An association sequence suitable for producing ground-state RbCs molecules in
optical lattices

Arpita Das1,$\ast$,${\ddagger}$, Philip D. Gregory3,$\dagger$, ${\ddagger}$,
Tetsu Takekoshi2, Luke Fernley3, Manuele Landini1, Jeremy M. Hutson4, Simon L.
Cornish3 and Hanns-Christoph Nägerl1

1 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck,
Austria.

2 Alpine Quantum Technologies GmbH, 6020 Innsbruck, Austria.

3 Department of Physics and Joint Quantum Centre (JQC) Durham-Newcastle,
Durham University, Durham DH1 3LE, United Kingdom.

4 Department of Chemistry and Joint Quantum Centre (JQC) Durham-Newcastle,
Durham University, Durham DH1 3LE, United Kingdom.

∗<EMAIL_ADDRESS>
†<EMAIL_ADDRESS>
‡ These authors contributed equally

## Abstract

We identify a route for the production of 87Rb133Cs molecules in the
$\textrm{X}\,^{1}\Sigma^{+}$ rovibronic ground state that is compatible with
efficient mixing of the atoms in optical lattices. We first construct a model
for the excited-state structure using constants found by fitting to
spectroscopy of the relevant
$\textrm{a}\,^{3}\Sigma^{+}\rightarrow\textrm{b}\,^{3}\Pi_{1}$ transitions at
181.5 G and 217.1 G. We then compare the predicted transition dipole moments
from this model to those found for the transitions that have been successfully
used for STIRAP at 181.5 G. We form molecules by magnetoassociation on a broad
interspecies Feshbach resonance at 352.7 G and explore the pattern of Feshbach
states near 305 G. This allows us to navigate to a suitable initial state for
STIRAP by jumping across an avoided crossing with radiofrequency radiation. We
identify suitable transitions for STIRAP at 305 G. We characterize these
transitions experimentally and demonstrate STIRAP to a single hyperfine level
of the ground state with a one-way efficiency of 85(4) %.

###### Contents

  1. 1 Introduction
  2. 2 Near-threshold levels
  3. 3 Choice of basis set for calculating transition dipole moments
  4. 4 Model for the excited state
  5. 5 Benchmarking the model on the STIRAP transitions at 181.5 G
  6. 6 Navigating the near-threshold levels after association at 352.7 G
  7. 7 STIRAP near 305 G
  8. 8 Conclusion
  9. A Transformation between coupled-atom basis and Hund’s case (a) basis and calculation of E1 matrix element
  10. B Molecular constants used in the ground-state calculations
  11. C State compositions of the accessible ground-state sublevels

##  1 Introduction

Arrays of ultracold polar molecules have promising applications for quantum
simulation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and quantum computation [11, 12,
13, 14, 15, 16, 17]. Long-range and anisotropic dipole-dipole interactions
engineered using dc or ac electric fields allow the exploration of complex
many-body Hamiltonians [18]. For single molecules pinned to the sites of an
optical lattice, these dipolar interactions are combined with extremely long
trap lifetimes. At present, the coldest and densest samples of polar molecules
[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] are produced in experiments
using a two-step indirect method. First, the constituent pre-cooled atoms are
associated to form weakly bound molecules by tuning the magnetic field across
an interspecies Feshbach resonance. These weakly bound molecules are then
transferred to the ground state using stimulated Raman adiabatic passage
(STIRAP) [31, 32, 33].

87Rb133Cs was the second polar molecule, after 40K87Rb [21], to be produced in
the ultracold regime [22, 23]. However, the production of large arrays of RbCs
molecules has proved to be difficult. This is primarily due to the scattering
properties of the constituent atoms; there is a large background interspecies
scattering length ($\sim 650\,a_{0}$), which renders the atomic clouds
immiscible at most magnetic fields. Nevertheless, Reichsöllner  _et al._ [34]
have demonstrated a protocol for efficient mixing of quantum-degenerate
samples of 87Rb and 133Cs. First, Bose-Einstein condensates (BECs) of each
species are prepared in spatially separated dipole traps. An optical lattice
potential is turned on across both samples, with parameters such that Cs
crosses the superfluid-to-Mott-insulator transition and Rb remains a
superfluid. The magnetic field is then tuned close to a broad interspecies
Feshbach resonance at 352.7 G, such that the interspecies scattering length
approaches zero. Finally, the Rb is moved to overlap with the Cs, and the trap
depth is increased such that Rb also crosses the superfluid-to-Mott-insulator
transition. Ideally, this creates an atom array with one Rb and one Cs atom
pinned to each site of the optical lattice, from which molecules may be formed
efficiently by associating pairs of atoms. So far, lattice filling fractions
for double occupancy exceeding 30% have been demonstrated [34].

Previous experiments with 87Rb133Cs have created the molecules by
magnetoassociation on a narrow interspecies Feshbach resonance at 197.1 G [22,
35]. From here the molecules were transferred to the absolute ground state
using STIRAP at 181.5 G with a one-way efficiency of $\sim 90\%$ [22, 23, 25].
Magnetoassociation following the new mixing protocol is straightforward on the
much broader resonance at 352.7 G. However, at this new field the weakly bound
states that the molecules can most easily populate are different and therefore
the available optical transitions for STIRAP to the singlet ground state are
also altered.

In this work, we demonstrate an efficient route for STIRAP to transfer RbCs to
the rovibronic ground state at 305 G that is compatible with the protocol for
mixing Rb and Cs in optical lattices. We construct a model for the hyperfine
structure of the excited state $\textrm{b}^{3}\Pi_{1},\,v^{\prime}=29,\\\
J^{\prime}=1$, where $v^{\prime}$ and $J^{\prime}$ are vibrational and
rotational quantum numbers. We use unprimed, primed and double-primed quantum
numbers for the Feshbach, excited and ground states, respectively. We
constrain the model using spectroscopy at magnetic fields of 181.5 G and 217.1
G. We combine this with coupled-channel wavefunctions of the weakly bound
states to calculate transition dipole moments (TDMs) for the STIRAP
transitions previously used at 181.5 G. We present new experiments to
characterise the weakly bound states of RbCs around 305 G and use the results
to identify suitable STIRAP transitions. We then demonstrate STIRAP to a
single hyperfine level of the rovibronic ground state.

The structure of this paper is as follows. In Sec. 2, we describe the near-
threshold levels that exist near 181.5 and 305 G and may be used as the
starting point for STIRAP. In Sec. 3, we describe the basis set we use to
calculate the TDMs. In Sec. 4, we describe our model for the excited state. In
Sec. 5, we verify our model by characterising both transitions used for STIRAP
at $181.5$ G. In Sec. 6, we characterise the weakly bound levels of RbCs near
305 G both experimentally and theoretically. In Sec. 7, we identify suitable
STIRAP transitions to the ground state and demonstrate STIRAP experimentally
at this new field. Finally, in Sec. 8 we summarize our work and discuss its
significance.

##  2 Near-threshold levels

Ground-state RbCs molecules have previously been produced by STIRAP at 181.5
G. Fig. 1(a) shows the weakly bound levels involved near this field, obtained
from coupled-channel bound-state calculations [36, 37, 38, 39] using the
interaction potential of ref. [40]. The states may be labelled by approximate
quantum numbers
$(n(f_{\textrm{Rb}},f_{\textrm{Cs}})L(m_{f_{\textrm{Rb}}},m_{f_{\textrm{Cs}}}))$;
here $f_{\textrm{Rb}}$ and $f_{\textrm{Cs}}$ are the total angular momenta of
the Rb and Cs atoms, $m_{f_{\textrm{Rb}}}$ and $m_{f_{\textrm{Cs}}}$ are their
projections onto the quantisation axis provided by the magnetic field
$\boldsymbol{B}$, $n$ is a vibrational quantum number, counted down from the
energy of the atom-pair state
$(f_{\textrm{Rb}},m_{f_{\textrm{Rb}}})$+$(f_{\textrm{Cs}},m_{f_{\textrm{Cs}}})$,
and $L$ is a quantum number for relative rotation of the atoms. The total
parity is $(-1)^{L}$ and is conserved in a collision, so only states with even
values of $L$ can cause resonances in s-wave scattering; values $L=0$, 2, 4,
etc. are indicated by labels s, d, g, etc. Ultracold Rb and Cs atoms are first
prepared in their absolute ground states, corresponding to the atom pair state
(1,1)+(3,3).

Figure 1:  Near-threshold states accessible following magnetoassociation on an
interspecies Feshbach resonance at (a) 197 G; (b) 352.7 G. In each case, the
molecules are in state s1 between the Feshbach resonance used for association
and the maximum magnetic field shown here. The filled circle shows the initial
state used for STIRAP at 181.5 G. Avoided crossings in (a) are labelled as
described in the main text. In (b), dashed (dotted) lines show states obtained
with only s-wave (only d-wave) basis functions.

Weakly bound molecules can be created by magnetoassociation on a Feshbach
resonance at 197.1 G. The molecules initially enter a very weakly bound state
s1 = $(-1(1,3)\textrm{s}(1,3))$, known as the least-bound state. This runs
almost parallel to the atomic state as a function of magnetic field and is
bound by only about 120 kHz. The molecules remain in s1 as the magnetic field
is lowered to 182 G, passing over two very narrow avoided crossings with
d-wave states on the way which are labelled as (i) and (ii) in Fig. 1(a). At
182 G there is an avoided crossing (iii) between s1 and d6′ =
$(-6(2,4)\textrm{d}(2,4))$. The molecules transfer adiabatically into d6′, and
then briefly into d2 = $(-2(1,3)\textrm{d}(0,3))$ using another avoided
crossing (iv), which allows separation of the molecules from the remaining
atoms using the Stern-Gerlach effect. The molecules are then transferred back
into d6′, which is the state used for STIRAP. It is important to note that
state d6′ has vibrational quantum number $n=-6$; it lies about 16 GHz below
the atomic threshold that supports it, so is a shorter-range state than s1 or
d2 and this gives it improved Franck-Condon overlap with the excited
electronic state used for STIRAP.

When molecules are formed at the Feshbach resonance at 352.7 G, they are again
initially in the least-bound state s1 and remain there as the magnetic field
is lowered to an avoided crossing near 315 G. We have carried out coupled-
channel calculations on the weakly bound states in this region, using a basis
set with $L_{\textrm{max}}=2$, and the results are shown in Fig. 1(b). There
are two bound states that undergo avoided crossings with s1 between 310 and
315 G, and the molecules enter the higher-field of these, labelled ds6 in Fig.
1(b). However, in this case the character of the states is strongly dependent
on field. There is strong mixing between two underlying states, shown as
dashed lines in Fig. 1(b). We designate these underlying states s6 and d6; s6
has character $(-6(2,4)\textrm{s}(1,3))$, while d6 has character
$(-6(2,4)\textrm{d}(0,3))$ near threshold but becomes mostly
$(-2(1,3)\textrm{d}(1,2))$ at fields below $\sim 305$ G. The dashed and dotted
lines on Fig. 1(b) are obtained from coupled-channel calculations that include
only s-wave or only d-wave channels, respectively. States s6 and d6 undergo a
strong avoided crossing centred near 307 G, at an energy about 14 MHz below
the threshold (1,1)+(3,3). The eigenstates, whose energies are shown as solid
lines, change character over the avoided crossing; the higher-field state ds6
is mostly of d6 character close to threshold but transitions to the lower-
field state sd6 which has dominant s6 character below about 20 MHz. Since
s-wave states typically have larger transition intensities than d-wave states
to the excited states used for STIRAP, we expect it to be most favourable to
perform STIRAP from the deeper part, sd6 in Fig. 1(b). This will be quantified
in Sec. 6 below.

Coupled-channel calculations can provide bound-state wavefunctions as well as
energies [41, 39]. In the present work we perform calculations in a coupled-
atom basis set, with basis functions
$|f_{\textrm{Rb}},m_{f_{\textrm{Rb}}};f_{\textrm{Cs}},m_{f_{\textrm{Cs}}};L,M_{L}\rangle$.
Separate calculations are carried out for each state at each magnetic field of
interest. The full coupled-channel wavefunction is expressed as

$\Psi=R^{-1}\sum_{j}\Phi_{j}\psi_{j}(R),$ (1)

where $R$ is the internuclear distance, $\Phi_{j}$ is one of the basis
functions above, and $j$ is a collective index representing
$f_{\textrm{Rb}},m_{f_{\textrm{Rb}}},f_{\textrm{Cs}},m_{f_{\textrm{Cs}}},L,M_{L}$.
There is a separate radial channel function $\psi_{j}(R)$, expressed pointwise
on a grid of $R$, for each basis function $j$. The quantum numbers used to
identify states are obtained by inspecting the wavefunctions expressed in this
basis set, and the wavefunctions themselves are used in the calculations of
TDMs described in Secs. 5 and 6 below.

