Feshbach resonances in $p$-wave three-body recombination within Fermi-Fermi mixtures of open-shell $^6$Li and closed-shell $^{173}$Yb atoms

We report on observations and modeling of interspecies magnetic Feshbach resonances in dilute ultracold mixtures of open-shell alkali-metal $^6$Li and closed-shell $^{173}$Yb atoms with temperatures just above quantum degeneracy for both fermionic species. Resonances are located by detecting magnetic-field-dependent atom loss due to three-body recombination. We resolve closely-located resonances that originate from a weak separation-dependent hyperfine coupling between the electronic spin of $^6$Li and the nuclear spin of $^{173}$Yb, and confirm their magnetic field spacing by ab initio electronic-structure calculations. Through quantitative comparisons of theoretical atom-loss profiles and experimental data at various temperatures between 1 $\mu$K and 20 $\mu$K, we show that three-body recombination in fermionic mixtures has a $p$-wave Wigner threshold behavior leading to characteristic asymmetric loss profiles. Such resonances can be applied towards the formation of ultracold doublet ground-state molecules and quantum simulation of superfluid $p$-wave pairing.


I. INTRODUCTION
Magnetic Feshbach resonances (MFRs) are valuable tools in ultracold bosonic and fermionic atomic gases, providing access to tunable interactions between atoms [1,2]. First observed two decades ago [3,4], they are now routinely used in few-and many-body physics. For example, they are used in the creation of ultracold molecules [5] and in studies of three-body physics [6]. Resonantly-enhanced three-body recombination leads to atom loss from an ultracold sample as well as the formation of tri-atomic Efimov states [7,8]. These processes have been studied around weak Feshbach resonances [9,10] and in gases of polar molecules [11], while their collision-energy dependence has been examined for bosons [12]. Finally, Feshbach resonances are used to elucidate collective phenomena in Bose-Einstein condensates [13] and fermionic superfluids [14].
Currently, the best studied Feshbach resonances are those between two alkali-metal atoms, each in their openshell 2 S ground state. Such resonances are a consequence of atomic hyperfine-and Zeeman-induced interactions between the electron spin singlet and triplet Born-Oppenheimer potentials. The magnetic field dependence of the zero-collision-energy scattering length is given by a(B) = a bg [1 − ∆/(B − B res )] where B is the magnetic field, B res is the resonance location and a bg is the background scattering length [15]. The resonance strength ∆ can be as large as hundreds of Gauss (tens of mT) [1].
Recently, Feshbach resonances have been observed in mixtures of bosonic alkali-metal and 1 S alkaline-earth * agreen13@uw.edu atomic gases (in samples of 87 Rb and 87,88 Sr with nearly equal masses) [16]. Their mixed-species interactions are controlled by a single Born-Oppenheimer potential and, at first glance, no resonances might be expected. However, theoretical predictions have indicated the existence of weak atom-separation-dependent interactions that lead to Feshbach resonances [17][18][19]. The strongest of these interactions corresponds to a separation-dependent hyperfine coupling between the electron spin of the 2 S atom and the nuclear spin of the 1 S atom. The resonances are narrow, with strengths well below one Gauss.
Our research collaboration focuses on ultracold mixtures of 2 S lithium and alkaline-earth-like 1 S ytterbium with an extreme mass-imbalance factor of approximately 30. Atomic quantum-degenerate mass-imbalanced mixtures in different isotopic combinations [20][21][22], and thus with specific statistical properties, are studied for several reasons. They enable observation of novel fermionic Efimov states in a Fermi-Fermi mixture [23,24] and may provide a platform for quantum simulation of such phenomena as the Kondo effect [25] and induced p-wave pairing and superfluidity in Fermi-Bose mixtures [23,26,27]. Impurity physics in Li and Yb mixtures, where one of the two atomic species is far more prevalent, also has promising basic research implications [27][28][29]. For example, in a Fermi liquid with impurities near a Feshbach resonance, polarons are an important elementary excitation whose properties will determine the stability of the liquid. Finally, Feshbach resonances among alkali-metal and closed-shell atoms open up the possibility of creating ultracold heteronuclear 2 Σ + molecules [16,19,[30][31][32][33]. With the additional degree of freedom from the unpaired electron, such molecules extend the scientific reach of ultracold molecules beyond that of the currently available bi-alkali molecules [5,[34][35][36][37][38] with unique roles  in quantum simulation of many-body systems, studies of quantum magnetism, fundamental symmetry tests and ultracold chemistry [39][40][41][42].
