Observation of Feshbach resonances between alkali and closed-shell atoms

TL;DR: Magnetic Feshbach resonances allow control of the interactions between ultracold atoms and can be used to convert pairs of atoms into molecules by ramping an applied magnetic field across a resonance as mentioned in this paper.

Abstract: Magnetic Feshbach resonances allow control of the interactions between ultracold atoms1. They are an invaluable tool in studies of few-body and many-body physics2,3, and can be used to convert pairs of atoms into molecules4,5 by ramping an applied magnetic field across a resonance. Molecules formed from pairs of alkali atoms have been transferred to low-lying states, producing dipolar quantum gases6. There is great interest in making molecules formed from an alkali atom and a closed-shell atom such as ground-state Sr or Yb. Such molecules have both a strong electric dipole and an electron spin; they will open up new possibilities for designing quantum many-body systems7,8, and for tests of fundamental symmetries9. The crucial first step is to observe Feshbach resonances in the corresponding atomic mixtures. Very narrow resonances have been predicted theoretically10,11,12, but until now have eluded observation. Here we present the observation of magnetic Feshbach resonances of this type, for an alkali atom, Rb, interacting with ground-state Sr. Magnetically tunable scattering resonances between strontium and rubidium atoms are observed in an ultracold experiment, opening the door to exploring quantum many-body physics with new designed molecules.

Summary

1 Introduction

• The idea of using pseudo one-dimensional fillers for materials reinforcement dates back to about 4000 BC, where materials, like straw, were used to reinforce mud bricks [1].
• Finally, the sol-gel method may also enhance the formation of interfacial bonds between the CNTs and the ceramic matrix through the use of functionalized CNTs and metallic hydroxides or alkoxides as precursors [23].
• Regarding the measurement of fracture toughness, the possible sources of disparity can be found, not only in the differences among the synthesis processes employed or the different precursors considered, but also in the characterization protocols and models considered for the toughness measurement.

2.1 Sample synthesis and preparation

• Samples were prepared following a procedure based on a report by Barrera-Solano et al. [15,21].
• Once the MWCNTs bundles could not be optically distinguished, NH3(aq) (Panreac, pH = 11.6) was added to the dispersion, and rapid homogeneous gelation occurred (~1-2 min.), minimizing possible MWCNT aggregation during gelation.
• With X describing the carbon content, the samples were named BSE-X. Fig. 1 illustrates the complete process.
• These parameters have previously been successfully used for the sintering of alumina [31].
• During the sintering process, shrinkage and temperature were recorded.

2.2 Mechanical characterization

• In order to assess the presence of toughening effects due to the addition of CNTs, a proper comparison of the mechanical properties in this set of samples was performed by Vickers indentation tests.
• These tests were performed in a Buehler Wilson® VH1150 Micro Vickers Hardness.
• More details of the indentation fracture resistance calculation can be seen in [32].
• Good resolution of the optical microscopes was remarked as a major feature for reliably assessing KIFR values by VIF tests [28,33].
• Indentations of depths between 200 and 800 nm were performed in a series of 25 indentations for statistics, using a load/unload ramp of 2 mN/s. Load-depth curves were analyzed using the Oliver and Pharr method [34], and they were used to determine the relative elastic modulus and nanoindentation hardness.

3.1 Densities

• The density of the synthesized sample series was measured by Archimedes’ method, using 3.985 and 2.1 g/cm3 as the reference theoretical bulk density for pure alumina [35] and the MWCNT content, respectively.
• The complete sample series exhibited low relative densities, around 85% of theoretical density regardless the MWCNTs content (see Table 1), probably due to the use of nanosized γ-Al2O3 as starting powder.
• This feature is of major relevance regarding the mechanical properties of bulk materials and should be taken into consideration when performing comparisons with other sample series.
• Though these relative low densities and high porosities may indicate that lower mechanical values should be expected, the influence of the presence of MWCNTs in the mechanical properties can be explored.