##  3 Choice of basis set for calculating transition dipole moments

For efficient STIRAP we must identify two strong transitions that couple the
state F of the Feshbach molecule to the rovibrational singlet ground state G,
which has $\textrm{X}\,^{1}\Sigma^{+}$ character. The state F has mostly
$\textrm{a}\,^{3}\Sigma^{+}$ character, because all the contributing states
have relatively high spin projections,
$M_{F}=m_{f_{\textrm{Rb}}}+m_{f_{\textrm{Cs}}}\geq 3$. Transitions between
pure singlet and triplet states are forbidden, so we exploit the singlet-
triplet mixing between the electronically excited states
$\textrm{A}\,^{1}\Sigma^{+}$ and $\textrm{b}\,^{3}\Pi$ to allow efficient
optical transfer between the initial and final states. Specifically we target
the intermediate state
$\textrm{E}=(\textrm{b}\,^{3}\Pi_{1},v^{\prime}=29,J^{\prime}=1)$, which has a
small admixture of $\textrm{A}\,^{1}\Sigma^{+}$ [42] and was used successfully
for STIRAP of RbCs at 181.5 G. We refer to the transitions that connect to
$\textrm{a}\,^{3}\Sigma^{+}$ as the ‘pump’ transitions and those that connect
to the singlet ground state G as the ‘Stokes’ transitions, as shown in Fig.
2(a). At 181.5 G, the transitions for STIRAP were found starting from a model
without hyperfine structure, and so required an exhaustive search through the
many available transitions by experiment [42]. Here we identify suitable
transitions by first constructing a model for the electronically excited
state, including hyperfine structure. This is used to calculate the relevant
energies and, together with the wavefunctions describing states F and G, the
TDMs for the candidate transitions.

The system $\textrm{A}\,^{1}\Sigma^{+}-\textrm{b}\,^{3}\Pi$ has previously
been investigated in many different alkali dimers [43, 44, 45, 46, 47, 48, 49,
50, 51, 52]. In the case of RbCs, the spin-orbit interaction is large enough
that the ratio of the fine-structure splitting $A$ (between the
$\textrm{b}\,^{3}\Pi_{0}$, $\textrm{b}\,^{3}\Pi_{1}$ and
$\textrm{b}\,^{3}\Pi_{2}$ levels) to the rotational energy
$B_{v^{\prime}}J^{\prime}(J^{\prime}+1)$ is very high
($A/B_{v^{\prime}}\approx 6300$ for small $J^{\prime}$) [47]. This makes
Hund’s case (a) a good description for these states. However, there is no good
description of the coupling of the nuclear spins to the other angular momenta
at the magnetic fields where we can produce Feshbach molecules. While the
physical result does not depend on the basis used for the Hamiltonian matrix,
we wish to choose a basis diagonal in most relevant quantum numbers in order
to simplify the description of the intermediate state. In addition, we wish to
choose a basis in which the initial state F and the final state G can be
expressed simply, as the selection rules apply only between molecular states
expressed in the same basis set.

Figure 2: (a) Electronic potential curves for RbCs, showing the STIRAP scheme
and corresponding transition wavelengths for our experiment. The initial
Feshbach state (mostly $\textrm{a}\,^{3}\Sigma^{+}$), the intermediate
(excited) state
($\textrm{A}\,^{1}\Sigma^{+}-\textrm{b}\,^{3}\Pi,\,v^{\prime}=29,\,J^{\prime}=1$)
and the ground state
($\textrm{X}\,^{1}\Sigma^{+},\,v^{\prime\prime}=0,\,J^{\prime\prime}=0$) are
labelled F, E, and G respectively. (b) Zeeman structure of the excited state
$\textrm{E}=(\textrm{b}\,^{3}\Pi_{1},\,v^{\prime}=29,\,J^{\prime}=1)$, which
has a small admixture of $\textrm{A}\,^{1}\Sigma^{+}$, with the best-fit
parameters from our model fitted to the observed pump transitions at $181.5$ G
and $217.1$ G. The intermediate state labelled ‘STIRAP1’ was used to perform
STIRAP at $181.5$ G. The intermediate state used at $305$ G is labelled
‘STIRAP2’.

For these reasons, we choose to express the wavefunctions for RbCs in terms of
Hund’s case (a) basis functions with uncoupled nuclear spins,

$|\Lambda;S,\Sigma;J,\Omega,M_{J};i_{\mathrm{Rb}},m_{i_{\mathrm{Rb}}};i_{\mathrm{Cs}},m_{i_{\mathrm{Cs}}}\rangle.$
(2)

Here, the quantum numbers $\Lambda$ and $\Sigma$ are the projections of the
total electronic orbital angular momentum ${L_{\mathrm{o}}}$ and spin angular
momentum ${S}$ onto the internuclear axis, with sum $\Omega=\Lambda+\Sigma$.
The quantity $J$ is the total angular momentum, including rotation of the
molecule, with projection along the quantisation axis $M_{J}$, which we choose
to be along the direction of the applied magnetic field. The nuclear spins for
the component nuclei are described by quantum numbers
$i_{\mathrm{Rb}},i_{\mathrm{Cs}}$ with corresponding angular momentum
projections $m_{i_{\mathrm{Rb}}},m_{i_{\mathrm{Cs}}}$. Finally,
$M_{F}=M_{J}+m_{i_{\mathrm{Rb}}}+m_{i_{\mathrm{Cs}}}$ is the projection of the
total angular momentum including nuclear spin onto the quantisation axis.
These basis functions do not include the dependence of the wavefunctions on
the internuclear distance $R$, which is handled separately.

The wavefunction of the Feshbach state F is obtained from coupled-channel
calculations as described in Sec. 2. The coupled-channel wavefunctions are
converted to the Hund’s case (a) basis as described in Appendix A. Since the
state F is of even parity, it can connect only to intermediate states of odd
parity. We therefore express the intermediate state in terms of parity-adapted
functions [53, 54],

$\begin{split}&\dfrac{1}{\sqrt{2}}\Bigl{\\{}|\Lambda^{\prime};S^{\prime},\Sigma^{\prime};J^{\prime},\Omega^{\prime},M_{J}^{\prime};i_{\mathrm{Rb}}^{\prime},m_{i_{\mathrm{Rb}}}^{\prime};i_{\mathrm{Cs}}^{\prime},m_{i_{\mathrm{Cs}}}^{\prime}\rangle\\\
&-(-1)^{J^{\prime}-S^{\prime}+s^{\prime}}|-\Lambda^{\prime};S^{\prime},-\Sigma^{\prime};J^{\prime},-\Omega^{\prime},M_{J}^{\prime};i_{\mathrm{Rb}}^{\prime},m_{i_{\mathrm{Rb}}}^{\prime};i_{\mathrm{Cs}}^{\prime},m_{i_{\mathrm{Cs}}}^{\prime}\rangle\Bigr{\\}},\end{split}$
(3)

where $s^{\prime}$ is even for $\Sigma^{+}$ (and higher $\Lambda$) and is odd
for $\Sigma^{-}$. The electric dipole (E1) matrix elements for the pump and
Stokes transitions, $\langle\textrm{E}|T_{q}^{1}(\vec{\mu})|\textrm{F}\rangle$
and $\langle\textrm{E}|T_{q}^{1}(\vec{\mu})|\textrm{G}\rangle$ respectively
are given in Appendix A.

##  4 Model for the excited state

To construct a model for the state
$\textrm{E}=(\textrm{b}\,^{3}\Pi_{1},\,v^{\prime}=29,\,J^{\prime}=1)$, we
treat the electronic, vibrational, and fine-structure parts of the calculation
as already solved. Our Hamiltonian ($H$) consists of three terms to describe
the rotational ($H_{\mathrm{rot}}$), Zeeman ($H_{\mathrm{Z}}$), and hyperfine
($H_{\mathrm{hf}}$) structure such that

$H=H_{\text{rot}}+H_{\text{Z}}+H_{\text{hf}}\,.$ (4)

Here, the rotational and Zeeman terms are

$H_{\text{rot}}=B_{v^{\prime}}\mathbf{J}^{2},$ (5)
$H_{\text{Z}}=g_{L}\mu_{\text{B}}\,\mathbf{B}\cdot\mathbf{L_{\mathrm{o}}}+g_{S}\mu_{\text{B}}\,\mathbf{B}\cdot\mathbf{S},$
(6)

where $\mathbf{B}$ is the vector describing the applied magnetic field,
$B_{v^{\prime}}$ is the rotational constant in the excited state, $g_{L}$ and
$g_{S}$ are $g$-factors associated with the electronic orbital and spin
angular momentum, respectively, and $\mu_{\mathrm{B}}$ is the Bohr magneton.
The remaining hyperfine term is

$\begin{split}H_{\text{hf}}&=a_{\mathrm{Rb}}\
\mathbf{i}_{\mathrm{Rb}}\cdot\mathbf{L_{\mathrm{o}}}+a_{\mathrm{Cs}}\
\mathbf{i}_{\mathrm{Cs}}\cdot\mathbf{L_{\mathrm{o}}}\\\
&+\delta\left(b_{f_{\mathrm{Rb}}}\
\mathbf{i}_{\mathrm{Rb}}\cdot\mathbf{S}+b_{f_{\mathrm{Cs}}}\
\mathbf{i}_{\mathrm{Cs}}\cdot\mathbf{S}\right),\end{split}$ (7)

where the first two terms represent the orbital magnetic dipole interaction
and the last two correspond to the Fermi contact interaction. The Fermi
contact interaction averages to zero in $\textrm{b}\,^{3}\Pi_{1}$ because the
electron spin precesses rapidly around the internuclear axis with no remaining
projection ($\Sigma^{\prime}=0$). Despite this, there is still a small
contribution of the Fermi contact term due to mixing of
$\textrm{b}\,^{3}\Pi_{0}$ with $\textrm{b}\,^{3}\Pi_{1}$ of an amplitude
$\delta$, as given in the supplemental material of ref. [47]. We therefore
include the Fermi contact term in the hyperfine interaction. Contributions
from the electron-nuclear-spin tensor hyperfine interaction and the nuclear
electric quadrupole interaction are insignificant compared to the wavemeter
measurement uncertainty of 30 MHz and are thus excluded.

The Zeeman terms in the Hamiltonian are off-diagonal in $J^{\prime}$. We
therefore include rotational states up to $J^{\prime}=3$ in our model to
construct the Hamiltonian, which then produces a $480\times 480$ matrix
representation. By diagonalizing this matrix, we find the energies and
wavefunctions of the rotational and hyperfine states.

To constrain our model, we carry out a least-squares fit to the observed pump
transitions from one-photon absorption spectra, taken at magnetic fields of
$181.5$ G and $217.1$ G using the apparatus for RbCs molecules at the
University of Innsbruck. The Fermi contact constants are fixed at
$b_{f_{\mathrm{Rb(Cs)}}}=A_{\text{hf}_{\mathrm{Rb(Cs)}}}/4$ [44], where
$A_{\text{hf}}$ is the hyperfine coupling constant for the atomic ground
state, and the rotational constant is taken from the supplemental material of
ref. [47]. There are thus only three fitting parameters: $a_{\mathrm{Rb}}$,
$a_{\mathrm{Cs}}$ and the overall frequency offset. These have best-fit values
$154.1(22)$ MHz $\times h$, $48.7(3)$ MHz $\times h$, and $192571.564(2)$ GHz,
respectively. The uncertainties in these fitted parameters are due to a
combination of the uncertainties in the parameters that are fixed during the
fitting and the uncertainty with which the transitions are resolved in the
experimental spectra. In principle, for pump light polarised parallel
(perpendicular) to the quantisation axis, $6\,(12)$ transitions are possible,
corresponding to different spin channels
$(M_{J}^{\prime},m_{i_{\textrm{Rb}}}^{\prime},m_{i_{\textrm{Cs}}}^{\prime})$,
but not all these transitions are resolved in the experiment.

In Fig. 2(b) we show the output of our model of the state E over a broad range
of magnetic field from 0 G to 375 G. The red dashed and solid lines represent
the states with $M^{\prime}_{F}=3$ and $M^{\prime}_{F}=5$ respectively, and
the blue solid lines represent the states with $M^{\prime}_{F}=4$. The red and
blue markers with error bars indicate observed transitions
$\textrm{F}\rightarrow\textrm{E}$ for pump light polarised perpendicular (red)
and parallel (blue) to the quantisation axis. There are three distinct
manifolds of states, which correspond to those with $M_{J}^{\prime}=-1,0,1$ as
labelled.

##  5 Benchmarking the model on the STIRAP transitions at 181.5 G

We first test our model on the transitions previously used for STIRAP at 181.5
G [22, 25] as these have been well characterised experimentally. The
intermediate state used in this transfer is labelled in Fig. 2(b) as
‘STIRAP1’. The initial state $\textrm{F}_{181.5\,\text{G}}$ for the transfer
is the state d6′ shown in Fig. 1(a), which has character
$(-6(2,4)\textrm{d}(2,4))$. To couple the intermediate state with the states
$\textrm{F}_{181.5\,\text{G}}$ and $\textrm{G}_{181.5\,\text{G}}$, the pump
light is polarised parallel and the Stokes light is polarised perpendicular to
the quantisation axis.

We first calculate the TDMs
$\mu_{\textrm{F}_{181.5\,\mathrm{G}},\textrm{E}_{181.5\,\mathrm{G}}}$ for each
pump transition following Eqs. (8) and (9) in Appendix A. This is then
multiplied by a vibrational matrix element, calculated separately for each
combination of a component of the Feshbach state with a component of the
intermediate state in the case (a) basis. We use vibronic wave functions for
the intermediate state from ref. [55] and coupled-channel wavefunctions for
the Feshbach state calculated as described in Sec. 2.