In this paper we report on first observations and a theoretical study of Feshbach resonances in mixtures of fermionic 6 Li and fermionic 173 Yb. Using atom-loss spectroscopy as a function of magnetic field with spinpolarized atomic mixtures, we observe a series of interspecies Feshbach resonances corresponding to different magnetic Zeeman states of 6 Li and 173 Yb. Our observations are in quantitative agreement with our theoretical expectations based on a recently determined Born-Oppenheimer potential [33] and a new electronic valencebond calculation of separation-dependent hyperfine interactions. We also report on the temperature dependence of the resonance line shapes in conjunction with a model of the resonant interspecies three-body loss processes. As two of the three fermionic atoms in this process are identical, p-wave collisions and their corresponding Wigner-threshold behavior play a crucial role, as demonstrated by our analysis. Previously, such temperature analysis has only been performed in single-species bosonic gases [43][44][45][46] and in mixtures of bosonic 7 Li and 87 Rb [47]. Feshbach resonances between ground-state 6 Li and 173 Yb in a magnetic field B are due to the coupling between an electronic Born-Oppenheimer (BO) poten-tial and weak interatomic-separation-dependent hyperfine couplings. The Hamiltonian for this system is H = and U s-i (R) describes the weak R-dependent hyperfine couplings. Here, R is the interatomic separation, µ is the reduced atomic mass, = h/(2π) and h is the Planck constant. The term V (R) represents the ground-state X 2 Σ + BO potential. The last two terms of Eq. 1 describe the individual atomic hyperfine and Zeeman Hamiltonians where the 6 Li total electron spin is s Li = 1 /2 and its nuclear spin is i Li = 1. The closed-shell 173 Yb has a nuclear spin i Yb = 5 /2. The 6 Li hyperfine coupling constant a Li has units of energy and g e Li , g nuc Li , and g nuc Yb are the dimensionless electronic and nuclear g factors of 6 Li and 173 Yb, respectively. Their values are found in Refs. [48,49]. Finally, µ B is the Bohr magneton.
The 6 Li atomic Hamiltonian H Li has magnetic fielddependent eigenstates |m s,Li , m i,Li ; B labeled by the projection of electron and nuclear spin quantum numbers along B. We call this the "high-field" basis, where the B field label will often be suppressed in states and kets for clarity. Eigenstates of H Yb are |m i,Yb labeled by the projection of the 173 Yb nuclear spin quantum number along the magnetic field B. The eigenvalues of H Li +H Yb as a function of B are shown in Fig. 1. The 173 Yb nuclear Zeeman splittings are only resolved in Fig. 1(c).
The eigenstates of the H 0 in Eq. 1 can be written as where spherical harmonics Y m (R) describe the rotation of the two atoms with relative orbital angular momentum and its projection m . The radial wave function φ(R) is either a scattering solution of the BO potential at collision energy E > 0 and partial wave or a bound state with energy E ν, < 0, where ν and are the vibrational and rotational quantum numbers respectively. Throughout this paper these two types of solutions and spin states are distinguished by superscripts At and Mol where necessary. We use the X 2 Σ + BO potential from Ref. [33], which was obtained by fitting to six experimentally-determined weakly-bound states of the isotopologue 174 Yb 6 Li. The most weakly-bound state of 173 Yb 6 Li in this potential has energy E −1,0 /h =  Resonances appear as B-dependent atom loss. The 6 Li atom loss is shown after the 2.8 µK mixture has been held for 3 s in the dipole trap. Red squares, purple circles, and blue triangles correspond to data for 6 Li in states |ms,Li, mi,Li = | − 1 /2, 1 , | − 1 /2, 0 , and | − 1 /2, −1 , respectively. The nuclear Zeeman state of 173 Yb is spin-polarized to |m i,Yb = | + 5 /2 in each case. Each point is the average of at least four measurements and the error bars are one-standard-deviation statistical uncertainties. Curves are best-fit Gaussians and only meant as a guide to the eye. Our full line shape analysis is found in Sec. VII.
Crossings between atomic and molecular states are visible, although the states are not coupled within H 0 . Once we include the weak U s-i (R) interactions they change into Feshbach resonances. We postpone the description of this mixing until Sec. V and first describe our experimental procedures to precisely locate the resonances.

