# Long-range states and Feshbach resonances in collisions between
ultracold alkali-metal diatomic molecules and atoms

Matthew D. Frye  Joint Quantum Centre (JQC) Durham-Newcastle, Department of
Chemistry, Durham University, South Road, Durham DH1 3LE, United Kingdom
Jeremy M. Hutson  Joint Quantum Centre (JQC) Durham-Newcastle, Department of
Chemistry, Durham University, South Road, Durham DH1 3LE, United Kingdom

###### Abstract

We consider the long-range states expected for complexes formed from an
alkali-metal diatomic molecule in a singlet state and an alkali-metal atom. We
explore the structure of the Hamiltonian for such systems, and the couplings
between the six angular momenta that are present. We consider the patterns and
densities of the long-range states, and the terms in the Hamiltonian that can
cause Feshbach resonances when the states cross threshold as a function of
magnetic field. We present a case study of 40K87Rb+87Rb. We show multiple
types of resonance due to long-range states with rotational and/or hyperfine
excitation, and consider the likelihood of them existing at low to moderate
magnetic fields.

##  I Introduction

Collisions are fundamentally important in ultracold physics. They not only
dictate the lifetimes of ultracold samples and the efficiency of evaporative
cooling, but also allow exquisite control of ultracold gases. For ultracold
atoms, a detailed understanding of two-body collisions has evolved through
close interplay between experiment and theory. In particular, a zero-energy
Feshbach resonance occurs whenever a molecular bound state is close in energy
to an atomic threshold, and is coupled to it by the interaction potential. A
_tunable_ Feshbach resonance occurs when the state can be tuned across the
threshold, most commonly with a magnetic field. At the lowest threshold, the
s-wave scattering length passes through a pole as a function of applied field.
This allows the effective interaction strength in a quantum gas to be tuned to
any desired value in the vicinity of a resonance. As a result, magnetically
tunable Feshbach resonances have become a mainstay of ultracold atomic physics
Chin:RMP:2010 , with applications that range from the study of Efimov physics
Naidon:2017  to investigations of the BCS-BEC crossover in degenerate Fermi
gases Bloch:2012 . Feshbach resonances have also been used for
magnetoassociation, in which pairs of ultracold atoms are converted to weakly
bound diatomic molecules by sweeping a magnetic field across a resonance
Regal:40K2:2003 ; Herbig:2003 ; Kohler:RMP:2006 . Several of these molecules
have been transferred to the absolute ground state, usually by stimulated
Raman adiabatic passage Ni:KRb:2008 ; Danzl:v73:2008 ; Lang:cruising:2008 ;
Danzl:ground:2010 ; Takekoshi:RbCs:2014 ; Molony:RbCs:2014 ; Park:NaK:2015 ;
Guo:NaRb:2016 ; Rvachov:2017 ; Seesselberg:2018 ; Yang:K_NaK:2019 ;
Voges:NaK:2020 ; Cairncross:2021 .

Many new opportunities will open up if molecular collisions can be controlled
in the same way as atomic collisions, through tunable Feshbach resonances.
However, ultracold molecule-molecule collisions have turned out to be
unexpectedly lossy Gregory:RbCs-collisions:2019 ; Bause:2021 ; Gersema:2021 ,
even in systems where there is no energetically allowed two-body reaction
between the colliding species Zuchowski:trimers:2010 . Indeed, most
experiments on diatomic molecules have observed loss rates close to the so-
called universal rate, corresponding to unit probability of loss for colliding
pairs that reach short range Idziaszek:PRL:2010 . This has prompted work to
engineer long-range potentials that prevent collisions reaching short range
Wang:dipolar:2015 ; Karman:shielding:2018 ; Matsuda:2020 ; Anderegg:2021 ;
Li:KRb-shield-3D:2021 ; Schindewolf:NaK-degen:2022 . The short-range loss is
usually explained in terms of the formation of long-lived collision complexes
Mayle:2012 ; Mayle:2013 , which cause a very dense (and so far unresolved)
mass of Feshbach resonances. The complexes can be destroyed in a variety of
ways, including by chemical reaction or via optical excitation by the trapping
light Christianen:laser:2019 . However, there is conflicting evidence on the
lifetimes of the complexes and the mechanisms of their loss Gregory:RbCs-
complex-lifetime:2020 ; Liu:2020 ; Gregory:atom-molecule:2021 ; Bause:2021 ;
Gersema:2021 .

Tunable Feshbach resonances have not yet been observed in molecule-molecule
collisions. However, ultracold atom-molecule collisions provide a middle
ground between the simplicity of atom-atom collisions and the complexity of
molecule-molecule collisions. Collisional losses are still fast in some cases,
but much slower in others. The first observations of atom-molecule Feshbach
resonances have recently been made. In particular, magnetic Feshbach
resonances have been observed in collisions between ultracold 40K atoms and
23Na40K molecules in singlet states Yang:K_NaK:2019 ; Wang:K_NaK:2021 ;
Su:elastic:2022 ; Yang:rf-association:2022 ; Yang:magnetoassociation:2022  and
between 23Na atoms and 6Li23Na molecules in triplet states Son:Na_NaLi:2022 .

We have recently investigated the theory of the triatomic complexes that can
be formed in collisions between alkali-metal diatomic molecules and atoms
Frye:triatomic-complexes:2021 . We focussed on the doublet states formed in
collisions of diatomic molecules in singlet states. We calculated densities of
states for atom-molecule pairs at short range and showed that, for a single
hyperfine manifold, the states are much more widely spaced than the range of
collision energies. These states are expected to be chaotic in nature
Mayle:2012 ; Frye:chaos:2016 . We also considered the role of hyperfine
coupling due to the Fermi contact interaction, and showed that it can couple
different hyperfine manifolds, producing broad resonances and background loss.
However, the short-range states are unlikely to be responsible for the
resonances observed in K+NaK; the patterns of the resonances observed suggest
that they are due to long-range states Wang:K_NaK:2021 , which are more weakly
coupled to the scattering channels and cause narrower resonances that are
potentially controllable with magnetic fields.

The purpose of the present paper is to investigate the long-range states of
the complexes that can be formed in collisions between alkali-metal diatomic
molecules and atoms, and the circumstances under which they can produce
tunable Feshbach resonances. The long-range states are relatively weakly
coupled to one another; they are likely to have more structured energy-level
patterns than the short-range states, and not to be chaotic in nature.
Nevertheless, they have complicated Zeeman and hyperfine Hamiltonians, with
many terms that can couple them to the incoming and (in some cases) inelastic
scattering channels. These states are shown schematically in Fig. 1.

Figure 1:  Schematic picture of the channels and states in an alkali-metal
atom+diatom collision. Short range states are expected to be chaotic, but the
asymptotic channels have atom+diatom character, and so do any states supported
by those channels in the long-range region. Examples of long-range states are
shown for one channel with rotational excitation (blue) and one channel with
hyperfine excitation (red) but similar states will also exist for other
channels. States that lie close to the incoming channel threshold can cause
magnetically tunable Feshbach resonances.

The structure of the paper is as follows. Section II explores the form of the
Hamiltonian for atom-diatom complexes, the ways that different angular momenta
are coupled together, and the selection rules rules for different coupling
terms. Section III considers the patterns and densities of near-threshold
states, and their consequences for Feshbach resonances in different alkali
molecule-atom systems. Section IV describes a case study for 40K87Rb+87Rb,
with specific examples of level-crossing diagrams for resonances due to
different types of long-range state. Finally, Section V presents our
conclusions.

##  II Hamiltonian and selection rules

There are 6 sources of angular momentum in a triatomic system AB+C formed from
a singlet molecule and an alkali-metal atom: the electron spin $S=1/2$, three
nuclear spins $i_{\textrm{A}}$, $i_{\textrm{B}}$, $i_{\textrm{C}}$, the
rotation $n$ of the diatomic molecule AB and the rotation $L$ of AB and C
about one another.

###  II.1 Atomic and molecular states

An alkali-metal atom C in a 2S state, with electron spin $S=1/2$ and nuclear
spin $i_{\textrm{C}}$, is characterized in zero field by its total spin
$f_{\textrm{C}}=i_{\textrm{C}}\pm\frac{1}{2}$. The two hyperfine states are
separated by the hyperfine splitting
$(i_{\textrm{C}}+\frac{1}{2})\zeta_{\textrm{C}}$, where $\zeta_{\textrm{C}}$
is the scalar hyperfine coupling constant that arises from the Fermi contact
interaction; the splitting varies from $228$ MHz for 6Li to 9.2 GHz for 133Cs.
In a small magnetic field $B$, each hyperfine state $f_{\textrm{C}}$ is split
into $2f_{\textrm{C}}+1$ Zeeman states labeled by the projection
$m_{f,\textrm{C}}$ onto the axis of the field. At higher fields, the Zeeman
effect mixes the two states of different $f_{\textrm{C}}$, and at very high
fields the good quantum numbers are $M_{S}$ and $m_{\textrm{C}}$ rather than
$f_{\textrm{C}}$, with $m_{f,\textrm{C}}=M_{S}+m_{\textrm{C}}$. The states at
any field may nevertheless be labeled
$(\tilde{f}_{\textrm{C}},m_{f,\textrm{C}})$, where $\tilde{f}_{\textrm{C}}$ is
the value of $f_{\textrm{C}}$ that the state correlates with at zero field.

