SU(N) magnetism with ultracold molecules
Bijit Mukherjee Jeremy M. Hutson Kaden R. A. Hazzard




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

j.m.hutson@durham.ac.uk Joint Quantum Centre (JQC)
Durham-Newcastle, Department of Chemistry, Durham University, South Road,
Durham, DH1 3LE, United Kingdom.

kaden.hazzard@gmail.com
Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
Smalley-Curl Institute, Rice University, Houston, Texas 77005, USA


             


 Quantum systems with SU(N) symmetry are paradigmatic settings for quantum many-body physics.  They have been studied for the insights they provide into complex materials and their ability to stabilize  exotic ground states. Ultracold alkaline-earth atoms were predicted to exhibit SU(N) symmetry for N=2I+1=1,2,…,10, where I is the nuclear spin.  Subsequent experiments have revealed rich many-body physics. However, alkaline-earth atoms realize this symmetry only for fermions with repulsive interactions. In this paper, we predict that ultracold molecules shielded from destructive collisions with static   electric fields or microwaves  exhibit SU(N) symmetry, which holds because deviations of the s-wave scattering length from the spin-free values are only about 3% for CaF with static-field shielding and  are estimated to be even smaller for bialkali molecules.  They open the door to N as large as 32 for bosons and 36 for fermions. They offer important  features unachievable with atoms, including bosonic systems and attractive interactions.   


SU(N)  magnetism with  ultracold molecules
    Kaden R. A. Hazzard
    April 29, 2024
==========================================



§ INTRODUCTION

 
The prediction and experimental realization  of  SU(N=2I+1) symmetry in ultracold alkaline-earth atoms <cit.> with nuclear spin I has allowed the study of a wealth of phenomena:  bosonization of high-spin fermions by measurements of collective modes <cit.>,  flavor-selective Mott  transitions in a lattice <cit.>, and the SU(N) Fermi-Hubbard model's equations of state <cit.> and short-ranged magnetic correlations <cit.>.   
The symmetry is also responsible for the temperature T=1 nK reached in Ref. <cit.>; this is the lowest temperature ever achieved  for fermions.  Unlike ordinary SU(2) spins, quantum fluctuations need not become classical for large spin.  Consequently, exotic behavior is predicted to occur in lattices as the temperature is lowered further. Predicted phases abound for different lattice geometries and N, including  chiral spin liquids, a topological phase of matter never before  observed <cit.>. The itinerant or doped SU(N) Hubbard models are   little explored and likely to show extremely rich phenomena. 

However, alkaline-earth-atom realizations of SU(N) physics have important limitations.  One constraint is that, to have  I 0 (N>1), such atoms   must be fermions to satisfy the “even-even" rule of nuclear physics <cit.>.
Another constraint is that, empirically, all the   interactions in the experimentally viable ultracold alkaline-earth atoms, Sr and Yb, are repulsive,  i.e., have a  positive scattering length a <cit.>. Moreover, the important tool of magnetic Feshbach resonances that is used to tune interactions in   alkali atoms is absent for ground-state alkaline-earth atoms, since they lack unpaired electrons.  

Rapidly advancing experiments with ultracold molecules offer exciting possibilities for many-body physics, quantum technologies, precision measurement, and studying chemical reactions <cit.>. A wide variety of ultracold molecules have  been produced <cit.>, including alkali-metal dimers   produced by assembly of ultracold atoms, and other species produced by direct laser cooling. Experiments revealed losses that occur when molecular collisions reach short range (R≲ 100 a_0),  which impede the creation of interesting many-body states.    
They occur  for both reactive and non-reactive molecules.  This discovery stimulated 
theoretical proposals to shield the molecules from short-range loss by creating intermediate-range repulsive interactions  using static electric fields <cit.> or microwave radiation <cit.>. These have now been demonstrated experimentally <cit.>. Notably, Bose-Einstein condensation has recently been achieved for NaCs using two-color microwave shielding    <cit.>.
  

In this paper, we show that shielded molecules can realize SU(N) systems that circumvent the constraints of alkaline-earth atoms, and introduce other interesting properties.  Fermionic and bosonic molecules are available, and the sign and magnitude of the scattering length can be tuned by varying the control fields that generate the shielding <cit.>. The dipolar interactions may also be tuned.
Experimentally available molecules can realize all N up to N=36, for Na^40K <cit.>, much larger than for Sr (N=10) and Yb (N=6).  