III. EXPERIMENTAL SETUP
We observe interspecies magnetic Feshbach resonances through enhanced atom loss over narrow ranges of magnetic field. Specifically, we measure the remaining fraction of atoms after an optically-trapped spin-polarized mixture is held for a fixed time at constant magnetic field. This atom-loss spectroscopy begins with ultracold atomic samples in a crossed optical dipole trap (ODT) described in earlier work [50,51]. Unpolarized lasercooled samples of atomic 173 Yb are loaded into the ODT first at a bias magnetic field of 1 G. Subsequently, 6 Li atoms are laser cooled, loaded into the ODT, and optically pumped into the two energetically-lowest hyperfine states by applying light resonant on the 2 S1 /2 → 2 P3 /2 transition. We then increase the bias magnetic field to 500 G in order to spectroscopically resolve the two remaining hyperfine states and subsequently remove atoms in the |m s,Li , m i,Li = |− 1 /2, +1 state with an additional resonant light pulse.
To prepare a particular spin-polarized heteronuclear mixture we use the following strategy (additional details can be found in the Supplemetary Material [52]). A first stage of evaporative cooling to 5.8 µK is performed by ODT depth reduction before the 173 Yb sam-ple is partially polarized through optical pumping into a spin mixture containing a majority of 173 Yb atoms in state |m i,Yb = m and a minority in a sacrificial state |m i,Yb = + 5 /2 or |m i,Yb = − 5 /2 . The sacrificial state is retained to increase the efficiency of further evaporative cooling. Subsequently, the sample temperature is either increased or decreased to the desired value by either increasing or decreasing the ODT depth [53]. The 173 Yb atoms in the sacrificial state are then removed with a resonant light pulse, resulting in a fully polarized 173 Yb sample in state |m i,Yb = m . Finally, the atoms in the 6 Li sample are transferred to the hyperfine ground state of interest through radiofrequency (RF) adiabatic rapid passage. We have verified from a separate diagnostic that we achieve > 90% spin polarization for each atomic species in the targeted spin state [52].
Once the desired spin-polarized heteronuclear mixture is prepared, we smoothly ramp the magnetic field to a specific value to perform loss spectroscopy. The spectroscopy phase consists of letting the atoms interact for a fixed hold time at this magnetic field after which we measure the remaining atom number through absorption imaging at 500 G. We then repeat the process for many magnetic field values. The magnetic field is generated by a pair of coils in Helmholtz configuration connected to a programmable power supply, and is calibrated through RF spectroscopy on the 6 Li atomic hyperfine transitions.
The temperature range explored in this work is between 1 µK and 20 µK. The differential gravitational potential in this highly mass-imbalanced system results in a partial separation of the two species at the lowest temperatures, causing a lengthening of interspecies thermalization time below 2 µK. In this work, the lowest 173 Yb ( 6 Li) temperature at the beginning of the loss spectroscopy phase is 1.0 µK (1.8 µK). This corresponds to T /T F = 3.3 (0.57), where T F is the Fermi temperature for each species. Under these conditions, the measured 173 Yb ( 6 Li) atom number at the beginning of the spectroscopy phase is 1.0×10 5 (1.3×10 5 ) with corresponding peak density of 2.6 × 10 12 cm −3 (6.1 × 10 12 cm −3 ). Here and elsewhere in this paper, the uncertainties in temperature, atom number and density are 10%, 10% and 18% respectively, mainly stemming from uncertainties in the imaging system. For higher temperatures, the two species are in thermal equilibrium with each other at the beginning of the loss spectroscopy phase.

IV. OBSERVATION OF MAGNETIC FESHBACH RESONANCES
For the magnetic field range investigated in this work three 6 Li ground hyperfine states exhibit interspecies Feshbach resonances with 173 Yb. These are |m s,Li , m i,Li = |− 1 /2, +1 , |− 1 /2, 0 , and |− 1 /2, −1 . Fig. 2 shows the experimental observation of their interspecies Feshbach resonances when the 173 Yb is prepared in |m i,Yb = | + 5 /2 . We have confirmed that the three loss features in Fig 6 Li 173 Yb mixture seen in the remaining 6 Li atom number as functions of B near the 640 G Feshbach resonance. 6 Li is prepared in hyperfine state |ms,Li, mi,Li = | − 1 /2, 0 and the 173 Yb sample is spin polarized in different nuclear Zeeman states. The data is normalized by the remaining 6 Li atom number away from the resonance and offset for clarity. From top to bottom black, red, yellow, green, blue, and pink markers correspond to data with 173 Yb prepared in nuclear spin state m i,Yb = − 5 /2, − 3 /2, − 1 /2, + 1 /2, + 3 /2, and + 5 /2, respectively. No resonance exists for m i,Yb = − 5 /2. The temperature is 1.8 µK (1.0 µK) for Li (Yb) and the hold time is 1.5 s. Curves are best-fit Gaussians. Our full line shape analysis is found in Sec. VII.