A diatomic molecule AB in a ${}^{1}\Sigma$ state, with vibrational and
rotational quantum numbers $v$ and $n$, is characterized in zero field by its
total angular momentum $f_{\textrm{AB}}$, which is the resultant of $n$,
$i_{\textrm{A}}$ and $i_{\textrm{B}}$. For the ground rotational state, $n=0$,
$i_{\textrm{A}}$ and $i_{\textrm{B}}$ are coupled only very weakly, with small
splittings between the zero-field states with different values of
$f_{\textrm{AB}}$. For the alkali-metal diatomic molecules, the spread is
typically only tens of kHz and is due to the scalar nuclear-spin–nuclear-spin
interaction Aldegunde:polar:2008 ; Aldegunde:singlet:2017 . Thus only a small
magnetic field (tens of Gauss) is needed to decouple the nuclear spins
Aldegunde:polar:2008 ; Aldegunde:singlet:2017 , such that the projections
$m_{\textrm{A}}$ and $m_{\textrm{B}}$ of $i_{\textrm{A}}$ and $i_{\textrm{B}}$
become nearly good quantum numbers. However, for $n>0$ there are much larger
zero-field splittings, usually dominated by nuclear quadrupole couplings, but
including nuclear-spin–rotation and scalar and tensor nuclear-spin–nuclear-
spin interactions; for these states a considerably larger magnetic field
(hundreds of Gauss) is needed to decouple $i_{\textrm{A}}$ and
$i_{\textrm{B}}$ from $n$ Ran:2010 , and $f_{\textrm{AB}}$ may remain a useful
quantum number at considerably higher fields. In an optical trap, states with
$n>0$ are further complicated by mixing due to the anisotropic ac Stark effect
Gregory:RbCs-AC-Stark:2017 .

###  II.2 The interaction operator

Colliding atoms and molecules are initially in eigenstates of their individual
Hamiltonians. Terms that are present in the Hamiltonians of the separated
species, and are independent of their separation $R$, do not provide coupling
between the colliding pair and the bound states of the triatomic complex. They
therefore cannot by themselves produce Feshbach resonances. It is convenient
to define an _interaction operator_ $\hat{V}(R,\xi)$ that includes all
$R$-dependent terms in the Hamiltonian of the colliding pair, excluding the
relative kinetic energy. Here $\xi$ represents all coordinates except $R$,
including spin coordinates. The interaction operator includes the
intermolecular potential, but also contains spin-dependent and field-dependent
terms as described below.

The complete wavefunction of the collision system or the triatomic complex may
be expanded

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

The _channel functions_ $\Phi_{j}(\xi)$ form a complete orthonormal basis set
for motion in the coordinates $\xi$. The component of the wavefunction in each
channel $j$ is described by a _radial channel function_ $\psi_{j}(R)$. A full
numerical solution of the resulting coupled-channel problem is beyond the
scope of this paper, but the formalism is nonetheless useful for considering
the various couplings and their effects.

When considering collisions and long-range states, it is convenient to choose
channel functions with quantum numbers
$\\{v,n,L,N,M_{N},m_{\textrm{A}},m_{\textrm{B}},\tilde{f}_{\textrm{C}},m_{f,\textrm{C}}\\}$,
collectively represented by $j$ in Eq. (1). Here $N$ is the total spin-free
angular momentum, which is the resultant of $n$ and $L$, and $M_{N}$ is its
projection onto the axis of the magnetic field. The channel functions are
defined more formally in Appendix A. It is useful to distinguish a quantum
number for the incoming channel, $X^{\textrm{in}}$, from the corresponding one
for a resonant bound state, $X^{\textrm{res}}$.

For each term $\hat{\Omega}$ in the interaction operator, we are interested in
the matrix elements $\langle
j^{\textrm{res}}|\hat{\Omega}|j^{\textrm{in}}\rangle$ and particularly in the
associated selection rules $\Delta X=X^{\textrm{res}}-X^{\textrm{in}}$
involving the different quantum numbers described above. We now consider these
selection rules for each term in the interaction operator.

####  II.2.1 Interaction terms diagonal in $N$

The strongest term in the interaction operator is the electrostatic
interaction potential $V(R,r,\theta)$. This is written here in Jacobi
coordinates: $\vec{r}$ is the interatomic vector of AB, of length $r$,
$\vec{R}$ is the vector from the center of mass of AB to atom C, of length
$R$, and $\theta$ is the angle between $\vec{r}$ and $\vec{R}$. For the
alkali-metal systems of interest here, $V(R,r,\theta)$ is deep (of order 50
THz) and highly anisotropic at short range. Such potentials have been
considered in depth elsewhere Martin:1978 ; Varandas:1982 ; Martins:1983 ;
Varandas:1986 ; vonBusch:1998 ; Hauser:2008 ; Zuchowski:trimers:2010 ;
Hauser:2010 ; Hauser:2015 ; Croft:K+KRb:2017 ; Kendrick:Li+LiNa:2021 , but
their details are not crucial for the present purposes. The important feature
of the short-range potential is that it provides strong coupling between
channels with different values of $v$, $n$ and $L$. However, it is diagonal in
$N$ and its projection $M_{N}$. It is also diagonal in all spin quantum
numbers. Ultracold collisions are usually dominated by the incoming s wave,
with $L^{\textrm{in}}=0$, which implies $N^{\textrm{in}}=n^{\textrm{in}}$. If
the molecule AB is initially in $n^{\textrm{in}}=0$, the electrostatic
potential couples the incoming wave only to other channels with
$N^{\textrm{res}}=0$, requiring $L^{\textrm{res}}=n^{\textrm{res}}$, and the
same spin quantum numbers as the incoming channel.

There are also terms in the interaction operator that depend on spins and/or
applied fields. All the hyperfine and Zeeman terms that exist in the separated
atom and molecule become functions of $R$ and $\theta$ when the two species
interact, and the differences from the monomer terms contribute to the
interaction operator. In addition, there are scalar and tensor interactions
that develop between the angular momenta on the atom and those on the
molecule. The magnetic moments are also modified, and contribute Zeeman terms
to the interaction operator.

The strongest spin-dependent term is the Fermi contact interaction. At long
range this couples $S$ to $i_{\textrm{C}}$ for the free atom to form
$f_{\textrm{C}}$. We showed in Ref. Frye:triatomic-complexes:2021  that the
Fermi contact interaction depends strongly on the geometry of the complex,
particularly at short range, where the electron spin that is originally on
atom C is distributed among all three atoms. The Fermi contact operator is of
the form Frye:triatomic-complexes:2021

$\hat{H}_{\textrm{Fc}}=\sum_{X=\textrm{A,B,C}}\zeta_{X}(R,r,\theta)\hat{\boldsymbol{i}}_{X}\cdot\hat{\boldsymbol{S}},$
(2)

where $\hat{\boldsymbol{S}}$ and $\hat{\boldsymbol{i}}_{X}$ are the vector
operators for the electron and nuclear spin angular momenta. Each term is the
product of a scalar spatial operator $\zeta_{X}(R,r,\theta)$, which can couple
states of different $v$, $n$ and $L$ while conserving $N$ and $M_{N}$, and a
spin operator $\hat{\boldsymbol{i}}_{X}\cdot\hat{\boldsymbol{S}}$. The latter
provides a coupling that can change $\tilde{f}_{\textrm{C}}$ and/or
$m_{f,\textrm{C}}$ by 0 or 1, while conserving the total spin projection
$m_{f,\textrm{tot}}=m_{\textrm{A}}+m_{\textrm{B}}+m_{f,\textrm{C}}$.

If all the external fields present share a common axis, the total projection
quantum number $M_{F}$ is conserved,

$M_{F}=M_{N}+m_{f,\textrm{tot}}=M_{N}+M_{S}+m_{\textrm{A}}+m_{\textrm{B}}+m_{\textrm{C}}.$
(3)

Conservation of $m_{f,\textrm{tot}}$ thus implies conservation of $M_{N}$, and
_vice versa_.

####  II.2.2 Interaction terms off-diagonal in $N$

An important question is the strength of the coupling between channels with
different values of $N$ and $M_{N}$. If such couplings are significant, bound
states with one value of $(N,M_{N})$ may cause Feshbach resonances in an
incoming channel with different values. In experiments that involve diatomic
molecules in their rotationless ground state, $n^{\textrm{in}}=0$, and are
dominated by s-wave scattering, $L^{\textrm{in}}=0$, the incoming channel has
$N^{\textrm{in}}=0$. Couplings off-diagonal in $N$ are needed for collision
complexes in any state $(N^{\textrm{res}},M^{\textrm{res}}_{N})$ other than
(0,0) to contribute in such cases.

