Journal ArticleDOI

# Interspecies thermalization in an ultracold mixture of Cs and Yb in an optical trap

14 Jul 2017-Physical Review A (American Physical Society)-Vol. 96, Iss: 1, pp 012704

Abstract: We present measurements of interspecies thermalization between ultracold samples of 133Cs and either 174Yb or 170Yb. The two species are trapped in a far-off-resonance optical dipole trap and 133Cs is sympathetically cooled by Yb. We extract effective interspecies thermalization cross sections by fitting the thermalization measurements to a kinetic model, giving σCs174Yb = (5 ± 2) × 10−13 cm2 and σCs170Yb = (18 ± 8) × 10−13 cm2. We perform quantum scattering calculations of the thermalization cross sections and optimize the CsYb interaction potential to reproduce the measurements. We predict scattering lengths for all isotopic combinations of Cs and Yb. We also demonstrate the independent production of 174Yb and 133Cs Bose-Einstein condensates using the same optical dipole trap, an important step toward the realization of a quantum-degenerate mixture of the two species.

### Introduction

• Durham Research Online Deposited in DRO: 17 July 2017 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Guttridge, A. and Hopkins, S.A. and Kemp, S.L. and Frye, M.D. and Hutson, J.M. and Cornish, S.L. (2017) 'Interspecies thermalization in an ultracold mixture of Cs and Yb in an optical trap.', Physical review A., 96 (1). 012704.
• Further information on publisher's website: https://doi.org/10.1103/PhysRevA.96.012704.
• The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.
• Interspecies thermalization in an ultracold mixture of Cs and Yb in an optical trap A. Guttridge,1,* S. A. Hopkins,1 S. L. Kemp,1 Matthew D. Frye,2,† Jeremy M. Hutson,2 and Simon L. Cornish1,‡ 1Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham University, South Road, Durham, DH1 3LE, United Kingdom 2Joint Quantum Centre (JQC) Durham-Newcastle, Department of Chemistry, Durham University, South Road, Durham, DH1 3LE, United Kingdom (Received 10 April 2017; published 14 July 2017).
• In addition, the long-range dipole-dipole interactions present between pairs of polar molecules make them useful in the study of dipolar quantum matter [35,36] and ultracold molecules confined in an optical lattice can simulate a variety of condensed-matter systems [37–39].

### II. EXPERIMENT

• A detailed description of their experimental apparatus can be found in Ref. [51], but the authors will summarize the main components here.
• During this stage 9 × 104 atoms are transferred into the ODT and the magnetic bias field is set to 22.3 G, corresponding to the Efimov minimum in the Cs three-body recombination rate [50].
• For each experimental run the number and temperature are determined by quickly turning off the ODT after a variable hold time and performing resonant absorption imaging of both species after a variable time of flight.
• The smaller initial number of Cs atoms is chosen to reduce the density of Cs such that the effects of three-body recombination play a relatively small role in the thermalization [55].
• Treatment of the number evolution of the Cs atoms requires careful attention due to the presence of Cs atoms both in the crossed-beam region of the trap and in the wings, where confinement is due to only a single ODT beam.

### III. RATE EQUATIONS FOR THERMALIZATION

• To model the thermalization results, the authors formulate a set of coupled equations that describe the number and temperature kinetics.
• Kbg is the background loss rate, Ki,3 is the three-body loss coefficient, ni(r) is the density, and 〈. . .〉sp represents a spatial average.
• Ti,H is the recombination heating term, which accounts for the increase in temperature due to the release of the molecular binding energy during recombination [59].
• The authors choose to neglect the three-body loss coefficient for Yb, KYb,3, because they do not observe any evidence of three-body loss on the experimental timescale in single-species Yb experiments.
• The effective intraspecies collision rate per atom for thermalization is γii = 〈ni〉spσii v̄ii , where σii is an effective energy-independent scattering cross section.

### IV. ANALYSIS OF RESULTS

• The coupled Eqs. (1) and (2) are solved numerically.
• The solid lines in Fig. 3 show the results of the fitted model, while the dashed lines show the results in the absence of interspecies collisions.
• The authors instead constrain the number of Cs atoms inside and outside this region using interpolating functions matched to the experimentally measured values.
• ṪCs,Heat = 4 ± 1 μK/s, it results from the sum of two heating mechanisms, recombination heating and heating from the optical potential.
• Due to the difference in the natural abundance (31.8% for 174Yb and 3.0% for 170Yb [67]) and the intraspecies scattering lengths (a174 = 105 a0 and a170 = 64 a0 [68]), the authors obtain a number of 170Yb atoms that is half that of 174Yb, leading to a greater final temperature for 170Yb.