3.2 Vickers hardness

• Additionally, the size of the prints and the length of the cracks at the corners were measured in order to evaluate the Vickers hardness and the indentation fracture resistance, respectively.
• As seen, hardness values are low in comparison to typical reported values, as expected.
• The decisive factor producing the low values for hardness is the lack of densification, which describes the presence of significant porosities that reduces the volume exposed to mechanical stresses.
• Using an empirical exponential model, the authors are able to estimate the expected hardness of fully dense, pure alumina: 𝐻 = 𝐻0𝑒 −𝑏𝑝 , (3) where H is the hardness for a given porosity fraction p, H0 is the hardness at porosity zero, and b is a material constant that should be adjusted from the study of different porosities.

3.3 Indentation fracture Resistance

• The measured values of indentation fracture resistance calculated by the VIF method and considering Shetty’s formula (1) are listed in Table 2.
• The authors are focused on the comparison of the fracture toughness between this sample series, in order to find its possible correlation with the presence of MWCNTs.
• Thus, in order to thoroughly assess the properties of the sample, the tests were performed in a repeated series of 25 indentations for statistics.
• As shown in Table 2 and Fig. 4, the hardness values observed by nanoindentation are higher than those obtained by Vickers tests in all cases.

3.5 Indentation Fracture Resistance recalculated

• As explained in the introduction section, the indentation fracture resistance of alumina-based CMCs is classically obtained by the VIF method, considering Young’s modulus of the pure alumina from the literature instead of the actual CMC’s modulus.
• Reliable and representative values of KIc are desirable to explore the bulk elastic modulus for each composite.
• Among the many different procedures to obtain Young’s modulus, in this study, the authors have measured this value for each sample by the nanoindentation technique (see Table 2).
• Thus, despite the possible controversy due to the different length scales at which the different parameters involved in the calculation are measured, namely hardness (H) at the micrometer scale and E at the nanometer scale, it is feasible to calculate the indentation fracture resistance of samples, obtaining the values shown in Table 2 and Fig.
• The pure alumina KIFR result is more realistic and closer to the previously reported values for alumina using several measurement methods [28].

3.6 Microstructure exploration

• The analyses of the cracks in the samples by electron microscopy revealed very interesting features.
• At first sight, the presence of grey dots in the polished surface (see Fig. 6) proves the high porosity mentioned previously, confirmed by the low densities (Table 1).
• Therefore, developing a tool for real time measurement and quantification of the quality of CNT dispersions during sample preparation is of paramount relevance, especially before and after gelation.
• This will ensure perfect dispersion and the absence of coils and bundles that will eventually act as defects at the macroscopic scale, affecting densification or fracture toughness.
• Another remarkable fact is that all CNTs found in the cracks seem to be almost perpendicular to the crack propagation direction due to the reorientation of exposed segments of MWCNTs, as indicated in the micrographs in Fig. 6 and sketched in Fig. 7.

4 Conclusions

• Sample preparation needs urgent revision to improve the density and to adequately incorporate the MWCNTs within the ceramic matrix in order to obtain the highest mechanical values.
• The measurement of every parameter present on an indentation fracture resistance equation is also vital for a realistic comparison, even with samples of the same series.
• Though typical values of the indentation fracture resistance calculated with the classical Young’s modulus of pure alumina from the literature are higher for all samples, the recalculated indentation fracture resistance reveals an increasing behavior when considering the actual elastic moduli of samples, obtained in this case by nanoindentation.
• Microstructural inspection by SEM reveals the presence of new toughening phenomena such as crack bridging and crack deflection.
• These mechanisms transfer tensile loads from alumina matrix to the reinforcement nanophase during the propagation of the cracks, which explain the increasing behavior of recalculated KIFR with MWCNT content.

Figures

Figure 1. Detection of Rb-Sr Feshbach resonances by field-dependent loss of Rb. The fraction of Rb atoms remaining in state (f,mf ) after loss at each observed Feshbach resonance, normalised to unity far from the loss feature. Eleven loss features are observed in 87Rb -87Sr mixtures and one in 87Rb -88Sr (lower right panel). The loss features are labelled by [f ,mf ]j, where j ∈{a,b,c} is an index used when losses due to several molecular states are observed at the same atomic threshold. Most loss features show a single dip in the atom number, whereas [1,0]a and [1,1]a show several. Each dip is fit by a Gaussian (black line), with results shown in Tab. I. The color and shape of symbols indicates the coupling mechanism for the Feshbach resonance: mechanism I (orange triangles), II (blue circles), or III (green squares). The resonance near 521 G also has a contribution from mechanism II. Error bars represent the standard error of three or more data points.