There are tensor hyperfine terms in the Hamiltonian, arising from dipolar
interactions between electron and nuclear spins. These terms are off-diagonal
in $m_{X}$ for the nucleus concerned by up to 1, in $M_{S}$ by up to 1, and in
$n$, $L$, $N$ and $M_{N}$ by up to 2, while conserving $M_{F}$. They depend on
the asymmetry of the spin density around the nucleus concerned. The tensor
coupling constants for Na3 at its equilibrium geometry have been obtained from
electronic structure calculations Hauser:2015  and found to be up to 142 MHz.
They are zero for free atoms and singlet molecules, so form part of the
interaction operator. The calculated value for Na3 is about 16% of the atomic
hyperfine coupling of Na; it seems reasonable to expect similar percentages
for the triatomic complexes of interest here.

There are additional hyperfine terms arising from the interaction of nuclear
electric quadrupole moments with the electric field gradient at the nucleus
concerned. These too are off-diagonal in $n$, $L$, $N$ and $M_{N}$ by up to 2.
However, the coupling constants are generally less than 10 MHz for alkali-
metal diatomic molecules Aldegunde:singlet:2017 , and are probably comparable
in the 3-atom complexes. They are again zero for free atoms, so there is some
dependence on $R$, but their absolute magnitude is small.

A magnetic field $B$ interacts mostly with the magnetic moment due to the
electron spin $S$ and to a much lesser extent with those due to the nuclear
spins $i_{X}$ and the rotational angular momenta $n$, $L$ and $N$. It is
likely that only the electron-spin contribution is significant in the
interaction operator. For a free atom, the matrix elements are of magnitude
$g_{S}\mu_{\textrm{B}}M_{S}B$, where $g_{S}$ is the g-factor for the electron
spin and $\mu_{\textrm{B}}$ is the Bohr magneton. The electron g-tensor for a
molecule is anisotropic and geometry-dependent, with components $g_{ij}$ in a
frame fixed in the molecule. The term that enters the interaction operator
depends on the deviation of $g_{ij}(R,r,\theta)$ from $g_{S}$; such deviations
are small (usually less than 1%) for molecules with well-separated electronic
states that contain only light atoms, but they may be enhanced if there is a
nearby excited state with a spin density spatially close to that of the state
of interest Lushington:1997 . For example, the shifts can be a substantial
fraction of $g_{S}$ for transition-metal species Abragam:1970 . Moreover, the
spin-orbit matrix elements responsible for them scale roughly as $Z^{4}$ with
nuclear charge $Z$. They might therefore be substantial for the alkali-metal
triatomic molecules. They are diagonal in $M_{S}$ and therefore off-diagonal
in $\tilde{f}_{\textrm{C}}$ and $m_{f,\textrm{C}}$. Since the g-tensor is
defined in a frame fixed in the triatomic complex, it must be rotated to
evaluate its matrix elements between space-fixed functions; as a result its
anisotropic part can mix states with different $N$ and $M_{N}$.

There are also spin-rotation terms arising from the interaction of the
electron spin with molecular rotation and internal rotation. These arise
mainly from the combined effects of spin-orbit and Coriolis mixing between
electronic states VanVleck:RMP:1951 . The spin-rotation coupling tensor is
approximately related to the g-tensor through Curl’s approximation Curl:1965 ,

$g_{ij}=g_{S}\delta_{ij}-\hbar^{-2}\sum_{k}\epsilon_{ik}I_{kj}.$ (4)

Here $I_{kj}$ is an element of the molecular inertial tensor, so that the
spin-rotation constants scale roughly with rotational constants as well as
g-tensor shifts. The isotropic part has matrix elements off-diagonal in
$M_{N}$, while the anisotropic part is off-diagonal in $N$ and $M_{N}$; both
parts conserve $M_{J}=M_{N}+M_{S}$. Such terms are known to be up to several
GHz for the ground states of molecules such as NF2, NO2 and ClO2 Curl:1965 ;
Tarczay:2010 , but values below 1 MHz have been obtained from electronic
structure calculations on Na3 at near-equilibrium geometries Hauser:2015 .

An electric field interacts strongly with an electric dipole moment to produce
matrix elements diagonal and off-diagonal in $N$. However, there are no
purposely applied electric fields in the experiments considered here.

In an optical trap, the light from the trapping laser can interact with the
molecular polarizability through the ac Stark effect Gregory:RbCs-AC-
Stark:2017 . The polarizability is a second-rank tensor that depends on the
geometry of the triatomic complex, so the AC Stark effect contributes to the
interaction operator. It has matrix elements that can change $N$ by 0 or 2 and
can also be off-diagonal in $v$, $n$ and $L$. It is diagonal in the spin
quantum numbers.

####  II.2.3 Calculation of magnetic properties

The Fermi contact interactions, tensor hyperfine couplings, spin-rotation
couplings and g-tensor components can in principle be obtained from electronic
structure calculations. For simple species, considerable success has been
achieved both with wavefunction-based methods Gauss:2009 ; Datta:2019  and
with density-functional theory vanLenthe:1998 ; Neese:2001 ; Neese:2005 ;
Autschbach:2011 . However, for the alkali-metal 3-atom complexes, the task is
complicated by the existence of two low-lying electronic states, with seams of
conical intersections between them as described in Ref. Frye:triatomic-
complexes:2021 . Hauser _et al._ Hauser:2015  have used a variety of
electronic structure methods to calculate some of these properties in Na3.
However, they focused on high-symmetry geometries around the path sampled by
pseudorotation in the lowest vibronic state, and obtained coupling constants
involving spin densities by a fairly coarse integration over electronic
coordinates. Extending their calculations to heteronuclear systems at a wide
range of geometries is beyond the scope of the present paper.

##  III Long-range states and Feshbach resonances

###  III.1 Structure of long-range states

The states of complexes formed in alkali-metal atom-diatom collisions may be
loosely separated into short-range and long-range states. The short-range
states are supported by high-lying closed channels, with large values of $n$
and/or $v$. They spend most of their time in regions where the interaction
potential $V(R,r,\theta)$ is deep and strongly anisotropic. Such states are
strongly coupled to one another and are likely to be chaotic Mayle:2012 ;
Frye:chaos:2016 . The resulting resonances were investigated in Ref.
Frye:triatomic-complexes:2021 : the states may have magnetic moments
significantly different from the colliding atom and molecule, so the
resonances are magnetically tunable, but the coupling between the bound states
and the continuum is very strong; for atom-diatom systems in which the levels
are chaotic the resulting resonances are generally very broad (typically
hundreds of Gauss).

Interaction potentials with asymptotic form $-C_{j}/R^{j}$ support states with
wavefunctions concentrated at long range LeRoy:1970 . The spatial overlap
between the long-range and short-range states decreases as threshold is
approached. The long-range states are thus relatively weakly coupled to the
short-range states by the potential anisotropy and other short-range
interactions; the longest-range states are likely to be separate from the
chaotic manifold, though they will exhibit avoided crossings with short-range
states as a function of magnetic field. States of this general character have
been observed experimentally in the chaotic spectrum of dysprosium dimers
Maier:universal:2015  and also appear close to threshold in calculations on
the chaotic system Li + CaH Frye:chaos:2016 . This picture contrasts with that
of Mayle _et al._ Mayle:2012 , who assumed complete statistical mixing for all
states without distinction.

The long-range states are approximately described by the quantum numbers of
the free atom C and molecule AB. To a first approximation, the long-range
states of the complex are characterized by $\tilde{f}_{\textrm{C}}$ and
$m_{f,\textrm{C}}$, together with the rotational quantum number $n$ of the
diatomic molecule AB and a vibrational quantum number $\eta$ for motion in the
atom-molecule separation $R$. For present purposes $\eta$ is most conveniently
counted downwards from threshold, such that the least-bound state in each
channel with $L=0$ is $\eta=-1$. The hyperfine and Zeeman structure of the
alkali-metal diatomic molecule is complicated Aldegunde:polar:2008 ,
particularly in the presence of strong laser fields Gregory:RbCs-AC-Stark:2017
, but in general $n$ couples relatively weakly to the nuclear spins
$i_{\textrm{A}}$ and $i_{\textrm{B}}$ with hyperfine and Zeeman splittings of
a few MHz or less.

For long-range complexes at zero field, $n$ couples to $L$ to form resultant
$N$, which couples to $f_{\textrm{C}}$ to form resultant $F_{\textrm{C}}$;
this couples to $i_{\textrm{A}}$ and $i_{\textrm{B}}$ to form the total
angular momentum $F$. However, long-range states of the same $(n,L,N)$ and
$f_{\textrm{C}}$ but different $F$ are unlikely to be split by more than a few
MHz, except in the vicinity of avoided crossings with short-range states. It
therefore takes a magnetic field of only a few G to decouple $M_{N}$ and the
nuclear spins of the diatomic molecule, so that the quantum numbers are
$(n,L,N,M_{N},\tilde{f}_{\textrm{C}},m_{f,\textrm{C}},m_{\textrm{A}},m_{\textrm{B}})$.
The Zeeman effects for these states are dominated by that of atom C, and at
fields more than a few G they form groups, spaced by no more than a few MHz,
with the same values of $(n,L,N,\tilde{f}_{\textrm{C}},m_{f,\textrm{C}})$ but
differing $(M_{N},m_{\textrm{A}},m_{\textrm{B}})$. The total projection
$M_{F}=M_{N}+m_{f,\textrm{C}}+m_{\textrm{A}}+m_{\textrm{B}}$ is conserved if
the trapping laser is polarized parallel to the magnetic field.