correspond to interspecies Feshbach resonances by repeating the spectroscopy phase with only 6 Li atoms. No atom loss features were then observed.
To investigate the effect of the 173 Yb nuclear spin on the MFRs, we repeated our trap-loss spectroscopy for each of the six m i,Yb states, preparing the 6 Li sample in |m s,Li , m i,Li = |− 1 /2, 0 for all cases. The results are shown in Fig. 3. The absence of a MFR for m i,Yb = − 5 /2 is expected for reasons outlined in Sec. V.
The experimental value of B res for each MFR is determined as the center value of a Gaussian fit to our lowest temperature data [54]. The locations of all observed resonances with spin-polarized heteronuclear mixtures are listed in Table I and are consistent with the predictions for Feshbach resonance locations due to the least bound state, i.e. ν = −1, = 0, of the 2 Σ + BO potential shown in Fig. 1. We present a detailed comparison of our theoretical analysis and experimental observations in Sec. VI.
The locations of the five resonances in Fig. 3 are also indicated in Fig. 4(a). From these locations we derive the magnetic moment of the 173 Yb nucleus in the ν = −1, = 0 173 Yb 6 Li molecule and note that its sign is opposite that of the free 173 Yb atom. In Sec. V and VI we show that this sign change originates from the Rdependent U s−i (R) hyperfine coupling.

V. SEPARATION-DEPENDENT HYPERFINE INTERACTIONS
We now define the weak interaction U s-i (R) that leads to coupling between eigenstates of H 0 and hence the Feshbach resonances. The interaction describes the effects of the modified electron spin densities at the nuclear positions of 6 Li and 173 Yb when the atoms are in close proximity. For our experiment the relevant coupling is where the hyperfine coupling coefficient ζ Yb (R) is obtained from an all-electron ab initio calculation based on the non-relativistic configuration interaction valencebond (CI-VB) method [55][56][57]. Fig. 4(b) shows ζ Yb (R) together with the real-valued radial wave function of the most weakly-bound state of the 2 Σ + BO potential as a function of interatomic separation. The ζ Yb (R) is on the order of a Li near the inner point, R ≈ 6a 0 , of the vibrational bound state wave function and then approaches zero rapidly when R → ∞. Other weaker coupling terms [17][18][19] are discussed in the Supplementary Material [52]. The weak U s-i (R) changes the crossings between the atomic and molecular levels in Fig. 1 into resonances. For this interaction to lead to a resonance the sum m s,Li + m i,Yb must be the same for the scattering and bound states. In particular, since the MFRs we investigate satisfy m At The weak U s-i (R) also modifies the energies of the hyperfine and Zeeman states of the ν = −1, = 0 bound state. The size of these energy shifts can only be observed over magnetic field intervals of less than 1 G as shown in Fig. 1(c) for resonances around 640 G. These resonances are labeled by the 6  To get an intuitive understanding of the molecular level splitting, we perturbatively study the energies of these six bound states. The states of different m i,Yb are split by the nuclear Zeeman interaction of 173 Yb, and a contribution from the diagonal matrix elements of Eq. 5 where for the last equality we used the fact that the magnetic field is large. The radial integral over ζ Yb (R) is −0.369 × h MHz for our CI-VB values. Both Eqs. 6 and 7 are proportional to m i,Yb and the two contributions have opposite signs. For B = 640 G, |∆E 1 | is about 30% larger than ∆E 0 , resulting in an overall change in the sign of the level shifts as compared to shifts of the free-atom state. Figure 1(c) shows the theoretical energies of the atomic levels crossing with the molecular bound states including the corrections ∆E 0 and ∆E 1 . The energies of scattering states are not affected by U s-i (R). Crossings with markers in Fig. 1(c) correspond to resonances satisfying the selection rules of U s-i (R). Without the correction of Eq. 7 in molecular state energies, all crossings would occur at the same magnetic field [19].