Figure 3: (a) Calculated TDMs for the transitions
$\textrm{F}_{181.5\,\mathrm{G}}\rightarrow\textrm{E}_{181.5\,\mathrm{G}}$ for
parallel (blue) and perpendicular (red) polarisation of the pump laser at
$181.5$ G. The transition frequencies are shown as a manifold of
$M_{J}^{\prime}$ from left to right. (b) Calculated TDMs for the transitions
$\textrm{E}_{181.5\,\mathrm{G}}$
(‘STIRAP1’)$\rightarrow\textrm{G}_{181.5\,\mathrm{G}}$ for perpendicular and
parallel polarisation of the Stokes laser as a function of Stokes detuning,
given with respect to the frequency of the transition to the state with
$M_{F}^{\prime\prime}=5$.

The calculated TDMs for all available pump transitions for
$\textrm{F}_{181.5\,\mathrm{G}}\rightarrow\textrm{E}_{181.5\,\mathrm{G}}$ for
both parallel (blue) and perpendicular (red) polarisation of the pump beam are
shown in Fig. 3(a). It can be seen that two of the transitions have
significantly greater TDMs than the others. The transition to state ‘STIRAP1’
was used in previous studies [22, 25]; this state has dominant nuclear spin
components
$(m^{\prime}_{i_{\mathrm{Rb}}},m^{\prime}_{i_{\mathrm{Cs}}})=(3/2,7/2)$. The
transition to this state can be driven with pump light polarised parallel to
the quantisation axis. The calculated E1 TDM for this transition is shown in
Table (1), along with the measured values from Innsbruck [22], and Durham
[25]. The values for the Stokes transitions are within about 50% of
experiment, but there is roughly a factor of two difference between the
calculated and measured values for the pump transitions.

The experimental TDMs are obtained by measuring the Rabi frequency on each
transition and normalising it to the intensity of the light. The dominant
source of uncertainty in the TDM is from the uncertainty in the intensity of
the light. We believe that it is unlikely that these measured values could be
incorrect by a factor of 2. Table 1 gives TDMs for the transitions at 181.5 G
measured in Durham and Innsbruck and the difference between these values gives
a reasonable estimate of the uncertainties present in the experiments.

The differences between the experimental and theoretical values of the TDMs
for the pump transitions are probably due to uncertainties in the electronic
wavefunctions for the excited states. The calculated TDMs depend strongly on
the electronic transition dipole functions, and this dependence is greater for
the pump transitions because there is substantial oscillatory cancellation in
the radial integrals.

Table 1: Calculated and measured TDMs for the pump and Stokes transitions at
$181.5$ G and $305.0$ G.

$|\mathbf{B}|$ / G Transition Initial/final state Excited state Calculated
Measured $/(10^{-4}~{}ea_{0})$ $/(10^{-4}~{}ea_{0})$ In Innsbruck In Durham
$181.5$ Pump $\textrm{a}\,^{3}\Sigma^{+}~{}(-6(2,4)\textrm{d}(2,4))$
$\textrm{b}\,^{3}\Pi_{1}~{}(-1,3/2,7/2)$ $4.0$ $8(3)$ [22] $8.1(1)$ [25]
Stokes $\textrm{X}\,^{1}\Sigma^{+}~{}(0,5)_{0}$ $31.0$ $35.0(9)$ [22]
$28.0(3)$ [25] $305.0$ Pump
$\textrm{a}\,^{3}\Sigma^{+}~{}(-6(2,4)\textrm{s}(1,3))$
$\textrm{b}\,^{3}\Pi_{1}~{}(+1,1/2,5/2)$ $3.1$ $--$ $7.2(1)$ [This work]
Stokes $\textrm{X}\,^{1}\Sigma^{+}~{}(0,4)_{1}$ $7.8$ $--$ $5.1(6)$ [This
work]

We next calculate the TDMs for the Stokes transitions available from the
intermediate state ‘STIRAP1’. To calculate the ground-state Zeeman structure
we use the Hamiltonian of ref. [56] with the hyperfine constants given in
Appendix B. This includes nuclear Zeeman, nuclear quadrupole, nuclear spin-
rotation, and scalar and tensor interactions between the nuclear spins.
Diamagnetic shifts and ac Stark shifts are neglected. The quantum numbers
$N^{\prime\prime}$ and $M_{F}^{\prime\prime}$ are not sufficient to identify
all the states uniquely, so we label the states
$(N^{\prime\prime},M_{F}^{\prime\prime})_{k}$, where $k$ is an index counting
up the states with given $N^{\prime\prime}$ and $M_{F}^{\prime\prime}$ in
order of increasing energy.

The transition $\textrm{b}\,^{3}\Pi_{1}\rightarrow\textrm{X}\,^{1}\Sigma^{+}$
is E1 electron-spin forbidden. The Stokes transition is therefore allowed only
due to mixing between the singlet $\textrm{A}\,^{1}\Sigma^{+}$ and triplet
$\textrm{b}\,^{3}\Pi_{1}$ states caused by the spin-orbit interaction.
Supplementary results published by Docenko  _et al._ [47] indicate that the
state $\textrm{b}\,^{3}\Pi_{1},\,v^{\prime}=29$ used for STIRAP has fractional
$\textrm{A}\,^{1}\Sigma^{+}$ character of $0.00029$. We calculate the E1
matrix elements for all the possible Stokes transitions
$\textrm{E}_{181.5\,\mathrm{G}}\rightarrow\textrm{G}_{181.5\,\mathrm{G}}$
using Eq. (9) in Appendix A. These are combined with vibrational matrix
elements as above. The resulting TDMs are shown in Fig. 3(b).

For Stokes light polarised perpendicular to the quantisation axis, there is
one strong transition, to the state $(0,5)_{0}$; this is the hyperfine ground
state at magnetic fields above 90 G and contains only the spin component
$(m^{\prime\prime}_{i_{\mathrm{Rb}}},m^{\prime\prime}_{i_{\mathrm{Cs}}})=(3/2,7/2)$.
For parallel polarisation, there is also only one strong transition, but to
the state $(0,4)_{1}$; this is a mixture of the spin components $(1/2,7/2)$
and $(3/2,5/2)$. The spin compositions of all accessible states
$(N^{\prime\prime},M_{F}^{\prime\prime})_{k}$ are given in Table (3) in
Appendix C. STIRAP has been performed successfully to both these ground states
[22, 25], but most characterisation has been done using the transition to
$(0,5)_{0}$. The calculated dipole moment is also given for the Stokes
transition in Table (1), along with the measured values from Innsbruck [22]
and Durham [25]. In this case, there is reasonable agreement between our
calculations and the experimental observations.

##  6 Navigating the near-threshold levels after association at 352.7 G

The largest transition intensities to the excited state targeted for STIRAP
are expected for Feshbach molecules prepared in the shortest-range s-wave
states. In this section, we perform spectroscopy of the pump transition, and
use it to map out the near-threshold bound states that are accessible
following magnetoassociation at 352.7 G. We then present our method for
preparing the molecules in a state suitable for efficient STIRAP.

The experiments presented from here onwards are performed using the RbCs
apparatus at Durham University. We begin with an ultracold mixture of
approximately $5\times 10^{5}$ 87Rb and $3\times 10^{5}$ 133Cs atoms in their
ground states, ($f_{\mathrm{Rb}}=1$, $m_{f_{\mathrm{Rb}}}=1$) and
($f_{\mathrm{Cs}}=3$, $m_{f_{\mathrm{Cs}}}=3$), respectively. The magnetic
field at the atoms is $21$ G. The mixture is confined in an optical dipole
trap operating at $\lambda=1550$ nm, and is levitated by a magnetic field
gradient $dB/dz=32$ G cm-1.

Figure 4: Navigation through the weakly bound states following
magnetoassociation at 352.74 G. Directly after formation, molecules occupy the
state s1. By decreasing the magnetic field, we access the states labelled ds6,
g2 and ds6’ shown in panel (a). Lines show bound-state energies from coupled-
channel calculations as described in the text. The markers indicate
measurements of the pump transition, with the binding energy inferred assuming
a linear Zeeman shift of the excited state. Filled circles show measurements
made by simply ramping the magnetic field, whereas empty circles show results
obtained after jumping the avoided crossing between ds6 and g2 at 308 G to
access sd6. (b) Scheme for transferring molecules to sd6. Dashed (solid) lines
show the avoided crossing with (without) a 400 kHz rf field applied. To
transfer molecules through the avoided crossing, we follow steps (i-iii) as
described in the text. (c) The number of molecules detected after transferring
through the avoided crossing once or twice as a function of the Rabi frequency
on the rf transition.

To form RbCs molecules, we perform magnetoassociation on an interspecies
Feshbach resonance at 352.74 G [40], following a scheme similar to that
developed in Innsbruck for the association of Rb and Cs in optical lattices
[34]. We jump the magnetic field above the resonance by increasing the
magnetic field from 21 G to $355$ G in $\sim$1 ms, and then form molecules by
sweeping down across the resonance at a rate of 2.5 G ms-1. The molecules are
initially formed in state s1, which runs approximately parallel to the free-
atom energy. We then ramp the magnetic field down rapidly ($\sim$0.5 ms) to an
adjacent Feshbach resonance at 314.74 G [40], where we adiabatically follow an
avoided crossing to transfer the molecules into the state ds6 shown in Fig.
4(a); this has principal component $(-6(2,4)\textrm{d}(0,3))$. We separate the
atoms from the molecules using the Stern-Gerlach effect [20] (discussed in
more detail later), and then image the molecules by ramping the magnetic field
back up above 353 G, where the molecules are dissociated and the resulting
atoms observed with absorption imaging. We form and detect up to 8000
molecules directly after the separation. For spectroscopy and STIRAP we turn
up the power of the optical trap over 20 ms and ramp off the magnetic field
gradient to transfer the molecules to a purely optical potential. We lose
around half the molecules during this procedure.

Light for spectroscopy and STIRAP is derived from a pair of external cavity
diode lasers (Toptica DL Pro) locked to a high-finesse ($10^{4}$) cavity with
an ultralow-expansion glass spacer as described in ref. [57]. The light is
delivered to the molecules in a beam that propagates perpendicular to the
magnetic field, and has a waist of 35 $\mu$m at the position of the molecules.

To perform spectroscopy of the pump transition, we expose the molecules to 11
mW of laser light for 500 $\mu$s and measure the number of molecules remaining
as a function of the laser frequency. For all experiments shown here, the pump
light is linearly polarised parallel to the magnetic field. During the
spectroscopy pulse, the optical dipole trap is switched off to avoid ac Stark
shifts of the optical transitions, which may vary spatially across the sample.
We have previously shown that turning off the dipole trap at $\lambda=1550$ nm
is crucial for efficient STIRAP [25]. We measure the centre frequency of the
pump transition as a function of magnetic field, with the molecules initially
occupying the state ds6. We measure relative frequency changes in the
transition with an uncertainty ($<160$ kHz) limited by the width of the loss
feature. We expect the excited state to shift linearly with magnetic field, so
by measuring the pump transition energy we experimentally map out the Feshbach
structure as shown by the points in Fig. 4(a). By comparing the measured
transition energies to the calculated energies of the near-threshold bound
states, we find that the magnetic moment of the excited state is 0.381(6)
$\mu_{\mathrm{B}}$; this agrees with the predicted magnetic moment [0.388(4)
$\mu_{\mathrm{B}}$] of the state ‘STIRAP2’, shown in Fig. 2(b), which has
$M^{\prime}_{J}=+1$ and dominant spin component $(1/2,5/2)$.

We originally expected that lowering the magnetic field would tune the
molecules directly from the d-wave state ds6 to the s-wave state sd6. This was
based on the structure shown in Fig. 1, obtained from coupled-channel
calculations using a basis set with $L_{\mathrm{max}}=2$. However, we
discovered an avoided crossing with an unexpected state that runs almost
parallel to threshold at a binding energy near $-17$ MHz. Additional coupled-
channel calculations using a basis set with $L_{\mathrm{max}}=4$ identified
this as a g-wave state with character $(-2(1,3)\textrm{g}(1,3))$. Because of
this, the molecules follow the path shown in Fig. 4 and end up in the d-wave
state ds6′. This state has dominant character $(-2(1,3)\textrm{d}(1,2))$ at
fields below $\sim$305 G, as described in Sec. 2.

We use the avoided crossing between ds6 and g2 to facilitate the separation of
the atoms and molecules using the Stern-Gerlach effect. In general, this
requires that the atoms and molecules possess a different ratio of magnetic
moment to mass. However, our current setup can apply only magnetic field
gradients that levitate high-field-seeking states (with negative magnetic
moment). To perform the separation while keeping the molecules levitated, we
therefore set the magnetic field close to the avoided crossing, where we can
tune the magnetic moment of the molecules between +1.1 $\mu_{\mathrm{B}}$ and
$-1.4\,\mu_{\mathrm{B}}$.

Figure 5:  (a) Calculated TDMs as a function of pump frequency for the
transition
$\textrm{F}_{305\,\mathrm{G}}\rightarrow\textrm{E}_{305\,\mathrm{G}}$ at $305$
G with the pump laser polarisation parallel (blue) and perpendicular (red) to
the quantisation axis. (b) Characterisation of the pump transition at 305 G.
The main panel shows the molecule number as a function of the duration of
exposure to resonant pump light. The inset shows the number of molecules
remaining in the state sd6 as a function of the pump laser frequency for a 100
$\mu$s pulse. The line is a Gaussian fit to the results.