Table II. Molecular states responsible for Feshbach resonances. Binding energies obtained from observed Feshbach resonances, EFRb , and from two-photon photoassociation, EPAb .

Table I. Properties of observed Feshbach resonances. For resonances with many components, the theoretical width is the largest calculated value.

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Barbe, V. and Ciamei, A. and Pasquiou, B. and Reichsollner, L. and Schreck, F. and

Zuchowski, P. S. and
Hutson, J. M. (2018) 'Observation of Feshbach resonances between alkali and closed-shell atoms.', Nature
physics., 14 . pp. 881-884.
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Observation of Feshbach resonances between alkali and closed-shell atoms
Vincent Barb´e,
1, ∗
Alessio Ciamei,
1, ∗
Benjamin Pasquiou,
1
Lukas
Reichs¨ollner,
1
Florian Schreck,
1
Piotr S.
˙
Zuchowski,
2
and Jeremy M. Hutson
3
1
Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam,
Science Park 904, 1098XH Amsterdam, The Netherlands
2
Institute of Physics, Faculty of Physics, Astronomy and Informatics,
Nicolaus Copernicus University, ul. Grudziadzka 5/7, 87-100 Torun, Poland
3
Joint Quantum Centre (JQC) Durham-Newcastle, Department of Chemistry,
Durham University, South Road, Durham, DH1 3LE, United Kingdom
(Dated: October 10, 2017)
Magnetic Feshbach resonances are an invalu-
able tool for controlling ultracold atoms and
molecules [1]. They can be used to tune atomic
interactions and have been used extensively to
explore few- and many-body phenomena [2, 3].
They can also be used for magnetoassociation, in
which pairs of atoms are converted into molecules
by ramping an applied magnetic field across a res-
onance [4, 5]. Pairs of open-shell atoms, such as
the alkalis, chromium [6], and some lanthanides
[7–9], exhibit broad resonances because the corre-
sponding molecule has multiple electronic states.
However, molecules formed between alkali and
closed-shell atoms have only one electronic state
and no broad resonances. Narrow resonances
have been predicted in such systems [10–12], but
until now have eluded observation. Here we
present the first observation of magnetic Fesh-
bach resonances in a system containing a closed-
shell atom, Sr, interacting with an alkali atom,
Rb. These resonances pave the way to creat-
ing an ultracold gas of strongly polar, open-shell
molecules, which will open up new possibilities for
designing quantum many-body systems [13, 14]
and for tests of fundamental symmetries [15].
A magnetic Feshbach resonance arises when a pair
of ultracold atoms couples to a near-threshold molecu-
lar state that is tuned to be close in energy by an ap-
plied magnetic field. Magnetoassociation at such a reso-
nance coherently transfers the atoms into the molecular
state [16, 17]. In a few cases, near-threshold molecules
formed in this way have been transferred to their absolute
ground states [18–20], allowing exploration of quantum