The  properties of  SU(N)  molecular systems open interesting paths for many-body physics. Attractively interacting SU(N) systems are predicted to have  rich pairing structures, for example transitions between color superfluid and trion phases <cit.>,  
with similarities to the  crossover from a confining hadronic phase to a parton gas  in high-energy physics <cit.>.  Bosonic SU(N) systems have garnered interest   as integrable systems  
<cit.>,   holographic duals <cit.>, and  ferromagnets with non-Abelian symmetry breaking <cit.>.  
Such examples offer a small glimpse of the possibilities offered by shielded ultracold molecules. 

Section <ref>   presents coupled-channel calculations for CaF collisions shielded with a static electric field. Even CaF, which has much larger spin couplings than  bialkali molecules, satisfies   SU(N) symmetry to about  3%   relative accuracy.  We develop  a model to estimate quantitatively  the nuclear-spin dependence of interactions for other molecules. This  shows that SU(N) symmetry will hold to even higher  accuracy for alkali dimers. 
Section <ref> derives  many-body models for these systems, especially the SU(N) Hubbard models that describe shielded molecules in an optical lattice.   Section <ref> summarizes, suggests experiments to verify the predictions, and outlines next steps for the field. 
 



§ SHIELDED INTERACTIONS OF MOLECULES


A pair of polar molecules, k=1,2, interact at long range via the dipole-dipole interaction 

    Ĥ_dd = -3(μ_1·R̂) (μ_2·R̂) - μ_1·μ_2/4πϵ_0 R^3,

where R is the intermolecular distance, R̂ is the corresponding unit vector, and μ_k is the dipole moment of molecule k which lies along the molecular axis.
For an s-wave collision, with relative angular momentum (partial-wave) quantum number  L=0, the dipole-dipole interaction averages to zero. However, it has matrix elements between L=0 and 2, both diagonal and off-diagonal in molecular pair state. The matrix elements off-diagonal only in L cause an effective long-range attraction proportional to d^4/R^4 <cit.>, where d is the space-fixed dipole moment of each molecule (for static-field shielding) or the rotating dipole (for microwave shielding).

Both static-field and microwave shielding operate by engineering a  field-dressed pair state to be a small energy Δ E below the initial state of interest.  
Matrix elements of Ĥ_dd that connect the two pair states  
produce a repulsive contribution to the interaction potential for the upper (initial) state. Shielding occurs when the repulsion is sufficient to prevent  pairs of colliding molecules coming close together. The repulsion is proportional to (d^4/Δ E)/R^6 at long range. 

The combination of long-range attraction and shorter-range repulsion produces a potential well at long range, whose depth and position depend on the molecule and the field applied. This allows considerable control over the scattering length, which for some molecules may be tuned from positive to negative values and even through poles <cit.>.



We focus here on static-field shielding, using coupled-channel scattering calculations on CaF as an example.  The methods used are described in Ref. <cit.> and their numerical convergence is described in Methods. We begin with spin-free calculations (Sec. <ref>) and then consider the effects of electron and nuclear spins (Sec. <ref>).   



 §.§ The spin-free case



We consider collisions between molecules in the state (ñ,m_n)=(1,0), for which static-field shielding is most effective. The single-molecule eigenstates are labeled by  hindered-rotor quantum numbers  ñ, which correlate at zero field with the free-rotor quantum number n, and m_n, its projection onto z.  

It is helpful to consider the effective potentials for scattering. We define these as adiabats that are the R-dependent eigenvalues U_i(R) of the pair Hamiltonian given in Methods.  Figure <ref> shows the spin-free adiabats for s-wave scattering for two CaF molecules in state (ñ,m_n)=(1,0) at two  values of an applied electric field F that produce effective shielding. 