For slightly deeper states with $N>0$, the anisotropy $V_{\textrm{aniso}}$ of
the interaction potential $V(R,r,\theta)$ may be sufficient to quantize $n$
along the intermolecular vector with projection $K$. In this case $(n,L,N)$ is
replaced by $(n,K,N)$. This recoupling occurs when $V_{\textrm{aniso}}$ is
large compared to the rotational constant of the complex, $\hbar^{2}/(2\mu
R^{2})$, in the same way as for Case 2 coupling in spin-free Van der Waals
complexes Hutson:AMVCD:1991 . For K+NaK, for example, it takes place near
$R\sim 100\,a_{0}$. As the anisotropy increases further at shorter range,
$V_{\textrm{aniso}}(R)$ becomes greater than the rotational constant of the
diatomic molecule, $\hbar^{2}/(2\mu_{\textrm{AB}}r^{2})$. The diatom
rotational quantum number $n$ is then replaced by a quantum number for a
bending vibration, as for Case 3 coupling in Van der Waals complexes with
strong anisotropy Hutson:AMVCD:1991 . Eventually, as the binding energy
increases, the couplings to the short-range states will become strong enough
that the long-range states will merge into the chaotic manifold, with level
spacings described by random-matrix theory as in Ref. Frye:triatomic-
complexes:2021 .

Feshbach resonances require coupling between the incoming scattering state and
a quasibound state. Although the quantum numbers
$(n,L,N,M_{N},\tilde{f}_{\textrm{C}},m_{f,\textrm{C}},m_{\textrm{A}},m_{\textrm{B}})$
described above are _approximately_ conserved for the long-range states, they
are not _fully_ conserved. The long-range states do mix with the short-range
states, and the mixed states have a component at short range; the couplings
described in Section II then provide the necessary coupling to the incoming
state. In particular, magnetically tunable Feshbach resonances occur when a
bound or quasibound state crosses the energy of the incoming state; they can
be observed only if the incoming state and the quasibound state have different
magnetic moments, and this will usually require differing values of
$\tilde{f}_{\textrm{C}}$ or $m_{f,\textrm{C}}$; since $M_{F}$ is conserved, a
change in $m_{f,\textrm{C}}$ requires a compensating change in $M_{N}$,
$m_{\textrm{A}}$ or $m_{\textrm{B}}$. As described in Section II, the Fermi
contact interaction can exchange angular momentum between $m_{f,\textrm{C}}$
and $m_{\textrm{A}}$ or $m_{\textrm{B}}$ while conserving
$m_{f,\textrm{tot}}$, with selection rule $\Delta m_{f,\textrm{C}}=\pm 1$. It
can also change $\tilde{f}_{\textrm{C}}$. Couplings off-diagonal in $N$ and/or
$M_{N}$ are generally weaker but can arise from tensor hyperfine coupling,
g-tensor anisotropy and spin-rotation coupling as described above.

The alkali molecule + atom systems have some similarities with systems in
which an alkali-metal atom interacts with a closed-shell atom, such as Rb+Sr
Zuchowski:RbSr:2010 ; Barbe:RbSr:2018  and alkali+Yb Brue:AlkYb:2013 ;
Yang:CsYb:2019 . In both cases the resonances are principally due to
$R$-dependent hyperfine coupling, and for Sr and Yb they are very narrow, with
widths in the mG range. However, the alkali triatomic systems differ because
the near-threshold states typically have much larger components at short
range, due to mixing with the chaotic manifold of short-range states. In
consequence, the resulting resonances will be substantially wider.

The long-range states will be much less susceptible to laser-induced loss than
short-range states. For large $R$, the electronically excited triatomic states
responsible for loss are much less strongly bound, so are likely to be
inaccessible with the laser frequencies used for trapping. In an alternative
viewpoint, the vibronically excited states responsible for loss are short-
range states, which have poor Franck-Condon overlap with the long-range
states. Because of this, the long-range states can be sharp enough to cause
well-defined Feshbach resonances Yang:K_NaK:2019 ; Wang:K_NaK:2021 .

Table 1: Energy and length scales and bin depths for of $AB+A$ alkali-metal systems.  System |  $\bar{a}$ (Å) |  $\bar{E}/h$ (MHz) |  $E^{L=0}_{\textrm{bin}}/h$ (MHz) |  $E^{L=0}_{\textrm{50\%}}/h$ (MHz) |  $E^{L=1}_{\textrm{bin}}/h$ (MHz) |  $E^{L=2}_{\textrm{bin}}/h$ (MHz)
---|---|---|---|---|---|---
7Li7Li+7Li | 21 | 245 | 8861 | 1505 | 15150 | 22909
7Li23Na+7Li | 20.8 | 205 | 7390 | 1255 | 12634 | 19104
7Li39K+7Li | 23.2 | 154 | 5579 | 947 | 9539 | 14424
7Li87Rb+7Li | 24.2 | 132 | 4784 | 812 | 8179 | 12368
7Li133Cs+7Li | 25.4 | 117 | 4226 | 718 | 7224 | 10924
7Li23Na+23Na | 26 | 57.3 | 2069 | 351 | 3537 | 5349
23Na23Na+23Na | 29 | 39.1 | 1411 | 240 | 2413 | 3649
23Na39K+23Na | 30.8 | 31.8 | 1148 | 195 | 1963 | 2968
23Na87Rb+23Na | 32.5 | 25.1 | 907 | 154 | 1551 | 2345
23Na133Cs+23Na | 34.3 | 21.4 | 774 | 131 | 1324 | 2001
7Li39K+39K | 36 | 18.5 | 667 | 113 | 1141 | 1725
23Na39K+39K | 37.8 | 14.8 | 535 | 90.8 | 915 | 1383
39K39K+39K | 41.7 | 11.2 | 404 | 68.6 | 691 | 1045
39K87Rb+39K | 42.6 | 9.34 | 337 | 57.3 | 577 | 872
39K133Cs+39K | 44.8 | 7.94 | 287 | 48.7 | 490 | 741
7Li87Rb+87Rb | 45.7 | 5.36 | 194 | 32.9 | 331 | 500
23Na87Rb+87Rb | 47.3 | 4.66 | 168 | 28.6 | 288 | 435
39K87Rb+87Rb | 50 | 3.93 | 142 | 24.1 | 242 | 367
87Rb87Rb+87Rb | 53.4 | 3.06 | 111 | 18.8 | 189 | 286
87Rb133Cs+87Rb | 55 | 2.68 | 96.9 | 16.5 | 166 | 251
7Li133Cs+133Cs | 55.7 | 2.39 | 86.3 | 14.7 | 148 | 223
23Na133Cs+133Cs | 57 | 2.17 | 78.2 | 13.3 | 134 | 202
39K133Cs+133Cs | 59.5 | 1.9 | 68.7 | 11.7 | 117 | 178
87Rb133Cs+133Cs | 61.9 | 1.59 | 57.5 | 9.76 | 98.3 | 149
133Cs133Cs+133Cs | 65.2 | 1.34 | 48.4 | 8.22 | 82.8 | 125

###  III.2 Expected positions of Feshbach resonances

The alkali-metal atom-molecule systems have interaction potentials of the form
$V(R)=-C_{6}/R^{6}$ at long range. If channel coupling is neglected, each
asymptotic channel labeled by quantum numbers
$(n,L,N,M_{N},\tilde{f}_{\textrm{C}},m_{f,\textrm{C}},m_{\textrm{A}},m_{\textrm{B}})$
supports a set of near-threshold bound states with a simple pattern of binding
energies given by quantum defect theory Gao:2000 . There is always exactly one
s-wave bound state ($L=0$) in a certain energy window immediately below
threshold. This window is known as the top bin and its width is determined by
only the asymptotic form of the interaction potential and the reduced mass
$\mu$. For an interaction potential $-C_{6}R^{-6}$, the width of the top bin
is approximately $36\bar{E}$, where $\bar{E}=\hbar^{2}/(2\mu\bar{a}^{2})$ is
the energy scale associated with the mean scattering length of Gribakin and
Flambaum Gribakin:1993 , $\bar{a}=(2\mu C_{6}/\hbar^{2})^{1/4}\times
0.4779888\dots$. Values of $\bar{a}$ and $\bar{E}$ are given in Table 1 for
all possible systems $AB+A$, where $A$ and $B$ are different alkali-metal
atoms, using the $C_{6}$ coefficients of Zuchowski:vdW:2013 . The concept of
bins can be extended to higher partial waves and deeper bound states, as
described by Gao Gao:2000 . Table 1 includes the depth of the top bin,
$E^{L}_{\textrm{bin}}$, for $L=0$, 1 and 2.