gases with strong dipolar interactions [21]. However, this
has so far been achieved only for molecules formed from
pairs of alkali atoms.
Mixtures of closed-shell alkaline-earth atoms with
open-shell alkali atoms have been studied in several labo-
ratories [22–25]. No strong coupling mechanism between
atomic and molecular states exists in systems of this type,
but theoretical work has identified weak coupling mech-
∗
RbSrFR@strontiumBEC.com;
These authors contributed equally to this work.
anisms that should lead to narrow Feshbach resonances,
suitable for magnetoassociation [10–12]. In this letter
we describe the detection of Feshbach resonances in mix-
tures of
87
Sr or
88
Sr with
87
Rb. The coupling between
atomic and molecular states arises from two mechanisms
previously predicted [10–12] and an additional, weaker
mechanism that we identify here. The energies of the
bound states responsible for the resonances are confirmed
by two-photon photoassociation spectroscopy.
The experimental signature of a Feshbach resonance is
field-dependent loss of Rb atoms. This may arise from
either 3-body recombination or inelastic collisions, both
of which are enhanced near a resonance. We perform loss
spectroscopy using an ultracold Rb-Sr mixture, typically
consisting of 5×10
4
Rb atoms mixed with 10
6 87
Sr or 10
7
88
Sr atoms at a temperature of 2 to 5 µK (see Methods).
Figure 1 shows the observed loss features, eleven aris-
ing in the
87
Rb -
87
Sr Bose-Fermi mixture and one in the
87
Rb -
88
Sr Bose-Bose mixture. Ten loss features consist
of a single, slightly asymmetrical dip with FWHM be-
tween 200 and 400 mG. The loss features labelled [1,0]a
and [1,1]a each consist of several dips with a width of 20
to 60 mG at a spacing of 80 mG. We fit each dip with a
Gaussian and give the resulting positions and widths in
Tab. I. None of these Rb loss features arises in the ab-
sence of Sr, proving that they depend on Rb-Sr interac-
tions. We also observe Rb loss features in the absence of
Sr, which coincide with known Rb Feshbach resonances
[26].
Both the atomic and molecular states are described
by the total angular momentum of the Rb atom, f, and
its projection m
f
onto the magnetic field. Where nec-
essary, atomic and molecular quantum numbers are dis-
tinguished with subscripts at and mol. In addition, the
molecule has a vibrational quantum number n, counted
down from n = −1 for the uppermost level, and a rota-
tional quantum number L, with projection M
L
.
88
Sr has
nuclear spin i
Sr
= 0, whereas
87
Sr has i
Sr
= 9/2 and a
corresponding projection m
i,Sr
.
The near-threshold molecular states lie almost paral-
lel to the Rb atomic states as a function of magnetic
field. This is because the presence of the Sr atom barely
changes the Rb hyperfine structure, and the Sr hyper-
fine energy is very small. We can therefore use the
Breit-Rabi formula for Rb to convert the resonance posi-
arXiv:1710.03093v1 [physics.atom-ph] 9 Oct 2017

