.7 that

$\lim_{g\to\infty}\bigg{\langle}U_{\boldkey{T}}\bigg{(}f\otimes\bigotimes_{j>n}1_{B_{j}}\bigg{)},g\otimes\bigotimes_{j>n}1_{B_{j}}\bigg{\rangle}=0.$

This implies that $\boldkey{T}$ is of 0-type.222It is worthy to note that at
this point we have proved completely Theorem A.

Our next claim is that upon a replacement of $\gamma^{\boldkey{B}}$ with an
equivalent probability measure, $\boldkey{T}$ is an IDPFT. For that, we choose
a sequence of reals $(\epsilon_{n})_{n=1}^{\infty}$ such that
$0<\epsilon_{n}<1$ for each $n$ and $\sum_{n=1}^{\infty}\epsilon_{n}<\infty$.
Then we define, for each $n\in\mathbb{N}$, a Borel function
$\phi_{n}:Y\to\mathbb{R}_{+}$ by setting

$\phi_{n}:=2\epsilon_{n}1_{X\setminus B_{n}}+2(1-\epsilon_{n})1_{B_{n}}.$

Since $\gamma(B_{n})=\frac{1}{2}$, a simple verification yields that
$\int_{Y}\phi_{n}d\gamma=1$. Denote by $\mu_{n}$ the probability measure on
$Y$ such that $\mu_{n}\sim\gamma$ and $\frac{d\mu_{n}}{d\gamma}:=\phi_{n}$. We
now recall that given two probability measures $\alpha$ and $\beta$ on
$(Y,\goth C)$ such that $\alpha\prec\gamma$ and $\beta\prec\gamma$, the
squared Hellinger distance between $\alpha$ and $\beta$ is

$H^{2}(\alpha,\beta):=\frac{1}{2}\int_{Y}\left(\sqrt{\frac{d\alpha}{d\gamma}}-\sqrt{\frac{d\beta}{d\gamma}}\right)^{2}d\gamma.$

A straightforward computation yields that

$H^{2}\left(\frac{1}{\gamma(B_{n})}\gamma\restriction
B_{n},\mu_{n}\right)=\frac{1}{2}\int_{Y}(\sqrt{2}\cdot
1_{B_{n}}-\sqrt{\phi_{n}})^{2}d\gamma=\frac{1}{2\sqrt{2}}((\sqrt{1-\epsilon_{n}}-1)^{2}+\epsilon_{n})$

and hence
$\sum_{n=1}^{\infty}H^{2}(\frac{1}{\gamma(B_{n})}\,\gamma\restriction
B_{n},\mu_{n})\leq+\infty.$ Therefore, by [Hi,  Theorems 3.9,  3.6],
$\gamma^{\boldkey{B}}\sim\bigotimes_{n=1}^{\infty}\mu_{n}$. Hence
$\boldkey{T}$ is an IDPFT, as claimed.

We now deduce from Proposition 1.9 that $\boldkey{T}$ is either sharply weak
mixing or totally dissipative. Therefore, to complete the proof of
(i)$\Rightarrow$(ii), it suffices to show that there exists a subsequence of
$\boldkey{B}$ such that the corresponding $G$-action $\boldkey{T}$ (determined
by this subsequence of $\boldkey{B}$) is conservative. Let
$(\Cal{P}_{n})_{n=1}^{\infty}$ be a refining sequence of finite partitions of
$Y$ into Borel subsets of equal measure such that
$\bigvee_{n=1}^{\infty}\Cal{P}_{n}$ is dense in $(\goth C,\gamma)$ and $B_{n}$
is $\Cal{P}_{n}$-measurable for each $n\in\mathbb{N}$. We set
$\Cal{P}_{n}^{\prime}:=\\{P\times B_{n+1}\times B_{n+2}\times\cdots\mid
P\in(\Cal{P}_{n})^{\otimes n}\\}$. Then $\Cal{P}_{n}^{\prime}$ is a finite
partition of $\boldkey{B}^{n}$ into subsets of equal measure for each
$n\in\mathbb{N}$. Moreover,
$\Cal{P}_{1}^{\prime}\prec\Cal{P}_{2}^{\prime}\prec\cdots$ and
$\bigvee_{n=1}^{\infty}\Cal{P}_{n}^{\prime}$ is dense in $(\goth
B,\gamma^{\boldkey{B}})$. Fix a sequence $(K_{n})_{n=1}^{\infty}$ of compact
subsets of $G$ such that $K_{1}\subset K_{2}\subset\cdots$ and
$\bigcup_{n=1}^{\infty}K_{n}=G$. Then for every $n>0$, there is a finite
subset $F_{n}\subset G\setminus K_{n}$ such that for each atom
$P\in(\Cal{P}_{n})^{\otimes n}$ there is a family of measured subsets
$(P_{f})_{f\in F_{n}}$ of $P$ satisfying the following conditions:

