# Tunable Feshbach resonances in collisions of ultracold molecules in
${}^{2}\Sigma$ states
with alkali-metal atoms

Robert C. Bird  Joint Quantum Centre (JQC) Durham-Newcastle, Department of
Chemistry, Durham University, South Road, Durham, DH1 3LE, United Kingdom.
Michael R. Tarbutt  Centre for Cold Matter, Blackett Laboratory, Imperial
College London, Prince Consort Road, London SW7 2AZ UK  Jeremy M. Hutson
<EMAIL_ADDRESS>Joint Quantum Centre (JQC) Durham-Newcastle,
Department of Chemistry, Durham University, South Road, Durham, DH1 3LE,
United Kingdom.

(August 28, 2024)

###### Abstract

We consider the magnetically tunable Feshbach resonances that may exist in
ultracold mixtures of molecules in ${}^{2}\Sigma$ states and alkali-metal
atoms. We focus on Rb+CaF as a prototype system. There are likely to be
Feshbach resonances analogous to those between pairs of alkali-metal atoms. We
investigate the patterns of near-threshold states and the resonances that they
cause, using coupled-channel calculations of the bound states and low-energy
scattering on model interaction potentials. We explore the dependence of the
properties on as-yet unknown potential parameters. There is a high probability
that resonances will exist at magnetic fields below 1000 G, and that these
will be broad enough to control collisions and form triatomic molecules by
magnetoassociation. We consider the effect of CaF rotation and potential
anisotropy, and conclude that they may produce additional resonances but
should not affect the existence of rotation-free resonances.

##  I Introduction

Ultracold molecules have many applications that are now emerging, ranging from
quantum simulation [1, 2], quantum computing [3, 4, 5], the study of novel
quantum phases [6, 7], and tests of fundamental physics [8, 9, 10]. Key to
most of these applications are polar molecules, which can have long-range
anisotropic interactions resulting from their permanent dipoles. Many such
molecules have been produced at microkelvin temperatures by association of
pairs of alkali-metal atoms, followed by laser transfer to the vibrational
ground state [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Another class of
molecules, exemplified by CaF and SrF, have been cooled directly by magneto-
optical trapping followed by sub-Doppler laser cooling [21, 22, 23, 24, 25,
26].

Elastic and inelastic collisions are at the heart of ultracold physics. For
ultracold atoms, it is often possible to control ultracold collisions by
adjusting an applied magnetic field close to a zero-energy Feshbach resonance
[27]. Such a resonance occurs whenever a molecular bound state can be tuned
across a scattering threshold as a function of applied field. The s-wave
scattering length then passes through a pole as a function of field, allowing
the effective interaction strength to be tuned to any desired value. This
control has been applied in many areas of ultracold physics, including
condensate collapse [28], soliton creation [29], Efimov physics [30] and
investigations of the BCS-BEC crossover in degenerate Fermi gases [31].
Feshbach resonances are also used for magnetoassociation, in which pairs of
ultracold atoms are converted to weakly bound diatomic molecules by sweeping a
magnetic field across the resonance [32, 33].

Much new physics will become accessible when molecular collisions can be
controlled with tunable Feshbach resonances. Such resonances have now been
observed in collisions between ultracold 40K atoms and 23Na40K molecules in
singlet states [18, 34, 35, 36, 37] and between 23Na atoms and 6Li23Na
molecules in triplet states [38]. These systems have also been investigated
theoretically [34, 39, 38, 40]. Resonances have not yet been observed in
collisions of laser-cooled molecules such as CaF and SrF, with ${}^{2}\Sigma$
ground states, but we have recently succeeded in making ultracold mixtures of
CaF molecules and Rb atoms, and studied their inelastic collisions in both
magnetic traps [41] and magneto-optical traps [42]. Several laser-coolable
molecules have been cooled to 5 $\mu$K [24, 25] and confined in optical traps
[23] and optical tweezers [43], opening the way to experiments in controlled
magnetic fields.

The purpose of the present paper is to investigate the resonances that are
expected in collisions between molecules in ${}^{2}\Sigma$ states and alkali-
metal atoms. These systems have strong similarities to pairs of alkali-metal
atoms, particularly for the long-range states that are most likely to cause
magnetically tunable Feshbach resonances. There are additional complications
and additional resonances that arise from the rotational structure of the
molecule and the anisotropy of the interaction potential, but we find that
these are unlikely to affect the general features of the scattering. We focus
on 87Rb+40Ca19F as a prototype system, but many of the features are
transferable to other molecules such as SrF and other alkali-metal atoms. In
the following we mostly omit isotopic masses and write Rb+CaF for
87Rb+40Ca19F.

The structure of the paper is as follows. Section II describes the underlying
theory, including monomer Hamiltonians, interaction potentials, and
computational methods. Section III.1 describes the near-threshold levels that
can exist for Rb+CaF and the Feshbach resonances they can cause, using a
simple model that omits rotational degrees of freedom. Section IV considers
the effects of CaF rotation and potential anisotropy. Section V considers the
possible effects of quantum chaos at short range. Section VI presents
conclusions and offers perspectives for future work to take advantage of the
resonances.

##  II Theory

###  II.1 Monomer Hamiltonians and levels

The Hamiltonian of an alkali-metal atom A in its ground 2S state is

$\hat{h}_{\textrm{A}}=\zeta_{\textrm{A}}\bm{\hat{i}}_{\textrm{A}}\cdot\bm{\hat{s}}_{\textrm{A}}+\left(g_{s,\textrm{A}}\bm{\hat{s}}_{\textrm{A},z}+g_{i}\hat{i}_{\textrm{A},z}\right)\mu_{\textrm{B}}B,$
(1)

