Magnetic Feshbach resonances in ultracold collisions between Cs and Yb atoms
B. C. Yang, Matthew D. Frye, A. Guttridge, Jesus Aldegunde, Piotr S. Żuchowski, Simon L. Cornish, Jeremy M. Hutson
MMagnetic Feshbach resonances in ultracold collisions between Cs and Yb atoms
B. C. Yang, Matthew D. Frye, A. Guttridge, Jesus Aldegunde, Piotr S. ˙Zuchowski, Simon L. Cornish, ∗ and Jeremy M. Hutson † Joint Quantum Centre (JQC) Durham-Newcastle, Department of Chemistry,Durham University, South Road, Durham, DH1 3LE, United Kingdom. Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics,Durham University, South Road, Durham, DH1 3LE, United Kingdom. Departamento de Quimica Fisica, Universidad de Salamanca, 37008 Salamanca, Spain Institute of Physics, Faculty of Physics, Astronomy and Informatics,Nicolaus Copernicus University, ul. Grudziadzka 5/7, 87-100 Torun, Poland (Dated: May 29, 2019)We investigate magnetically tunable Feshbach resonances in ultracold collisions between ground-state Yb and Cs atoms, using coupled-channel calculations based on an interaction potential recentlydetermined from photoassociation spectroscopy. We predict resonance positions and widths for allstable isotopes of Yb, together with resonance decay parameters where appropriate. The resonancepatterns are richer and more complicated for fermionic Yb than for spin-zero isotopes, because thereare additional level splittings and couplings due to scalar and tensorial Yb hyperfine interactions. Weexamine collisions involving Cs atoms in a variety of hyperfine states, and identify resonances thatappear most promising for experimental observation and for magnetoassociation to form ultracoldCsYb molecules.
I. INTRODUCTION
Magnetic Feshbach resonances are a valuable tool fortuning the scattering length by varying an external mag-netic field, and have found a wide range of applicationsin studying and controlling ultracold gases [1]. Greatprogress has been achieved in the exploration of Feshbachresonances for pairs of alkali-metal atoms. One signifi-cant achievement is the formation of ultracold moleculesby adiabatically ramping the magnetic field across a zero-energy Feshbach resonance [2–10], known as magnetoas-sociation [11, 12]. The resulting weakly bound dimerscan be transferred to their absolute ground states bystimulated Raman adiabatic passage [13–20]. These ul-tracold molecules promise diverse applications in fieldsfrom ultracold chemistry to precision measurement, dueto their rich internal degrees of freedom and complexinteractions compared to ultracold atoms [21, 22]. Inparticular, the inherent electric dipole moment of ultra-cold polar molecules makes them valuable in studyingquantum dipolar matter [23, 24] and for applications inquantum computation and simulation [25, 26].There is currently great interest in ultracold mixturesof alkali-metal and closed-shell ( S) atoms [27–38]. Themolecules formed from these atoms have Σ ground stateswith unpaired electron spin. They therefore have bothelectric and magnetic dipole moments, and provide a newplatform for studying lattice spin models in many-bodyphysics [39]. They may also be valuable in searches forthe electric dipole moment of the electron [40]. How-ever, magnetoassociation in such mixtures will be chal-lenging because the Feshbach resonances are expected to ∗ [email protected] † [email protected] be narrow. This is because the lack of structure of a S atom removes the strong couplings that cause manywide resonances in alkali+alkali systems; the strongestsource of coupling in alkali+ S systems is the weak de-pendence of hyperfine coupling on interatomic distance[32, 33]. Nevertheless, Feshbach resonances have