We calculate the TDM expected for pump transitions from each of the states
accessible in Fig. 4(a). We find that only the state
$\textrm{F}_{305\,\mathrm{G}}=\textrm{sd6}$, which has dominant component
$(-6(2,4)\textrm{s}(1,3))$, can couple strongly to the excited state. The TDM
of the pump transitions from the state
$\textrm{sd6}=\textrm{F}_{305\,\mathrm{G}}$ to the various hyperfine levels of
$\textrm{E}_{305\,\mathrm{G}}$ are shown in Fig. 5(a) for parallel (blue) and
perpendicular (red) polarisation. The strongest coupling to the excited state
is achieved for pump light polarised parallel to the quantisation axis, with
the transition expected at a frequency of 192573.1 GHz. The transition to this
new state [‘STIRAP2’ in Fig. 2(b)] has a TDM comparable to that used in the
previous STIRAP at 181.5 G.

To enter the state sd6, we must jump over the avoided crossing between the
states ds6 and g2. We achieve this using the hybrid transfer method developed
by Lang $et$ $al$. [58], as shown in Fig. 4(b). We first tune the magnetic
field to 300 mG above the avoided crossing and then switch on a radiofrequency
(rf) field at 400 kHz with a Rabi frequency of $\sim$38 kHz. This is blue-
detuned with respect to the width of the avoided crossing between the states
ds6 and g2. We then follow a three-step process to complete the transfer: (i)
With the rf on, we ramp the magnetic field to the centre of the avoided
crossing. This efficiently transfers the population from one side of the
avoided crossing to the other by adiabatically following an additional rf-
induced avoided crossing between the rf-dressed states. (ii) We switch the rf
field off, closing the rf-induced avoided crossing while leaving the molecular
state unperturbed. (iii) We continue ramping the magnetic field down,
completing the transfer of molecules into sd6. In Fig. 4(c) we show the number
of molecules detected after jumping the avoided crossing once and twice. The
empty markers in Fig. 4(a) show the binding energy of the molecules inferred
from the pump spectroscopy after performing this rf transfer. At 305 G we
measure the absolute pump transition frequency to be 192573.4(1) GHz, where
the uncertainty is limited by the precision of our wavemeter (Bristol 621A).

We measure the Rabi frequency at which we drive the pump transition for a
magnetic field of 305 G, starting from the state sd6. For this, we pulse on
the pump light for a variable time, and measure the number of molecules
remaining in the Feshbach state as shown in Fig. 5(b). Fitting the oscillation
yields a frequency of 632(16) kHz, which corresponds to an intensity-
normalised Rabi frequency of 0.8(1) kHz
$\sqrt{I_{\mathrm{p}}/(\mathrm{mW}\,\mathrm{cm}^{-2})}$. Within uncertainty,
this is the same coupling strength as for the transition used for STIRAP at
$181.5$ G. The measured value of the TDM, derived from the intensity-
normalised Rabi frequency, is shown in Table (1) along with the calculated
value using our model.

##  7 STIRAP near 305 G

Figure 6:  Number of molecules remaining (with empty and filled circles
corresponding to the primary $y$-axis) after the round-trip STIRAP pulse shown
as inset in (a). We vary the Stokes detuning for light that is linearly
polarised (a) parallel and (b) perpendicular to the magnetic field. The line
in (a) is a Gaussian fit to the results, with the fitted centre defining zero
detuning, which is assumed to correspond to the location of the state
$(0,4)_{1}$. The line in (b) is a sum of two Gaussians fitted to the data to
guide the eye. The vertical lines in each plot indicate the accessible
hyperfine states of the ground state $\textrm{X}^{1}\Sigma^{+}$
($v^{\prime\prime}=0,N^{\prime\prime}=0$). The height of each line indicates
the TDM for that transition corresponding to the secondary $y$-axis. Figure 7:
Identification of the state accessed via STIRAP with parallel Stokes
polarisation. We perform microwave spectroscopy of the strongest three
transitions between $N^{\prime\prime}=0$ and $N^{\prime\prime}=1$, as shown in
(a-c). (d,e) show the calculated TDMs in units of the molecule-frame dipole
moment ($d_{0}=1.2$ D [22, 23]) for each of the available transitions from
either (d) the higher-energy state $(0,4)_{1}$ or (e) the lower-energy state
$(0,4)_{0}$. Blue, red, and green colour codings indicate transitions to
states with $N^{\prime}=1$ and $M_{F}^{\prime\prime}=3,4,5$ respectively. The
centre frequencies of each of the transitions found in (a-c) are indicated by
the vertical dotted lines. The microwave spectra observed indicate that the
molecules occupy the higher-energy state $(0,4)_{1}$.

To find the Stokes transition, we apply the STIRAP pulse sequence with peak
laser powers of 11 mW for the pump light and 7.6 mW for the Stokes light, with
ramp timings optimised for STIRAP at 181.5 G [25]. We first apply the Stokes
light for 5 $\mu$s before a sinusoidal ramp turns the Stokes light off and the
pump light on over 20 $\mu$s. After a 5 $\mu$s hold with the pump light on,
the sequence is reversed. When the Stokes light is off-resonant, the pump
light removes molecules from the Feshbach state, so we observe a background of
no molecules. When the Stokes light is on or near resonance, the population
experiences round-trip STIRAP to and from the ground state, so the loss is
suppressed. We find the Stokes transition to the rotational ground state at a
laser frequency of 306831.2(1) GHz. We show the variation in molecule number
as a function of the Stokes laser frequency in Fig. 6, for linear polarisation
both parallel and perpendicular to the magnetic field. By comparing the
initial number of molecules with the number remaining after round-trip STIRAP
at the Stokes detunings indicated, we measure maximum one-way efficiencies of
85(4)% and 92(7)% for parallel and perpendicular polarisation, respectively.

Angular momentum selection rules limit the sublevels of the ground state that
we can access to those with $M_{F}^{\prime\prime}=4$ for parallel polarisation
and $M_{F}^{\prime\prime}=3,5$ for perpendicular polarisation. To identify the
states that are populated during STIRAP, we perform microwave spectroscopy
[59] of the strongest $\pi$, $\sigma^{+}$, and $\sigma^{-}$ transitions from
$N^{\prime\prime}=0$ to $N^{\prime\prime}=1$ and compare the results with the
hyperfine structure and TDMs calculated using the codes of Ref. [60]. We focus
on the state populated with parallel polarisation, as there are only two
states with $M_{F}^{\prime\prime}=4$ that might be accessible. To perform the
spectroscopy, the microwave pulse parameters are set to approximate a $\pi$
pulse when close to resonance. The three strongest transitions are shown in
Fig. 7 and align well with the transitions expected from the higher-energy
state $(0,4)_{1}$. Moreover, we find that we can drive Rabi oscillations on
each of the available transitions with 100% contrast; this indicates that the
molecules occupy just the single hyperfine state $(0,4)_{1}$ following STIRAP.

We have calculated the TDMs for the available Stokes transitions from the
intermediate state ‘STIRAP2’ identified in section 6 to all accessible
sublevels of the rovibronic ground state. These are shown as the vertical bars
in Fig. 6. We see that the detunings where we observe high STIRAP efficiencies
broadly agree with the locations of the four strongest transitions. We do not
see evidence for STIRAP to the lower-energy state $(0,4)_{0}$, even though the
predicted TDM for the transition to this state is substantial. Interference
effects that depend on the relative signs of the Stokes matrix elements
between nearby transitions can suppress or enhance the STIRAP efficiency [22,
61]. The measurements presented here have 100 kHz spacing between detunings,
which is probably too broad to resolve the narrow features that interference
effects would cause. This may explain the lack of a second peak in STIRAP
efficiency at the expected transition to $(0,4)_{0}$.

Figure 8:  Rabi oscillations on the Stokes transition at 305 G. We pulse on
the Stokes light between transfer to the ground state and return to the
Feshbach state as shown in the schematic inset. The number of molecules
detected after round-trip STIRAP is shown as a function of the duration of the
Stokes pulse.

To measure the Rabi frequency for the Stokes transition, we pulse on the
Stokes light for a variable time between STIRAP pulses. The resulting damped
Rabi oscillations are shown in Fig. 8. We extract a Rabi frequency of 250(7)
kHz from the oscillations, corresponding to an intensity-normalised Rabi
frequency of 0.40(5) kHz
$\sqrt{I_{\mathrm{p}}/(\mathrm{mW}\,\mathrm{cm}^{-2})}$. This is around a
factor of 5 times lower than for the Stokes transition previously used at
181.5 G; the difference arises because the current transition is from a spin
sublevel of the intermediate state different from the original. The TDM
derived from the intensity-normalised Rabi frequency is shown in Table (1)
alongside the calculated TDM corresponding to the state $(0,4)_{1}$.

##  8 Conclusion

We have found an efficient route to produce 87Rb133Cs molecules in the
rovibronic ground state, compatible with a recently developed protocol for
efficient mixing of the atomic species in an optical lattice. To do this, we
have constructed a model for the intermediate excited state involved in
STIRAP, and used this to calculate TDMs for both pump and Stokes transitions.
We have combined this with new calculations and experiments on the weakly
bound states of RbCs that exist near 305 G. We encountered an avoided crossing
with an unexpected g-wave state at a binding energy near 17 MHz, which
interferes with transfer to the s-wave state that is most favourable for
STIRAP. We have found a way to jump over this state to reach the target s-wave
state. We have demonstrated STIRAP near 305 G, and observed one-way
efficiencies of 85(4)% to the $(0,4)_{0}$ sublevel of the rovibronic ground
state at 305 G for parallel polarisation of both pump and Stokes lasers. This
is comparable to the efficiency achieved in earlier work with STIRAP at 181.5
G [22, 23, 25]. Our calculations of TDMs show generally good agreement with
the experimental observations, and are able to predict accurately the
strongest transitions. This work will allow the production of large, ordered
arrays of ultracold polar RbCs molecules.

## Data access statement

The data that support the findings of this study are openly available from
Zenodo at [62], together with the codes for calculating the Zeeman structure
of the excited state and the TDMs. These codes are also available from GitHub
at [63].

## Rights retention statement

For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission.

## Acknowledgements

### Funding information

The Innsbruck team acknowledges funding by the DFG-FWF Forschergruppe FOR2247
under the FWF project number I4343-N36, via a Wittgenstein prize grant under
project number Z336-N36, and by the European Research Council (ERC) under
project number 789017. The Durham authors’ work was supported by UK
Engineering and Physical Sciences Research Council (EPSRC) Grants
EP/P01058X/1, EP/P008275/1 and EP/W00299X/1, UK Research and Innovation (UKRI)
Frontier Research Grant EP/X023354/1, the Royal Society and Durham University.

##  Appendix A Transformation between coupled-atom basis and Hund’s case (a)
basis and calculation of E1 matrix element

The transformation between the coupled-atom basis
$|f_{\mathrm{Rb}},m_{f_{\mathrm{Rb}}};f_{\mathrm{Cs}},m_{f_{\mathrm{Cs}}};L,M_{L}\rangle$
and the Hund’s case (a) basis is

$\centering\begin{gathered}\innerproduct{\Lambda;f_{\mathrm{Rb}},m_{f_{\mathrm{Rb}}};f_{\mathrm{Cs}},m_{f_{\mathrm{Cs}}};L,M_{L}}{\Lambda;f_{\mathrm{Rb}},m_{f_{\mathrm{Rb}}};f_{\mathrm{Cs}},m_{f_{\mathrm{Cs}}};L,M_{L}}{\Lambda;S,\Sigma;J,M_{J},\Omega;m_{i_{\mathrm{Rb}}},m_{i_{\mathrm{Cs}}}}=\innerproduct{L,\Lambda;S,\Sigma}{J,\Omega}\innerproduct{L,M_{L};S,M_{S}}{J,M_{J}}\\\
\times\sum_{M_{S},m_{s_{\mathrm{Rb}}},m_{s_{\mathrm{Cs}}}}\innerproduct{S,M_{S}}{s_{\mathrm{Rb}},m_{s_{\mathrm{Rb}}};s_{\mathrm{Cs}},m_{s_{\mathrm{Cs}}}}\innerproduct{s_{\mathrm{Rb}},m_{s_{\mathrm{Rb}}};i_{\mathrm{Rb}},m_{i_{\mathrm{Rb}}}}{f_{\mathrm{Rb}},m_{f_{\mathrm{Rb}}}}\innerproduct{s_{\mathrm{Cs}},m_{s_{\mathrm{Cs}}};i_{\mathrm{Cs}},m_{i_{\mathrm{Cs}}}}{f_{\mathrm{Cs}},m_{f_{\mathrm{Cs}}}},\\\
\end{gathered}\@add@centering$ (8)

where $\Lambda=0$ for the Feshbach state.

The first Clebsch-Gordan coefficient converts from Hund’s case (a) to Hund’s
case (b) [53] and the remainder recouple the electron and nuclear spins.