To a first approximation, a near-threshold bound state retains the character
of the threshold that supports it. If channel mixing is neglected, it shares
the magnetic moment of the threshold and runs below it and parallel to it as a
function of magnetic field. Consider incoming and resonant channels with
threshold energies $E^{\textrm{in}}(B)$ and $E^{\textrm{res}}(B)$ as a
function of magnetic field. A bound state with binding energy
$E^{\textrm{res}}_{\textrm{b}}$ relative to $E^{\textrm{res}}(B)$ crosses the
incoming channel and may cause a Feshbach resonance near the field where
$E^{\textrm{in}}(B)=E^{\textrm{res}}(B)-E^{\textrm{res}}_{\textrm{b}}$. In the
simple case where the Zeeman effect is linear, as for heavier alkali-metal
atoms at low fields, and the channels correlate with the same zero-field
level, $E^{\textrm{res}}(B)-E^{\textrm{in}}(B)=-g_{S}\mu_{\textrm{B}}B\Delta
m_{f,\textrm{C}}/(i_{\textrm{C}}+1/2)$.

The positions of states within their bins are governed by the scattering
length $a$ and may be obtained from quantum defect theory Gao:QDT:1998 ;
Gao:C6:1998 ; Gao:2001 ; Gao:AQDTroutines . An s-wave bound state exists close
to the top of the bin if the scattering length is large and positive,
$a\gg\bar{a}$. Similarly, a p-wave state ($L=1$) exists just below threshold
if $a$ is slightly less than $2\bar{a}$ and a d-wave state ($L=2$) exists just
below threshold if $a$ is slightly less than $\bar{a}$. For a potential with
unknown scattering length, it is possible to calculate the _probability_ that
a state with $L=0$, 1 or 2 exists in a particular range of binding energies.
For example, there is a 50% probability of an s-wave state bound by less than
$E^{L=0}_{\textrm{50\%}}=6.1\bar{E}$; these values are included in Table 1.

Figure 2:  Cumulative probability of a bound state within energy $E$ of
threshold for $L=0$ (black solid line), 1 (red dashed line), and 2 (blue
dotted line). Panel (a) shows this on a linear energy scale to the depth of
the first bin, and panel (b) shows it as a function of $(E/\bar{E})^{1/3}$ to
larger binding energies. Probabilities above 1 indicate that there is at least
one state, in the top bin, and a probability of a second state.

Fig. 2(a) shows the cumulative probability of a state with $L=0$, 1 or 2
existing within an energy range $E_{\textrm{b}}/\bar{E}$ below threshold. Fig.
2(b) show the same quantity as a function of $(E_{\textrm{b}}/\bar{E})^{1/3}$,
which approaches linearity away from threshold. This may be interpreted as the
probability of a bound state of this $L$ causing a resonance in an incoming
channel at a field _below_ that where the differential Zeeman effect,
$E^{\textrm{res}}(B)-E^{\textrm{in}}(B)$, is equal to
$E^{\textrm{res}}_{\textrm{b}}$. To estimate the probability of a resonance
occurring below a field $B$, calculate the differential Zeeman shift between
the two atomic states, divide by $\bar{E}$ from Table 1, and read off the
probabilities.

When the couplings that produce resonances are short-range in character, the
widths of resonances are proportional to $E_{\textrm{b}}^{2/3}$
Brue:AlkYb:2013 . States very close to threshold may therefore produce
resonances that are too narrow to observe. Under these circumstances the
lowest-field resonances that are actually observable may be those at the top
of the second bin, at slightly higher fields than implied by Table 1. This was
the case, for example, for analogous resonances for
$\tilde{f}_{\textrm{Rb}}=1$ in 87Rb+87Sr Barbe:RbSr:2018 .

##  IV Case study: Long-range states of KRb+Rb

Figure 3:  Example level crossing diagrams for different types of resonances
due to long-range states for 40K87Rb+87Rb. Heavy black lines show thresholds,
and thin colored lines show bound states. Crossings that can cause Feshbach
resonances are marked by symbols. All examples show single-channel bound
states for $a=\infty$, for which there is a state at the bottom of each s-wave
bin. (a) States
$(n^{\textrm{res}},\tilde{f}^{\textrm{res}}_{\textrm{C}},L^{\textrm{res}})=(0,1,0)$.
(b) States $(0,2,0)$. (c) States $(0,1,0)$ are shown with blue solid lines and
circles, (1,1,1) with green dashed lines and squares, and (2,1,2) with red
dotted lines and diamonds. (d) States $(0,1,0)$ are shown with solid lines and
circles, (0,1,1) with dashed lines and squares, and (0,1,2) with dotted lines
and diamonds. Also shown in red are the bottom of the bins for
$L^{\textrm{res}}=1$ and 2.

The bound states that cause Feshbach resonances in an incoming channel labeled
by
$(n^{\textrm{in}}=0,L^{\textrm{in}}=0,\tilde{f}^{\textrm{in}}_{\textrm{C}},m^{\textrm{in}}_{f,\textrm{C}})$
may belong to parent closed channels with differing values of
$n^{\textrm{res}}$, $\tilde{f}^{\textrm{res}}_{\textrm{C}}$ or
$m^{\textrm{res}}_{f,\textrm{C}}$ (or any combination of them). It is useful
to consider a specific example here, and we choose 40K87Rb+87Rb with the Rb
atom in its lowest state
$(\tilde{f}^{\textrm{in}}_{\textrm{C}},m^{\textrm{in}}_{f,\textrm{C}})=(1,1)$
and the KRb molecule in the excited Zeeman state
$(n^{\textrm{in}},m^{\textrm{in}}_{\textrm{A}},m^{\textrm{in}}_{\textrm{B}})=(0,-4,1/2)$.

Closed channels with
$(n^{\textrm{res}},\tilde{f}^{\textrm{res}}_{\textrm{C}})=(0,1)$ are almost
degenerate with the incoming channel at zero field; each such channel must
have a least-bound state with $\eta=-1$ somewhere in the top 142 MHz. The
Fermi contact term $\hat{H}_{\textrm{Fc}}$ has selection rule $\Delta
m_{f,\textrm{C}}=0,\pm 1$, but only $-1$ is relevant for this set of
resonances. Since the differential magnetic moment between adjacent Zeeman
states for Rb$(f_{\textrm{C}}=1)$ is 0.7 MHz/G at low field, the corresponding
resonances are expected at fields below 200 G, depending on where the least-
bound state lies within its bin. We show an illustrative example of a level-
crossing diagram for these resonances in Fig. 3(a). This is for the case
$a=\infty$, for which there is a state in each channel at the bottom of each
bin; the binding energies are again calculated using quantum defect theory.
Corresponding examples for other values of $a$ are given in Supplemental
Material supp:long-range . The crossings for $a=\infty$ are not predictions of
specific resonance positions, but instead predict a resonance due to a state
with $\eta=-1$ somewhere between $B=0$ and the crossing marked at 200 G. There
will also be a bound state in the second bin, with $\eta=-2$. This will lie
between the two zero-field levels shown in Fig. 3(a). It will cause a
resonance above 200 G; the upper limit is more complicated due to non-linear
Zeeman effects, and depends on $m_{f,\textrm{C}}$, but is readily calculated
from the depth of bottom of the second bin, which is 980 MHz for this system.
These resonances are analogous to Type II resonances that can occur in systems
such as LiYb, RbSr and CsYb Brue:LiYb:2012 ; Barbe:RbSr:2018 ; Yang:CsYb:2019
.

The change $\Delta m_{f,\textrm{C}}=-1$ requires a compensating change in
$m_{\textrm{A}}$ or $m_{\textrm{B}}$. For the chosen incoming channel,
resonances first-order in $\hat{H}_{\textrm{Fc}}$ are expected from two closed
channels:
$(m^{\textrm{res}}_{f,\textrm{C}},m^{\textrm{res}}_{\textrm{A}},m^{\textrm{res}}_{\textrm{B}})=(0,-3,1/2)$
and $(0,-4,3/2)$. Resonances from both these channels occur in the same region
of 0 to 200 G, but there are spin-dependent shifts due to the diagonal portion
of $\hat{H}_{\textrm{Fc}}$; depending on the magnitude of these, the states
may be either close together or shifted far from each other within this
region.

Additional resonances may occur due to closed channels with
$\tilde{f}^{\textrm{res}}_{\textrm{C}}=2$. This state is excited by 6.83 GHz
at zero field; to cause resonances accessible in magnetic fields below 600 G,
the states for
$(\tilde{f}^{\textrm{res}}_{\textrm{C}},m^{\textrm{res}}_{f,\textrm{C}})=(2,1)$
would need to be bound by between 6.83 and 7.81 GHz. The relative magnetic
moment is larger in this case, but the states are much sparser in energy this
far below threshold. It is therefore relatively unlikely (probability $\sim
19\%$) that resonances due to states with
$\tilde{f}^{\textrm{res}}_{\textrm{C}}=2$ will exist at such fields. This
likelihood is considerably larger when the free atom has a substantially
smaller hyperfine splitting, as for isotopes of Li, Na or K. An example of
such crossings is shown in Fig. 3(b), showing that they will occur at low
fields only if there is a state near the boundary between the fourth and fifth
bins. Resonances may also exist due to states in channels with
$(\tilde{f}^{\textrm{res}}_{\textrm{C}},m^{\textrm{res}}_{f,\textrm{C}})=(2,0)$
or (2,2). These resonances are analogous to Type I and II resonances that can
occur in systems such as RbSr and CsYb Zuchowski:RbSr:2010 ; Brue:AlkYb:2013 ;
Barbe:RbSr:2018 ; Yang:CsYb:2019 .