We use our coupled-channels code to determine the strength and resonance locations of the MFRs. Specifically, we compute the zero-energy s-wave scattering length a(B) as a function of magnetic field for all relevant scattering channels |m At s,Li , m At i,Li . Near each resonance, we fit to a(B) = a bg [1 − ∆/(B − B res )] as defined in the introduction [58]. ). We note that the observed and theoretical locations are consistent, as the 3.4 MHz uncertainty in the binding energy of the ν = −1 and = 0 state and the approximately 2µ B magnetic moment difference between the bound and scattering states leads to a 1.3 G uncertainty in the theoretical resonance location. All observed locations occur at a larger magnetic field value than those of the theoretical predictions, indicating that the binding energy |E −1,0 | is slightly underestimated.

VII. FERMIONIC FEATURES IN RESONANT ATOM-LOSS LINE SHAPES
Our atom loss measurements also confirm the fermionic statistical properties of our mixture. The requirement of an anti-symmetric scattering wavefunction under interchange of identical fermions leads to a three-body loss rate coefficient that has a "p-wave Wigner threshold character". Specifically, we will show that the locations of the maxima of the three-body loss rates as functions of B shift linearly with increasing temperature in a way that can only be explained by the fermionic nature of the scattering atoms. Our data are also consistent with the observation that the maximum event rate coefficient must also be independent of temperature.
We start by noting that atom loss from our mixture at temperature T is described by the two-coupled equations  6 Li Feshbach resonance locations and corresponding theoretical predictions and assignments based on s-wave coupled-channels calculations. The first two columns describe the quantum numbers of the scattering states and the projection of the total angular momentum, respectively. The third and fourth columns give the observed and predicted resonance locations. Finally, the last column gives the resonance strength from the coupled-channels calculations. The error in the observed locations is the onestandard-deviation uncertainty from the quadrature sum of the statistical error in the Gaussian fit and the systematic error in the field calibration. The theoretical locations of the resonances have a 1.3 G one-standard-deviation uncertainty due to the uncertainty of the binding energy of the most weakly-bound state. The differences between neighboring resonances are not affected by this uncertainty.
where atom numbers N a for a = Li or Yb are time dependent. Rates Γ a describe one-body background-collisioninduced losses, while event rates γ 1 and γ 2 describe the three-body recombination processes starting from 6 Li+ 6 Li+ 173 Yb and 6 Li+ 173 Yb+ 173 Yb collisions, respectively. For both processes two of the three atoms are identical fermions. Finally, γ i = K i (B, T )/V i with i =1 or 2 and temperature-dependent hypervolumes where the time-independent ρ a ( x) are unit-normalized spatial density profiles. The event rate coefficients K i (B, T ) will be discussed below. Several assumptions have gone into deriving Eqs. 8 from two coupled Boltzmann equations for the singleparticle phase space densities f a ( x, p, t) [59,60] with momentum p.
We assume that both fermionic 6 Li and 173 Yb gases are in thermal equilibrium at a temperature above degeneracy and f a ( x, p, t) ∝ N a (t)ρ a ( x) exp[−p 2 /(2m a k B T )], where m a is the mass of atom a and k B is the Boltzmann constant. This is justified as the mean time between thermalizing elastic Yb+Li collisions is much smaller than the time scales of atom loss due to three-body recombination [20,33,61]. Even though the two species are held in the same dipole trap, their spatial density profiles ρ a ( x) are distinct as their dynamic polarizabilities and gravitational potentials are different. In this section, however, the small differences in temperature and spatial density profiles between the 6 Li and 173 Yb gases will be ignored. In fact, differences are only significant for our smallest measured temperature, where quantum degeneracy is almost reached and the thermalization times are longest. Finally, losses from two-body Li+Yb collisions are negligible as confirmed by our coupled-channels calculations. Other two-and threebody losses are suppressed by the fermionic nature of the 6 Li and 173 Yb atoms.