2
3 9 6 . 8 3 9 7 . 2 3 9 7 . 62 9 4 . 9 2 9 5 . 3 2 9 5 . 7 4 3 2 . 0 4 3 2 . 4 4 3 2 . 85 2 1 . 0 5 2 1 . 4 5 2 1 . 8
0 . 0
0 . 5
1 . 0
3 6 5 . 4 3 6 5 . 8 3 6 6 . 22 7 8 . 1 2 7 8 . 3
0 . 0
0 . 5
1 . 0
4 2 0 . 6 4 2 1 . 0 4 2 1 . 42 9 5 . 0 2 9 5 . 2
3 6 6 . 8 3 6 7 . 2 3 6 7 . 6
0 . 0
0 . 5
1 . 0
4 7 4 . 6 4 7 5 . 0 4 7 5 . 43 9 9 . 6 4 0 0 . 0 4 0 0 . 4 4 3 5 . 4 4 3 5 . 8 4 3 6 . 2
M a g n e t i c f i e l d ( G )
[ 1 , 0 ] b[ 1 , - 1 ] a [ 1 , + 1 ] b[ 1 , - 1 ] c
8 7
R b
8 8
S r [ 1 , + 1 ]
[ 1 , 0 ] a [ 1 , - 1 ] b[ 1 , + 1 ] a
R e m a i n i n g R b f r a c t i o n
[ 2 , - 2 ] [ 2 , + 1 ][ 2 , - 1 ] [ 2 , 0 ]
Figure 1. Detection of Rb-Sr Feshbach resonances by field-dependent loss of Rb. The fraction of Rb atoms
remaining in state (f, m
f
) after loss at each observed Feshbach resonance, normalised to unity far from the loss feature. Eleven
loss features are observed in
87
Rb -
87
Sr mixtures and one in
87
Rb -
88
Sr (lower right panel). The loss features are labelled
by [f,m
f
]j, where j ∈{a,b,c} is an index used when losses due to several molecular states are observed at the same atomic
threshold. Most loss features show a single dip in the atom number, whereas [1,0]a and [1,1]a show several. Each dip is fit by
a Gaussian (black line), with results shown in Tab. I. The color and shape of symbols indicates the coupling mechanism for the
Feshbach resonance: mechanism I (orange triangles), II (blue circles), or III (green squares). The resonance near 521 G also
has a contribution from mechanism II. Error bars represent the standard error of three or more data points.
tions into zero-field binding energies E
b
for the molecu-
lar states, which are given in Tab. II. The crossing atomic
and molecular levels are shown in Figs. 2 and 3, with filled
symbols where we observe loss features.
To verify the bound-state energies and validate our
model of Feshbach resonances, we use two-photon pho-
toassociation (PA) spectroscopy. We detect the two
n = −2 states (with L = 0 and 2) below the lower (f = 1)
threshold of
87
Rb-
87
Sr (states E and F in Tab. II) at al-
most exactly the energies deduced from the resonance
positions. All the states observed through Feshbach res-
onances (B to F) also arise to within 2 MHz in a more
complete model of the Rb-Sr interaction potential, as de-
scribed below.
Three different coupling mechanisms are responsible
for the observed loss features. The first mechanism was
proposed in ref. [10] and relies on the change of the Rb hy-
perfine splitting when the Rb electron distribution is per-
turbed by an approaching Sr atom. Its coupling strength
is proportional to the magnetic field in the field region ex-
plored here [12]. Since only states of equal m
f
and L are
coupled, it leads to Feshbach resonances only at crossings
between atomic states with Rb in f = 1 and molecular
states with L = 0 that correlate with f = 2. We observe
one such resonance with each of
87
Sr and
88
Sr.
The second mechanism involves hyperfine coupling of
the Sr nucleus to the valence electron of Rb and was first
proposed in ref. [11]. Since only fermionic
87
Sr has a nu-
clear magnetic moment, this can occur only in Rb -
87
Sr
collisions. This coupling conserves L and m
f
+ m
i,Sr
,
with the selection rule m
f,at
− m
f,mol
= 0, ±1. Crossings
that fulfil these conditions occur also for molecular states
with the same f value as the atomic state, which makes
them much more abundant than crossings obeying the se-
lection rules of the first mechanism. Feshbach resonances
belonging to different m
i,Sr
are slightly shifted with re-
spect to one another because of the weak Zeeman effect
on the Sr nucleus and the weak Sr hyperfine splitting.
However, since the shift is only ∼ 10 mG for neighboring
m
i,Sr
, much smaller than the width of the loss features
of typically 300 mG, we do not resolve this splitting.
The third mechanism is the anisotropic interaction
of the electron spin with the nucleus of either Rb or
fermionic Sr. This mechanism can couple the s-wave
atomic state to molecules with rotational quantum num-
ber L = 2. As usual, the total angular momentum projec-
tion (now m
f
+m
i,Sr
+M
L
) is conserved. If the Sr nucleus
is involved, an additional selection rule is ∆m
f
= ±1. By