Passing to a subsequence in $\boldkey{B}=(B_{n})_{n=1}^{\infty}$ we may assume
without loss of generality that $\max_{f\in
F_{n}}\prod_{k>n}\frac{\gamma(T_{f}B_{k}\cap B_{k})}{\gamma(B_{k})}>0.5$ for
each $n$. Take an atom $P^{\prime}\in\Cal{P}_{n}^{\prime}$ for some
$n\in\mathbb{N}$. Then $P^{\prime}=P\times B_{n+1}\times B_{n+2}\times\cdots$
for an atom $P\in(\Cal{P}_{n})^{\otimes n}$. We now set
$P_{f}^{\prime}:=P_{f}\times(B_{n+1}\cap T_{f}^{-1}B_{n+1})\times(B_{n+2}\cap
T_{f}^{-1}B_{n+2})\times\cdots$ for each $f\in F_{n}$. Then
$(P_{f}^{\prime})_{f\in F_{n}}$ are mutually disjoint Borel subsets of
$P^{\prime}$ and $(\boldkey{T}_{f}P_{f}^{\prime})_{f\in F_{n}}$ are also
mutually disjoint Borel subsets of $P^{\prime}$. Moreover,

$\displaystyle\gamma^{\lx@ams@boldsymbol@{B}}\bigg{(}\bigsqcup_{f\in
F_{n}}P_{f}^{\prime}\bigg{)}$ $\displaystyle=\sum_{f\in
F_{n}}\frac{\gamma^{\otimes
n}(P_{f})}{\prod_{j=1}^{n}\gamma(B_{j})}\prod_{k>n}\frac{\gamma(T_{f}B_{k}\cap
B_{k})}{\gamma(B_{k})}$ $None$ $\displaystyle>\frac{1}{2}\sum_{f\in
F_{n}}\frac{\gamma^{\otimes n}(P_{f})}{\prod_{j=1}^{n}\gamma(B_{j})}$
$\displaystyle=\frac{\gamma^{\otimes n}(\bigsqcup_{f\in
F_{n}}P_{f})}{2\prod_{j=1}^{n}\gamma(B_{j})}$
$\displaystyle>\frac{\gamma^{\lx@ams@boldsymbol@{B}}(P)}{4}.$

It follows from this and Proposition 2.1(i) that $\lx@ams@boldsymbol@{T}$ is
conservative, as desired.

To prove (1)$\Rightarrow$(3), consider the dynamical system $(X,\goth
B,\gamma^{\boldkey{B}},\boldkey{T})$ constructed above. Since $T$ is not
strongly ergodic, we have that for each $\alpha\in(0,1)$, there exists a
$T$-asymptotically invariant sequence $(A_{n})_{n=1}^{\infty}$ such that
$\gamma(A_{n})=\alpha$ for each $n\in\mathbb{N}$. Hence, using the standard
diagonalization argument, we can select an asymptotically invariant sequence
$(A_{n})_{n=1}^{\infty}$ such that
$\gamma(A_{n})=\prod_{j=1}^{n}\gamma(B_{j})$ for each $n\in\mathbb{N}$. We now
set $\boldkey{A}^{n}:=Y^{n-1}\times A_{n}\times B_{n+1}\times
B_{n+2}\times\cdots\subset X.$ Then

$\displaystyle\gamma^{\boldkey{B}}(\boldkey{A}^{n})$
$\displaystyle=\frac{\gamma(A_{n})}{\prod_{j=1}^{n}\gamma(B_{j})}=1\qquad\text{
and}$
$\displaystyle\gamma^{\boldkey{B}}(\boldkey{T}_{g}\boldkey{A}^{n}\cap\boldkey{A}^{n})$
$\displaystyle=\Bigg{(}\,{\prod_{j=1}^{n-1}\gamma(B_{j})^{-1}}\Bigg{)}\frac{\gamma(A_{n}\cap
T_{g}A_{n})}{\gamma(B_{n})}\prod_{j>n}\frac{\gamma(B_{j}\cap
T_{g}B_{j})}{\gamma(B_{j})}.$