The E1 matrix elements between case (a) basis functions are

$\centering\begin{gathered}\matrixelement{\Lambda^{\prime};S^{\prime},\Sigma^{\prime};J^{\prime},M^{\prime}_{J},\Omega^{\prime}}{T^{1}_{q}(\vec{\mu})}{{\Lambda;S,\Sigma;J,M_{J},\Omega}}=\\\
\mu^{S}_{\Lambda^{\prime}\Lambda}(R)\delta_{\Sigma^{\prime},\Sigma}\delta_{S^{\prime},S}(-1)^{M_{J}^{\prime}-\Sigma^{\prime}-\Lambda^{\prime}}\sqrt{(2J^{\prime}+1)(2J+1)}\begin{pmatrix}J^{\prime}&1&J\\\
-M_{J}^{\prime}&q&M_{J}\\\ \end{pmatrix}\begin{pmatrix}J^{\prime}&1&J\\\
-\Omega^{\prime}&\Lambda^{\prime}-\Lambda&\Omega\end{pmatrix},\end{gathered}\@add@centering$
(9)

and are diagonal in the nuclear spin quantum numbers $m_{i_{\mathrm{Rb}}}$ and
$m_{i_{\mathrm{Cs}}}$. Here $\mu^{S}_{\Lambda^{\prime}\Lambda}$ is an
$R$-dependent electronic transition-dipole matrix element [42]. The delta
functions and the two 3-$j$ symbols give the E1 selection rules, with the
laser polarisation determining $q=-1,\,0,\,1$. In our calculations, we
consider driving transitions only with light that is linearly polarised
parallel ($q=0$) or perpendicular ($q=\pm 1$) to the quantisation axis.

##  Appendix B Molecular constants used in the ground-state calculations

Table 2: The molecular constants [56, 59] used in the ground-state Hamiltonian for 87Rb133Cs.  Nuclear spin of Rb ($I_{\mathrm{Rb}}$) |  3/2
---|---
Nuclear spin of Cs ($I_{\mathrm{Cs}}$) |  7/2
Nuclear g-factor of Rb ($g_{\mathrm{Rb}}$) |  1.8295
Nuclear g-factor of Cs ($g_{\mathrm{Cs}}$) |  0.7331
Rotational g-factor ($g_{\mathrm{r}}$) |  0.0062
Rotational constant ($B_{0}$ / MHz) |  490.17
Electric quadrupole coupling constant of Rb ($(eQq)_{\mathrm{Rb}}$ / MHz) |  $-0.809$
Electric quadrupole coupling constant of Cs ($(eQq)_{\mathrm{Cs}}$ / MHz) |  0.059
Nuclear spin-rotation coefficient of Rb ($c_{1}$ / Hz) |  98.4
Nuclear spin-rotation coefficient of Cs ($c_{2}$ / Hz) |  194.2
Tensor nuclear spin-spin rotation coefficient ($c_{3}$ / Hz) |  192.4
Scalar nuclear spin-spin rotation coefficient ($c_{4}$ / Hz) |  19018.96
Isotropic shielding factor of Rb ($\sigma_{\mathrm{Rb}}$ / ppm) |  3531
Isotropic shielding factor of Cs ($\sigma_{\mathrm{Cs}}$ / ppm) |  6367

##  Appendix C State compositions of the accessible ground-state sublevels

Table 3: Accessible sublevels $(N^{\prime\prime},M_{F}^{\prime\prime})_{k}$ of the vibronic ground state $\textrm{X}\,^{1}\Sigma^{+},v=0$ at 181.5 G and 305.0 G, with spin components indicated as $|m_{i_{\textrm{Rb}}}^{\prime\prime},m_{i_{\textrm{Cs}}}^{\prime\prime}\rangle$.  Magnetic field / G |  Spin state
---|---
$181.5$ |  $(0,5)_{0}\equiv|3/2,7/2\rangle$
$(0,4)_{0}\equiv-0.322|1/2,7/2\rangle+0.947|3/2,5/2\rangle$
$\begin{multlined}(0,3)_{0}\equiv+0.074|-1/2,7/2\rangle-0.366|1/2,5/2\rangle+0.928|3/2,3/2\rangle\end{multlined}(0,3)_{0}\equiv+0.074|-1/2,7/2\rangle-0.366|1/2,5/2\rangle+0.928|3/2,3/2\rangle$
$(0,4)_{1}\equiv+0.947|1/2,7/2\rangle+0.322|3/2,5/2\rangle$
$\begin{multlined}(0,3)_{1}\equiv-0.433|-1/2,7/2\rangle+0.826|1/2,5/2\rangle+0.360|3/2,3/2\rangle\end{multlined}(0,3)_{1}\equiv-0.433|-1/2,7/2\rangle+0.826|1/2,5/2\rangle+0.360|3/2,3/2\rangle$
$\begin{multlined}(0,3)_{2}\equiv+0.898|-1/2,7/2\rangle+0.427|1/2,5/2\rangle+0.098|3/2,3/2\rangle\end{multlined}(0,3)_{2}\equiv+0.898|-1/2,7/2\rangle+0.427|1/2,5/2\rangle+0.098|3/2,3/2\rangle$
$305.0$ |  $(0,5)_{0}\equiv|3/2,7/2\rangle$
$(0,4)_{0}\equiv-0.190|1/2,7/2\rangle+0.982|3/2,5/2\rangle$
$\begin{multlined}(0,3)_{0}\equiv+0.026|-1/2,7/2\rangle-0.228|1/2,5/2\rangle+0.973|3/2,3/2\rangle\end{multlined}(0,3)_{0}\equiv+0.026|-1/2,7/2\rangle-0.228|1/2,5/2\rangle+0.973|3/2,3/2\rangle$
$(0,4)_{1}\equiv+0.982|1/2,7/2\rangle+0.190|3/2,5/2\rangle$
$\begin{multlined}(0,3)_{1}\equiv-0.242|-1/2,7/2\rangle+0.943|1/2,5/2\rangle+0.228|3/2,3/2\rangle\end{multlined}(0,3)_{1}\equiv-0.242|-1/2,7/2\rangle+0.943|1/2,5/2\rangle+0.228|3/2,3/2\rangle$
$\begin{multlined}(0,3)_{2}\equiv+0.970|-1/2,7/2\rangle+0.241|1/2,5/2\rangle+0.031|3/2,3/2\rangle\end{multlined}(0,3)_{2}\equiv+0.970|-1/2,7/2\rangle+0.241|1/2,5/2\rangle+0.031|3/2,3/2\rangle$