where $\bm{\hat{s}}_{\textrm{A}}$ and $\bm{\hat{i}}_{\textrm{A}}$ are vector
operators for the electron and nuclear spin, $\hat{s}_{\textrm{A},z}$ and
$\hat{i}_{\textrm{A},z}$ are their components along the $z$ axis defined by
the magnetic field $B$, $\zeta_{\textrm{A}}$ is the hyperfine coupling
constant, and $g_{s,\textrm{A}}$ and $g_{i}$ are the g-factors for the
electron and nuclear spins 111In writing basis sets for pairs of atoms, it is
necessary to distinguish between quantum numbers for the individual atoms and
those for the pair. We adopt the widely used convention of using lower-case
letters for the individual atoms and upper-case letters for the pair.. The
nuclear spins vary from 1 for 6Li to 9/2 for 40K, and the hyperfine splittings
$A_{\textrm{hfs}}=\zeta_{\textrm{A}}(i_{\textrm{A}}+\frac{1}{2})/h$ vary from
228 MHz for 6Li to 9.19 GHz for 133Cs. We focus here on 87Rb, $i=3/2$ and
$A_{\textrm{hfs}}\approx 6.83$ GHz. The resulting levels are well known, but
are shown in Fig. 1(a) for convenience. At zero field the levels are labeled
by total angular momentum $f_{\textrm{Rb}}=1$ and 2. When a field is applied,
each level splits into $2f_{\textrm{Rb}}+1$ sublevels, color-coded according
to the projection $m_{f,\textrm{Rb}}$. At sufficiently high field, pairs of
levels with $f_{\textrm{Rb}}=1$ and 2 but the same value of
$m_{f,\textrm{Rb}}$ mix sufficiently that the levels are better described by
$m_{s,\textrm{Rb}}$ and $m_{i,\textrm{Rb}}$ than by $f_{\textrm{Rb}}$. For
87Rb this transition is still incomplete at 2000 G, but it occurs at much
lower fields for alkali-metal atoms with small hyperfine splittings, such as
Li and Na.

Figure 1: Energies as a function of magnetic field for (a) 87Rb atom in ground
2S state; (b) Lowest two rotational levels of CaF, with expanded views of
$n=0$ and 1 in (c) and (d), respectively; (e) Scattering thresholds of
87Rb+CaF, with expanded views of $(f_{\textrm{Rb}},n)=(1,0)$, (2,0), (1,1) and
(2,1) in (f) and (g), (h) and (i), respectively. All level energies are shown
relative to the ground state at zero field and are color-coded as shown in the
legend according to $m_{f,\textrm{Rb}}$, $m_{f,\textrm{CaF}}$ or
$M_{F}=m_{f,\textrm{Rb}}+m_{f,\textrm{CaF}}$, as appropriate; negative values
are indicated by dashed lines.

The CaF or SrF molecule may be treated at different levels of complexity. The
stable isotopes 40Ca, 88Sr, 86Sr and 84Sr all have zero nuclear spin, while
87Sr has $i=9/2$; only the spin-zero isotopes will be considered here. The
simplest useful approximation is to neglect the molecular rotation, and in
this case the molecular Hamiltonian $\hat{h}_{\textrm{CaF}}^{n=0}$ is the same
as Eq. 1, with $i_{\textrm{F}}=1/2$ for 19F in CaF. However, when rotation is
included, several extra terms are needed. The ones important here are

$\hat{h}_{\mathrm{CaF}}^{\mathrm{rfhf}}=b_{0}\hat{\bm{n}}^{2}+\gamma\hat{\bm{s}}_{\textrm{CaF}}\cdot\hat{\bm{n}}+t\sqrt{6}T^{2}(C)\cdot
T^{2}(\hat{\bm{i}}_{\textrm{F}},\hat{\bm{s}}_{\textrm{CaF}}),$ (2)

where $\hat{\bm{n}}$ is the vector operator for the molecular rotation. The
first term represents the rotational energy, the second represents the
electron spin-rotation interaction, and the third accounts for the anisotropic
interaction between electron and nuclear spins:
$T^{2}(\bm{\hat{i}},\bm{\hat{s}})$ is the rank-2 spherical tensor formed from
$\bm{\hat{i}}$ and $\bm{\hat{s}}$, and $T^{2}(C)$ is a spherical tensor whose
components are the Racah-normalized spherical harmonics
$C^{2}_{q}(\theta,\phi)$ involving the orientation of the molecular axis.
Values of $b_{0}/h\approx 10.3$ GHz, $\gamma/h\approx 40$ MHz,
$\zeta_{\textrm{F}}/h\approx 120$ MHz and $t/h\approx 14$ MHz are taken from
ref. [45] 222Ref. [45] uses the notation of Frosch and Foley [78], where our
$b_{0}$, $\gamma$, $\zeta_{\textrm{F}}$ and $t$ are $B$, $\gamma$, $b+c/3$ and
$c/3$, respectively.. A more complete version of Eq. 2, including additional
contributions of the order of kHz that are unimportant here, has been given in
ref. [47].

The full Hamiltonian for CaF is
$\hat{h}_{\mathrm{CaF}}=\hat{h}_{\mathrm{CaF}}^{n=0}+\hat{h}_{\mathrm{CaF}}^{\mathrm{rfhf}}$.
The resulting level diagram is shown as a function of magnetic field in Fig.
1(b), with expanded views for $n=0$ and 1 in Figs. 1(c) and (d). There are
only very small matrix elements that are off-diagonal in $n$, so the levels
for $n=0$ are very similar to those of an alkali-metal atom with
$i_{\textrm{F}}=1/2$. The hyperfine splitting is small, so $i_{\textrm{F}}$
and $s_{\textrm{CaF}}$ are mostly decoupled by 50 G. At higher field, the
states are well described by $m_{s,\textrm{CaF}}$ and $m_{i,\textrm{F}}$.

In a rotating molecule at low field, $i_{\textrm{F}}$ and
$s_{\textrm{CaF}}=1/2$ couple to give a resultant $g=0$ or 1, and $g$ couples
to the rotational angular momentum $n$ to produce the total molecular angular
momentum $f_{\textrm{CaF}}$. For $n=1$, there are zero-field levels with
$f_{\textrm{CaF}}=0$, 1, 1, 2, as labeled on Fig. 1(d). The lower level with
$f=1$ is predominantly $g=0$ and the remaining three are predominantly $g=1$.
In a magnetic field, however, $i_{\textrm{F}}$, $s_{\textrm{CaF}}$ and $n$ are
again mostly decoupled by 50 G; at higher fields, the states are better
described by $m_{s,\textrm{CaF}}$, $m_{i,\textrm{F}}$ and $m_{n}$ than by $g$
and $f_{\textrm{CaF}}$. States of different $m_{s,\textrm{CaF}}$ are well
separated; within the group for a particular value of $m_{s,\textrm{CaF}}$,
there are 2 subgroups with $m_{i,\textrm{F}}=\pm\frac{1}{2}$, with splitting
about $\zeta/2=60$ MHz, and each subgroup is further divided into states with
different $m_{n}$, with adjacent states separated by about $\gamma/2=20$ MHz.
The projection quantum numbers are not fully conserved, but these qualitative
arguments help to understand the general patterns at high field.