The different values of $\Delta m_{f,\textrm{C}}=0,\pm 1$ require different
combinations $(m^{\textrm{res}}_{\textrm{A}},m^{\textrm{res}}_{\textrm{B}})$
to produce resonances at each of the three crossings. For our chosen incoming
state, for $\Delta m_{f,\textrm{C}}=-1$ we once again expect first-order
resonances due to states with $(-3,1/2)$ and $(-4,3/2)$; for $\Delta
m_{f,\textrm{C}}=0$ we expect $(-4,1/2)$; and for $\Delta m_{f,\textrm{C}}=+1$
we expect $(-4,-1/2)$. The states that cause these resonances are more deeply
bound than those shown in Fig. 3(a), so their wavefunctions will have greater
density at short range, sample more of the regions where the Fermi contact
interaction is strong, and thus be spread out over a larger range of energy
and field by spin-dependent shifts.

Further classes of resonances may exist in molecular collisions, due to closed
channels with $n^{\textrm{res}}>0$ but $N^{\textrm{res}}=0$. For KRb the $n=1$
state is located 2.23 GHz above $n=0$ Ospelkaus:hyperfine-control:2010 , and
the $n=2$ state is located 6.68 GHz above. The molecule+atom threshold for
$(n,f_{\textrm{C}})=(2,1)$ is thus below that for $(0,2)$ by just 150 MHz at
zero field. The states that can cause resonances at low fields must be bound
by slightly more than the rotational splittings. This is similar to the case
of $(0,2)$, but with smaller relative magnetic moments. This means that the
energy windows needed to produce resonances below 600 G are narrower: for
$n^{\textrm{res}}=1$, the states must be bound between 2.23 and 2.62 GHz, with
a probability of 15%; for $n^{\textrm{res}}=2$, the states must be bound
between 6.68 and 7.07 GHz, with a probability of 8%. These states are coupled
to the incoming channel by the part of the Fermi contact interaction that is
anisotropic in the sense that it depends on the Jacobi coordinate $\theta$. In
Fig. 3(c) we show examples of these states with (1,1) and (2,1), and
$N^{\textrm{res}}=0$ such that $L^{\textrm{res}}=n^{\textrm{res}}$. This shows
that the states for $(2,1)$ will cause resonances only if they lie near the
boundary between the fourth and fifth bins, as for $(0,2)$, and those for
$(1,1)$ if they lie in the middle of the third bin. The components
$(m^{\textrm{res}}_{\textrm{A}},m^{\textrm{res}}_{\textrm{B}})$ that can cause
resonances are the same as for $n^{\textrm{res}}=0$.

There can also be resonances from states supported by higher thresholds. These
include thresholds with $n^{\textrm{res}}>2$, or with $n^{\textrm{res}}>0$ and
$\tilde{f}^{\textrm{res}}_{\textrm{C}}=2$. However these states must be even
more deeply bound, so are more sparse. At sufficient depth, such states become
increasingly strongly coupled to short-range states and are expected to become
part of the chaotic manifold(s). No current theories exist to describe how and
for what states this happens.

One special feature of this system occurs due to the near-degeneracy,
mentioned above, between the rotational excitation to $n=2$ and the hyperfine
splitting. As seen in Fig. 3(c), a bound state in the top bin for a channel
with $(n,\tilde{f}_{\textrm{C}},m_{f,\textrm{C}})=(2,1,-1)$ crosses the
threshold with $(0,2,-2)$ between 70 and 140 G, and the corresponding states
in the second bin crosses between 140 and 580 G. This will produce an
additional type of resonance at thresholds with
$\tilde{f}^{\textrm{in}}_{\textrm{C}}=2.$

Finally, there can be resonances due to states with $N^{\textrm{res}}>0$.
These are coupled to the incoming state by only the couplings described in
Sec. II.2.2. We focus on states with
$(n^{\textrm{res}},\tilde{f}^{\textrm{res}}_{\textrm{C}},L^{\textrm{res}})=(0,1,1)$
and (0,1,2) (with $N^{\textrm{res}}=1$ and 2, respectively), which are those
most likely to cause low-field resonances. Examples of these states are shown
in Fig. 3(d). These are again shown for $a=\infty$, but for
$L^{\textrm{res}}=1$ and 2 these states are not at the bin boundary. We
therefore also show the bin boundaries as discussed in Sec. III.2. This shows
that such resonances are expected below 360 and 560 G for $L^{\textrm{res}}=1$
and 2, respectively. There are many more components
$(m_{\textrm{A}}^{\textrm{res}},m_{\textrm{B}}^{\textrm{res}})$ that can cause
resonances in this case, because the sum
$m_{f,\textrm{tot}}=m_{\textrm{A}}+m_{\textrm{B}}+m_{f,\textrm{C}}$ can
change, compensated by $M_{N}$ to conserve $M_{F}$. The components that cause
the strongest resonances will depend on which of the couplings discussed in
Sec. II.2.2 are most important. We have previously attributed the magnetic
Feshbach resonances observed for 40K+NaK to states of this type
Wang:K_NaK:2021 .

##  V Conclusions

We have provided a framework for understanding long-range states of alkali-
metal atom-diatom complexes, and the types of magnetic Feshbach resonances
they can produce. We first explored the terms that exist in the Hamiltonian
for such triatomic systems and the ways in which they couple the six sources
of angular momentum present. We separated the couplings into those that
conserve the total spin-free angular momentum $N$ and those that can change
it. The former category includes the electronic interaction potential and the
Fermi-contact hyperfine interaction. The couplings in the latter category
include spin-rotation, tensor hyperfine, and anisotropic Zeeman interactions.
These terms are all expected to be significantly weaker than the Fermi-contact
interaction, but they allow coupling to a wider variety of channels and thus
may still be crucial to the dynamics.

We then considered the nature of the long-range states themselves. In contrast
to the short-range states of the complex considered previously Mayle:2012 ;
Frye:chaos:2016 , these are expected to retain much of the character of the
separated atom and diatomic molecule. They are likely to have structured (non-
chaotic) patterns of energy levels; the levels exist in ‘bins’ of energy below
each threshold that are determined by the reduced mass and the long-range
potential.

Finally, through a case study of KRb+Rb, we considered the types of magnetic
Feshbach resonances that can occur. These split into a number of broad
categories: those which involve only Zeeman splittings, with no hyperfine or
rotational excitation; those which involve hyperfine excitation but no
rotation; those which involve rotational excitation (with or without hyperfine
excitation) but conserve the total spin-free angular momentum; and those that
change the spin-free angular momentum. For each type we considered the ranges
of field where they occur and the conditions that need to be met for the
resonances to exist at experimentally accessible fields.

The work described here paves the way for future studies of Feshbach
resonances in alkali-metal atom–diatom systems and for their use to control
collisions and form ultracold triatomic molecules.

###### Acknowledgements.

This work was supported by the U.K. Engineering and Physical Sciences Research
Council (EPSRC) Grants No. EP/P01058X/1 and EP/W00299X/1.

##  Appendix A Channel functions

The channel functions used in the coupled-channel expansion (1) are based on,
but not equivalent to, the eigenfunctions of the separated atom and molecule.
They are

$\Phi_{j}(\xi)=\psi^{\textrm{C}}_{\tilde{f},m_{f,\textrm{C}}}\psi^{\textrm{AB}}_{m_{\textrm{A}},m_{\textrm{B}}}\psi^{\textrm{ABC}}_{vnLNM_{N}}(\vec{r},\hat{R}).$
(5)

Here $\psi^{\textrm{C}}_{\tilde{f},m_{f,\textrm{C}}}$ is an eigenstate of the
free atom in a magnetic field,
$\psi^{\textrm{AB}}_{m_{\textrm{A}},m_{\textrm{B}}}$ is a spin state of the
free diatomic molecule, and the coupled spatial function is

$\displaystyle\psi^{\textrm{ABC}}_{vnLNM_{N}}(\vec{r},\hat{R})$
$\displaystyle=\sum_{m_{n},M_{L}}\langle nm_{n}LM_{L}|NM_{N}\rangle$
$\displaystyle\times\psi_{vn}(r)Y_{nm_{n}}(\beta_{r},\alpha_{r})Y_{LM_{L}}(\beta_{R},\alpha_{R}),$
(6)

where $\psi_{vn}(r)$ is a vibrational function for the diatomic molecule, the
functions $Y$ are spherical harmonics whose arguments are the polar
coordinates of unit vectors $\hat{r}$ and $\hat{R}$ along $\vec{r}$ and
$\vec{R}$, and $\langle nm_{n}LM_{L}|NM_{N}\rangle$ is a Clebsch-Gordan
coefficient. These channel functions are eigenfunctions of the Hamiltonian of
the free atom and the vibration-rotation and nuclear Zeeman terms in the
Hamiltonian of the free diatomic molecule. However, there are small off-
diagonal terms arising from the hyperfine, rotational Zeeman and ac Stark
terms for the free molecule, and from the interaction operator
$\hat{V}(R,\xi)$.