recentlybeen observed in an ultracold Rb+Sr mixture [41]. Thereis now great hope that it will be possible to form ultra-cold open-shell molecules by magnetoassociation at theseresonances.We have studied ultracold Cs+Yb mixtures both ex-perimentally and theoretically [34, 36–38, 42]. The mer-its of this system include the existence of seven stableisotopes of Yb, including five spin-zero bosons and twofermions. Because of the large mass of Cs, significantvariation of the atom-pair reduced mass can be achievedby choosing different isotopes of Yb. This produces awide variety of Feshbach resonances with substantiallydifferent properties for different isotopes [34]. However,the predictions of Ref. [34] were limited because, atthat time, the ground-state interaction potential was notknown accurately enough to predict scattering lengthsfor specific isotopic combinations.In recent work, we have measured the binding energiesof near-threshold bound states for several isotopologs ofCsYb and determined the ground-state electronic poten-tial [43]. This allows us to make specific predictions forthe positions and widths of Feshbach resonances. Fig-ure 1 shows the atomic thresholds for Cs as a functionof magnetic field, and the near-threshold energy levelsof CsYb predicted for the ground-state potential of Ref.[43]. Because there is only a single electronic state andthe hyperfine coupling is weakly dependent on distance,the molecular levels are essentially parallel to the thresh-old that supports them [32, 34]. Feshbach resonances dueto the Cs hyperfine coupling are predicted at the cross- a r X i v : . [ phy s i c s . a t o m - ph ] M a y B (Gauss) -9-8-7-6-5-4-3-2 E n e r g y ( G H z ) -9-6-303 E n e r g y ( G H z ) n = − n = − n = − n = − YbCs+
YbCs+
YbCs+
YbCs+
YbCs+
YbCs+
YbCs+ Yb n = − (a) f = 4 n = − m f = − f = 3 2 1 m f = 3 0 − − − − − (b) FIG. 1. (Color online) Near-threshold bound states (thin col-ored lines) crossing atomic thresholds (thick black lines) as afunction of magnetic field. The solid circles mark the Fesh-bach resonances caused by the dependence of the Cs hyper-fine coupling on the internuclear distance. (a) The hyperfine+ Zeeman splittings of the atomic threshold and molecularbound states for the example of Cs+
Yb. The atomic lev-els are labeled by quantum numbers f and m f as discussedin the text. Only molecular levels from the upper hyperfinemanifold ( f = 4) are shown, labeled by the vibrational num-ber n . (b) molecular levels for n = − − ings indicated by colored dots. The figure also shows howdifferent choices of Yb isotope shift the near-thresholdstates and strongly affect the resonance positions.In this paper we perform coupled-channel calculationsto identify, locate, and characterize Feshbach resonancesin ultracold collisions between Cs and Yb atoms. Ourmain focus is to understand the physics behind the prop-erties of Feshbach resonances in this system and to es-tablish which Feshbach resonances are promising for ex-perimental observation and molecule formation. In Sec.II we introduce the underlying theory of these Feshbachresonances: the coupling terms in the Hamiltonian whichcause them, the methods we use to characterize them, and the framework we use to understand the results. InSec. III we use Cs+ Yb as an example system to discussthe effects of the different coupling mechanisms and thegeneral characteristics of the different resonances theycause. In Sec. IV we identify promising resonances forobservation and magnetoassociation for various isotopiccombinations of Cs+Yb, taking account of experimentalconsiderations. Comprehensive results for resonances ofall isotopic combinations are provided in SupplementalMaterial.