### OF SCATTERING LENGTHS

• Except near narrow Feshbach resonances, CsYb collisions can be treated as those of two structureless particles with an interaction potential V (R), which behaves at long range as 012704-4 −C6R−6.
• For CsYb, with ∼70 bound states [49], changes in the Yb isotope alter vD by less than 1, so that the scattering lengths for all possible isotopologs may be placed on a single curve.
• At the lowest temperatures, thermalization is governed by the elastic cross section σel = 4πa2.
• The authors vary the magnitude of its short-range part by a factor λ to vary the scattering length, while keeping the long-range part −C6R−6 fixed, with C6 taken from Ref. [49].
• The authors obtain optimal values of the potential scaling factor λ by least-squares fitting to the experimental cross sections.

### VI. TOWARD A DOUBLY DEGENERATE MIXTURE

• In their current system, the authors can independently create BoseEinstein condensates of Cs and the two Yb isotopes with positive intraspecies scattering lengths and workable abundance, 174Yb and 170Yb.
• Note that the intraspecies scattering length for 172Yb is large and negative [68], precluding the creation of a large condensate.
• For 174Yb, the initial steps are the same as in preparing the 174Yb gas for the thermalization measurements (see Fig. 2), where the sample is loaded into the ODT at high power and evaporatively cooled.
• The authors then transfer 9 × 105 of these atoms into the ODT used previously for thermalization measurements (and for 174Yb BEC); this is also known as the dimple trap.
• Additionally, the two species require very different traps for efficient evaporative cooling.

### VII. CONCLUSIONS

• The authors have used a kinetic model to determine the cross sections for interspecies thermalization, taking account of additional heating effects that prevent complete thermalization of the two species.
• The authors have used the resulting interaction potential to calculate scattering lengths for all isotopologs of CsYb.
• The authors have cooled both 174Yb and Cs to degeneracy in the same apparatus, but cooling both species to degeneracy in the same optical trap will be challenging, as illustrated by the contrasting routines they use to produce independent BECs of the two species.
• The optimized CsYb potential will assist direct measurements of the CsYb binding energies using two-photon photoassociation spectroscopy [68,79].
• The data presented in this paper are available online [80].