## References

  * [1] L. Santos, G. V. Shlyapnikov, P. Zoller and M. Lewenstein,  _Bose-Einstein condensation in trapped dipolar gases_ ,  Phys. Rev. Lett. 85, 1791 (2000),  10.1103/PhysRevLett.85.1791.
  * [2] R. Barnett, D. Petrov, M. Lukin and E. Demler,  _Quantum magnetism with multicomponent dipolar molecules in an optical lattice_ ,  Phys. Rev. Lett. 96, 190401 (2006),  10.1103/PhysRevLett.96.190401.
  * [3] A. Micheli, G. Brennen and P. Zoller,  _A toolbox for lattice-spin models with polar molecules_ ,  Nat. Phys. 2, 341 (2006),  10.1038/nphys287.
  * [4] H. P. Büchler, E. Demler, M. Lukin, A. Micheli, N. Prokof’ev, G. Pupillo and P. Zoller,  _Strongly correlated 2D quantum phases with cold polar molecules: Controlling the shape of the interaction potential_ ,  Phys. Rev. Lett. 98, 060404 (2007),  10.1103/PhysRevLett.98.060404.
  * [5] A. V. Gorshkov, S. R. Manmana, G. Chen, E. Demler, M. D. Lukin and A. M. Rey,  _Quantum magnetism with polar alkali-metal dimers_ ,  Phys. Rev. A 84, 033619 (2011),  10.1103/PhysRevA.84.033619.
  * [6] M. A. Baranov, M. Dalmonte, G. Pupillo and P. Zoller,  _Condensed matter theory of dipolar quantum gases_ ,  Chem. Rev. 112, 5012 (2012),  10.1021/cr2003568.
  * [7] I. Bloch, J. Dalibard and S. Nascimbène,  _Quantum simulations with ultracold quantum gases_ ,  Nat. Phys. 8, 267 (2012),  10.1038/nphys2259.
  * [8] A. Macià, D. Hufnagl, F. Mazzanti, J. Boronat and R. E. Zillich,  _Excitations and stripe phase formation in a two-dimensional dipolar Bose gas with tilted polarization_ ,  Phys. Rev. Lett. 109, 235307 (2012),  10.1103/PhysRevLett.109.235307.
  * [9] S. R. Manmana, E. M. Stoudenmire, K. R. A. Hazzard, A. M. Rey and A. V. Gorshkov,  _Topological phases in ultracold polar-molecule quantum magnets_ ,  Phys. Rev. B 87, 081106 (2013),  10.1103/PhysRevB.87.081106.
  * [10] A. V. Gorshkov, K. R. Hazzard and A. M. Rey,  _Kitaev honeycomb and other exotic spin models with polar molecules_ ,  Mol. Phys. 111, 1908 (2013),  10.1080/00268976.2013.800604.
  * [11] D. DeMille,  _Quantum computation with trapped polar molecules_ ,  Phys. Rev. Lett. 88, 067901 (2002),  10.1103/PhysRevLett.88.067901.
  * [12] S. F. Yelin, K. Kirby and R. Côté,  _Schemes for robust quantum computation with polar molecules_ ,  Phys. Rev. A 74, 050301 (2006),  10.1103/PhysRevA.74.050301.
  * [13] J. Zhu, S. Kais, Q. Wei, D. Herschbach and B. Friedrich,  _Implementation of quantum logic gates using polar molecules in pendular states_ ,  J. Chem. Phys. 138, 024104 (2013),  10.1063/1.4774058.
  * [14] F. Herrera, Y. Cao, S. Kais and K. B. Whaley,  _Infrared-dressed entanglement of cold open-shell polar molecules for universal matchgate quantum computing_ ,  New J. Phys. 16, 075001 (2014),  10.1088/1367-2630/16/7/075001.
  * [15] K. K. Ni, T. Rosenband and D. D. Grimes,  _Dipolar exchange quantum logic gate with polar molecules_ ,  Chem. Sci. 9, 6830 (2018),  10.1039/C8SC02355G.
  * [16] M. Hughes, M. D. Frye, R. Sawant, G. Bhole, J. A. Jones, S. L. Cornish, M. R. Tarbutt, J. M. Hutson, D. Jaksch and J. Mur-Petit,  _Robust entangling gate for polar molecules using magnetic and microwave fields_ ,  Phys. Rev. A 101, 062308 (2020),  10.1103/PhysRevA.101.062308.
  * [17] R. Sawant, J. A. Blackmore, P. D. Gregory, J. Mur-Petit, D. Jaksch, J. Aldegunde, J. M. Hutson, M. R. Tarbutt and S. L. Cornish,  _Ultracold polar molecules as qudits_ ,  New J. Phys. 22, 013027 (2020),  10.1088/1367-2630/ab60f4.
  * [18] J. A. Blackmore, L. Caldwell, P. D. Gregory, E. M. Bridge, R. Sawant, J. Aldegunde, J. Mur-Petit, D. Jaksch, J. M. Hutson, B. E. Sauer, M. R. Tarbutt and S. L. Cornish,  _Ultracold molecules for quantum simulation: rotational coherences in CaF and RbCs_ ,  Quantum Sci. Technol. 4, 014010 (2018),  10.1088/2058-9565/aaee35.
  * [19] C. A. Regal, C. Ticknor, J. L. Bohn and D. S. Jin,  _Creation of ultracold molecules from a Fermi gas of atoms_ ,  Nature 424, 47 (2003),  10.1038/nature01738.
  * [20] J. Herbig, T. Kraemer, M. Mark, T. Weber, C. Chin, H.-C. Nägerl and R. Grimm,  _Preparation of a pure molecular quantum gas_ ,  Science 301, 1510 (2003),  10.1126/science.1088876.
  * [21] K. K. Ni, S. Ospelkaus, M. H. De Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin and J. Ye,  _A high phase-space-density gas of polar molecules_ ,  Science 322, 231 (2008),  10.1126/science.1163861.
  * [22] T. Takekoshi, L. Reichsöllner, A. Schindewolf, J. M. Hutson, C. R. Le Sueur, O. Dulieu, F. Ferlaino, R. Grimm and H.-C. Nägerl,  _Ultracold dense samples of dipolar RbCs molecules in the rovibrational and hyperfine ground state_ ,  Phys. Rev. Lett. 113, 205301 (2014),  10.1103/PhysRevLett.113.205301.
  * [23] P. K. Molony, P. D. Gregory, Z. Ji, B. Lu, M. P. Köppinger, C. R. Le Sueur, C. L. Blackley, J. M. Hutson and S. L. Cornish,  _Creation of ultracold 87Rb133Cs molecules in the rovibrational ground state_,  Phys. Rev. Lett. 113, 255301 (2014),  10.1103/PhysRevLett.113.255301.
  * [24] J. W. Park, S. A. Will and M. W. Zwierlein,  _Ultracold dipolar gas of Fermionic 23Na40K molecules in their absolute ground state_,  Phys. Rev. Lett. 114, 205302 (2015),  10.1103/PhysRevLett.114.205302.
  * [25] P. K. Molony, P. D. Gregory, A. Kumar, C. R. Le Sueur, J. M. Hutson and S. L. Cornish,  _Production of ultracold 87Rb133Cs in the absolute ground state: Complete characterisation of the stimulated Raman adiabatic passage transfer_,  ChemPhysChem 17, 3811 (2016),  10.1002/CPHC.201600501.
  * [26] M. Guo, B. Zhu, B. Lu, X. Ye, F. Wang, R. Vexiau, N. Bouloufa-Maafa, G. Quéméner, O. Dulieu and D. Wang,  _Creation of an ultracold gas of ground-state dipolar ${}^{23}\mathrm{Na}^{87}\mathrm{Rb}$ molecules_,  Phys. Rev. Lett. 116, 205303 (2016),  10.1103/PhysRevLett.116.205303.
  * [27] T. M. Rvachov, H. Son, A. T. Sommer, S. Ebadi, J. J. Park, M. W. Zwierlein, W. Ketterle and A. O. Jamison,  _Long-lived ultracold molecules with electric and magnetic dipole moments_ ,  Phys. Rev. Lett. 119, 143001 (2017),  10.1103/PhysRevLett.119.143001.
  * [28] F. Seeßelberg, N. Buchheim, Z.-K. Lu, T. Schneider, X.-Y. Luo, E. Tiemann, I. Bloch and C. Gohle,  _Modeling the adiabatic creation of ultracold polar ${}^{23}{\mathrm{Na}}^{40}\mathrm{K}$ molecules_,  Phys. Rev. A 97, 013405 (2018),  10.1103/PhysRevA.97.013405.
  * [29] K. K. Voges, P. Gersema, M. Meyer zum Alten Borgloh, T. A. Schulze, T. Hartmann, A. Zenesini and S. Ospelkaus,  _Ultracold gas of Bosonic ${}^{23}\mathrm{Na}^{39}\mathrm{K}$ ground-state molecules_,  Phys. Rev. Lett. 125, 083401 (2020),  10.1103/PhysRevLett.125.083401.
  * [30] I. Stevenson, A. Z. Lam, N. Bigagli, C. Warner, W. Yuan, S. Zhang and S. Will,  _Ultracold gas of dipolar NaCs ground state molecules_ ,  Phys. Rev. Lett. 130, 113002 (2023),  10.1103/PhysRevLett.130.113002.
  * [31] U. Gaubatz, P. Rudecki, S. Schiemann and K. Bergmann,  _Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. a new concept and experimental results_ ,  J. Chem. Phys. 92, 5363 (1998),  10.1063/1.458514.
  * [32] K. Bergmann, H. Theuer and B. W. Shore,  _Coherent population transfer among quantum states of atoms and molecules_ ,  Rev. Mod. Phys. 70, 1003 (1998),  10.1103/RevModPhys.70.1003.
  * [33] K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit _et al._ ,  _Roadmap on STIRAP applications_ ,  J. Phys. B - At. Mol. Opt. 52, 202001 (2019),  10.1088/1361-6455/ab3995.
  * [34] L. Reichsöllner, A. Schindewolf, T. Takekoshi, R. Grimm and H.-C. Nägerl,  _Quantum engineering of a low-entropy gas of heteronuclear Bosonic molecules in an optical lattice_ ,  Phys. Rev. Lett. 118, 073201 (2017),  10.1103/PhysRevLett.118.073201.
  * [35] M. P. Köppinger, D. J. McCarron, D. L. Jenkin, P. K. Molony, H.-W. Cho, S. L. Cornish, C. R. Le Sueur, C. L. Blackley and J. M. Hutson,  _Production of optically trapped ${}^{87}\mathrm{RbCs}$ Feshbach molecules_,  Phys. Rev. A 89, 033604 (2014),  10.1103/PhysRevA.89.033604.
  * [36] J. M. Hutson,  _Coupled-channel methods for solving the bound-state Schrödinger equation_ ,  Comput. Phys. Commun. 84, 1 (1994),  10.1016/0010-4655(94)90200-3.
  * [37] J. M. Hutson, E. Tiesinga and P. S. Julienne,  _Avoided crossings between bound states of ultracold cesium dimers_ ,  Phys. Rev. A 78, 052703 (2008),  10.1103/PhysRevA.78.052703.
  * [38] J. M. Hutson and C. R. Le Sueur,  _bound and field: programs for calculating bound states of interacting pairs of atoms and molecules_,  Comp. Phys. Comm. 241, 1 (2019),  10.1016/j.cpc.2019.02.017.
  * [39] J. M. Hutson and C. R. Le Sueur,  _molscat , bound and field, version 2022.0_,  https://github.com/molscat/molscat (2022).
  * [40] T. Takekoshi, M. Debatin, R. Rameshan, F. Ferlaino, R. Grimm, H.-C. Nägerl, C. R. Le Sueur, J. M. Hutson, P. S. Julienne, S. Kotochigova and E. Tiemann,  _Towards the production of ultracold ground-state RbCs molecules: Feshbach resonances, weakly bound states, and the coupled-channel model_ ,  Phys. Rev. A 85, 032506 (2012),  10.1103/PhysRevA.85.032506.
  * [41] A. E. Thornley and J. M. Hutson,  _Bound-state wavefunctions from coupled-channel calculations using log-derivative propagators – application to spectroscopic intensities in Ar-HF_ ,  J. Chem. Phys. 101, 5578 (1994),  10.1063/1.467345.
  * [42] M. Debatin, T. Takekoshi, R. Rameshan, L. Reichsöllner, F. Ferlaino, R. Grimm, R. Vexiau, N. Bouloufa, O. Dulieu and H.-C. Nägerl,  _Molecular spectroscopy for ground-state transfer of ultracold RbCs molecules_ ,  Phys. Chem. Chem. Phys. 13, 18926 (2011),  10.1039/c1cp21769k.
  * [43] A. N. Drozdova, A. V. Stolyarov, M. Tamanis, R. Ferber, P. Crozet and A. J. Ross,  _Fourier transform spectroscopy and extended deperturbation treatment of the spin-orbit-coupled $A\,^{1}\Sigma_{u}^{+}$ and $b\,^{3}{\Pi}_{u}$ states of the Rb2 molecule_,  Phys. Rev. A 88, 022504 (2013),  10.1103/PhysRevA.88.022504.
  * [44] J. Bai, E. H. Ahmed, B. Beser, Y. Guan, S. Kotochigova, A. M. Lyyra, S. Ashman, C. M. Wolfe, J. Huennekens, F. Xie, D. Li, L. Li _et al._ ,  _Global analysis of data on the spin-orbit-coupled $A\,^{1}\Sigma_{u}^{+}$ and $b\,^{3}\Pi_{u}$ states of Cs2_,  Phys. Rev. A 83, 032514 (2011),  10.1103/PhysRevA.83.032514.
  * [45] C. Lisdat, O. Dulieu, H. Knöckel and E. Tiemann,  _Inversion analysis of k 2 coupled electronic states with the fourier grid method_ ,  The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics 17, 319 (2001),  https://doi.org/10.1007/s100530170006.
  * [46] K. Alps, A. Kruzins, M. Tamanis, R. Ferber, E. A. Pazyuk and A. V. Stolyarov,  _Fourier-transform spectroscopy and deperturbation analysis of the spin-orbit coupled $A\,^{1}\Sigma^{+}$ and $b\,^{3}\Pi$ states of KRb_,  J. Chem. Phys. 144, 144310 (2016),  10.1063/1.4945721.
  * [47] O. Docenko, M. Tamanis, R. Ferber, T. Bergeman, S. Kotochigova, A. V. Stolyarov, A. De Faria Nogueira and C. E. Fellows,  _Spectroscopic data, spin-orbit functions, and revised analysis of strong perturbative interactions for the $A\,^{1}\Sigma^{+}$ and $b\,^{3}\Pi$ states of RbCs_,  Phys. Rev. A 81, 042511 (2010),  10.1103/PhysRevA.81.042511.
  * [48] J. Zaharova, M. Tamanis, R. Ferber, A. N. Drozdova, E. A. Pazyuk and A. V. Stolyarov,  _Solution of the fully-mixed-state problem: Direct deperturbation analysis of the $A\phantom{\rule{1.99997pt}{0.0pt}}^{1}\Sigma^{+}-b\phantom{\rule{1.99997pt}{0.0pt}}^{3}\Pi$ complex in a NaCs dimer_,  Phys. Rev. A 79, 012508 (2009),  10.1103/PhysRevA.79.012508.
  * [49] H. Harker, P. Crozet, A. J. Ross, K. Richter, J. Jones, C. Faust, J. Huennekens, A. V. Stolyarov, H. Salami and T. Bergeman,  _Experimental and theoretical studies of the coupled $A\phantom{\rule{1.60004pt}{0.0pt}}{}^{1}{\mathrm{\Sigma}}^{+}$ and $b\phantom{\rule{1.60004pt}{0.0pt}}^{3}\mathrm{\Pi}$ states of NaK_,  Phys. Rev. A 92, 012506 (2015),  10.1103/PhysRevA.92.012506.
  * [50] P. Kowalczyk, W. Jastrzebski, J. Szczepkowski, E. Pazyuk and A. Stolyarov,  _Direct coupled-channels deperturbation analysis of the A $\,{}^{1}\Sigma^{+}$–b$\,{}^{3}\Pi$ complex in LiCs with experimental accuracy_,  J. Chem. Phys. 142, 234308 (2015),  10.1063/1.4922610.
  * [51] O. Docenko, M. Tamanis, R. Ferber, E. A. Pazyuk, A. Zaitsevskii, A. V. Stolyarov, A. Pashov, H. Knöckel and E. Tiemann,  _Deperturbation treatment of the $A\phantom{\rule{1.99997pt}{0.0pt}}^{1}\Sigma^{+}-b\phantom{\rule{1.99997pt}{0.0pt}}^{3}\Pi$ complex of NaRb and prospects for ultracold molecule formation in $X\phantom{\rule{1.99997pt}{0.0pt}}^{1}\Sigma^{+}(v=0;J=0)$_,  Phys. Rev. A 75, 042503 (2007),  10.1103/PhysRevA.75.042503.
  * [52] A. Kruzins, I. Klincare, O. Nikolayeva, M. Tamanis, R. Ferber, E. A. Pazyuk and A. V. Stolyarov,  _Fourier-transform spectroscopy of $(4)\,^{1}\Sigma^{+}\rightarrow A\,^{1}\Sigma^{+}-b\,^{3}\Pi$, $A\,^{1}\Sigma^{+}-b\,^{3}\Pi\rightarrow X\,^{1}\Sigma^{+}$, and $(1)\,^{3}\Delta\rightarrow b\,^{3}\Pi_{0^{\pm}}$ transitions in KCs and deperturbation treatment of $A\,^{1}\Sigma^{+}$ and $b\,^{3}\Pi$ states_,  J. Chem. Phys. 139, 244301 (2013),  10.1063/1.4844275.
  * [53] J. M. Brown and A. Carrington,  _Rotational spectroscopy of diatomic molecules_ ,  Cambridge University Press,  ISBN 0521810094 (2003).
  * [54] J. K. G. Watson,  _Hönl–London factors for multiplet transitions in Hund’s case a or b_ ,  J. Mol. Spectrosc. 252, 5 (2008),  10.1016/j.jms.2008.04.014.
  * [55] R. Vexiau, N. Bouloufa and O. Dulieu,  _private communication_ (2012).
  * [56] J. Aldegunde, B. A. Rivington, P. S. Zuchowski and J. M. Hutson,  _Hyperfine energy levels of alkali-metal dimers: Ground-state polar molecules in electric and magnetic fields_ ,  Phys. Rev. A 78, 033434 (2008),  10.1103/PhysRevA.78.033434.
  * [57] P. D. Gregory, P. K. Molony, M. P. Köppinger, A. Kumar, Z. Ji, B. Lu, A. L. Marchant and S. L. Cornish,  _A simple, versatile laser system for the creation of ultracold ground state molecules_ ,  New J. Phys. 17, 055006 (2015),  10.1088/1367-2630/17/5/055006.
  * [58] F. Lang, P. Straten, B. Brandstätter, G. Thalhammer, K. Winkler, P. Julienne, R. Grimm and J. Hecker Denschlag,  _Cruising through molecular bound-state manifolds with radiofrequency_ ,  Nat. Phys. 4, 223 (2008),  10.1038/nphys838.
  * [59] P. D. Gregory, J. Aldegunde, J. M. Hutson and S. L. Cornish,  _Controlling the rotational and hyperfine state of ultracold 87Rb133Cs molecules_,  Phys. Rev. A 94, 041403 (2016),  10.1103/PhysRevA.94.041403.
  * [60] J. A. Blackmore, P. D. Gregory, J. M. Hutson and S. L. Cornish,  _Diatomic-py: A Python module for calculating the rotational and hyperfine structure of ${}^{1}\Sigma$ molecules_,  Comput. Phys. Commun. 282, 108512 (2023),  https://doi.org/10.1016/j.cpc.2022.108512.
  * [61] L. Liu, D.-C. Zhang, Y.-X. Liu, J. Nan, J. Rui, B. Zhao and J.-W. Pan,  _Observation of interference between resonant and detuned STIRAP in the adiabatic creation of 23Na40K molecules._,  Phys. Rev. Lett. 122, 253201 (2019),  10.1103/PhysRevLett.122.253201.
  * [62] Data set is available from Zenodo at doi: 10.5281/zenodo.8412500.
  * [63] Codes are available from Github at : arpitadas1/STIRAP_RbCs.git.