II. THEORETICAL BACKGROUND
We consider the ultracold scattering between
Cs( S)and Yb( S) atoms. The Hamiltonian ˆ H can be written[33]ˆ H = (cid:126) µ (cid:34) − R d dR R + ˆ L R (cid:35) + ˆ H Cs + ˆ H Yb + ˆ U ( R ) , (1)where R is the internuclear distance, µ is the reducedmass, and (cid:126) is the reduced Planck constant. ˆ L is thetwo-atom rotational angular momentum operator, withquantum number L and projection M L . ˆ H Cs and ˆ H Yb arethe Hamiltonians for the separated single atoms, whichare independent of R and contain hyperfine coupling andZeeman terms,ˆ H Cs = ζ Cs ˆ i Cs · ˆ s + ( g Cs ˆ i Cs ,z + g s ˆ s z ) µ B B, (2)ˆ H Yb = g Yb ˆ i Yb ,z µ B B. (3)Here, B is a magnetic field oriented along the z axis, and µ B is the Bohr magneton. ζ Cs is the hyperfine couplingconstant for the Cs atom. ˆ i Cs , ˆ i Yb and ˆ s are the nuclearand electron spin operators, with projections on the z axis ˆ i Cs ,z , ˆ i Yb ,z , and ˆ s z ; their corresponding g factors are g Cs , g Yb , and g s , respectively. The specific values of ζ Cs , g Cs , and g s in this work are taken from Ref. [44] andthose for g Yb are obtained from the shielded magneticmoments (without diamagnetic correction) of Ref. [45].The interaction operator ˆ U ( R ) is divided into the elec-tronic interaction potential V elec ( R ) and spin-dependentterms ˆ V spin ( R ). The electronic potential is by far thestrongest interaction, and almost completely determinesthe bound states and non-resonant scattering in eachchannel. However, it cannot change the electron or nu-clear spins and so does not couple different channels orcause Feshbach resonances. We use the ground-state in-teraction potential fitted to two-photon spectroscopy inRef. [43], which provides an accurate representation ofthe near-threshold bound states that produce Feshbachresonances. The scattering length and the energy of thehighest bound state are given in Table I for Cs interact-ing with each isotope of Yb on this potential (withoutany internal structure on either collision partner). TABLE I. Scattering length and energy of the highest boundstate for Cs interacting with each isotope of Yb on the poten-tial V elec ( R ) of Ref. [43].Mixture a ( a ) E b ( n = − /h (MHz)Cs+ Yb 165.98 3.70Cs+
Yb 96.24 15.6Cs+
Yb 69.99 25.8Cs+
Yb 41.03 39.5Cs+
Yb 1.0 57.0Cs+ Yb − . Yb 798 0.0513
A. Spin-dependent terms
The systems considered here lack the strong couplingsthat cause wide Feshbach resonances for pairs of alkali-metal atoms, due to differences between singlet andtriplet potentials and electron spin-spin couplings. In-stead, couplings between different channels are causedby the change in hyperfine interactions due to the prox-imity of the two atoms. The operator ˆ V spin ( R ) may bewritten [46],ˆ V spin ( R ) = ∆ ζ Cs ( R )ˆ i Cs · ˆ s + ∆ ζ Yb ( R )ˆ i Yb · ˆ s + t Yb ( R ) √ T (ˆ i Yb , ˆ s ) · T ( C )+ t Cs ( R ) √ T (ˆ i Cs , ˆ s ) · T ( C )+ e Q Yb · q Yb ( R )+ e Q Cs · q Cs ( R ) + γ ( R )ˆ s · ˆ L. (4)The first two terms represent the scalar contact interac-tion between the electron and nuclear spins, while thethird and fourth terms represent the corresponding dipo-lar interaction. Here T indicates a spherical tensor ofrank 2; T ( C ) has components C q ( θ, φ ), where C is arenormalised spherical harmonic and θ, φ are the polarcoordinates of the internuclear vector. The fifth and sixthterms represent the interaction between the nuclear elec-tric quadrupole tensor e Q j of nucleus j and the distance-dependent electric field gradient tensor q j ( R ) at the nu-cleus, due to the electrons. The final term represents theinteraction between the electron spin and the molecularrotation.The first three terms in Eq. 4 are the ones that are prin-cipally