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Deposited in DRO:
17 July 2017
Version of attached le:
Published Version
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Peer-reviewed
Citation for published item:
Guttridge, A. and Hopkins, S.A. and Kemp, S.L. and Frye, M.D. and Hutson, J.M. and Cornish, S.L. (2017)
'Interspecies thermalization in an ultracold mixture of Cs and Yb in an optical trap.', Physical review A., 96
(1). 012704.
Further information on publisher's website:
https://doi.org/10.1103/PhysRevA.96.012704
Reprinted with permission from the American Physical Society: Physical Review A 96, 012704
c
(2017) by the
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PHYSICAL REVIEW A 96, 012704 (2017)
Interspecies thermalization in an ultracold mixture of Cs and Yb in an optical trap
A. Guttridge,
1,*
S. A. Hopkins,
1
S. L. Kemp,
1
Matthew D. Frye,
2,
Jeremy M. Hutson,
2
and Simon L. Cornish
1,
1
Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham University,
South Road, Durham, DH1 3LE, United Kingdom
2
Joint Quantum Centre (JQC) Durham-Newcastle, Department of Chemistry, Durham University,
South Road, Durham, DH1 3LE, United Kingdom
(Received 10 April 2017; published 14 July 2017)
We present measurements of interspecies thermalization between ultracold samples of
133
Cs and either
174
Yb or
170
Yb. The two species are trapped in a far-off-resonance optical dipole trap and
133
Cs is sympathetically cooled
by Yb. We extract effective interspecies thermalization cross sections by ﬁtting the thermalization measurements
to a kinetic model, giving σ
Cs
174
Yb
= (5 ± 2) × 10
13
cm
2
and σ
Cs
170
Yb
= (18 ± 8) × 10
13
cm
2
. We perform
quantum scattering calculations of the thermalization cross sections and optimize the CsYb interaction potential
to reproduce the measurements. We predict scattering lengths for all isotopic combinations of Cs and Yb. We
also demonstrate the independent production of
174
Yb and
133
Cs Bose-Einstein condensates using the same
optical dipole trap, an important step toward the realization of a quantum-degenerate mixture of the two
species.
DOI: 10.1103/PhysRevA.96.012704
I. INTRODUCTION
The realization of ultracold atomic mixtures [112] has
opened up the possibility of exploring new regimes of few-
and many-body physics. Such mixtures have been used to
study Eﬁmov physics [1315], probe impurities in Bose gases
[16], and entropically cool gases conﬁned in an optical lattice
[17]. Pairs of atoms in the mixtures can be combined using
magnetically or optically tunable Feshbach resonances to
create ultracold molecules [ 1826]. These ultracold molecules
have a wealth of applications, such as tests of fundamental
physics [2729], realization of novel phase transitions [3032],
and the study of ultracold chemistry [33,34]. In addition, the
long-range dipole-dipole interactions present between pairs
of polar molecules make them useful in the study of dipolar
quantum matter [35,36] and ultracold molecules conﬁned in
an optical lattice can simulate a variety of condensed-matter
systems [3739].
Although the large majority of work on ultracold molecules
has focused on bi-alkali systems, there is burgeoning interest
in pairing alkali-metal atoms with divalent atoms such as Yb
[4045]orSr[46]. The heteronuclear
2
molecules formed
in these systems have both an electric and a magnetic dipole
moment in the ground electronic state. The extra magnetic
degree of freedom opens up new possibilities for simulating a
range of Hamiltonians for spins interacting on a lattice and for
topologically protected quantum information processing [47].
One of the challenging aspects of creating molecules in
these systems is that the Feshbach resonances tend to be
narrow and sparse. They are narrow because the main coupling
responsible for them is the weak distance dependence of the
alkali-metal hyperﬁne coupling, caused by the spin-singlet
atom at short range [48]. They are sparse because only
molecular states with the same value of the alkali-metal
*
alexander.guttridge@durham.ac.uk
matthew.frye@durham.ac.uk
s.l.cornish@durham.ac.uk
magnetic quantum number M
F
as the incoming atomic channel
can cause resonances. The resonance positions are determined
by the (often unknown) background scattering length [48,49]
and for some systems may be at impractically high magnetic
ﬁelds. Among the various alkali-Yb combinations, CsYb
has been proposed as the most favorable candidate because
the high mass of Cs facilitates a higher density of bound
states near threshold and its large hyperﬁne coupling constant
increases the resonance widths [49]. However, the short-range
part of the molecular ground-state potential is not known
accurately enough to predict background scattering lengths,
so experimental characterization is essential before accurate
predictions of resonance positions and widths can be made.
Here we present simultaneous optical trapping of Cs and
Yb and ﬁrst measurements of the scattering properties of
133
Cs +
174
Yb and
133
Cs +
170
Yb. We measure interspecies
thermalization in the optical dipole trap and use a kinetic model
to extract effective thermalization cross sections. We model
these cross sections using quantum scattering calculations,
taking full account of the anisotropy of differential cross
sections and thermal averaging. We obtain an optimized
interaction potential and use it to make predictions of the
scattering lengths for all accessible isotopologs. For all
isotopes except
176
Yb, binary quantum-degenerate mixtures of
Cs and Yb are expected to be miscible at ﬁelds around 22 G,
where the Eﬁmov minimum in the three-body recombination
rate allows efﬁcient evaporation of Cs to quantum degeneracy
[50].
II. EXPERIMENT
A detailed description of our experimental apparatus can
be found in Ref. [51], but we will summarize the main
components here. Cs and Yb magneto-optical traps (MOTs)
are s equentially loaded from an atomic beam that effuses
from a dual-species oven and is slowed by a dual-species
Zeeman slower [52]. The Cs atomic beam is slowed and
trapped in the MOT using the
2
S
1/2
2
P
3/2
transition at
λ = 852 nm. For Yb we use the broad
1
S
0
1
P
1
transition
2469-9926/2017/96(1)/012704(10) 012704-1 ©2017 American Physical Society