## References

  * (1) C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Rev. Mod. Phys. 82, 1225 (2010).
  * (2) P. Naidon and S. Endo, “Efimov physics: a review,” Rep. Prog. Phys. 80, 056001 (2017).
  * (3) I. Bloch, J. Dalibard, and S. Nascimbène, “Quantum simulations with ultracold quantum gases,” Nature Physics 8, 267 (2012).
  * (4) 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).
  * (5) 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).
  * (6) T. Köhler, K. Góral, and P. S. Julienne, “Production of cold molecules via magnetically tunable Feshbach resonances,” Rev. Mod. Phys. 78, 1311 (2006).
  * (7) K.-K. Ni, S. Ospelkaus, M. H. G. 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 in the rovibrational ground state,” Science 322, 231 (2008).
  * (8) J. G. Danzl, E. Haller, M. Gustavsson, M. J. Mark, R. Hart, N. Bouloufa, O. Dulieu, H. Ritsch, and H.-C. Nägerl, “Quantum gas of deeply bound ground state molecules,” Science 321, 1062 (2008).
  * (9) F. Lang, P. van der Straten, B. Brandstätter, G. Thalhammer, K. Winkler, P. S. Julienne, R. Grimm, and J. Hecker Denschlag, “Cruising through molecular bound state manifolds with radio frequency,” Nature Physics 4, 223 (2008).
  * (10) J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, R. Hart, J. Aldegunde, J. M. Hutson, and H.-C. Nägerl, “An ultracold, high-density sample of rovibronic ground-state molecules in an optical lattice,” Nature Physics 6, 265 (2010).
  * (11) 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).
  * (12) 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).
  * (13) 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).
  * (14) 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 23Na87Rb molecules,” Phys. Rev. Lett. 116, 205303 (2016).
  * (15) 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).
  * (16) 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 23Na40K molecules,” Phys. Rev. A 97, 013405 (2018).
  * (17) H. Yang, D.-C. Zhang, L. Liu, Y.-X. Liu, J. Nan, B. Zhao, and J.-W. Pan, “Observation of magnetically tunable Feshbach resonances in ultracold 23Na40K + 40K collisions,” Science 363, 261 (2019).
  * (18) 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).
  * (19) W. B. Cairncross, J. T. Zhang, L. R. Picard, Y. Yu, K. Wang, and K.-K. Ni, “Assembly of a rovibrational ground state molecule in an optical tweezer,” Phys. Rev. Lett. 126, 123402 (2021).
  * (20) P. D. Gregory, M. D. Frye, J. A. Blackmore, E. M. Bridge, R. Sawant, J. M. Hutson, and S. L. Cornish, “Sticky collisions of ultracold RbCs molecules,” Nature Comm. 10, 3104 (2019).
  * (21) R. Bause, A. Schindewolf, R. Tao, M. Duda, X.-Y. Chen, G. Quéméner, T. Karman, A. Christianen, I. Bloch, and X.-Y. Luo, “Collisions of ultracold molecules in bright and dark optical dipole traps,” Phys. Rev. Research 3, 033013 (2021).
  * (22) P. Gersema, K. K. Voges, M. Meyer zum Alten Borgloh, L. Koch, T. Hartmann, A. Zenesini, S. Ospelkaus, J. Lin, J. He, and D. Wang, “Probing photoinduced two-body loss of ultracold nonreactive bosonic ${}^{23}\mathrm{Na}^{87}\mathrm{Rb}$ and ${}^{23}\mathrm{Na}^{39}\mathrm{K}$ molecules,” Phys. Rev. Lett. 127, 163401 (2021).
  * (23) P. S. Żuchowski and J. M. Hutson, “Reactions of ultracold alkali metal dimers,” Phys. Rev. A 81, 060703(R) (2010).
  * (24) Z. Idziaszek and P. S. Julienne, “Universal rate constants for reactive collisions of ultracold molecules,” Phys. Rev. Lett. 104, 113202 (2010).
  * (25) G. Wang and G. Quéméner, “Tuning ultracold collisions of excited rotational dipolar molecules,” New J. Phys. 17, 035015 (2015).
  * (26) T. Karman and J. M. Hutson, “Microwave shielding of ultracold polar molecules,” Phys. Rev. Lett. 121, 163401 (2018).
  * (27) K. Matsuda, L. D. Marco, J.-R. Li, W. G. Tobias, G. Valtolina, G. Quéméner, and J. Ye, “Resonant collisional shielding of reactive molecules using electric fields,” Science 370, 1324 (2020).
  * (28) L. Anderegg, S. Burchesky, Y. Bao, S. S. Yu, T. Karman, E. Chae, K.-K. Ni, W. Ketterle, and J. M. Doyle, “Observation of microwave shielding of ultracold molecules,” Science 373, 779 (2021).
  * (29) J.-R. Li, W. G. Tobias, K. Matsuda, C. Miller, G. Valtolina, L. D. Marco, R. R. W. Wang, L. Lassablière, G. Quéméner, J. L. Bohn, and J. Ye, “Tuning of dipolar interactions and evaporative cooling in a three-dimensional molecular quantum gas,” Nature Physics 17, 1144 (2021).
  * (30) A. Schindewolf, R. Bause, X.-Y. Chen, M. Duda, T. Karman, I. Bloch, and X.-Y. Luo, “Evaporation of microwave-shielded polar molecules to quantum degeneracy,” Nature 607, 677 (2022).
  * (31) M. Mayle, B. P. Ruzic, and J. L. Bohn, “Statistical aspects of ultracold resonant scattering,” Phys. Rev. A 85, 062712 (2012).
  * (32) M. Mayle, G. Quéméner, B. P. Ruzic, and J. L. Bohn, “Scattering of ultracold molecules in the highly resonant regime,” Phys. Rev. A 87, 012709 (2013).
  * (33) A. Christianen, M. W. Zwierlein, G. C. Groenenboom, and T. Karman, “Photoinduced two-body loss of ultracold molecules,” Phys. Rev. Lett. 123, 123402 (2019).
  * (34) P. D. Gregory, J. A. Blackmore, S. L. Bromley, and S. L. Cornish, “Loss of ultracold 87Rb133Cs molecules via optical excitation of long-lived two-body collision complexes,” Phys. Rev. Lett. 124, 163402 (2020).
  * (35) Y. Liu, M.-G. Hu, M. A. Nichols, D. D. Grimes, T. Karman, H. Guo, and K.-K. Ni, “Photo-excitation of long-lived transient intermediates in ultracold reactions,” Nature Physics 16, 1132 (2020).
  * (36) P. D. Gregory, J. A. Blackmore, M. D. Frye, L. Fernley, S. L. Bromley, J. M. Hutson, and S. L. Cornish., “Molecule-molecule and atom-molecule collisions with ultracold RbCs molecules,” New J. Phys. 23, 125004 (2021).
  * (37) X.-Y. Wang, M. D. Frye, Z. Su, J. Cao, L. Liu, D.-C. Zhang, H. Yang, J. M. Hutson, B. Zhao, C.-L. Bai, and J.-W. Pan, “Magnetic Feshbach resonances in collisions of 23Na40K with 40K,” New J. Phys. 23, 115010 (2021).
  * (38) Z. Su, H. Yang, J. Cao, X.-Y. Wang, J. Rui, B. Zhao, and J.-W. Pan, “Resonant control of elastic collisions between ${}^{23}\mathrm{Na}^{40}\mathrm{K}$ molecules and ${}^{40}\mathrm{K}$ atoms,” Phys. Rev. Lett. 129, 033401 (2022).
  * (39) H. Yang, X.-Y. Wang, Z. Su, J. Cao, D.-C. Zhang, J. Rui, B. Zhao, C.-L. Bai, and J.-W. Pan, “Evidence for the association of triatomic molecules in ultracold 23Na40K + 40K mixtures,” Nature 602, 229 (2022).
  * (40) H. Yang, J. Cao, Z. Su, J. Rui, B. Zhao, and J.-W. Pan, “Creation of an ultracold gas of triatomic molecules from an atom–diatomic molecule mixture,” Science 378, 1009 (2022).
  * (41) H. Son, J. J. Park, Y.-K. Lu, A. O. Jamison, T. Karman, and W. Ketterle, “Control of reactive collisions by quantum interference,” Science 375, 1006 (2022).
  * (42) M. D. Frye and J. M. Hutson, “Complexes formed in collisions between ultracold alkali-metal diatomic molecules and atoms,” New J. Phys. 23, 125008 (2021).
  * (43) M. D. Frye, M. Morita, C. L. Vaillant, D. G. Green, and J. M. Hutson, “Approach to chaos in ultracold atomic and molecular physics: Statistics of near-threshold bound states for Li+CaH and Li+CaF,” Phys. Rev. A 93, 052713 (2016).
  * (44) J. Aldegunde, B. A. Rivington, P. S. Żuchowski, 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).