###  II.2 Calculations of bound states and scattering

The Hamiltonian for an alkali-metal atom interacting with a CaF molecule is

$\hat{H}=\frac{\hbar^{2}}{2\mu}\left(-R^{-1}\frac{d^{2}}{dR^{2}}R+\frac{\hat{\bm{L}}^{2}}{R^{2}}\right)+\hat{h}_{\textrm{A}}+\hat{h}_{\textrm{CaF}}+\hat{V}_{\textrm{int}},$
(3)

where $R$ is the intermolecular distance, $\mu$ is the reduced mass,
$\hat{\bm{L}}^{2}$ is the operator for relative rotation of the pair and
$\hat{V}_{\textrm{int}}$ is the interaction operator described below. We carry
out calculations of both bound states and scattering using coupled-channel
methods [48, 49, 27]. The total wavefunction is expanded as

$\Psi(R,\xi)=R^{-1}\sum_{j}\Phi_{j}(\xi)\psi_{j}(R).$ (4)

Here $\\{\Phi_{j}(\xi)\\}$ is a set of basis functions that span all
coordinates except $R$, including the relative rotation; these coordinates are
collectively designated $\xi$. In the coupled-channel calculations describe in
Sec. III.1, $\xi$ includes only electron and nuclear spins. However, in more
complete treatments, it may also include basis functions for overall rotation
of the collision complex and rotation and vibration of CaF.

Substituting the expansion (4) into the total Schrödinger equation produces a
set of coupled differential equations that are solved by propagation with
respect to the internuclear distance $R$. The coupled equations are identical
for bound states and scattering, but the boundary conditions are different.

Scattering calculations are performed with the molscat package [50, 51]. Such
calculations produce the scattering matrix $\bm{S}$, for a single value of the
collision energy and magnetic field each time. The complex s-wave scattering
length $a(k_{0})$ is obtained from the diagonal element of $\bm{S}$ in the
incoming channel, $S_{00}$,

$a(k_{0})=\frac{1}{ik_{0}}\left(\frac{1-S_{00}(k_{0})}{1+S_{00}(k_{0})}\right),$
(5)

where $k_{0}$ is the incoming wavenumber, related to the collision energy
$E_{\textrm{coll}}$ by $E_{\textrm{coll}}=\hbar^{2}k_{0}^{2}/(2\mu)$. The
scattering length $a(k_{0})$ becomes constant at sufficiently low
$E_{\textrm{coll}}$, with limiting value $a$. In the present work, s-wave
scattering lengths are calculated at $E_{\textrm{coll}}/k_{\textrm{B}}=10$ nK,
which is low enough to neglect the dependence on $k_{0}$.

A zero-energy Feshbach resonance occurs where a bound state of the atom-
molecule pair (triatomic molecule) crosses a scattering threshold as a
function of applied field. At the lowest threshold, or in the absence of
inelastic processes, the scattering length is real. Near a resonance, $a(B)$
passes through a pole, and is approximately

$a(B)=a_{\textrm{bg}}\left(1-\frac{\Delta}{B-B_{\textrm{res}}}\right),$ (6)

where $B_{\textrm{res}}$ is the position of the resonance, $\Delta$ is its
width, and $a_{\textrm{bg}}$ is a slowly varying background scattering length.
In the presence of inelastic processes, $a(B)$ is complex and the pole is
replaced by an oscillation [52]. molscat can converge on Feshbach resonances
automatically and characterize them to obtain $B_{\textrm{res}}$, $\Delta$ and
$a_{\textrm{bg}}$ (and the additional parameters needed in the presence of
inelasticity) as described in ref. 53.

Coupled-channel bound-state calculations are performed using the packages
bound and field [54, 55], which converge upon bound-state energies at fixed
field, or bound-state fields at fixed energy, respectively. The methods used
are described in ref. 56.

In the present work, the coupled equations for both scattering and bound-state
calculations are solved using the fixed-step symplectic log-derivative
propagator of Manolopoulos and Gray [57] from $R_{\textrm{min}}=3\ a_{0}$ to
$R_{\textrm{mid}}=15\ a_{0}$, with an interval size of $0.001\ a_{0}$, and the
variable-step Airy propagator of Alexander and Manolopoulos [58] between
$R_{\textrm{mid}}$ and $R_{\textrm{max}}$, where $R_{\textrm{max}}=300\ a_{0}$
for bound and field and $3,000\ a_{0}$ for molscat.

###  II.3 The interaction operator

Rb(2S) and CaF(${}^{2}\Sigma$) interact to give two electronic surfaces of 1A′
and 3A′ symmetry. These are to some extent analogous to the singlet and
triplet curves of alkali-metal dimers: the singlet surface is expected to be
deep, and the triplet surface much shallower. The surfaces have not been
characterized in any detail, either experimentally or theoretically, but both
of them are expected to be strongly anisotropic at short range. We designate
them $V^{S}(R,\theta)$, with $S=0$ for the singlet and $S=1$ for the triplet.
Here $\theta$ is the angle between the CaF bond and the intermolecular axis in
Jacobi coordinates. The interaction operator is

$\hat{V}_{\textrm{int}}=V^{0}(R,\theta)\hat{\cal
P}^{0}+V^{1}(R,\theta)\hat{\cal P}^{1}+\hat{V}^{\textrm{d}},$ (7)

where $\hat{\cal P}^{0}$ and $\hat{\cal P}^{1}$ are projection operators onto
the singlet and triplet spin spaces, respectively, and $\hat{V}^{\textrm{d}}$
is a small electron spin-spin term described below.