3
Energy/h (MHz)
0
-400
400
-800
Energy/h (MHz)
0
-400
400
800
0
Magnetic field (G)
100 200 300 400 500
A
B
C
D
E
F
[1,-1]a
[1,-1]b
[1,0]a
[1,0]b
[1,+1]a
[2,-2]
[2,-1]
[2,0]
[2,+1]
[1,-1]c
[1,+1]b
Figure 2. Origin of the
87
Rb -
87
Sr Feshbach reso-
nances. Energies of atomic (red) and molecular (orange)
states as functions of magnetic field, shown with respect to
the zero-field atomic level with f = 1 or 2 as appropriate.
Molecular states are labelled as in Tab. II and shown dashed
if rotationally excited (L = 2). Observed Feshbach resonances
are labelled as in Fig. 1 and marked by filled symbols (orange
triangles, blue circles or green squares for coupling mechanism
I, II or III, respectively). Predicted but unobserved Feshbach
resonances are marked by hollow symbols.
contrast, if the Rb nucleus is involved, the selection rule
is ∆m
f
= −∆M
L
. These loss features are made up of
many (m
f
, M
L
) components, split by several hyperfine
terms [27]; in some cases the components separate into
groups for different values of M
L
. Three loss features are
attributed to this mechanism and two of them ([1,1]a and
[1,0]a) indeed show a structure of two or three dips.
Table I includes a theoretical width ∆, obtained from
the Golden Rule approximation [12]. However, this is a
physically different quantity from the experimental width
δ, and for narrow resonances there is no simple link be-
tween them. We have also searched for further resonances
predicted by our model, marked by hollow symbols in
Fig. 2, but did not observe them.
We have previously carried out electronic structure cal-
culations of the RbSr ground-state potential [28]. We
have used the binding energies from two-photon photoas-
sociation, supplemented by the Feshbach resonance po-
sitions measured here, to determine a short-range cor-
rection to this potential [29]. This allows us to esti-
mate that the interspecies scattering length of
87
Rb -
87
Sr is a
87,87
> 1600(+600, −450) a
0
and that of
87
Rb -
88
Sr is a
87,88
= 170(20) a
0
, where a
0
is the Bohr ra-
dius. The large positive scattering length for
87
Rb -
87
Sr will produce a molecular state with binding energy
h× 25(15) kHz, which would lead to Feshbach resonances
Energy/h (MHz)
-400
400
-800
0
Magnetic field (G)
100 200 300 400 500
0
[1,+1]
G
Figure 3. Origin of the
87
Rb -
88
Sr Feshbach resonance.
Energies of atomic (red) and molecular (orange) states as
functions of magnetic field, shown with respect to the zero-
field f = 1 atomic level. Only one Feshbach resonance has
been observed, produced by coupling mechanism I. Since
88
Sr
has zero nuclear spin, mechanism II is absent.
at low magnetic field. We searched for such resonances
between 10 mG and 1 G, but did not find any.
There are several factors that affect resonance widths
and hence observability. First, the amplitude of the
atomic scattering function at short range depends on the
background scattering length a; it is largest when a is
large, and the resonance widths are proportional to a in
this regime [12]. This effect enhances all the resonance
widths for
87
Rb -
87
Sr. However, bound states very near
dissociation exist mostly at long range, and the widths
also depend on the binding energy as |E
b
|
2/3
[12]. This
latter effect may explain our failure to observe the low-
Table I. Properties of observed Feshbach resonances.
For resonances with many components, the theoretical width
is the largest calculated value.
[f, m
f
]j (mol. state, B δ ∆ cpl.
m
f
, M
L
) (G) (mG) (mG) mech.
87
Rb -
87
Sr
[2,+1] (B, +2, 0) 474.9(4) 373(7) 0.04 II
[2,0] (B, +1, 0) 435.9(4) 378(7) 0.07 II
[2,−1] (B, 0, 0) 400.0(4) 247(4) 0.07 II
[2,−2] (B, −1, 0) 367.1(4) 260(5) 0.04 II
[1,−1]a (D, −2, 0) 295.4(4) 372(10) 0.33 II
[1,−1]b (C, −2, mix) 420.9(4) 386(11) 0.002 III
[1,−1]c (D, −1, 0) 521.5(4) 366(3) 2.4 I,II
[1,0]a (E, −1, −1) B
1
= 278.2(4) 30(3) 0.00009 III
(E, −1, −2) B
1
+ 0.081(2) 58(4) 0.00011 III
[1,0]b (F, −1, 0) 397.3(4) 207(4) 0.02 II
[1,+1]a (E, 0, 0) B
2
= 295.0(4) 24(3) 0.00002 III
(E, 0, −1) B
2
+ 0.083(2) 35(3) 0.00009 III
(E, 0, −2) B
2
+ 0.162(2) 30(1) 0.00011 III
[1,+1]b (F, 0, 0) 432.5(4) 213(6) 0.02 II
87
Rb -
88
Sr
[1,+1] (G, +1, 0) 365.8(4) 105(2) 0.05 I