The three-body recombination event rate coefficient K i (B, T ) has a Lorentzian form as a function of B describing the formation of a resonant trimer followed by breakup into a weakly bound dimer and a free atom. This is given by [62,63] where · · · is the thermal average, the relative kinetic energy E = 2 k 2 /(2µ 3 ), µ 3 defines the three-body reduced mass [62] and k is the relative wave number. For our low temperatures, only the lowest-allowed total three-body angular momentum J contributes to atom loss. We have J = 1 for our fermion-fermion mixture. The square of the dimensionless S-matrix is [44,62,63] with E 0 = δµ(B − B 0 ) where B 0 and δµ are the threebody resonance location and the relative magnetic moment, respectively. A priori, B 0 and the two-body B res resonance locations need not be the same. We will conservatively assume that B 0 and B res agree to within the experimental uncertainty. This uncertainty is much larger than the strength ∆ of the resonance. The stimulated width is given by Γ(E, J) = A(E/E ref ) 2+J with scaled width A and reference energy E ref . Finally, Γ br is the breakup width. The parameters B 0 , δµ, A, and Γ br are determined by fitting the line shapes at different temperatures. We now make several simplifications of Eqs. 8 consistent with our experimental system parameters. We use that the initial atom number and peak density of the two species are the same to good approximation and that the one-body loss rates satisfy Γ Li Γ Yb = Γ bg [64]. Similarly, for the three-body rate we assume that K 1 (B, T ) = K 2 (B, T ) = K(B, T ) and 1/V 1 = 1/V 2 = ρ 2 Li ( x = 0) as both processes involve fermions that have roughly the same phase-space density and x = 0 is the center of the trap. Then Eq. 8 becomes for both a = Li and Yb. This differential equation has an analytic solution. We note that with Eq. 11 we have opted for the simplest possible model that still captures the relevant physics. In particular, there exists no formal justification for our choice K 1 (B, T ) = K 2 (B, T ). We are also unable to experimentally distinguish the two processes. In a recent experiment with a fermion-fermion mixture [24], the two processes could also not be distinguished. Reports on three-body recombination in boson-boson [65,66] and boson-fermion [67][68][69] mixtures showed that the light+heavy+heavy process is much faster than the light+light+heavy one. Figure 5 shows atom-loss spectra and theoretical line shapes based on Eq. 11 and the three-body event rate coefficient derived from Eq. 9 for a mixture with 6 Li prepared in |m s,Li , m i,Li = | − 1 /2, 0 and 173 Yb prepared in |m i,Yb = | + 5 /2 . Data is shown for four temperatures between 1.8 µK and 16.1 µK. While the three-body recombination process also causes heating of the atomic clouds, the measured temperature growth remains within 20% during the first second of hold time when the threebody process is most dominant. The loss features are asymmetric and shift and broaden with increasing temperature. The parameters in the theoretical line shapes are the same for all temperatures except the one-body loss rate at our lowest temperature of 1.8 µK, when some of our theoretical assumptions begin to break down as noted earlier.
The satisfactory agreement between experimental data and theoretical line shapes enables us to extract the pwave character of the rate coefficients. For our parame-ters the inequalities Γ(E, J) Γ br k B T hold and for B > B 0 the rate coefficient simplifies to with the dimensionless parameter given by = δµ(B − B 0 )/k B T .
Then K(B, T ) has a maximum at B = B 0 + 3k B T /δµ, and a maximum value that is independent of temperature as shown in Fig. 5(e). We find that the magnetic moment of the trimer resonance is δµ = 2.8µ B , which can be compared to the 2µ B magnetic moment of the diatomic 173 Yb 6 Li resonance. We expect that the trimer magnetic moment lies between 2µ B and 4µ B , corresponding to a superposition state of only one 173 Yb 6 Li pair in the dimer resonant state and two 173 Yb 6 Li pairs in the resonant state. The agreement proves the p-wave character of our three-body recombination process and also shows that the width of the atom loss features is thermally limited even for our lowest temperature.
We contrast our observation in this Fermi-Fermi mixture with a similar analysis of Bose gases and bosonboson mixtures which predicts that the maximum loss rate scales as 1/(k B T ) [44]. Finally, Fig. 5(e) shows that the maximum three-body loss rate coefficient is relatively small for resonant processes, on the order of a few times 10 −27 cm 6 /s, and consistent with recently reported values for the fermionic 40 K and 162 Dy mixture [24].

VIII. CONCLUSION AND OUTLOOK
We have experimentally and theoretically studied the resonant scattering of ultracold fermionic 6 Li and 173 Yb atoms in a magnetic field. Using spin-polarized samples, we located several narrow magnetic Feshbach resonances between 580 G and 700 G by detecting enhanced three-body recombination near these resonances. We showed that their locations can be quantitatively explained based on the most-accurate Born-Oppenheimer potential in the literature and our own ab initio calculation of a separation-dependent hyperfine coupling between the electron spin of 6 Li and the nuclear spin of 173 Yb.