J

Lan [a]Chia-HaoLiu [a, b]QiangDu [a, c]Simon J<EMAIL_ADDRESS>[a]Department of Applied Physics and Applied Mathematics, Columbia University,
New York, NY 10027, United States [b]Data Science Institute, Columbia
University, New York, NY 10027, United States [c]Condensed Matter Physics and
Materials Science Department, Brookhaven National Laboratory, Uptown, NY
11973, United States

# Robustness test of the spacegroupMining model for determining space groups
from atomic pair distribution function data

Ling

###### Abstract

Machine learning models based on convolutional neural networks have been used
for predicting space groups of crystal structures from their atomic pair
distribution function (PDF). However, the PDFs used to train the model are
calculated using a fixed set of parameters that reflect specific experimental
conditions, and the accuracy of the model when given PDFs generated with
different choices of these parameters is unknown. In this paper, we report
that the results of the top-1 accuracy and top-6 accuracy are robust when
applied to PDFs of different choices of experimental parameters
$r_{\text{max}}$, $Q_{\text{max}}$, $Q_{\text{damp}}$ and atomic displacement
parameters.

###### keywords:

Robustness Test, Machine Learning, Data Mining, Space Group, Pair Distribution
Function

##  1 Introduction

Recently it was shown [liu;aca19] that a convolutional neural network (CNN)
machine learning model could predict the space group of a material from its
atomic pair distribution function (PDF) [billi;b;itoch19, egami;b;utbp12] with
good accuracy. This model is called spacegroupMining and was recently deployed
as a web application on the pdfitc.org website [yang;aca21].

The atomic pair distribution function (PDF) method is a total scattering
technique for determining local order in nanostructured materials.
Theoretically, the PDF gives the scaled probability of finding two atoms in a
material a distance $r$ apart and is related to the density of atom pairs in
the material [billi;b;itoch19, egami;b;utbp12].

The model of [liu;aca19] was trained, as shown in the red section in Figure 1,
using calculated PDFs, denoted by $G(r,\Omega)$ here, where $\Omega$ indicates
the set of parameters that define experimental details of the measurement and
the sample. These model experimental parameters that affect the quality of the
data, such as the maximum range of $Q$, $Q_{\text{max}}$, where $Q$ is the
modulus of the scattering vector [egami;b;utbp12], the $r$-range of the input
PDF, $r_{\text{max}}$, and so on [proff;jac99]. These are listed in full in
Table 1. Although a specific set of values were used to train the model, in
general, different parameter values might be relevant in a scientist’s
measured PDFs. We denote these as $G(r,\Omega^{\prime})$, where the prime on
the $\Omega$ indicates that some experimental values in the set are different
from the ones used in the training. A natural and important question is
whether the trained model with $G(r,\Omega)$ could provide reasonable
predictions on materials associated with $G(r,\Omega^{\prime})$. If the
accuracy of the model predictions on $G(r,\Omega^{\prime})$ is close to its
performance on the PDFs, $G(r,\Omega)$, that the model learned from, we
believe that the model is robust. To be more explicit, this robustness test
investigates how input data quality translates to performance, while the input
data distribution does not shift, i.e., the materials we use to train and test
are not varying. In this paper we assess how well the model performs when it
is tested on PDFs that were calculated using experimental parameters different
from those for the training set (blue section in Figure 1). We conclude that
overall it performs quite well with respect to $r_{\text{max}}$,
$Q_{\text{max}}$, $Q_{\text{damp}}$ and atomic displacement factor (ADP), or
$U_{\text{iso}}$, of the measurement, hence providing evidence to the
robustness of the CNN machine learning model developed in [liu;aca19].

Figure 1: $G(r,\Omega)$ are the calculated PDFs w.r.t. the original
experimental parameters used to train the CNN model, while
$G(r,\Omega^{\prime})$ are PDFs with varying parameters. $p$ and $p^{\prime}$
are the corresponding model outputs from which we could measure the accuracy
of the model predictions. The red (blue) section depicts the training
(testing) process respectively.

##  2 Method

###  2.1 Data and Model

Our main objective is to test the robustness of the originally trained model.
However, we are not able to identify the exact datasets that constituted the
training set in the original training. In order to avoid testing robustness by
inadvertently using a dataset that might be part of the original model’s
training data, our first step is to rebuild the model again.

The input PDF data are calculated from $98,830$ structures in the 45 most
heavily represented space groups in the ICSD [belsky;acb02] structural
database. The PDFs are calculated from crystallographic information framework
(CIF) [hall;aca91] files obtained from ICSD using the diffpy-cmi [juhas;aca15]
package with parameters $\Omega$ defined in Table 1. The parameters are the
same as the ones used in [liu;aca19], except that the grid size is $\pi/40$ in
our experiment (the paper used $\pi/23$), so that we could calculate the PDF
with higher $Q_{\text{max}}$’s. $80\%$ of the data is considered as training
samples, and the rest is treated as test samples. The choice of
$r_{\text{min}}$, $r_{\text{max}}$, and $r_{\text{grid}}$ in Table 1
discretizes the input PDFs to 1D signal sequences of dimension $209\times
1\times 1$. We further normalize the PDF input, $G(r,\Omega)$, to ensure that
it lies between 0 and 1 for each entry.

Table 1: Experimental parameters used to calculate the PDFs used for training, validation and testing of the model. Here $Q_{\text{min}}$ and $Q_{\text{max}}$ define the range of $Q$ that was included in the Fourier transform to obtain the PDF ($Q$ is the modulus of the scattering vector), $r_{\text{min}}$ and $r_{\text{max}}$ are the minimum and maximum range of $r$ of the PDF and $r_{\text{grid}}$ was the size of the bins in the numerical determination of $G$. $U_{\text{iso}}$ is the atomic displacement factor of the atoms. $Q_{\text{damp}}$ and $Q_{\text{broad}}$ are parameters that determine the $r$-dependence of the damping of the PDF signal, mostly due to the $Q$-space resolution of the measurement. All these parameters are defined in detail in [egami;b;utbp12, proff;jac99]. $r_{\text{min}}$ (Å) | 1.5
---|---
$r_{\text{max}}$ (Å) | 30.0
$r_{\text{grid}}$ (Å) | $\pi/40$
$Q_{\text{min}}$ (Å-1) | 0.5
$Q_{\text{max}}$ (Å-1) | 23.0
$U_{\text{iso}}$ (Å2) | 0.008
$Q_{\text{damp}}$ (Å-1) | 0.04
$Q_{\text{broad}}$ (Å-1) | 0.01

To rebuild the model, we use the architecture based on the convolutional
neural network (CNN) used in Liu’s paper [liu;aca19]. The output, $p$, of the
model is a $45\times 1$ vector, which represents the probability of the input
PDF being in each of the $45$ space groups considered in our study. We use
weighted categorical cross entropy loss,

$\displaystyle\text{Loss}=-\sum_{i=1}^{45}w_{i}\cdot{p_{true}}_{i}\cdot\log
p_{i},$ (1)

to mitigate the effects of unbalanced data, where the weight $w_{i}$ is
defined as the number of structures in the training set over the number of
structures of each space group in the training dataset. Adaptive moment
estimation (Adam) with a mini-batch size of $64$ is used to train the model.
Furthermore, we modify the learning rate as an exponential decay, $l=5\times
10^{-4}e^{-0.025\times\text{epoch}}$. The model is trained using Keras on a
single Nvidia Tesla P100 GPU.

An accuracy of $67.7\%$ from top-1 prediction and $90.2\%$ from top-6
predictions is achieved. The performance of our reconstructed model is similar
to the one shown in the original paper, which was $70.0\%$ top-1 accuracy and
$91.9\%$ top-6 accuracy. The model rebuilt here is used, without any further
retraining, in subsequent robustness tests on datasets involving PDFs having
different parameter values, as illustrated schematically in Fig. 1.

###  2.2 Robustness Test

In order to test the robustness, we consider four experimental parameters that
are used to calculate the PDFs from the structural CIFs, which are
$Q_{\text{max}}$, $r_{\text{max}}$, $Q_{\text{damp}}$, and $U_{\text{iso}}$.
The other parameters in Table 1 are not expected to affect the accuracy
greatly and were not explicitly tested. Variations in $Q_{\text{min}}$ produce
no effect until low-$Q$ Bragg peaks are lost and then result in long-
wavelength damped sinusoidal oscillations in the PDF that appear like an
oscillating background to the signal. The data are interpolated onto a
different $r_{\text{grid}}$ during the process and so the user
$r_{\text{grid}}$ will not affect the outcome, and most users are expected to
have data calculated to an arbitrarily small $r_{\text{min}}$. Finally,
$Q_{\text{broad}}$ has a very small effect on the width of peaks in rather
high $r$-regions that are unlikely to be uploaded.

To carry out the tests we randomly choose structures from the testing set
($10\%$ of the testing samples are chosen), and compute their PDFs while
varying each of these parameters separately between limits that are chosen to
bracket values that are experimentally reasonable. These calculated PDFs are
then given to the trained model, without model retraining despite the changes
in parameter values of the input PDFs, to predict the space group, and the
model accuracy is computed as a function of the experimental parameter value.

First we consider the robustness against a variation in $r_{\text{max}}$. The
model was trained with an $r_{\text{max}}$ of 30 Å and we want to test its
performance when given PDFs computed (or measured) over a narrower $r$-range,
from 10 Å to 30 Å every 2 Å. Variations in $r_{\text{max}}$ will change the
length of the PDF vector, which is not allowed in our model. Since we are only
considering $r$-ranges that are shorter than 30 Å, to keep the dimension of
all input PDFs consistent, the data are padded with zero’s up to the value of
$r_{\text{max}}$ $=30$ Å before being interpolated on to the $209\times
1\times 1$ grid using quadratic interpolation.

To test the $Q_{\text{max}}$ sensitivity, computed PDFs in the range of
$12\leq$ $Q_{\text{max}}$ $\leq 30$ Å-1 in steps of 3 Å-1 were tested against
the trained model. For $Q_{\text{damp}}$, we tested on computed PDFs in the
range of $0\leq$ $Q_{\text{damp}}$ $\leq 0.15$ Å-1 in steps of $0.03$ Å-1.
Finally, for the ADP, $U_{\text{iso}}$, from $0.005\leq$ $U_{\text{iso}}$
$\leq 0.01$ Å2 in steps of 0.001 Å2, where the model was trained on values
$Q_{\text{max}}$ $=23$ Å-1, $Q_{\text{damp}}$ $=0.04$ Å-1 and $U_{\text{iso}}$
$=0.008$ Å2, respectively.

##  3 Results

###  3.1 Robustness with respect to $r_{\text{max}}$

Figure 2 shows the top-6 accuracy against a variation in $r_{\text{max}}$ from
10 Å to 30 Å. When $r_{\text{max}}$ is larger than 20 Å, top-6 accuracy is
always above $87.1\%$, which is close to the optimal value of $90.2\%$. It is
recommended to give the model a PDF with a $r_{max}\geq 30$ Å. However, the
robustness test shows that if the signal is from data over a narrower range,
such as a nanoparticle whose signal dies on a shorter length-scale, the model
can still be categorized into space group with reasonably good accuracy,
though the performance drops off more quickly below an $r_{\text{max}}$ of 20
Å or so.

Figure 2: The black dots represent the top-6 accuracy as $r_{\text{max}}$ is
varied on the testing datasets. The value of $r_{\text{max}}$ $=30$ Å used to
train the model is shown as a vertical green dashed line. The box plot at each
$r_{\text{max}}$ value shows the uncertainties on the top-6 accuracy (see text
for details). The box extends from the first quartile (Q1) to the third
quartile (Q3) of the distribution, with a green horizontal line at the median.
The whiskers from the box extend by 1.5 times the inter-quartile range (IQR).
The hollow circles indicate samples that lie outside that range.

###  3.2 Robustness with respect to $Q_{\text{max}}$

Next we consider the robustness of the model when PDFs are generated using
different $Q_{\text{max}}$ values. As shown in Figure 3, when $Q_{\text{max}}$
is larger than 18 Å-1, top-6 accuracy is above $81.1\%$. The bump around 23
Å-1 makes sense, as the model favors the $Q_{\text{max}}$ value that it is
trained on. But the performance with $Q_{\text{max}}$ values deviated from 23
Å-1 is still fairly good over the entire range of values considered, the
accuracy never falls below $77.7\%$, and so the model is quite robust against
variations in $Q_{\text{max}}$.

Figure 3: The black dots represent the top-6 accuracy as $Q_{\text{max}}$ is
varied. The value of $Q_{\text{max}}$ $=23$ Å-1 used to train the model is
shown as a vertical green dashed line. The box plot at each $Q_{\text{max}}$
value is plotted in the same way as the ones in Fig. 2.

###  3.3 Robustness with respect to $Q_{\text{damp}}$

Figure 4 shows the top-6 accuracy against a variation in $Q_{\text{damp}}$ in
the range of $0\leq Q_{\text{damp}}\leq 0.15$ Å-1 in steps of $0.03$ Å-1.