The Feshbach resonances of interest here depend mostly on the properties of
near-threshold states. These are bound by amounts comparable to the hyperfine
and Zeeman splittings of Rb and CaF and (to a lesser extent) the low-lying
rotational states of CaF. The most important states are those with binding
energies less than about 30 GHz below their respective thresholds; this is
considerably less than 0.1% of the expected singlet well depth. The binding
energies of these states are dependent mostly on long-range dispersion and
induction forces, which are the same for the singlet and triplet surfaces. The
leading term is of the form

$V^{S}(R,\theta)=\left[-C_{6}^{(0)}-C_{6}^{(2)}P_{2}(\cos\theta)\right]R^{-6},$
(8)

with $C_{6}^{(0)}\approx 3084\ E_{\textrm{h}}a_{0}^{6}$ [59]. For
$C_{6}^{(2)}$ there is substantial cancelation between the dispersion and
induction contributions; we estimate $C_{6}^{(2)}\approx 100(20)\
E_{\textrm{h}}a_{0}^{6}$. For Rb+CaF, the outer turning point at a binding
energy of 30 GHz is near 30 $a_{0}$.

Potential terms that are the same for the singlet and triplet surfaces cannot
cause couplings between orthogonal spin states. They are therefore unlikely to
cause magnetically tunable Feshbach resonances. The most important
interactions that mix different spin states are spin-exchange interactions,
due to the difference between the singlet and triplet surfaces. Julienne _et
al._ [60] have shown that, for a pair of atoms, spin-exchange interactions can
cause nonadiabatic transitions between coupled channels at distances
$R_{\textrm{X}}$ where the interaction approximately matches the asymptotic
energy difference between the channels concerned. For 87Rb this occurs around
$22\ a_{0}$ [60]. The strength of the interaction is modulated by overall
phases due to the short-range parts of the potentials for the channels
concerned, and (if the long-range potentials are identical from
$R_{\textrm{X}}$ to $\infty$) is smallest when the two channels have the same
scattering length.

There is also a spin-spin term $\hat{V}^{\textrm{d}}$ in the interaction
operator that results from magnetic dipole-dipole interactions between the
electron spins on Rb and CaF, supplemented at short range by second-order
spin-orbit terms that have the same overall dependence on spin coordinates.
This term is important for heavy alkali-metal atoms such as Cs [61], and may
cause additional weak resonances in Rb+CaF as discussed below, but its effect
is not considered in detail in the present work.

###  II.4 Thresholds

Figure 1(e) shows the scattering thresholds for 87Rb+CaF, which are simply
sums of energies of Rb and CaF. Figures (f) to (i) show expanded views of each
group. The thresholds are color-coded according to
$M_{F}=m_{f,\textrm{Rb}}+m_{f,\textrm{CaF}}$, because this quantity is
conserved in collisions if anisotropic terms in $V_{\textrm{int}}$ are
neglected.

The importance of the thresholds lies in the fact that near-threshold levels
lie approximately parallel to them, within well-defined energy intervals known
as bins. This concept will be used extensively in discussing the patterns of
near-threshold levels and the resulting resonances in the following sections.

###  II.5 Near-threshold levels

Each scattering threshold $j$ supports a series of levels of the collision
complex that have binding energies $E^{\textrm{b}}_{j\eta}(B)$ below the
threshold concerned. Here $\eta$ is a vibrational quantum number, defined so
that the least-bound rotationless state below each threshold is labelled
$\eta=-1$ and successively deeper levels are labeled $-2$, $-3$, etc. To a
first approximation, the near-threshold levels retain the character of the
threshold that supports them. Because of this, each level lies approximately
parallel to the threshold that supports it and may be described in a single-
channel approximation. There are nevertheless interactions between levels
supported by different channels $j$, which cause $B$-dependent shifts and
avoided crossings between levels. These interactions, and the strengths of the
resulting avoided crossings, generally become larger as $|\eta|$ increases;
these will be discussed below.

For a single-channel system with an asymptotic potential $-C_{6}R^{-6}$, the
least-bound s-wave state (with $L=0$ and $\eta=-1$) lies within $\sim
36\bar{E}$ of threshold, where $\bar{E}=\hbar^{2}/(2\mu\bar{a}^{2})$ and
$\bar{a}$ is the mean scattering length of Gribakin and Flambaum [62],
$\bar{a}=(2\mu C_{6}/\hbar^{2})^{1/4}\times 0.4779888\dots$. We refer to this
energy interval as the top bin. The position of the bound state within this
bin depends on the background scattering length $a_{\textrm{bg}}$ for the
channel concerned, neglecting resonances (which themselves arise from
couplings between channels). Each subsequent level $(\eta=-2$, $-3$, etc.)
lies within its own bin, with successive bins becoming wider and bin
boundaries at energies roughly proportional to $|\eta+\frac{1}{8}|^{3}$ [63,
64]. For Rb+CaF, $\bar{a}=67.3\ a_{0}$, $\bar{E}/h=11.4$ MHz, and the first 5
bin boundaries are at about 410, 2900, 9100, 21000 and 40000 MHz. These values
may be shifted by the influence of terms beyond $-C_{6}R^{-6}$. In general,
the levels lie near the top of their bins when $a_{\textrm{bg}}\gg\bar{a}$ and
towards the bottom of the bins for $a_{\textrm{bg}}\ll\bar{a}$.

Figure 2:  Near-threshold levels of Rb+CaF with $M_{F}=1$, neglecting
anisotropy, shown relative to the energy of the lowest threshold, for four
representative combinations of the singlet and triplet scattering lengths.
Solid black lines show results from coupled-channel calculations. Dashed (dot-
dashed) lines show uncoupled states parallel to thresholds with
$f_{\textrm{Rb}}=1$ (2). Values of $m_{s,\textrm{CaF}}$ are encoded with red
(blue) for $\frac{1}{2}$ ($-\frac{1}{2})$, with darker (lighter) colors for
$m_{i,\textrm{F}}=\frac{1}{2}$ ($-\frac{1}{2}$). $m_{f,\textrm{Rb}}$ is given
by $M_{F}-m_{s,\textrm{CaF}}-m_{i,\textrm{F}}$. Above each plot of energies is
the corresponding plot of scattering length, with Feshbach resonances where
states cross threshold.