responsible for Feshbach resonances in CsYb andsimilar systems. We refer to them as mechanisms I, II,and III, respectively; each can be written as the productof a purely R -dependent term ω x ( R ) and a purely spin-dependent term ˆΩ x that is different for each of x = I, II,and III.Mechanism I is due to the variation in hyperfine cou-pling on the Cs atom, ˆΩ I = ˆ i Cs · ˆ s . This arises because theapproaching Yb atom pulls electron-spin density awayfrom the Cs nucleus, thereby reducing the strength ofthe hyperfine interaction. This coupling mechanism wasfirst proposed by ˙Zuchowski, Aldegunde, and Hutson [32] for Rb+Sr and was investigated extensively by Brue andHutson for alkali-metal + Yb systems [34]. As it reliesonly on the Cs nuclear spin, it exists for all isotopic com-binations of Cs+Yb.Mechanism II is due to the variation in hyperfine cou-pling on the Yb atom, ˆΩ II = ˆ i Yb · ˆ s . This mechanism iscomplementary to mechanism I: as electron-spin densityis pulled away from the Cs nucleus, some of it comes intocontact with the Yb nucleus, where it can interact witha nuclear spin. This mechanism was first proposed byBrue and Hutson [33]. It exists only for Yb isotopes witha non-zero nuclear spin, so only for Yb and
Yb.Mechanism III is due to the tensor, or anisotropic, hy-perfine coupling on the Yb atom, ˆΩ
III = √ T (ˆ i Yb , ˆ s ) · T ( C ). The approach of the Cs atom breaks the sphericalsymmetry of the electron density around the Yb nucleusand allows a dipolar coupling that can cause resonancesdue to L = 2 bound states in s-wave scattering. Thismechanism was briefly considered by Brue and Hutson[33] but they ultimately neglected it; nevertheless, res-onances caused by this mechanism were later observedin Rb+ Sr [41]. Like mechanism II, this mechanism re-lies on the Yb nuclear spin so exists only for
Yb and
Yb.The fourth term in Eq. 4, involving t Cs , is analogous tothe third and may formally be considered as contribut-ing to mechanism III. However, it is very weak in CsYb,as discussed below. The quadrupole term involving Q Yb does not generally produce resonances, but may causesignificant level shifts for levels of Cs Yb and Cs
Ybwith
L >
0, as described in section III C. The quadrupoleterm involving Q Cs can in principle cause resonances dueto L = 2 bound states, but is very weak in CsYb. Thespin-rotation term γ ( R )ˆ s · ˆ L has no matrix elements in-volving L = 0 states so does not cause resonances in s-wave scattering. All terms except that involving t Cs areincluded where applicable in the coupled-channel calcu-lations described below. B. Electronic structure calculations ofspin-dependent coefficients
We have calculated values of the scalar hyperfine cou-pling coefficients ζ Cs ( R ) and ζ Yb ( R ), the correspond-ing tensor coefficients t Cs ( R ) and t Yb ( R ), and the nu-clear quadrupole coupling coefficients ( eQq ) Cs ( R ) and( eQq ) Yb ( R ). We have also calculated the electron g -tensor anisotropy ∆ g ⊥ ( R ), which is related to the spin-rotation coefficient γ ( R ) [46]. We carried out density-functional (DFT) calculations using the Amsterdam Den-sity Functional (ADF) package [47, 48] as described inRef. [46], at 40 distances from R = 3 . Yb are obtained from those for
Yb byscaling using nuclear g -factors, nuclear quadrupole mo-ments and molecular rotational constants as described inRef. [49].Aldegunde and Hutson [46] concluded that the B3LYP TABLE II. Parameters for the R -dependence of the spin-dependent coefficients. A (MHz) a (˚A − ) R c (˚A)∆ ζ ( Cs) −
241 0 .
154 3 . ζ ( Yb) −
126 0 .
144 3 . ζ ( Yb) 457 0 .
144 3 . eQq ( Cs) 0 .
227 0 .
256 3 . eQq ( Yb) −
601 0 .
249 3 . A (MHz) b (˚A − ) σ (˚A) t ( Yb) − . .
953 4.1 t ( Yb) 88 . .
953 4.1 γ . .