A. GUTTRIDGE et al. PHYSICAL REVIEW A 96, 012704 (2017)
FIG. 1. Optical layout of the science chamber in the horizontal
plane. The Cs (Yb) imaging beam is combined with the Cs (Yb)
MOT beam using a polarizing beam splitter (dichroic mirror) and
then separated after the chamber and aligned onto a CCD camera.
The Raman lattice beams used for DRSC are split using a polarizing
beam splitter, with one arm retroreﬂected and the other arm dumped
after the ﬁrst pass. The ODT used for thermalization measurements is
referred to as the “dimple trap” to distinguish it from the large-volume
“reservoir trap” that is used for the preparation of Cs BEC.
at λ = 399 nm (/2π = 29 MHz) for Zeeman slowing and
absorption imaging, and the narrow
1
S
0
3
P
1
transition at
λ = 556 nm (/2π = 182 kHz) for laser cooling in the MOT.
The optical layout of our science chamber is shown in Fig. 1.
The thermalization measurements presented here take place
in an optical dipole trap (ODT) formed from the output of a
broadband ﬁber laser (IPG YLR-100-LP) with a wavelength
of 1070 ± 3 nm. The ODT consists of two beams crossed at
an angle of 40
with waists of 33 ± 3 μm and 72 ± 4 μm,
respectively. The intensity of each beam is independently
controlled by a water-cooled acousto-optic modulator. Yb has
a moderately low polarizability at the trapping wavelength
[α
Yb
(1070 nm) = 150 a
3
0
], so that, for the powers used in
the thermalization measurements, Yb atoms are trapped only
in the part of the potential where the axial conﬁnement is
provided by the second ODT beam. Cs, on the other hand,
has a much larger polarizability at the trapping wavelength
[α
Cs
(1070 nm) = 1140 a
3
0
], creating a trap deep enough that Cs
atoms are conﬁned both inside and outside the crossed-beam
region of the ODT. Some Cs atoms thus experience a trapping
potential dominated by just a single ODT beam.
A summary of the experimental sequence used for the
thermalization measurements is shown in Fig. 2.Thetwo
species are sequentially loaded into the dipole trap to avoid
unfavorable inelastic losses from overlapping MOTs [53].
We choose to prepare the Yb ﬁrst due to the much longer
ﬁelds. We ﬁrst load the Yb MOT for 10 s, preparing 5 × 10
8
atoms at T = 140 μK[54], before ramping the power and
detuning the MOT beams to cool the atoms to T = 40 μK.
7
atoms into the ODT with a trap depth
of U
Yb
= 950 μK. We then evaporatively cool the atoms by
exponentially reducing the trap depth to U
Yb
= 5 μKin7s,
producing a sample of 1 × 10
6
Yb atoms at a temperature of
T = 550 nK. At this stage the Yb trap frequencies as measured
by center-of-mass oscillations are 240 Hz radially and 40 Hz
axially.
Once the Yb is prepared in the dipole trap, the Cs MOT
is loaded for 0.15 s, at which point the MOT contains
1 × 10
7
atoms. The Cs MOT is then compressed via ramps
in the magnetic ﬁeld, laser intensity, and detuning before
it is overlapped with the ODT using magnetic bias coils.
The Cs atoms are then further cooled by optical molasses
before transfer into a near-detuned lattice with P = 100 mW,
where the atoms are then polarized in the |F = 3,m
F
=+3
state and cooled to T = 2 μK with 8 ms of degenerate Raman
sideband cooling (DRSC). During this stage 9 × 10
4
atoms
are transferred into the ODT and the magnetic bias ﬁeld is
set to 22.3 G, corresponding to the Eﬁmov minimum in the
Cs three-body recombination rate [50]. During the transfer the
atoms are heated to T = 5 μK. The heating and poor efﬁciency
of the transfer into the ODT are due to the poor mode matching
of the DRSC-cooled cloud and the deep ODT (U
Cs
= 85 μK).
This huge ratio of trap depths U
Cs
/U
Yb
= 15.5 is greater
than the (still large) ratio of the polarizabilities α
Cs
Yb
= 7.2
due to the effect of gravity on the weak Yb trap. The ratio
Yb MOT
10 s
Cooling
Ramp
300 ms
ODT
750 ms
Evaparation
Ramp
7s
Dipole Trap
7G/cm
40 W
2.5 G/cm
Cs MOT
100 ms
Compression
+ Molasses
100 ms
Rethermalization
Variable Hold
t
Dual-Species
Imaging
DRSC
8ms
t
7.5 G/cm
20 G/cm
U
Yb
U'
Yb
U
U
600 mW
FIG. 2. Simpliﬁed experimental sequence. The Yb MOT is loaded, then cooled and compressed to facilitate subsequent loading into an
ODT. The Yb is then evaporated in the ODT by ramping the trap depth until a temperature of T = 550 nK is reached. The displaced Cs MOT
is loaded before it is compressed, cooled, and transferred into a near-detuned optical lattice for DRSC. The DRSC stage loads Cs into the ODT,
where it is held with Yb for a variable time t before the trap is switched off and the atoms are destructively imaged after a variable time of
ﬂight using dual-species absorption imaging.
012704-2