##  III Bound states and resonances in the absence of anisotropy

###  III.1 Bound states below the lowest threshold

The coupling between CaF rotational levels is fairly small at long range. It
is driven mostly by the anisotropic part of the long-range interaction
potential, characterized by $C_{6}^{(2)}$. The effects of the anisotropy will
be considered in Section IV. In this section we will consider a simpler model,
with the anisotropy neglected. This is expected to be a reasonably good
approximation for collisions involving CaF ($n=0$), though it will neglect
some additional resonances considered later.

If anisotropy is neglected, the scattering is largely controlled by the
isotropic dispersion coefficient $C_{6}^{(0)}$ and by scattering lengths
$a_{\textrm{s}}$ and $a_{\textrm{t}}$ that characterize overall phases due to
the short-range parts of the singlet and triplet potentials. These scattering
lengths are completely unknown for Rb+CaF, so we explore the pattern of near-
threshold bound states, and the resulting Feshbach resonances, for a
representative sample of values of them.

Scattering lengths take values from $-\infty$ to $+\infty$, but some values
are more likely than others [62]. The most likely value is the mean scattering
length $\bar{a}$ defined above, and for a randomly chosen potential curve that
decays as $-C_{6}R^{-6}$ at long range there is a 50% probability of a
scattering length between $0$ and $2\bar{a}$. To a good approximation,
different interaction potentials that produce the same $a_{\textrm{s}}$ and
$a_{\textrm{t}}$, and have the same value of $C_{6}$, have the same low-energy
scattering properties and near-threshold bound states.

We use singlet and triplet potential curves based on those for Cs [61], but
with the value of $C_{6}$ replaced with $C_{6}^{(0)}$ for Rb+CaF. These
potentials are then adjusted at short range to give the desired scattering
length as described in ref. [61]. As an initial sample, we pick 3 values
$a_{\textrm{s}}=-79$, 71 and 242 $a_{0}$ and $a_{\textrm{t}}=-47$, 86 and 297
$a_{0}$. These are purposely not exact multiples of $\bar{a}$, because such
values can produce shape resonances at atypically low energy, and are slightly
different for $a_{\textrm{s}}$ and $a_{\textrm{t}}$, because
$a_{\textrm{s}}=a_{\textrm{t}}$ is a special case that produces unusually weak
interchannel couplings [60]. We consider all 9 combinations of these values of
$a_{\textrm{s}}$ and $a_{\textrm{t}}$.

The solid lines in Figure 2 shows the near-threshold energy levels for 4
combinations of $a_{\textrm{s}}$ and $a_{\textrm{t}}$, obtained from coupled-
channel calculations using the package bound. In this case we use a basis set
of fully uncoupled functions [65], including only rotationless functions,
$n=0$ and $L=0$. All energies are shown with respect to the (field-dependent)
energy of the lowest threshold, which has approximate quantum numbers
$(f_{\textrm{Rb}},m_{f,\textrm{Rb}},m_{s,\textrm{CaF}},m_{i,\textrm{F}})=(1,1,-\frac{1}{2},\frac{1}{2})$
at fields above 50 G. All states shown have $M_{F}=1$, which is the same as
the lowest threshold, because spin-exchange interactions cannot change
$M_{F}$. Also shown are dashed and dot-dashed lines, parallel to thresholds
but offset from them: these represent hypothetical states that would exist in
the absence of interchannel couplings; the real states may be interpreted in
terms of these, but with shifts and avoided crossings of various strengths due
to the couplings.

The simplest case is that in Fig. 2(a), for $a_{\textrm{s}}=71\ a_{0}$ and
$a_{\textrm{t}}=86\ a_{0}$. Here the real states lie close to the uncoupled
ones, with only small shifts and weak avoided crossings. $a_{\textrm{s}}$ and
$a_{\textrm{t}}$ are close to one another, so the interchannel coupling is
weak, and they are comparable to $\bar{a}$, so each state lies fairly high in
its bin. The near-horizontal states are those supported by the lowest
threshold in the first and second bins. There is also a pair of states that
originate near $-1.0$ GHz at zero field; the thresholds that support these
have approximate quantum numbers $(1,0,\frac{1}{2},\frac{1}{2})$ (upper) and
$(1,1,\frac{1}{2},-\frac{1}{2})$ (lower). In first order the spin-exchange
coupling can change $f_{\textrm{Rb}}$, $m_{f,\textrm{Rb}}$ and
$m_{s,\textrm{CaF}}$ by $\pm 1$ while conserving
$m_{f,\textrm{Rb}}+m_{s,\textrm{CaF}}$, but cannot change $m_{i,\textrm{F}}$.
There is therefore a much wider avoided crossing between the near-threshold
horizontal state and the upper state of the sloping pair, which is
predominantly $m_{i,\textrm{F}}=\frac{1}{2}$, than with the lower one, which
is predominantly $m_{i,\textrm{F}}=-\frac{1}{2}$.

A case with somewhat stronger coupling is shown in Fig. 2(b), for
$a_{\textrm{s}}=-79\ a_{0}$ and $a_{\textrm{t}}=-47\ a_{0}$. Here
$a_{\textrm{s}}$ and $a_{\textrm{t}}$ are negative, so the states lie much
deeper in their bins than in (a). The real states still lie close to the
uncoupled ones, but there is a strong avoided crossing between the states
shown as red dashed and blue dot-dashed lines. States approximately parallel
to the thresholds with $f_{\textrm{Rb}}=1$ can again be identified, with the
states in the second bin now originating from around $-2.3$ GHz at zero field.
These are echoed by similar states in the top bin. However, there are two
further pairs of states; these are supported by thresholds with
$f_{\textrm{Rb}}=2$, and lie in the third bin beneath their thresholds. The
pair originating near $-1.6$ GHz have approximate quantum numbers
$(2,1,-\frac{1}{2},\frac{1}{2})$ (lower, involving ground-state CaF but
excited Rb) and $(2,2,-\frac{1}{2},-\frac{1}{2})$ (upper), while the pair
originating near $-1.2$ GHz have $(2,0,\frac{1}{2},\frac{1}{2})$ (lower) and
$(2,1,\frac{1}{2},-\frac{1}{2})$ (upper). Once again the strong avoided
crossings are those between states with the same values of $m_{i,\textrm{F}}$.