58 4.1 functional [50, 51] gives good accuracy for hyperfinecoupling coefficients in Σ molecules, and that spin-unrestricted calculations are slightly more accurate thanrestricted calculations when the two results are similar.We obtained similar results from restricted and unre-stricted calculations, so we report the unrestricted re-sults here. The one exception to this is the coefficient t Cs ( R ), which is so small that the differences betweenthe restricted and unrestricted results are comparable totheir absolute magnitude. We consider these results tobe consistent with zero, so do not report t Cs ( R ) and ex-clude the corresponding term from our coupled-channelcalculations.For all the coefficients, the values from DFT calcu-lations behave irregularly inside the zero-energy innerturning point σ , which is at 4.1 ˚A for CsYb. The irregu-larities probably occur because different electronic statesmix strongly in the region of the repulsive wall. Sincethe resonance properties of interest here are insensitiveto the behavior of the couplings inside the inner turningpoint, we have fitted functional forms to the points at R ≥ . ζ Cs ( R ) = ζ Cs ( R ) − ζ Cs and ∆ ζ Yb ( R ) = ζ Yb ( R ) are both negativefor all R > σ , but both of them show positive curvatureslightly outside σ . The same is true for the quadrupolecoupling coefficients ( eQq ) Cs ( R ) and ( eQq ) Yb ( R ). Forconsistency with Brue and Hutson [34], we have chosento represent these coefficients with Gaussian functions, A exp[ − a ( R − R c ) ]. However, there is no sign of suchcurvature for t Yb or γ ( R ), and for these we have usedsimple decaying exponentials A exp( − b ( R − σ )), with σ fixed at 4.1 ˚A. The resulting parameters are given inTable II.The calculated function ∆ ζ Cs ( R ) predicts that the f = 4, n = − Yb is bound by 11 MHzmore than the corresponding f = 3 level. This may becompared with an experimental shift of 10 ± C. Magnetic Feshbach resonances
A magnetic Feshbach resonance occurs when a molec-ular bound state is tuned across an atomic scatteringthreshold by varying an applied magnetic field. For anisolated resonance without inelastic decay, the scatteringlength a ( B ) has a characteristic pole at the resonanceposition B res [53], a ( B ) = a bg (cid:18) − ∆ B − B res (cid:19) , (5)where a bg is the background scattering length. The res-onance width ∆ can be artificially large if a bg is par-ticularly small. A better measure of the strength of theresonant pole is the product a bg ∆, which provides a mea-sure of the observability of the resonance in 3-body lossspectroscopy and is proportional to the rate of the fieldsweep needed to achieve adiabatic passage in magnetoas-sociation [54, 55]. However, a bg ∆ has inconvenient di-mensions for qualitative discussion; in order to maintaina measure with dimensions of magnetic field, we define anormalized width ¯∆ = a bg ∆ / ¯ a, (6)where ¯ a = (2 µC / (cid:126) ) / × . . . . is the mean scat-tering length of Gribakin and Flambaum [56] and rangesfrom 83.62 a for Cs+ Yb to 84.05 a for Cs+ Yb.When inelastic decay occurs, the scattering length be-comes complex, with the imaginary part describing in-elastic loss [57]. Near a resonance, the scattering lengthno longer has a pole, but instead both real and imaginaryparts show an oscillation; this may be written [58] a ( B ) = a bg + a res B − B res ) / Γ inel B + i , (7)where a res is a resonant scattering length that charac-terizes the oscillation. In general, both a bg and a res arecomplex, but the weak background inelasticity for CsYbmeans that they are nearly real and we will neglect theircomplex parts. If | a res | is large then the oscillation inthe real part of the scattering length is large and pole-like, similar to the case without decay. Γ inel B is a de-cay width in field; the decay width in energy is givenby Γ inel E = Γ inel B δµ , where δµ is the magnetic momentof the bare resonant state relative to the atomic state.When inelasticity is present, the molecule formed bymagnetoassociation can decay (predissociate) with life-time τ = (cid:126) / Γ inel E . We define the width ∆ for a decayedresonance through a bg ∆ = − a res Γ inel B / . (8)This gives the same behavior in the wings as for an un-decayed resonance of the same width [59]. D. Coupled-channel calculations