4
field resonances due to the state at E
b
≈ h × 25(15) kHz.
Our model enables us to predict the background scat-
tering lengths and Feshbach resonance positions for all
isotopic Rb-Sr mixtures [29]. For example, we predicted
the position of the
87
Rb -
88
Sr resonance after calibrating
the model on
87
Rb -
87
Sr Feshbach resonances and pho-
toassociation results for three isotopic mixtures. This
resonance was subsequently observed within 10 G of the
prediction.
In summary, we have observed Feshbach resonances
in mixtures of Rb alkali and Sr alkaline-earth atoms.
Similar resonances will be ubiquitous in mixtures of al-
kali atoms with closed-shell atoms, particularly when the
closed-shell atom has a nuclear spin. Magnetoassocia-
tion using resonances of this type offers a path towards
a new class of ultracold molecules, with electron spin
and strong electric dipole moment, which are expected
to have important applications in quantum computation,
many-body physics and tests of fundamental symmetries.
ACKNOWLEDGMENTS
We thank the European Research Council (ERC) for
funding under Project No. 615117 QuantStro. B.P.
thanks the NWO for funding through Veni grant No.
680-47-438. P.S.
˙
Z. thanks the Foundation for Polish Sci-
ence for funding of Homing Plus project No. 2011-3/14
(co-financed by the EU European Regional Development
Fund). J.M.H. thanks the UK Engineering and Physical
Sciences Research Council for support under Grant No.
EP/P01058X/1.
Table II. Molecular states responsible for Feshbach res-
onances. Binding energies obtained from observed Fesh-
bach resonances, E
FR
b
, and from two-photon photoassocia-
tion, E
PA
b
.
label n F L E
FR
b
/h E
PA
b
/h
(MHz) (MHz)
87
Rb -
87
Sr
A −2 2 2 - -
B −2 2 0 288.2(4) -
C −4 2 2 5992(1) -
D −4 2 0 6234(1) -
E −2 1 2 200.0(3) 200.0(3)
F −2 1 0 287.3(3) 287.3(2)
87
Rb -
88
Sr
G −4 2 0 7401.0(7) -
I. METHODS
Sample preparation. We prepare ultracold
87
Rb -
87
Sr Bose-Fermi mixtures by methods similar to those
in our previous work [30]. We transfer Rb and
88
Sr
from magneto-optical traps into a horizontal “reservoir”
dipole trap with a waist of 63(2) µm and a wavelength
of 1070 nm. After Rb laser cooling we optically pump
Rb into the f = 1 hyperfine state. By laser cooling Sr
in the dipole trap on the narrow
1
S
0
-
3
P
1
line we sympa-
thetically cool Rb. We then transfer between 5×10
4
and
1 × 10
5
Rb atoms into the crossed-beam “science” dipole
trap described below. We then ramp off the reservoir
trap, discard
88
Sr atoms and transfer between 1 × 10
6
and 2 × 10
6 87
Sr atoms in a mixture of all ten nuclear
spin states into the science trap. The final temperature
is typically 2 to 5 µK. In order to prepare Rb in an equal
mixture of all three f = 1 m
f
states we then randomize
the distribution by non-adiabatic radiofrequency sweeps
at a magnetic field of 130 G. To prepare Rb in the f = 2
hyperfine states we instead use optical pumping, which
directly produces a nearly homogeneous distribution of
Rb over the f = 2 m
f
states. To prepare
87
Rb -
88
Sr
Bose-Bose mixtures we do not discard
88
Sr after transfer-
ring the gas into the science trap and we skip the loading
of
87
Sr.
Science dipole trap. The science trap consists of
two copropagating horizontal beams and one vertical
beam, all with coinciding foci. The first horizontal beam
has a wavelength of 1064 nm and a waist of 313(16) µm
(19(1) µm) in the horizontal (vertical) direction. The sec-
ond horizontal beam has a wavelength of 532 nm and a
waist of 219(4) µm (19(1) µm). The vertical beam has a
wavelength of 1070 nm and a waist of 78(2) µm. The hori-
zontal 1064-nm beam is typically used at a power of 5.7 W
to 6.2 W and dominates the trap potential. The 532-nm
beam is operated at 0.2 W to 0.4 W. The vertical beam
is operated at 0.7(1) W to measure loss feature [1,−1]b
and is off otherwise. These operating conditions result
in typical trap depths of 40 µK×k
B
for Sr and 95 µK×k
B
for Rb, taking account of gravitational sag.
Loss spectroscopy. We observe Feshbach resonances
through field-dependent loss of Rb atoms. We submit
the Rb-Sr mixture to a magnetic field of up to 550 G for
a hold time of 1 to 10 s. Close to a Feshbach resonance,
the rate of 2-body inelastic collisions or 3-body recom-
bination is increased and atoms are lost. After the hold
time we lower the magnetic field to near zero in 200 ms.
During the next 10 ms, we ramp off the horizontal 532-
nm beam and the vertical beam and decrease the power
of the horizontal 1064-nm beam. This decrease lowers
the evaporation threshold for Sr significantly, while Rb
stays well trapped because its polarizability at 1064 nm
is a factor of three higher. During the next 100 ms, Sr
evaporates and cools Rb, which is advantageous for the
subsequent imaging process.
At the end of the cooling stage, the science trap is
switched off and a magnetic field gradient is applied to