Figs. 2(c) and (d) show further examples for cases with much stronger
coupling, with $a_{\textrm{s}}$ and $a_{\textrm{t}}$ substantially different.
For these cases the identification of the dashed and dotted lines is less
certain, because real states are substantially shifted from the uncoupled
states by interchannel couplings. Plausible assignments are shown with the
same coding as in (a) and (b).

Additional bound states exist with $M_{F}\neq 1$. These are not connected to
the lowest incoming threshold by spin-exchange coupling. However, there are
additional small couplings due to the spin-spin interaction
$\hat{V}^{\textrm{d}}$. This has matrix elements off-diagonal in
$f_{\textrm{Rb}}$, $m_{f,\textrm{Rb}}$ and $m_{s,\textrm{CaF}}$ by $\pm 1$,
but can change $m_{f,\textrm{Rb}}+m_{s,\textrm{CaF}}$ by up to $\pm 2$, with
$M_{L}$ changing by up to $\mp 2$ to conserve $M_{\textrm{tot}}=M_{F}+M_{L}$.
Rotationally excited states with $L=2$ and $M_{F}$ from $-1$ to 3 can
therefore cause additional resonances at the lowest threshold. These are
expected to be narrow, and are not included in the present calculations
because there is no information available on the strength of second-order
spin-orbit coupling for Rb+CaF. States with other values of $L$ and $M_{F}$
might in principle cause resonances, but with higher-order coupling via
$\hat{V}^{\textrm{d}}$, so the resonances will be even narrower.

###  III.2 Resonances

Each bound state with $M_{F}=1$ causes a magnetically tunable Feshbach
resonance where it crosses threshold as a function of $B$. For all the cases
considered, several such resonances exist at fields below 1000 G. However,
their widths vary greatly. Figure 2 includes a panel above each energy-level
plot that shows the variation of scattering length with magnetic field. In
addition, we have characterized the resonances to extract $B_{\textrm{res}}$,
$\Delta$ and $a_{\textrm{bg}}$ for all resonances below 1000 G for all 9 of
our representative combinations of $a_{\textrm{s}}$ and $a_{\textrm{t}}$,
using the method of ref. [53], and the results are given in Table 1.

The resonance widths may be rationalized using the same arguments about
interchannel couplings used to interpret the strength of avoided crossings in
Section III.1. First, the resonances are generally broadest in cases where
$a_{\textrm{s}}$ and $a_{\textrm{t}}$ are substantially different, providing
strong spin-exchange coupling. Secondly, for any given combination of
$a_{\textrm{s}}$ and $a_{\textrm{t}}$, the strongest resonances are those
where the bound state causing the resonance has a substantial component with
the same value of $m_{i,\textrm{F}}$ as the incoming channel, which for the
lowest threshold is dominated by $m_{i,\textrm{F}}=\frac{1}{2}$ at fields
above 50 G. The specific uncoupled states that cause the widest resonances are
$(1,0,\frac{1}{2},\frac{1}{2})$ $(2,1,-\frac{1}{2},\frac{1}{2})$ and
$(2,0,\frac{1}{2},\frac{1}{2})$, though in some cases their character is
spread across more than one real state.

It is noteworthy that, even when $a_{\textrm{s}}\approx a_{\textrm{t}}$ and
spin-exchange coupling is weak, there are resonances that are wide enough to
use to control collisions or form triatomic molecules by magnetoassociation.

Table 1: Feshbach resonance positions, widths and background scattering lengths for different combinations of $a_{\textrm{s}}$ and $a_{\textrm{t}}$. The approximate quantum numbers of the uncoupled state that causes the resonance are given in each case. Asterisks indicate cases where this uncoupled state is substantially mixed with the least-bound state in the incoming channel where it crosses threshold. $a_{\textrm{s}}\ (a_{0})$ | $a_{\textrm{t}}\ (a_{0})$ |  $B_{\textrm{res}}$ (G) |  $\Delta$ (G) | $a_{\textrm{bg}}\ (a_{0})$ | $f_{\textrm{Rb}}$ | $m_{f,\textrm{Rb}}$ | $m_{s,\textrm{CaF}}$ | $m_{i,\textrm{F}}$ | $\eta$ |
---|---|---|---|---|---|---|---|---|---|---
$-79$ | $-47$ | 50 | $-6.8\times 10^{-4}$ | $-47$ | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-1$ |
|  | 81 | $-2.3\times 10^{-4}$ | $-47$ | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ |
|  | 319 | $-8.2\times 10^{-3}$ | $-46$ | 2 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-3$ |
|  | 375 | $-4.8\times 10^{-1}$ | $-46$ | 2 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-3$ |
|  | 658 | $-1.7\times 10^{-2}$ | $-46$ | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ |
|  | 692 | $-1.0\times 10^{-2}$ | $-46$ | 2 | 2 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-3$ |
|  | 843 | $-8.6\times 10^{-4}$ | $-45$ | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
|  | 914 | $-1.1$ | $-46$ | 2 | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-3$ |
$-79$ | 86 | 164 | 16 | 112 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ | *
|  | 188 | 3.3 | 27 | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
|  | 688 | 49 | 87 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-3$ | *
|  | 806 | $3.0\times 10^{-2}$ | 51 | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-3$ |
$-79$ | 297 | 72 | 24 | 577 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-1$ | *
|  | 124 | 1.8 | 273 | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ |
|  | 599 | 88 | 383 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ | *
|  | 725 | $5.7\times 10^{-2}$ | 117 | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
|  | 934 | $9.5\times 10^{-2}$ | 294 | 2 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-3$ |
|  | 953 | $3.8\times 10^{-1}$ | 292 | 2 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-3$ |
71 | $-47$ | 99 | $-32$ | $-82$ | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-1$ |
|  | 134 | $-2.4\times 10^{-1}$ | $-144$ | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ |
|  | 343 | $-1.9\times 10^{-1}$ | $-49$ | 2 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
|  | 455 | $-27$ | $-51$ | 2 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ |
|  | 689 | $-3.2$ | $-50$ | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ |
|  | 776 | $-3.9\times 10^{-2}$ | $-49$ | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
71 | 86 | 312 | $1.1\times 10^{-1}$ | 85 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ |
|  | 413 | $9.3\times 10^{-4}$ | 85 | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
71 | 297 | 172 | 18 | 202 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ | *
|  | 285 | $9.5\times 10^{-2}$ | 185 | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
|  | 860 | $2.9\times 10^{-2}$ | 279 | 2 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-3$ |
|  | 952 | $3.2\times 10^{-1}$ | 294 | 2 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-3$ |
242 | $-47$ | 61 | $-2.8$ | $-65$ | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-1$ |
|  | 99 | $-1.8\times 10^{-1}$ | $-67$ | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ |
|  | 331 | $-9.6\times 10^{-2}$ | $-50$ | 2 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-3$ |
|  | 414 | $-10$ | $-54$ | 2 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-3$ |
|  | 678 | $-1.4$ | $-50$ | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ |
|  | 766 | $-4.3\times 10^{-2}$ | $-48$ | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
242 | 86 | 248 | 4.0 | 93 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ | *
|  | 302 | $1.4\times 10^{-1}$ | 86 | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
242 | 297 | 134 | $3.3\times 10^{-3}$ | 300 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-2$ |
|  | 187 | $1.9\times 10^{-4}$ | 200 | 1 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-2$ |
|  | 838 | $7.8\times 10^{-4}$ | 302 | 2 | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-3$ |
|  | 951 | $1.3\times 10^{-1}$ | 301 | 2 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-3$ |
|  | 990 | $1.8\times 10^{-4}$ | 300 | 1 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-3$ |