INTERSPECIES THERMALIZATION IN AN ULTRACOLD . . . PHYSICAL REVIEW A 96, 012704 (2017)
FIG. 3. Results of thermalization experiments. (a, b) The evolution of the Cs number and temperature as a function of hold time t.(c,d)
The
174
Yb number and temperature as a function of the same hold time. Filled symbols indicate the presence of both Cs and
174
Yb in the ODT,
whereas open symbols indicate the presence of only one species in the trap. For the Cs number, triangles indicate the number in the single-beam
region of the trap and circles the number in the crossed-beam region, while dotted lines show the interpolating functions used to constrain the
Cs number in the model. The dashed line shows the result of our kinetic model with only one species trapped and the solid line shows the result
for the two-component mixture.
of the mean trap frequencies between the two species is
ω
Cs
Yb
= 3.1.
The thermalization measurements thus begin with a mixture
of 1 × 10
6
Yb atoms in their spin-singlet ground state
1
S
0
and
9 × 10
4
Cs atoms in their absolute ground state
2
S
1/2
|3, + 3.
For each experimental run the number and temperature are
determined by quickly turning off the ODT after a variable
hold time and performing resonant absorption imaging of both
species after a variable time of ﬂight.
Figure 3 shows the number and temperature evolution
of Cs and
174
Yb atoms, with and without the other species
present. The smaller initial number of Cs atoms is chosen
to reduce the density of Cs such that the effects of three-body
recombination play a relatively small role in the thermalization
[55]. Treatment of the number evolution of the Cs atoms
requires careful attention due to the presence of Cs atoms both
in the crossed-beam region of the trap and in the wings, where
conﬁnement is due to only a single ODT beam. Although the
Cs atoms in the crossed- and single-beam regions are in thermal
equilibrium, the atoms have different density distributions due
to the different potentials experienced. This is an important
effect to consider when calculating the spatial overlap of the
Cs and Yb atoms. We observe an increase in the number of
Cs atoms trapped in the crossed-beam region of the trap in the
presence of Yb, which we attribute to interspecies collisions
aiding the loading of this region. We do not observe any Cs
atoms loaded into the crossed-beam portion of the trap in
the absence of Yb, so the number is not plotted in this case.
For the Cs atoms in the single-beam region, we estimate the
axial trapping frequency to be the same as for a single-beam
trap, 5 Hz, and the radial frequencies to be the same as in the
crossed-beam region.
We observe a decay of the Yb number throughout the
thermalization. The timescale of this decay is much shorter
than the single-species 1/e background lifetime of 15 s and
we attribute the number loss to sympathetic evaporation [2].
The small change in the Yb temperature is explained in part by
the evaporation of hotter atoms and also by the large number
ratio N
Yb
/N
Cs
, which causes the ﬁnal mean temperature of the
sample to be close to the initial Yb temperature. In contrast
to Yb, we observe a large change in the temperature of the
Cs atoms for short times due to elastic collisions with the Yb
atoms. However, for l onger times we see the two species reach
a steady state at two distinct temperatures. The higher ﬁnal
temperature for Cs results from a Cs heating rate that balances
the sympathetic cooling rate.
III. RATE EQUATIONS FOR THERMALIZATION
To model the thermalization results, we formulate a set of
coupled equations that describe the number and temperature
kinetics. We expand upon the usual treatment [41,42,56,57]
by including terms for evaporation [58] and single-species
three-body recombination [59] as described in the Appendix.
The coupled equations for the number N
i
and temperature T
i
of the two species are
˙
N
i
=−N
i
γ
ii
η
i
exp(η
i
) K
bg
N
i
K
i,3
n
2
i
sp
N
i
, (1)
˙
T
i
=η
i
exp(η
i
)γ
ii
1
η
i
+ κ
i
3
T
i
+ K
i,3
n
2
i
sp
(T
i
+ T
i,H
)
3
±
ξ
CsYb
T (t)
3N
i
+
˙
T
i,ODT
, (2)
where i ={Yb,Cs}
i
= U
i
/k
B
T
i
, and κ
i
= (η
i
5)/
(η
i
4) [60]. K
bg
is the background loss rate, K
i,3
is the
three-body loss coefﬁcient, n
i
(r) is the density, and ...
sp
represents a spatial average. T
i,H
is the recombination
heating term, which accounts for the increase in temperature
due to the release of the molecular binding energy during
recombination [59]. We choose to neglect the three-body loss
coefﬁcient for Yb, K
Yb,3
, because we do not observe any
evidence of three-body loss on the experimental timescale
in single-species Yb experiments. The Cs three-body loss
coefﬁcient is measured to be K
Cs,3
= 1
+1
0.9
× 10
26
cm
6
/s
at the bias ﬁeld used in the measurements. In addition to
012704-3