There are two sources of 2-body loss. First, the repulsion does not extend all the way to R=0. There is a repulsive barrier, and some colliding pairs may tunnel through it to reach short range (R≲ 100 a_0), where loss may occur. Secondly, there can be  inelastic collisions that produce molecules in lower field-dressed states, particularly in the pair state just below the initial one.  The solid lines in  Fig. <ref>(a) show the calculated rate coefficients for spin-free elastic scattering and total (inelastic + short-range) loss for CaF as a function of F <cit.>. The overall effectiveness of shielding may be characterized by the ratio γ of rate coefficients for elastic scattering and loss; this can be up to 10^7 for CaF and much larger for molecules such as NaCs <cit.>. The calculations also provide the complex  s-wave scattering length, a(F)= α(F) - iβ(F), where β arises due to loss and L-changing collisions and is small when shielding is effective. The real part α(F) is of principal interest here. Figure <ref>(a) shows α(F) for CaF from spin-free coupled-channel calculations.



 §.§ Dependence on spin state


SU(N) symmetry exists when 
the shielded interactions are   almost independent of  molecular  spin state and diagonal in it. Here we describe an approach to estimate the dependence on spin state  and present quantum scattering calculations for CaF that validate the estimate. We then give estimates of the spin dependence for a variety of ultracold molecules of current experimental interest.
 
  Electron  and nuclear spins are described by the Hamiltonian for fine and hyperfine structure, ĥ_fhf, given in Methods. For CaF in a strong electric field, the electron spin s=1/2 and the nuclear spin i=1/2 of ^19F couple to form  g, with projection m_g; these  are approximately conserved and can take values (g,m_g)=(0,0), (1,0) and (1,±1).

We first diagonalize the single-molecule Hamiltonian, including ĥ_fhf.    We calculate the space-fixed dipole moments d_j=⟨ j|μ_z|j⟩ for  j=(ñ,m_n,g,m_g) with (ñ,m_n)=(1,0).
It is convenient to define fractional changes Δ d_j=(d_j-d_0)/d_0 from the spin-free value d_0. These  depend only weakly on F: for CaF at F=23 kV/cm, the values are -1.9×10^-5, 8.3×10^-4 and -4.4×10^-4 for the states (g,m_g)=(0,0), (1,0) and (1,±1), respectively.
The coefficient of the long-range attraction is proportional to d_j^4 and thus differs
by at most 0.33% from d_0^4. We may expect that the effective potentials will differ by about this amount.



To test this simple model of the interactions, we perform full coupled-channel calculations   including electron and nuclear spins for both molecules. Details are in Methods.  The calculations produce scattering lengths a_jj' and rate coefficients for elastic scattering and loss, as before, but now for each  pair of spin states j=(g,m_g) and j'=(g',m'_g). In zero magnetic field there are 7 distinct pairs, because  pairs with (m_g,m'_g)=(0,±1) are equivalent,  as are   the pairs (±1,±1), though the latter are different from (±1,∓1).

Shielding remains effective for  all spin states, even for distinguishable pairs.  However,  there are additional inelastic transitions for some spin states due to spin-changing collisions. In particular, a molecule initially in (g,m_g)= (1,1)  can  undergo a  transition to (1,0). The rate coefficient for this process in a collision of two  such molecules is shown as a blue dashed line in Fig. <ref>(a). The elastic and short-range loss are almost unaffected, but the total inelastic loss is modified as shown by the dashed green line.  The spin-changing rate coefficients are no larger than 10^-14 cm^3 s^-1 and the ratio γ remains above ∼10^5. The spin-changing rates are similar for other collisions involving a molecule in state (1,1), and  otherwise  very small.  This satisfies the requirement that the interactions are diagonal in spin state.

The real part of the scattering length α(F)  depends only weakly   on spin state. The solid lines in Fig. <ref>(b) show the differences δα_jj'(F)=α_jj'(F)-α_0(F) between the scattering lengths and the   spin-free value α_0(F) of Fig. <ref>(a) as a function of field. The values  α_jj' not shown are close to  (α_jj+α_j'j')/2.  This demonstrates that the scattering lengths are independent of spin state to within about 5% for shielded CaF.