A comparison of experimental and theoretical line profiles of the three-body recombination process at various temperatures has shown that recombination is controlled by p-wave scattering of the three-atom entrance channel. The observed temperature independence of the loss rate coefficient is unique to the fermionic quantum statistics of the collision partners and contrasts with the temperature dependent behavior for s-wave and d-wave bosonic scattering [44]. The analysis has also shown that the maximum recombination rate coefficient is small compared to those found for Feshbach resonances in bosonic gases.
Our observed MFRs endow the highly massimbalanced 173 Yb- 6 Li Fermi-Fermi mixture with strong interactions for potential applications in few-and manybody physics, and are also expected to exist in other Yb-Li isotopologues involving 173 Yb or 171 Yb. In particular, such resonances may aid in the pursuit of p-wave superfluidity in 173,171 Yb- 7 Li mixtures [22,26].
Our results also provide a launching pad for the production of ultracold doublet ground-state molecules. This exciting prospect may be facilitated by first producing low entropy samples of 6 Li and 173 Yb in a threedimensional optical lattice [70,71] and then using one of the observed MFRs to coherently create YbLi molecules using magnetic field sweeps across the resonance.

Supplemental Material Feshbach resonances in p-wave three-body recombination within Fermi-Fermi
mixtures of open-shell 6 Li and closed-shell 173 Yb atoms

IX. SPIN STATE PREPARATION OF THE ATOMIC MIXTURE
To prepare a particular spin-polarized heteronuclear mixture we rely on a set of distinct manipulation and diagnostic tools for each species.
As described in the main text, the 6 Li sample is prepared in the |m s,Li , m i,Li = |− 1 /2, 0 state prior to evaporative cooling and we can subsequently transfer the spinpolarized 6 Li sample to the hyperfine state of interest at any point in the evaporative cooling process through radiofrequency (RF) adiabatic rapid passage.
The 6 Li spin polarization is probed by spin-dependent absorption imaging at a magnetic field of 500 G which relies on the fact that the electronic ground hyperfine states are separated from each other at this field by an energy that is much greater than the linewidth of the 2 S1 /2 → 2 P3 /2 probe transition. In the course of our trap-loss spectroscopy experiments, the efficiency of the RF adiabatic rapid passage is routinely checked through this spin-dependent imaging.
We prepare the desired spin polarization of 173 Yb by optical pumping on the 1 S 0 (f Yb = 5 /2) |m i,Yb = m ± 1 states within the ground manifold. Using a sequence of such pulses, we can prepare a fully polarized sample in the targeted ground Zeeman state.
The electronic structure of the 173 Yb ground state makes spin-dependent imaging infeasible. Instead, we measure the spin polarization of 173 Yb samples with an optical Stern-Gerlach (OSG) method [S1, S2], utilizing a circularly polarized laser beam which is +860 MHz blue detuned from the zero-field 1 S 0 (f Yb = 5 /2) → 3 P 1 (f Yb = 7 /2) transition frequency to create a spindependent force on the atoms. The OSG is performed in a 7 G bias field with σ + polarization on an atomic sample at 500 nK. We achieve full spatial separation between the six atomic clouds corresponding to distinct nuclear spin states in absorption images taken after time-of-flight (see Fig. 6). In the course of our atom-loss spectroscopy, we routinely check the efficiency of the optical pumping process by performing an OSG diagnostic experiment.
We have verified that we achieve > 90% spin polarization for each atomic species in the targeted spin state.

X. SEPARATION-DEPENDENT HYPERFINE INTERACTIONS
We briefly expand upon our calculation of the separation-dependent hyperfine interactions between the electron and nuclear spins of 6 Li and the nuclear spin of 173 Yb. For s-wave 173 Yb+ 6 Li collisions two hyperfine coupling mechanisms are relevant. As the two atoms move closer, electron spin density is pulled away from the 6 Li nucleus, which reduces the hyperfine interaction strength between the electron and nuclear spin of 6 Li. Simultaneously, some of this electron spin density comes into contact with the 173 Yb nucleus and, thus, couples to its nuclear spin. These hyperfine interactions can be written as with strengths ζ Li (R) and ζ Yb (R) that both approach zero for large R. Following our definitions in the main text the asymptotic 6 Li hyperfine interaction a Li s Li · ı Li is accounted in the zeroth-order Hamiltonian. The second term in Eq. (13) was already introduced in the main text.