##  IV The role of CaF rotation

There can also be resonances due to states supported by rotationally excited
thresholds. This section will consider the structure of such states and the
likelihood that they produce resonances at experimentally accessible fields.

The thresholds for CaF ($n=1$) are from 20 to 30 GHz above the lowest
threshold, so states that can cause Feshbach resonances must be bound by about
this amount. The outer turning point at this depth is at around $R=30\ a_{0}$.
The potential anisotropy at this distance, due to dispersion and induction, is
around 1 GHz. This is substantially less than the CaF rotational spacing, so
will cause only weak mixing between different CaF rotational states at this
distance. However, it is substantially larger than the rotational constant of
the triatomic complex, $B=\hbar^{2}/(2\mu R^{2})$, which is about 60 MHz at
this distance. It is also larger than the spin-rotation coupling constant,
$\gamma\approx 40$ MHz. The long-range anisotropy is thus sufficient to
quantize $n$ along the intermolecular axis, with projection $K$, instead of
along the axis of the field. This is exactly analogous to the situation for
Van der Waals complexes in coupling case 2 [66].

For each CaF rotational level $(n,K)$ there will be a set of spin states,
labeled at fields above 50 G by
$(f_{\textrm{Rb}},m_{f,\textrm{Rb}},m_{s,\textrm{CaF}},m_{i,\textrm{F}})$.
Each such set $(n,K)$ will sample the short-range singlet and triplet
potentials over a different range of Jacobi angles $\theta$, so each group
will be characterized by different singlet and triplet scattering lengths
$a_{\textrm{s}}(n,K)$ and $a_{\textrm{t}}(n,K)$. These will probably be
unrelated to the corresponding quantities for the channels with $n=0$,
$a_{\textrm{s}}(0,0)$ and $a_{\textrm{t}}(0,0)$ (designated simply
$a_{\textrm{s}}$ and $a_{\textrm{t}}$ in Sec. III.1). For a particular
interaction potential, the sets of spin states for $n>0$ may therefore lie at
quite different depths within their bins from those for $n=0$. The patterns of
levels will nevertheless be characterized by $a_{\textrm{s}}(n,K)$ and
$a_{\textrm{t}}(n,K)$ and by quantum numbers
$(f_{\textrm{Rb}},m_{f,\textrm{Rb}},m_{s,\textrm{CaF}},m_{i,\textrm{F}})$, in
a similar way to those for the states with $n=0$ described above.

For Rb+CaF, the spin-exchange interaction may be characterized in terms of an
anisotropic surface $V^{-}(R,\theta)$ that is half the difference between the
singlet and triplet surfaces,

$V^{-}(R,\theta)=\textstyle{\frac{1}{2}}\left[V^{0}(R,\theta)-V^{1}(R,\theta)\right].$
(9)

This may be expanded in Legendre polynomials,

$V^{-}(R,\theta)=\sum_{\lambda}V^{-}_{\lambda}(R)P_{\lambda}(\cos\theta).$
(10)

Such a potential is diagonal in $K$, but each term in the expansion can couple
$(n,K)=(0,0)$ to $(\lambda,0)$. The term $V^{-}_{1}(R)$ can thus couple an
incoming state at the lowest threshold to states with $(n,K)=(1,0)$. The spin
selection rules are the same as for $n=0$, so the strongest resonances will be
those due to states dominated by $m_{i,\textrm{F}}=\frac{1}{2}$. As for $n=0$,
there are 3 such uncoupled states, with quantum numbers
$(f_{\textrm{Rb}},m_{f,\textrm{Rb}},m_{s,\textrm{CaF}},m_{i,\textrm{F}})=(1,0,\frac{1}{2},\frac{1}{2})$
$(2,1,-\frac{1}{2},\frac{1}{2})$ and $(2,0,\frac{1}{2},\frac{1}{2})$.
$V^{-}(R,\theta)$ is strongly anisotropic at short range, so there will always
be some intermolecular distance $R$ where it matches the separation between
the incoming and resonant thresholds, where nonadiabatic couplings can occur
by extension of the theory of ref. [60].