The coupled-channel calculations provide adiabats  as  
in Fig. <ref>, but now for each spin combination. They are almost indistinguishable on the scale of Fig. <ref>. However, they cross zero at slightly different inner turning points R_tjj', so to show  their  differences  we define

    Δ U_jj'(R) = U_jj'(R+R_tjj'-R_t0)-U_0(R)/U_0(R),

 shifting U_jj'(R)  slightly in R so that its turning point matches  R_t0.
Fig. <ref> shows  Δ U_jj'(R) for all spin combinations of CaF at 23 kV/cm. The differences between the adiabats including spin and the spin-free adiabat are  no more  than 2% over the entire  classically allowed  range of R.  
The effective  potential for interaction of molecules in spin states j and j'  has long-range form 

    U_jj'(R) = -4ħ^2 D_jj'^2/15μ_redR^4,

where D_jj'=d_j d_j'μ_red/(4πϵ_0ħ^2)  is the dipole length  for space-fixed dipoles d_j and d_j'.
At large R, the ratios of the adiabats are accurately given by the corresponding ratios of D_jj'^2.  

We have developed a model for the effective potentials and scattering lengths for shielded collisions, which allows simple estimates of the spin-dependence of the  scattering length  for CaF and other molecules. The model is derived in Methods and gives  

    dα/dD≈    (α-2R_t)/D - 2 Φ√(8/15)^2(Φ-π/4),

where Φ is a phase integral involving the spin-free potential and R_t is its zero-energy turning point. These quantities are shown for CaF in Fig. <ref>(b), as a function of field. 

The dashed lines in Fig. <ref> show the results of the model for δα_jj', compared to the coupled-channel results including spin for CaF (solid lines). The model captures the overall behavior well.   Details of the model for CaF at 23 kV/cm are given   in Extended Data Table I, including the spin combinations not shown in Fig. <ref>. At this field, Φ for CaF is slightly less than 3π/4; this is close to a pole in α as a function of Φ, with α large and negative. Here   the second term in Eq. <ref> dominates, and small changes in Φ   cause  large fractional changes in α. Nevertheless, even for CaF at 23 kV/cm, where d_j varies by up to 0.13% between spin states, the values of δα_jj' are no more than 3% of α. For systems where Φ is not close to a pole, there will be less amplification  of  dα/dD by the last term in Eq. <ref>.



CaF has only 4 spin states, so can  be used realize SU(N) up to N=4. Other molecules, particularly alkali dimers, can reach much  greater N: singlet molecules with two spin-3/2 nuclei, such as Na^87Rb or  Na^39K, can reach N=16, while bosonic molecules with larger spins, such as NaCs, can reach N=32.  Fermionic Na^40K can reach N=36. 



Alkali dimers have a hyperfine Hamiltonian <cit.>  similar to that for  CaF, but with the electron spin replaced by a second nuclear spin  and additional terms arising from nuclear quadrupole coupling.  Coupled-channel   calculations that fully include spin are challenging for these molecules, because the spin space is so much larger: for NaCs, for example, the number of spin functions for the pair is 64 times larger than for CaF, and the computer time scales  as the cube of this. Nevertheless, we can estimate the spin dependence, using the understanding  gained from CaF. The changes in scattering length from the field-free value are

    δα_jj'= (D_jj'-D_0) dα/dD≈ D_0 (Δ d_j + Δ d_j') dα/dD.

Table <ref> summarizes the range of Δ d_j across spin states, for several ultracold molecules of current interest, for the field-dressed level (ñ,m_n)=(1,0) that  can be  shielded with a static electric field. All the alkali dimers have ranges substantially smaller than CaF.  

NaCs is particularly interesting, because α can be tuned close to zero at F ≈ 2.395 kV/cm <cit.>,  in the region  where shielding is effective. It has 32 hyperfine states and a particularly small range of dipoles, because NaCs has unusually small nuclear quadrupole coupling constants <cit.>. Figure <ref>(a) shows R_t0(F) and α_0(F) from spin-free coupled-channel calculations on NaCs. Figure <ref>(b) shows δα_jj(F) from Eq. <ref> for the hyperfine states with the largest and smallest values of d_j. The variation in α_jj' between spin states is only about 0.1% at most fields, and only about 3 a_0 around the zero in α.