Hence for a compact subset $K\subset G$, we obtain that $\sup_{g\in
K}\gamma^{\boldkey{B}}(\boldkey{T}_{g}\boldkey{A}^{n}\cap\boldkey{A}^{n})\to
1$ as $n\to\infty$, as desired. ∎

###### Demonstration Proof of Corollary C

The “if” part follows from [Ch–Va,  Theorem~2.2.2]. We prove the “only if”
part. Let $G$ has the Haagerup property. Consider the dynamical system
$(X,\goth B,\gamma^{\boldkey{B}},\boldkey{T})$ constructed in the proof of
Theorem B. Let $(\boldkey{A}^{n})_{n=1}^{\infty}$ be the $\boldkey{T}$-Følner
sequence such that $\gamma^{\boldkey{B}}(\boldkey{A}^{n})=1$ for each
$n\in\mathbb{N}$. Denote by
$(X^{*},(\gamma^{\boldkey{B}})^{*},\boldkey{T}^{*})$ the Poisson suspension of
$(X,\gamma^{\boldkey{B}},\boldkey{T})$. For each $n\in\mathbb{N}$, we set
$[\boldkey{A}^{n}]_{0}:=\\{\omega\in X^{*}\mid\omega(\boldkey{A}^{n})=0\\}$.
Then

$(\gamma^{\boldkey{B}})^{*}([\boldkey{A}^{n}]_{0})=e^{-\gamma^{\boldkey{B}}(\boldkey{A}^{n})}=e^{-1}$

and

$\displaystyle(\gamma^{\boldkey{B}})^{*}(\boldkey{T}^{*}_{g}[\boldkey{A}^{n}]_{0}\cap[\boldkey{A}^{n}]_{0})$
$\displaystyle=(\gamma^{\boldkey{B}})^{*}([\boldkey{T}_{g}\boldkey{A}^{n}]_{0}\cap[\boldkey{A}^{n}]_{0})$
$\displaystyle=(\gamma^{\boldkey{B}})^{*}([\boldkey{T}_{g}\boldkey{A}^{n}\cup\boldkey{A}^{n}]_{0})$
$\displaystyle=e^{-\gamma^{\boldkey{B}}(\boldkey{T}_{g}\boldkey{A}^{n}\cup\boldkey{A}^{n})}.$

Since for each compact subset $K\subset G$,

$\sup_{g\in
K}|\gamma^{\boldkey{B}}(\boldkey{T}_{g}\boldkey{A}^{n}\cup\boldkey{A}^{n})-\gamma^{\boldkey{B}}(\boldkey{A}^{n})|\to
0,$

we obtain that

$\sup_{g\in
K}|(\gamma^{\boldkey{B}})^{*}\boldkey{T}^{*}_{g}[\boldkey{A}^{n}]_{0}\cap[\boldkey{A}^{n}]_{0})-(\gamma^{\boldkey{B}})^{*}([\boldkey{A}^{n}]_{0})|\to
0$

as $n\to\infty$. Thus, the sequence $([\boldkey{A}^{n}]_{0})_{n=1}^{\infty}$
is nontrivial and $\boldkey{T}^{*}$-asymptotically invariant. Thus
$\boldkey{T}^{*}$ is not strongly ergodic. Since $\boldkey{T}$ is of 0-type,
it follows from Fact 1.12(iii) that $U_{\boldkey{T}^{*}}(g)\restriction
L^{2}_{0}\big{(}X^{*},(\gamma^{\boldkey{B}})^{*}\big{)}\to 0$ weakly as
$g\to\infty$. Hence $\boldkey{T}^{*}$ is mixing. ∎

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some $\beta\in G_{0}$, since $G$ is flag-transitive on $\mathcal{D}$, but this
contradicts our assumption.
Assume that $B\subseteq\left\langle u\right\rangle_{GF(p)}$ for some non-zero
vector $u$ of $V$. Then $G_{B}\leqslant G_{\left\langle u\right\rangle}$ and
hence $S_{3}\cong(G_{B})^{\left\langle u\right\rangle}\leqslant
G_{\left\langle u\right\rangle}^{\left\langle u\right\rangle}\cong
AGL_{1}(p)$, since $G$ acts $2$-transitively on $V$. Thus $p=3$,
$B=\left\langle u\right\rangle_{GF(3)}$ and hence the assertion follows. ∎

Now, Theorem 7.1 follows from Proposition 7.3 and Theorems 7.4 and 7.5.
Finally, Theorem 6.1 follows from Theorem 7.1.

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