As seen in Sec. III.1, the states that can cause strong resonances traverse
about 3 GHz of binding energy between zero field and 1000 G. Since the
thresholds with $(n,f_{\textrm{Rb}})=(1,1)$ lie about 20 GHz above $(0,1)$,
the zero-field binding energy of a state must be between 20 and 23 GHz if it
is to cause a Feshbach resonance below 1000 G. For an unknown potential, there
is only about a 19% probability that the background scattering length for a
single channel will lie in a range to give such a binding energy. The
corresponding probability for $n=2$ is about 9%, and the probabilities
decrease for successively higher $n$, because the bins are correspondingly
wider at the required binding energy.

The overall conclusion of this section is that there _may_ be resonances due
to states involving rotationally excited CaF, but that they will occur at
fields below 1000 G for a fairly small subset of possible interaction
potentials. In any case, the mixing between rotational states of CaF due to
long-range anisotropy is weak enough that it will not affect the likelihood of
resonances due to the ground rotational state.

##  V Potential effects of chaos

The interaction potentials for Rb+CaF are very strongly anisotropic at short
range, and provide strong coupling between CaF rotational and vibrational
states. It is quite likely that Rb+CaF will possess short-range states that
exhibit quantum chaos, in the same way as alkali-metal 3-atom [67, 39] and
4-atom systems [68, 69]. The onset of chaos has also been studied in Li+CaH
and Li+CaF [70].

For Rb+CaF, the density of short-range singlet vibrational states at the
energy of the lowest threshold has been estimated as 4 K-1 [41], corresponding
to a mean spacing of 5 GHz. If these states are fully chaotic, it is likely to
produce structure in the singlet scattering length on this energy scale.
However, the hyperfine couplings in singlet states will be small, probably
dominated by nuclear electric quadrupole couplings of no more that a few MHz,
which is tiny compared to the state separations. Furthermore, Zeeman shifts
are very small for singlet states. At most, the presence of chaos at short
range might make the singlet scattering length different for collisions
involving Rb($f=1$) and Rb($f=2$). This would affect the details of the level
structure, but not the probabilities of observing Feshbach resonances.

The density of short-range triplet states at threshold is likely to be much
smaller, perhaps by an order of magnitude. This corresponds to a mean spacing
of order 50 GHz. The difference arises because the density of states for an
atom-diatom system scales approximately with $D^{3/2}$ [39], where $D$ is the
well depth, and the triplet surface of Rb+CaF is expected to be substantially
shallower than the singlet surface, as for the alkali-metal dimers. The
hyperfine couplings for triplet states will be comparable to those for the
separated atom and molecule (6.8 GHz for Rb, 120 MHz for CaF) but these are
still substantially smaller than the likely spacings between short-range
triplet states. Zeeman effects are also much larger for triplet states than
for singlet states, but are still only a few GHz at fields below 1000 G, so
will not cause substantial mixings between short-range triplet states.

It thus appears that the qualitative arguments in this paper about the
patterns of energy levels and likelihood of Feshbach resonances will remain
valid even if the short-range levels of Rb+CaF exhibit quantum chaos.

##  VI Conclusions

We have investigated magnetically tunable Feshbach resonances that may be
expected in collisions between molecules in ${}^{2}\Sigma$ states and alkali-
metal atoms, focussing on the prototype system Rb+CaF. The details of the
short-range interaction potential are unknown, but expected to have minor
influence, except to determine singlet and triplet scattering lengths
$a_{\textrm{s}}$ and $a_{\textrm{t}}$. We have carried out coupled-channel
calculations of the near-threshold bound states and scattering properties for
a variety of values of these scattering lengths. We find that the large
majority of plausible interaction potentials produce multiple resonances at
magnetic field below 1000 G, which are likely to be experimentally accessible.
In each case, at least some of these resonances are wide enough to be
experimentally useful for tuning scattering lengths or for forming triatomic
molecules by magnetoassociation.

The patterns of bound states may be understood in terms of underlying
uncoupled states that lie parallel to atom-molecule thresholds as a function
of magnetic field. There are varying degrees of coupling between these states,
which depend on the values of $a_{\textrm{s}}$ and $a_{\textrm{t}}$. The
coupling is weakest when $a_{\textrm{s}}$ and $a_{\textrm{t}}$ are similar.
The widths of the resonances may be explained in terms of the nature of the
states that cross threshold, together with effects due to the scattering
lengths.

We have considered the effect of potential anisotropy, which causes coupling
between CaF rotational states. This coupling is very strong at short range.
Even at long range, it is sufficient to quantize the CaF rotation along the
intermolecular axis instead of along the magnetic field. It is likely that
each rotational state of CaF will be characterized by different values of the
singlet and triplet scattering lengths. We have found that there is a small
but significant probability of additional wide resonances due to states
supported by rotationally excited thresholds. We have also considered the
potential influence of chaotic behavior for short-range states of Rb+CaF. We
expect that, even if present, it will have limited effects on the long-range
states that are principally responsible for the resonances and will not change
the qualitative conclusions.

This work indicates that atom-molecule systems such as Rb+CaF will have a rich
spectrum of magnetically tunable Feshbach resonances at experimentally
accessible magnetic fields. The resonances can be used to form a more detailed
understanding of the atom-diatom potential energy surfaces. Much new physics
will be accessible when these resonances are located. For example, a resonance
can be used to tune the s-wave scattering length for interspecies collisions.
In this way we can expect to find favorable conditions for sympathetic
cooling, which can greatly increase the phase-space density of the molecular
gas. The resonances may also be used to form polyatomic molecules by
magnetoassociation. Many applications have already been identified for such
molecules. They have unique advantages for probing interactions beyond the
Standard Model that violate time-reversal symmetry [71, 72] and for testing
theories of ultralight dark matter [73]. Their usefulness for quantum
information processing has been highlighted [74, 75], and the very large
number of stable, accessible internal states make them interesting as qudits
[76]. They can also be used to explore a rich diversity of many-body phenomena
such as quantum magnetism [77].

## Data availability statement

The data presented in this work are available from Durham University [URL to
be supplied]

## Acknowledgement

We are grateful to Matthew Frye for valuable discussions and to Ruth Le Sueur
for software support. This work was supported by the U.K. Engineering and
Physical Sciences Research Council (EPSRC) Grant Nos. EP/P01058X/1,
EP/W00299X/1, EP/V011499/1 and EP/V011677/1.

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