§ MANY-BODY PHYSICS


Molecular systems with an SU(N) symmetry offer vast new possibilities for quantum simulation and many-body physics. The large spin degeneracy and the high symmetry enhance quantum fluctuations, stabilize exotic states of matter such as chiral spin liquids <cit.>, and produce interesting dynamics, such as controllable prethermalization <cit.>. Experiments  will fall into two categories: experiments in continuous space with just a trap, and optical lattice experiments.

In continuum experiments, the use of molecules will enrich the SU(N) phenomena studied with alkaline-earth atoms and also allow exploration of totally new areas. Ref. <cit.> has demonstrated quantum gas microscopy, which remains in development for fermionic alkaline-earth atoms. The large number of hyperfine states
will allow exploration of repulsive SU(N) models with much larger N than for alkaline-earth atoms; this will enhance quantum fluctuations and topological order <cit.>.  
The larger N may also allow even lower temperatures than in alkaline-earth atoms, which already reach record-low  temperatures for fermions <cit.>. 

Attractive and bosonic systems are also rich areas. Attractive gases, both with and without an optical lattice, may allow experiments to explore the formation of energetically favorable clusters and their ordering, with connections to both condensed <cit.> and high-density nuclear matter <cit.>. Bosonic SU(N) systems have been considered, for example as integrable systems <cit.>, non-Abelian ferromagnets <cit.> and holographic duals <cit.>.
  
To explore these areas efficiently and connect experiments to models studied in many-body physics, it is necessary to reduce the  coupled-channel results to an effective interaction. 
This is analogous to replacing the complicated interatomic potential  for atoms  with a delta-function. Due to the range of thousands of bohr and 1/R^4 tail, a contact potential is probably adequate only for  very  dilute gases. 
The strength of the delta-function interaction can be determined from the scattering length in the coupled-channel calculations.  Higher densities  will probably require more accurate effective potentials  based on the complete adiabats.  
 
Optical lattice experiments with  molecules  provide another wide-ranging arena for SU(N)  many-body physics. In a sufficiently deep lattice, with temperatures and interactions small compared to the band gap, the system is described by an SU(N) Hubbard Hamiltonian,  

    Ĥ   = -t ∑_⟨ i,j ⟩, σ( ĉ_i σ^†ĉ_j σ^† + h.c.) + U/2∑_i,σ, τn̂_i σn̂_i τ
       + ∑_ij;στV_ij/2 n_iσ n_jτ + ∑_iσ(ϵ_σ-μ_σ) n̂_iσ,
 
where c_iσ^† and c_iσ^† are annihilation and creation operators at site i for hyperfine state σ, and n_iσ=c^†_iσ c_iσ^†, V_ij = C_3(1-3cos^2(Θ_ij))/|r⃗_i - r⃗_j|^3 with Θ_ij the angle between the intermolecular separation and the electric field is the dipole interaction between molecules i and j, and ϵ_σ and μ_σ are the number operator, single-molecule energy and chemical potential for component σ.
The tunneling energy  t is

    t = -∫ d^3 r   w^*(r) ( -ħ^2/2m+V(r) ) w(r+d),
  
where d is a nearest-neighbor lattice vector, m is the molecular mass, V(r) is the lattice potential of a molecule at center of mass position r⃗, and w(r) is the lowest-band Wannier function obtained from the single-particle band structure. 
 When  the spread of the Wannier functions  is much larger than the interaction range,  

    U = 4πħ^2 a/m∫d^3 r   |w(r)|^4 ,

where a is the scattering length. 
Due to the large spatial extent of the interaction potential, Eq. <ref>  may provide only a rough estimate of U. Quantitative calculations of U can be performed by solving the two-body problem numerically. This is a challenging calculation, but is tractable when the lattice is deep enough. 
Although the single-molecule energies in the last term of Eq. (<ref>)  apparently break SU(N) symmetry, they are irrelevant because N̂_σ=∑_i n̂_iσ, is conserved. 
 



§ OUTLOOK 


Realization of SU(N) symmetry in ultracold molecules will unlock a broad range of new physics, with strong connections to condensed matter and other areas of many-body physics. 