A. GUTTRIDGE et al. PHYSICAL REVIEW A 96, 012704 (2017)
the above terms,
˙
T
i,ODT
is added as an independent heating
term to account for any heating from the trapping potential,
such as off-resonant photon scattering [61] or additional
heating effects due to the multimode nature of the trapping
laser [6265]. The heating rate for Yb alone is found to
be zero within experimental error, so
˙
T
Yb,ODT
is ﬁxed at 1
nK/s, which is the predicted heating rate due to off-resonant
photon scattering. Equation (2) uses the fact that the average
energy transferred in a hard-sphere collision is ξk
B
T , where
ξ = 4m
Cs
m
Yb
/(m
Cs
+ m
Yb
)
2
, m
i
is the mass of species i, and
T = T
Cs
T
Yb
.
The effective intraspecies collision rate per atom for
thermalization is γ
ii
=n
i
sp
σ
ii
¯v
ii
, where σ
ii
is an effective
energy-independent scattering cross section. In a hard-sphere
model, the effective total interspecies collision rate is
CsYb
=
¯
n
CsYb
σ
CsYb
¯v
CsYb
, where the mean thermal velocity ¯v
ij
is
¯v
ij
=
8k
B
π
T
i
m
i
+
T
j
m
j
, (3)
and the spatial overlap
¯
n
CsYb
is found by integrating the density
distributions of the two species,
¯
n
CsYb
=
[n
Cs,single
(r) + n
Cs,cross
(r)]n
Yb
(r)d
3
r
= N
Yb
m
3/2
Yb
ω
3
Yb
2πk
B
N
Cs,single
T
Yb
+ β
2
single
T
Cs
3/2
+
N
Cs,cross
T
Yb
+ β
2
cross
T
Cs
3/2
. (4)
Here
ω
Yb
=
3
ω
x
ω
y
ω
z
is the mean Yb trap frequency and β
2
j
is deﬁned by β
2
j
m
Yb
ω
2
Yb
= m
Cs
ω
2
Cs,j
.Herej ={single,cross}
denotes the different cases for Cs atoms trapped in the crossed-
and single-beam regions.
Due to the large difference in trapping potentials between
the two species, Yb experiences a greater gravitational sag than
the tightly trapped Cs. For the case of two clouds spatially
separated by z, the spatial overlap must be reduced by a
factor F
z
(z), where
F
z
(z) = exp
m
Yb
ω
2
Yb,z
z
2
2k
B
(T
Yb
+ β
2
cross
T
Cs
)
. (5)
IV. ANALYSIS OF RESULTS
The coupled Eqs. (1) and (2) are solved numerically. We
perform least-squares ﬁts to the experimental results to obtain
optimal values of the parameters σ
CsYb
, T
Cs,H
, and
˙
T
Cs,ODT
.The
solid lines in Fig. 3 show the results of the ﬁtted model, while
the dashed lines show the results in the absence of interspecies
collisions. Figure 3(a) does not include model results, because
our analysis does not include the kinetics of Cs atoms entering
and leaving the crossed-beam region. We instead constrain
the number of Cs atoms inside and outside this region using
interpolating functions (dotted lines in ﬁgure) matched to the
experimentally measured values.
Since the origin of the heating present on long timescales
is unknown, we initially ﬁtted both T
Cs,H
and
˙
T
Cs,ODT
.We
found that these two parameters are strongly correlated, with
a correlation coefﬁcient of 0.99 [66]. We therefore choose
to extract the parameter
˙
T
Cs,Heat
corresponding to the total
heating rate from both recombination heating and heating due
to the ODT. As shown in Fig. 3, the best ﬁt, corresponding
to σ
Cs
174
Yb
= (5 ± 2) × 10
13
cm
2
and
˙
T
Cs,Heat
= 4 ± 1 μK/s,
describes the dynamics of t he system well. The large fractional
uncertainty in the value of the elastic cross section is primarily
due to the large uncertainty in the spatial overlap. We have
investigated the effect of systematic errors in the measured