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Observation of feshbach resonances between alkali and closed-shell atoms" ?

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Q2. Why is the fractional Sr loss in the hold time small?

Because of this atom number imbalance, and since most mi,Sr states can contribute to a given loss feature, the fractional Sr loss during the hold time is small, which keeps the Rb loss rate high. 

Q3. What is the effect of the recombination of Rb on the atoms?

The authors submit the Rb-Sr mixture to a magnetic field of up to 550 G for a hold time of 1 to 10 s. Close to a Feshbach resonance, the rate of 2-body inelastic collisions or 3-body recombination is increased and atoms are lost. 

Q4. What is the primary coil used to observe?

A third pair of coils is used to supplement the primary coils to observe the [1,-1]c loss feature, creating a bias field of 57 G.The authors calibrate the magnetic field produced by the primary coils up to 290 G by spectroscopy on the narrow 1S03P1 line of 88Sr. 

Q5. What is the sensitivity to Rb atom number fluctuations?

To reduce sensitivity to Rb atom number fluctuations, the authors normalize the atom number of the Rbmf state of interest by the total atom number in mf states that are not lost. 

Q6. How do the authors prepare Rb in a hyperfine state?

The authors then ramp off the reservoir trap, discard 88Sr atoms and transfer between 1 × 106 and 2 × 106 87Sr atoms in a mixture of all ten nuclear spin states into the science trap. 

Q7. How do the authors prepare Rb in the f = 2 hyperfine states?

To prepare Rb in the f = 2 hyperfine states the authors instead use optical pumping, which directly produces a nearly homogeneous distribution of Rb over the f = 2 mf states. 

Q8. What is the effect of the science trap on the Rb mf states?

At the end of the cooling stage, the science trap is switched off and a magnetic field gradient is applied to5 perform Stern-Gerlach separation of the Rb mf states. 

Q9. What is the temperature of the Rb-Sr mixture before the hold time?

The temperatures of the Rb-Sr mixture before the hold time are respectively 5(1)µK, 3.2(2)µK and 3.0(2)µK.The [1,+1] resonance observed in the Rb-88Sr mixture exhibits a high loss rate, because the authors typically load one order of magnitude more 88Sr atoms than 87Sr atoms into the science trap due to the naturally higher abundance of 88Sr. 

Q10. What is the peak densities for Rb?

The peak densities for Rb are 2(1) × 1012 cm−3, 4(2) × 1011 cm−3 and 3(2) × 1011 cm−3 for features [1,−1]b, [1,0]a and [1,+1]a respectively. 

Q11. How do the authors calibrate the secondary coils?

The authors calibrate the secondary coils by recording one of the three [1,+1]a loss features for different values of the field from the primary coils. 

Q12. Who is the author of the paper?

The authors thank the European Research Council (ERC) for funding under Project No. 615117 QuantStro. B.P. thanks the NWO for funding through Veni grant No. 680-47-438. P.S.Ż. thanks the Foundation for Polish Science for funding of Homing Plus project No. 2011-3/14 (co-financed by the EU European Regional Development Fund). 

Q13. What is the power of the horizontal beam?

During the next 10 ms, the authors ramp off the horizontal 532- nm beam and the vertical beam and decrease the power of the horizontal 1064-nm beam.