A first step will be to confirm and quantify the degree of symmetry experimentally. Initial characterizations may be performed by measuring the kinetics of evaporation or cross-dimensional thermalization, already measured for one spin component of ^40KRb <cit.>.  For  a system with SU(N) symmetry,  these are  independent of hyperfine state.  Spectroscopy can provide more accurate measurements, analogous to those for SU(N) symmetry with alkaline-earth atoms <cit.>.  Two-photon microwave or Raman spectroscopy can measure the difference in interaction energy when the spin state is changed. A useful limit is a deep lattice where tunneling is negligible, so photons will drive  one-molecule hyperfine transitions on doubly-occupied sites. Amplitude-modulation spectroscopy  has also been used to measure  interactions of atoms in lattices to high precision <cit.>. With the SU(N) symmetry confirmed, experiments can begin probing the many-body  phases of matter and dynamics  offered. 

This work also opens areas of theoretical research in both  molecular collisions and many-body physics. Although the treatment of the alkali dimers in Sec. <ref> is sufficient to estimate the degree of SU(N) symmetry, full coupled-channel calculations are needed for quantitative results. This challenges current methods due to the large number of nuclear states. Although we have focused here on shielding with static   electric fields, we expect that microwave shielding will provide interactions with a comparable degree of SU(N) symmetry. Coupled-channel calculations are needed to verify this.

The new many-body physics offered by shielded ultracold molecules includes gases, Hubbard models, and Heisenberg models with large N. Such systems can have positive or negative scattering lengths, bosonic or fermionic or constitutents, and dipolar interactions. These offer fertile ground to study new phases and dynamics of quantum matter.

 



§ RIGHTS RETENTION STATEMENT


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



§ DATA AVAILABILITY STATEMENT


Data supporting this study are openly available from Durham University <cit.>.


  
K.R.A.H. acknowledges support from the National Science Foundation (PHY-1848304), the Robert A. Welch Foundation (C-1872), and the W. M. Keck Foundation (Grant No. 995764). K.R.A.H. benefited from discussions at the the Aspen Center for Physics, which is supported in part by the National Science Foundation (PHY-1066293). B.M. and J.M.H. are grateful to Matthew Frye and Ruth Le Sueur for valuable discussions and acknowledge support from the U.K. Engineering and Physical Sciences Research Council (EPSRC) Grant Nos. EP/W00299X/1, and EP/V011677/1.

 



§ METHODS




 §.§ Hamiltonian for the spin-free case



The Hamiltonian of spin-free CaF in an electric field is

    ĥ = bn̂^2 - μ·F.

The molecule is treated as a rigid rotor with rotational constant b in an electric field F along the z axis; n̂ is the operator for molecular rotation.  
For ^40Ca^19F, b/h ≈ 10.267 GHz and |μ|=3.07 D. The corresponding Hamiltonian for a pair of molecules is

    ĥ_1 + ĥ_2 + L̂^2/2μ_red + Ĥ_dd - C_6^elec/R^6,

where L̂ is the   operator for relative rotation of the pair and μ_red is the reduced mass. C_6^elec accounts for the electronic dispersion interaction between the molecules <cit.>, but has only small effects on the results.



 §.§ Basis sets used in coupled-channel calculations


The methodology used for the coupled-channel calculations on CaF is as described in Ref. <cit.>, except that the present calculations use different basis sets.

As in Ref. <cit.>, we use basis sets that are constructed from symmetrized products of field-dressed rotor functions |ñ,m_n⟩ and spin functions |g,m_g⟩, together with functions for the partial-wave quantum number L and its projection M_L. We include rotor functions with ñ up to 5. For CaF, there are four spin functions for each monomer rotor state. 
However, the resulting number of pair basis functions, N_pair, is too large (∼10000 for each L, M_L) to be used directly in coupled-channel calculations. We therefore include only a relatively small number of “class 1” pair functions explicitly in the basis set, with the remaining “class 2” functions taken into account through Van Vleck transformations as described in ref. <cit.>.
 
It is the number of class 1 functions that determines the overall computational cost. To achieve a manageable basis-set size, we include only energetically nearby rotor pairs in class 1 and move the remainder to class 2. In the present work, we  
include 14 rotor pairs in class 1: (ñ,m_n) = (0,0)+(1,0),  (1,-1)+(1,-1), (1,1)+(1,-1), (1,1)+(1,1), (1,0)+(1,-1), (1,0)+(1,1), (1,0)+(1,0), (0,0)+(2,-1), (0,0)+(2,1), (0,0)+(2,0), (1,-1)+(2,-1), (1,-1)+(2,1), (1,1)+(2,-1) and (1,1)+(2,1). Inclusion of all spin functions for each of these rotor pairs gives a total number of symmetrized pair states N_pair=206. We refer to basis sets based on this as spin-N206.