  * (45) J. Aldegunde and J. M. Hutson, “Hyperfine structure of alkali-metal diatomic molecules,” Phys. Rev. A 96, 042506 (2017).
  * (46) H. Ran, J. Aldegunde, and J. M. Hutson, “Hyperfine structure in the microwave spectra of ultracold polar molecules,” New J. Phys. 12, 043015 (2010).
  * (47) P. D. Gregory, J. A. Blackmore, J. Aldegunde, J. M. Hutson, and S. L. Cornish, “ac Stark effect in ultracold polar 87Rb133Cs molecules,” Phys. Rev. A 96, 021402(R) (2017).
  * (48) R. L. Martin and E. R. Davidson, “Electronic structure of the sodium trimer,” Mol. Phys. 35, 1713 (1978).
  * (49) A. J. C. Varandas and V. M. F. Morais, “Semi-empirical valence bond potential energy surfaces for homonuclear alkali trimers,” Mol. Phys. 47, 1241 (1982).
  * (50) J. L. Martins, R. Car, and J. Buttet, “Electronic properties of alkali trimers,” J. Chem. Phys. 78, 5646 (1983).
  * (51) A. J. C. Varandas, V. M. F. Morais, and A. A. C. C. Pais, “Semiempirical valence bond potential energy surfaces for the alkali trimers,” Mol. Phys. 58, 285 (1986).
  * (52) H. von Busch, V. Dev, H.-A. Eckel, S. Kasahara, J. Wang, W. Demtröder, P. Sebald, and W. Meyer, “Unambiguous proof for Berry’s phase in the sodium trimer: Analysis of the transition $A^{2}E^{\prime}\leftarrow X^{2}E^{\prime}$,” Phys. Rev. Lett. 81, 4584 (1998).
  * (53) A. W. Hauser, C. Callegari, P. Soldán, and W. E. Ernst, “On the doublet states of the potassium trimer,” J. Chem. Phys. 129, 044307 (2008).
  * (54) A. W. Hauser, C. Callegari, P. Soldán, and W. E. Ernst, “A Jahn-Teller analysis of K3 and Rb3 in the electronic states $1^{2}{E}^{\prime}$ and $1^{2}{E}^{\prime\prime}$,” Chem. Phys. 375, 73 (2010).
  * (55) A. W. Hauser, J. V. Pototschnig, and W. E. Ernst, “A classic case of Jahn-Teller effect theory revisited: Ab initio simulation of hyperfine coupling and pseudorotation in the 12E′ state of Na3,” Chem. Phys. 460, 2 (2015).
  * (56) J. F. E. Croft, C. Makrides, M. Li, A. Petrov, B. K. Kendrick, N. Balakrishnan, and S. Kotochigova, “Universality and chaoticity in ultracold K+KRb chemical reactions,” Nature Communications 8, 15897 (2017).
  * (57) B. K. Kendrick, H. Li, M. Li, S. Kotochigova, J. F. E. Croft, and N. Balakrishnan, “Non-adiabatic quantum interference in the ultracold Li + LiNa $\to$ Li2 + Na reaction,” Phys. Chem. Chem. Phys. 23, 5096 (2021).
  * (58) G. H. Lushington and F. Grein, “Multireference configuration interaction calculations of electronic $g$-tensors for NO2, H2O+, and CO+,” J. Chem. Phys. 106, 3292 (1997).
  * (59) A. Abragam and B. Bleaney, _Electron paramagnetic resonance of transition ions_ (Oxford University Press, Oxford, 1970).
  * (60) J. H. Van Vleck, “The coupling of angular momentum vectors in molecules,” Rev. Mod. Phys. 23, 213 (1951).
  * (61) R. F. Curl Jr., “The relationship between electron spin rotation coupling constants and g-tensor components,” Mol. Phys. 9, 585 (1965).
  * (62) G. Tarczay, P. G. Szalay, and J. Gauss, “First-principles calculation of electron spin-rotation tensors,” J. Phys. Chem. A 114, 9246 (2010).
  * (63) J. Gauss, M. Kállay, and F. Neese, “Calculation of electronic g-tensors using coupled cluster theory,” J. Phys. Chem. A 113, 11541 (2009).
  * (64) D. Datta and J. Gauss, “Accurate prediction of hyperfine coupling tensors for main group elements using a unitary group based rigorously spin-adapted coupled-cluster theory,” J. Chem. Theory Comput. 15, 1572 (2019).
  * (65) E. van Lenthe, A. van der Avoird, and P. E. S. Wormer, “Density functional calculations of molecular hyperfine interactions in the zero order regular approximation for relativistic effects,” J. Chem. Phys. 108, 4783 (1998).
  * (66) F. Neese, “Prediction of electron paramagnetic resonance g values using coupled perturbed Hartree-Fock and Kohn-Sham theory,” J. Chem. Phys. 115, 11080 (2001).
  * (67) F. Neese, “Efficient and accurate approximations to the molecular spin-orbit coupling operator and their use in molecular g-tensor calculations,” J. Chem. Phys. 122, 034107 (2005).
  * (68) J. Autschbach, S. Patchkovskii, and B. Pritchard, “Calculation of hyperfine tensors and paramagnetic NMR shifts using the relativistic zeroth-order regular approximation and density functional theory,” J. Chem. Theor. Comput. 7, 2175 (2011).
  * (69) R. J. Le Roy and R. B. Bernstein, “Dissociation energy and long-range potential of diatomic molecules from vibrational spacings of higher levels,” J. Chem. Phys. 52, 3869 (1970).
  * (70) T. Maier, I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Pfau, K. Jachymski, and P. S. Julienne, “Broad universal Feshbach resonances in the chaotic spectrum of dysprosium atoms,” Phys. Rev. A 92, 060702(R) (2015).
  * (71) J. M. Hutson, “An introduction to the dynamics of Van der Waals molecules,” in _Advances in Molecular Vibrations and Collision Dynamics_ (JAI Press, Greenwich, Connecticut, 1991), vol. 1A, pp. 1–45.
  * (72) P. S. Żuchowski, J. Aldegunde, and J. M. Hutson, “Ultracold RbSr molecules can be formed by magnetoassociation,” Phys. Rev. Lett. 105, 153201 (2010).
  * (73) V. Barbé, A. Ciamei, B. Pasquiou, L. Reichsöllner, F. Schreck, P. S. Żuchowski, and J. M. Hutson, “Observation of Feshbach resonances between alkali and closed-shell atoms,” Nature Physics 14, 881 (2018).
  * (74) D. A. Brue and J. M. Hutson, “Prospects of forming molecules in ${}^{2}\Sigma$ states by magnetoassociation of alkali-metal atoms with Yb,” Phys. Rev. A 87, 052709 (2013).
  * (75) B. C. Yang, M. D. Frye, A. Guttridge, J. Aldegunde, P. S. Żuchowski, S. L. Cornish, and J. M. Hutson, “Magnetic Feshbach resonances in ultracold collisions between Cs and Yb atoms,” Phys. Rev. A 100, 022704 (2019).
  * (76) B. Gao, “Zero-energy bound or quasibound states and their implications for diatomic systems with an asymptotic van der Waals interaction,” Phys. Rev. A 62, 050702(R) (2000).
  * (77) G. F. Gribakin and V. V. Flambaum, “Calculation of the scattering length in atomic collisions using the semiclassical approximation,” Phys. Rev. A 48, 546 (1993).
  * (78) P. S. Żuchowski, M. Kosicki, M. Kodrycka, and P. Soldán, “Van der Waals coefficients for systems with ultracold polar alkali-metal molecules,” Phys. Rev. A 87, 022706 (2013).
  * (79) B. Gao, “Quantum-defect theory of atomic collisions and molecular vibration spectra,” Phys. Rev. A 58, 4222 (1998).
  * (80) B. Gao, “Solutions of the Schrödinger equation for an attractive $1/r^{6}$ potential,” Phys. Rev. A 58, 1728 (1998).
  * (81) B. Gao, “Angular-momentum-insensitive quantum-defect theory for diatomic systems,” Phys. Rev. A 64, 010701 (2001).
  * (82) B. Gao, “Routines to calculate the AQDT parameters for an attractive $1/r^{6}$ potential, Version 2,” (2003), university of Toledo, Ohio.
  * (83) See Supplemental Material at [URL will be inserted by publisher] for alternative example crossing diagrams equivalent to Fig. 3 for $a=0$, $\bar{a}$, and $2\bar{a}$, and for tabulations of the corresponding crossings.
  * (84) D. A. Brue and J. M. Hutson, “Magnetically tunable Feshbach resonances in ultracold Li-Yb mixtures,” Phys. Rev. Lett. 108, 043201 (2012).
  * (85) S. Ospelkaus, K.-K. Ni, G. Quéméner, B. Neyenhuis, D. Wang, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Controlling the hyperfine state of rovibronic ground-state polar molecules,” Phys. Rev. Lett. 104, 030402 (2010).