parameters of our model and found that the uncertainty in the
trap frequency is dominant, and is larger than the statistical
error. Inclusion of the correction of Eq. (5) is important because
the weaker conﬁnement of Yb produces a vertical separation
between the two species, reducing the spatial overlap. Initially,
F
z
(z) 0.75. Over the timescale of the measurement the
spatial overlap reduces further due to the decreasing width of
the Cs cloud as it cools. The ﬁnal value of F
z
(z) 0.6.
Although the total heating rate is large,
˙
T
Cs,Heat
= 4 ±
1 μK/s, it results from the sum of two heating mechanisms,
recombination heating and heating from the optical potential.
The value for recombination heating is reasonable because the
Cs trap depth of 85 μK is large enough to trap some of the
products of the three-body recombination event. For our
scattering length, a
CsCs
250 a
0
,T
Cs,H
is still within the range
from 2/9to/3 proposed by the simple model in Ref. [59],
where = ¯h
2
/m
Cs
(a
CsCs
¯
a)
2
with
¯
a = 95.5 a
0
for Cs. We
also cannot rule out any heating effects due to the broadband,
multimode nature of the trapping laser [6265], which may
inﬂate the value of
˙
T
Cs,Heat
above the simple estimate of
60 nK/s based upon off-resonant scattering of photons. We
ﬁnd that varying the value of the total trap heating rate
˙
T
Cs,Heat
over a large range changes the extracted cross section by less
than its error.
For the measurements presented in Fig. 3, we deliberately
use a low initial density of Cs atoms to avoid three-body recom-
bination collisions dominating the thermalization. This neces-
sitates use of the weakest possible trap and restricts the number
of Cs atoms to 9 × 10
4
. However, due to the large ratio of po-
larizabilities between Cs and Yb (and the effect of gravity), this
results in a very shallow trap for Yb. Preparation of Yb atoms
in this shallow trap requires that the intraspecies scattering
length be favorable for evaporation, currently limiting the Yb
isotopes we can study to
170
Yb and
174
Yb. In Fig. 4 we present
our thermalization measurements for
170
Yb alongside those
for
174
Yb. From the ﬁt to the temperature proﬁle we extract
an effective cross section σ
Cs
170
Yb
= (18 ± 8) × 10
13
cm
2
and
˙
T
Cs,Heat
= 5 ± 2 μK/s. The larger interspecies cross section
allows Cs to be cooled to a lower equilibrium temperature
than with
174
Yb. Due to the difference in the natural abundance
(31.8% for
174
Yb and 3.0% for
170
Yb [67]) and the intraspecies
scattering lengths (a
174
= 105 a
0
and a
170
= 64 a
0
[68]), we
obtain a number of
170
Yb atoms that is half that of
174
Yb,
leading to a greater ﬁnal temperature for
170
Yb.
V. CALCULATED CROSS SECTIONS AND EXTRACTION
OF SCATTERING LENGTHS
Except near narrow Feshbach resonances, CsYb collisions
can be treated as those of two structureless particles with an
interaction potential V (R), which behaves at long range as
012704-4

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### Cites background from "Interspecies thermalization in an u..."

• ...Mixtures of closed-shell alkaline-earth atoms with open-shell alkali atoms have been studied in several laboratories [40, 41, 208, 39, 244]....

[...]

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