The spin-dependence of scattering lengths, characterized by δα_jj', converges very fast with respect to the rotor basis, and is much better than 1% for spin-N206. The convergence of loss rates is slower, and varies with field because colliding pairs are more likely to reach short range when shielding is poor. Nevertheless, at 23 kV/cm, spin-N206 gives convergence of loss rates to within 1%.

The basis set of partial waves is also important. For each pair function (rotor plus spin), we include partial waves L up to 6 and refer to the resulting basis sets as spin-N206-L6. For each spin combination, calculations are performed for only a single value of M_tot=m_n,1+m_n,2+m_g,1+m_g,2+M_L, such that the s-wave channel for the initial state is included in the basis set. The total number of coupled channels in class 1 varies from 556 to 652.
Based on comparisons between spin-N206-L6 and spin-N206-L4, we estimate that spin-N206-L6 gives δα_jj' converged to better than 1% and loss rates converged within 10%.



 §.§ Hamiltonian for fine and hyperfine structure


For a single CaF molecule, the Hamiltonian for fine and hyperfine structure is

    ĥ_fhf = γŝ·n̂ + ζ_Fî·ŝ + t√(6)T^2(C) · T^2(î,ŝ) + c_Fî·n̂.

Here the first term represents the electron spin-rotation interaction, while the second and third terms account for the isotropic and anisotropic interactions between electron and nuclear spins. T^2(î,ŝ) denotes the rank-2 spherical tensor formed from î and ŝ, and T^2(C) is a spherical tensor whose components are the Racah-normalized spherical harmonics C^2_q(θ,ϕ). The last term represents the nuclear spin-rotation interaction, which is typically three orders of magnitude smaller than the others.
The values of the constants b, γ, ζ_F, t and c_F for CaF are the same as in Ref. <cit.>.



 §.§ Effective-potential model of spin dependence


In a semiclassical approximation <cit.>, the real part α of the s-wave scattering length for a potential with long-range part proportional to R^-4 may be written in terms of a phase integral Φ,

    α = R_t - √(8/15) D tan(Φ-π/4).

Here R_t is the inner turning point at zero collision energy, with U(R_t)=U(∞)=0,  D is the dipole length, and

    Φ = ∫_R_t^∞ k(R) dR,

where k(R)=(2μ_red|U(R)|/ħ^2)^1/2. 
The first term in Eq. <ref>, R_t, is omitted in Ref. <cit.>, but can be a substantial fraction of D for shielded collisions; for CaF at 23 kV/cm, with D≈ 1290 a_0, R_t contributes about 380 a_0 to α.

If we take the spin-free adiabat U_0(R) as a reference, with phase k_0(R) and phase integral Φ_0, the integral Φ_jj' for a slightly shifted potential U_jj'(R) for the interaction between species in spin states j and j' is

    Φ_jj' =   ∫_R_tjj'^∞ k_jj'(R) dR 
    ≈   ∫_R_t0^∞ k_0(R) [1 + 1/2Δ U_jj'(R)] dR,

where Δ U_jj'(R) is the difference between adiabats defined by Eq. <ref> of the main text.

As shown in Extended Data Figure 1, Φ_jj' scales approximately with D_jj'^2 and R_t scales approximately with D_jj'^-1.
These dependences apply to varying isotopic combination at constant field, but not to varying field, which also changes the separation of the field-dressed states.

Differentiating Eq. <ref> with these dependences on D gives

    dα/dD≈    -R_t/D - √(8/15)tan(Φ-π/4) 
       - 2 Φ√(8/15)^2(Φ-π/4)
    
    ≈    (α-2R_t)/D - 2 Φ√(8/15)^2(Φ-π/4).

The last term is large near any poles in α; it has minima near Φ/π=integer + 1/4, which is close to the zeroes in α, but is nevertheless usually the largest term under shielding conditions.



§ EXTENDED DATA










 