The existence of these R-dependent interactions was first proposed for RbSr [S3] and YbLi [S4] dimers. In 2018 they were confirmed by detecting Feshbach resonances in ultracold Rb+Sr mixtures [S5]. We will denote effects induced by the first or second terms in Eq. (13) by mechanism I and II, respectively. Mechanism I leads to resonances where the projection m f,Li = m s,Li + m i,Li must be the same for the scattering and the resonant bound state, while mechanism II leads to resonances where m s,Li + m i,Yb remains unchanged.
We have used the non-relativistic configurationinteraction valence-bond (CI-VB) method [S6] to calculate the R-dependent ζ Li (R)+a Li and ζ Yb (R) of 173 Yb 6 Li (The CI-VB method does not compute ζ Li (R) directly.) The basic idea is to construct electronic molecular wave functions from superpositions of determinants of atomic electron orbitals localized at the nuclear positions. Consequently, molecular wave functions approach a "pure" atomic form for large R, which automatically leads to the correct molecular dissociation limits. At small internuclear separations orbitals around different centers have considerable overlap and are non-orthogonal, which leads to large "exchange effects" that creates bonds. We use numerical Hartree-Fock (HF) atomic electron orbitals that avoid the need for large basis sets, as they have the correct radial behavior near their nucleus and for large separations between the electron and nucleus. Furthermore, the numerical orbitals have the correct number of nodes and are orthogonal with respect to other HF orbitals localized at the same center.
We find that it is sufficient to use a single HF orbital for the inner shells of Li and Yb and use only a few additional excited orbitals to describe valence electrons. The valence orbitals are either occupied or unoccupied HF orbitals or so-called Sturmian functions, e.g. functions that are solutions of Hartree-Fock equations of the Coulomb problem, where the energy is fixed and the strength of the Coulomb potential plays the role the (generalized) eigenvalue. Sturmian functions form a complete basis with similar asymptotic behaviour and orbital size as the occupied valence orbitals. Finally, our VB approach is an all-electron ab initio calculation, which enables us to evaluate electron densities at the nuclear sites and, thus, to calculate hyperfine structure constants as functions of nuclear separations [S7, S8].
The calculated CI-VB ζ Li (R)+a Li and ζ Yb (R) as functions of R are shown in Fig. 7. Our asymptotic value ζ Li + a Li → h × 152.147 MHz is in excellent agreement with the experimental value of h × 152.137 MHz for atomic 6 Li [S9]. The strengths ζ Li (R) and ζ Yb (R) have been fit to the Gaussian A 0 exp(−β[R−R c ] 2 ) and parameters values for A 0 , β, and R c are given in Table II. Small deviations between the numerical values and the fit are noticeable for R < 6a 0 . The deviations, however, have a negligible effect on the observable vibrationally-averaged hyperfine strengths ζ a (R) as the inner turning point of the most-weakly-bound s-wave bound state of 173 Yb 6 Li is located at ≈ 5.6a 0 . Finally, we note that ζ Li (R) is three times smaller than ζ Yb (R) for R ∈ [5a 0 , 10a 0 ], where the strengths have a significant value.
We can compare our hyperfine strengths with those of Ref. [S4] obtained using density-functional theory. Their asymptotic value of a Li differs by more than 5% from the experimental value. For completeness, the Gaussian parameters from Ref. [S4] are repeated in Table II. We have performed theoretical coupled-channels calcu-  lations to locate Feshbach resonances in the 173 Yb+ 6 Li collision. The calculations are based on the isotropic Hamiltonian using the X 2 Σ + potential of Ref. [S10]. Table III gives the complete list of predicted resonance location and strengths, ordered by magnetic field strength and grouped by resonance mechanism. Predictions for Feshbach resonances near a magnetic field of 640 G have been discussed in the main text. These resonances originate from mechanism II and have all been observed in this work. Two of the additional mechanism-II resonances have been observed as shown by a comparison with the data in the table in the main text. Resonance strengths ∆ for mechanism II are more than one order of magnitude larger than those for mechanism I, consistent with a larger ζ a (R) and making them easier to observe.