Figure 4: The black dots represent the top-6 accuracy as $Q_{\text{damp}}$ is
varied. The value of $Q_{\text{damp}}$ $=0.04$ Å-1 used to train the model is
shown as a vertical green dashed line. The box plot at each $Q_{\text{damp}}$
value is plotted in the same way as the ones in Fig. 2.

When $Q_{\text{damp}}$ is smaller than $0.06$ Å-1, the top-6 accuracy is
always above $86.1\%$. However, the performance drops off fairly quickly above
$Q_{\text{damp}}$ $=0.06$ Å-1 or so. When $Q_{\text{damp}}$ $=0.15$ Å-1, the
PDF signal practically vanishes in the region above $r=20$ Å and so we might
expect the accuracy to be similar to that of $r_{\text{max}}$ $=20$ Å. We find
that the accuracy of $Q_{\text{damp}}$ $=0.15$ Å-1 falls to $57.8\%$, which is
significantly lower than the value of $87.1\%$ of $r_{\text{max}}$ $=20$ Å.
This is presumably because $Q_{\text{damp}}$ damps the signal progressively
over the entire range and therefore the model is more sensitive to
$Q_{\text{damp}}$ variations than $r_{\text{max}}$. However, we note that the
accuracy with $Q_{\text{damp}}$ values deviated from $0.06$ Å-1 never falls
below $57.8\%$, which can still give acceptable results in many cases.

###  3.4 Robustness with respect to Atomic Displacement Parameter,
$U_{\text{iso}}$

Finally (Fig. 5), we consider robustness against variations in
$U_{\text{iso}}$. The results are even less sensitive to the choice of ADP.
When $U_{\text{iso}}$ of the PDFs were in the range 0.005 Å2 to 0.01 Å2, the
top-6 accuracy is always above $87.3\%$.

Figure 5: The black dots represent the top-6 accuracy as $U_{\text{iso}}$ is
varied. The value of $U_{\text{iso}}$ $=0.008$ Å2 used to train the model is
shown as a vertical green dashed line. The box plot at each $U_{\text{iso}}$
value is plotted in the same way as the ones in Fig. 2.

The numbers from all the robustness tests are reproduced in the supplementary
information associated with this paper.

##  4 Discussion

The main goal of this work is to investigate the model performance when given
PDFs computed using different experimental parameters than the model was
trained with. We found that the model predictions are quite robust for
reasonable ranges of parameters, giving hope that the tool can be used by
experimentalists when they have PDFs measured under different conditions. Here
we speculate on some more detailed aspects of the findings.

Two conditions must be satisfied for the model to perform well. First, the
input data must contain sufficient information to do the differentiation by
space group. Second, the values of the learned CNN model parameters must be
able to correctly classify based on this information, even when it has been
distorted by the use of different experimental parameters. We briefly discuss
the effect of each experimental parameter on the information content of the
PDF. These parameters are discussed in detail in [egami;b;utbp12]. Lowering
$Q_{\text{max}}$ can result in a loss of information from the missing high-$Q$
region, and results in a broadening of the peaks in the PDF. Increases in
$U_{\text{iso}}$ also broaden the PDF peaks but are coming from increased
static or thermal disorder in the sample itself. Because of peak overlap in
the PDF, especially in the high-$r$ region, broadened peaks always result in a
loss of structural information. Clearly, lowering the range of data used
($r_{\text{min}}$ to $r_{\text{max}}$) decreases the information content of
the supplied data. These are parameters under the control of the experimenter
though, as they are parameters that they can set during the data analysis to
produce the PDF. A lower $Q$-space resolution of the measurement (resulting in
a higher $Q_{\text{damp}}$) can also lower the information content of the data
due to Bragg peak overlap, especially in the high-$Q$ region of the data
before it is Fourier transformed to obtain the PDF. In the PDF this appears as
an approximately Gaussian fall off in the structural signal with
increasing-$r$. Any intrinsic nanocrystallinity in the sample, such as finite
nanoparticles or loss of structural coherence in the form of smaller
crystallites or domains, has a similar effect as $Q_{\text{damp}}$ on the PDF.
The fall-off in the signal will have a different functional form in this case
(for example a power-law in the case of spherical domains/particles) but
roughly speaking appears in a rather similar way as the Gaussian dropoff
modelled by $Q_{\text{damp}}$ and so we did not explicitly separate these
factors in this robustness test. We tested $Q_{\text{damp}}$ over a range of
values that simulated structural coherence down to a $\sim 2$ nm diameter.
Finally, $Q_{\text{min}}$ and $Q_{\text{broad}}$ are expected to have only a
very small effect on the accuracy. $Q_{\text{broad}}$ is only relevant for
data with very asymmetric Bragg peaks, for example, coming from time-of-flight
neutron data, and even then, only at very high values of $r$ that tend to be
higher than the values we have been giving to the CNN. $Q_{\text{min}}$ is
often determined by the shadow of the beamstop in an experiment and will only
affect the data if any low-angle signal is lost due to this. In that case, it
results in very long wavelength undulations in the background of the resulting
PDF that will not affect the model’s ability to classify by spacegroup.

The observed robustness indicates that measured PDFs generally contain
sufficient information to make this space-group determination, even when the
data content is degraded somewhat by reduced real-space resolution (lower
$Q_{\text{max}}$ higher $U_{\text{iso}}$) and a more limited $r$-range of the
data (lower $r_{\text{max}}$ and higher $Q_{\text{damp}}$). The accuracy falls
off more rapidly when there is a loss of information in the PDF (lower
$Q_{\text{max}}$, $r_{\text{max}}$, higher $Q_{\text{damp}}$,
$U_{\text{iso}}$); however, we note that the accuracy also falls off when we
give the model a dataset with higher resolution or $r$-range, and therefore
increased information content. The fall-off in accuracy in these cases must be
due to the less than optimal learned CNN parameter values. This could be
addressed by retraining the model with a wider range of experimental
parameters, but it seems that it may not be required, except perhaps for the
case of small nanoparticles (represented by large $Q_{\text{damp}}$ values
above 0.08 in this study). $r_{\text{min}}$ and $r_{\text{max}}$ is largely
under the control of the experimentalist (it is a setting in most PDF data
analysis programs), but more importantly, the range of $r$-that the signal
persists over depends on the crystallite/domain size of the sample and whether
it is nanocrystalline. This suggests that training a new model suitable for
small nanoparticles (i.e., data signal ranges up to 1 or 2 nm) may be
warranted. We will look into this in the future and consider deploying it at
the PDFitc website.

Another way that information in data is degraded is the presence of noise.
Noise may be random or correlated. We have not systematically tested the
robustness of the model to the presence of added noise in the data because it
is difficult to reliably mimic the actual errors that are present in real
data. A more meaningful measure of this is to establish how well the model
works on actual datasets from known materials. This was reported in the
original paper [liu;aca19]. There it was found that of 15 experimental PDFs
the model gave a correct prediction in the top-6 from 12 cases. This is not a
large sample, but is an 80% accuracy. Given that the datasets were already
obtained with experimental parameters that are not necessarily those that the
model was trained with, this is comparable, if somewhat degraded, performance
to the test data without noise that we report here. The spacegroupMining model
is apparently also quite robust against the effects of measurement noise.

##  5 Conclusions

The use of deep learning to do complex classifications from data is a
potentially useful approach that is becoming more widespread in materials
science, crystallography and diffraction. Inherent in the process is that the
model was trained on a particular set of data and its applicability to do the
classification on data that is, in some way, different, for example, measured
with different resolutions or over different ranges, might limit its ability
to make accurate predictions. In general, the model may be retrained on a
wider set of data that incorporates cases of different ranges, resolutions and
so on. However, here, for the case of the spacegroupMining model that is
deployed on pdfitc.org, we simply explored its robustness in making accurate
predictions on different range and resolution data without retraining the
model. The main result is that the model is quite robust and performs well
without having to be retrained in most cases. Modest reductions in prediction
accuracy were observed, but it still performed well given a rather wide, but
reasonable, range of resolution and range parameters, suggesting that it is
not of great urgency to retrain it. We note that retraining it with a more
diverse set of training data, whilst increasing accuracy for parameter values
away from the original training values, it may decrease the prediction
accuracy for PDFs with the original set of parameter values, where those
values were chosen as being somewhat representative of values in many rapid
acquisition x-ray PDF studies. Through this work, it has been shown that,
without additional retraining, the spacegroupMining@pdfitc model still
performs with reasonable accuracy for a relatively wide range of experimental
parameters, and can thus be used as a robust computational tool.

Funding information

This work in the Billinge group was supported by the U.S. National Science
Foundation through grant DMREF-1922234 and CCF-1704833.

## References

  * [1] [Belsky _et al._]Belsky, Hellenbrandt, Karen Luksch2002belsky;acb02 Belsky, A., Hellenbrandt, M., Karen, V. L. Luksch, P. 2002\.  _Acta Crystallographica Section B Structural Science_ , 58(3), 364–369.
  * [2] Billinge, S. J. L.2019billi;b;itoch19 Billinge, S. J. L. 2019.  In _Nanometre-Scale Structure from Powder Diffraction: Total Scattering and Atomic Pair Distribution Function Analysis_ , edited by C. Gilmore _et al._ , vol. H. Buffalo, NY, USA: International Union of Crystallography.
  * [3] Egami Billinge2012egami;b;utbp12 Egami, T. Billinge, S. J. L. 2012.  _Underneath the Bragg Peaks: Structural Analysis of Complex Materials_.  No. 16 in Pergamon Materials Series. Amsterdam: Elsevier, 2nd ed.
  * [4] [Hall _et al._]Hall, Allen Brown1991hall;aca91 Hall, S. R., Allen, F. H. Brown, I. D. 1991.  _Acta Crystallographica Section A: Foundations of Crystallography_ , 47(6), 655–685.
  * [5] [Juhás _et al._]Juhás, Farrow, Yang, Knox Billinge2015juhas;aca15 Juhás, P., Farrow, C., Yang, X., Knox, K. Billinge, S. 2015\.  _Acta Crystallographica Section A: Foundations and Advances_ , 71(6), 562–568.
  * [6] [Liu _et al._]Liu, Tao, Hsu, Du Billinge2019liu;aca19 Liu, C.-H., Tao, Y., Hsu, D., Du, Q. Billinge, S. J. L. 2019\.  _Acta Crystallographica Section A: Foundations and Advances_ , 75(4), 633–643.
  * [7] Proffen Billinge1999proff;jac99 Proffen, T. Billinge, S. J. L. 1999.  _J. Appl. Crystallogr._ 32, 572–575.
  * [8] [Yang _et al._]Yang, Culbertson, Thomas, Vuong, Kjær, Jensen, Tucker Billinge2021yang;aca21 Yang, L., Culbertson, E. A., Thomas, N. K., Vuong, H. T., Kjær, E. T. S., Jensen, K. M. Ø., Tucker, M. G. Billinge, S. J. L. 2021\.  _Acta Crystallographica Section A: Foundations and Advances_ , 77(1), 2–6.
  * [9]

Supplemental Materials

Table 2: Top-6 accuracy and top-1 accuracy when $r_{\text{max}}$ is chosen from 10 Å to 30 Å. $r_{\text{max}}$ (Å) | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | 30
---|---|---|---|---|---|---|---|---|---|---|---
Top-6 accuracy | 0.648 | 0.717 | 0.747 | 0.778 | 0.820 | 0.871 | 0.882 | 0.890 | 0.901 | 0.901 | 0.902
Top-1 accuracy | 0.285 | 0.367 | 0.433 | 0.449 | 0.511 | 0.552 | 0.600 | 0.617 | 0.652 | 0.671 | 0.677
Table 3: Top-6 accuracy and top-1 accuracy when $Q_{\text{max}}$ is chosen from 12 Å-1 to 30 Å-1. $Q_{\text{max}}$ (Å-1) | 12 | 15 | 18 | 21 | 23 | 24 | 27 | 30
---|---|---|---|---|---|---|---|---
Top-6 accuracy | 0.777 | 0.795 | 0.811 | 0.822 | 0.902 | 0.900 | 0.805 | 0.84
Top-1 accuracy | 0.516 | 0.591 | 0.597 | 0.604 | 0.677 | 0.663 | 0.598 | 0.610
Table 4: Top-6 accuracy and top-1 accuracy when $Q_{\text{damp}}$ is chosen from 0 Å-1 to $0.15$ Å-1. $Q_{\text{damp}}$ (Å-1) | 0 | 0.03 | 0.04 | 0.06 | 0.09 | 0.12 | 0.15
---|---|---|---|---|---|---|---
Top-6 accuracy | 0.861 | 0.897 | 0.902 | 0.879 | 0.741 | 0.648 | 0.578
Top-1 accuracy | 0.602 | 0.659 | 0.677 | 0.579 | 0.390 | 0.294 | 0.234
Table 5: Top-6 accuracy and top-1 accuracy when ADP is chosen from 0.005 Å2 to 0.01 Å2. adp (Å2) | 0.005 | 0.006 | 0.007 | 0.008 | 0.009 | 0.01
---|---|---|---|---|---|---
Top-6 accuracy | 0.873 | 0.895 | 0.904 | 0.902 | 0.900 | 0.900
Top-1 accuracy | 0.618 | 0.649 | 0.664 | 0.677 | 0.666 | 0.641