To locate and characterize the Feshbach resonances,we use coupled-channel bound-state and scattering cal-culations. The wavefunction is expanded in an uncoupledbasis set | s, m s (cid:105)| i Cs , m i, Cs (cid:105)| i Yb , m i, Yb (cid:105)| L, M L (cid:105) . (9)Here s , i Cs and i Yb are quantum numbers for the elec-tron and nuclear spin angular momenta and m s , m i, Cs and m i, Yb are the corresponding projections onto theaxis of the magnetic field. The only conserved quan-tum numbers are the total angular momentum projection M tot = m s + m i, Cs + m i, Yb + M L and the total parity( − L . Basis sets are constructed including all functionsof the required M tot and parity +1, including functionsup to L max . Different situations require L max = 0, 2 or4, as described below.The 5 bosonic isotopes of Yb all have zero nuclear spin, i Yb = 0, and the two fermions, Yb and
Yb, have i = 1 / i = 5 /
2, respectively. For the Csatom, i Cs = 7 / s = 1 /
2; these can be coupled togive a resultant f = 3 or 4. The corresponding projection m f is conserved by ˆ H Cs , but f is not except at B = 0.Nonetheless, we label states by the value of f that theycorrelate with at B = 0 [60].The resulting coupled equations are constructed andsolved for bound states using the bound and field pro-grams [61, 62] and for scattering using the molscat pro-gram [62, 63]. In the short-range region, 3 . < R <
25 ˚A, solutions are propagated using the diabatic log-derivative method of Manolopoulos [64, 65] with a fixedstep size 0.001 ˚A; in the long-range region, 25 ˚A ≤ R ≤ R max , the log-derivative Airy propagator of Alexanderand Manolopoulos is applied with a variable step size[66]. This allows efficient propagation to very large val-ues of R max . The calculations are converged with respectto integration range and step size.For scattering calculations, log-derivative solutions arepropagated outwards from short range to a distance R max = 50 ,
000 ˚A at long range. Since the basis func-tions (9) are not eigenfunctions of the separated-atomHamiltonian, the resulting log-derivative matrix at R max is transformed to the separated-atom basis set and thenmatched to asymptotic boundary conditions to obtainthe K matrix and then the scattering S matrix. The scat-tering length is obtained as a ( k ) = ( ik ) − (1 − S ) / (1 + S ), where k = √ µE/ (cid:126) is the wavevector and S isthe diagonal S-matrix element in the incoming channel.The kinetic energy in the incoming channel is set to be E = 100 nK × k B , where k B is the Boltzmann constant;this energy is low enough that the resulting scatteringlength has essentially reached its zero-energy value. Weuse the algorithms of Frye and Hutson [59] to locate andcharacterize the Feshbach resonances in the calculatedscattering lengths.For bound-state calculations, one log-derivative solu-tion Y out ( R ) is propagated outwards from short range, and another Y in ( R ) is propagated inwards from R max =200 ˚A, until both reach a matching point R match inthe classically allowed region. At a bound-state energy,the matching matrix Y out ( R match ) − Y in ( R match ) haszero determinant and one of its eigenvalues is zero [67]. bound locates eigenenergies by varying the energy of thecalculation at fixed magnetic field until the matching con-dition is met. field operates similarly, but varies themagnetic field at fixed energy relative to threshold. Thisapproach allows us to converge efficiently and accuratelyon bound-state energies and on magnetic fields at whichbound states cross threshold. E. Fermi’s golden rule
Accurate Feshbach resonance widths can be obtainedfrom coupled-channel scattering calculations, but suchcalculations do not provide much insight. We thereforeuse an analysis based on Fermi’s golden rule to under-stand our results. This gives an expression for the reso-nance width in terms of the matrix element of the cou-pling operator ˆ V spin ( R ) between the single-channel scat-tering state | αk (cid:105) , which is labeled by a channel index α and wavevector k and normalised to a δ -function ofenergy, and the bound state | α (cid:48) n (cid:105) , where n is the vibra-tional quantum number relative to threshold. Brue andHutson showed that the width can be written [34]∆ = πka bg δµ res (cid:104) n | ω x ( R ) | k (cid:105) R (cid:104) α (cid:48) | ˆΩ x | α (cid:105) , (10)where the matrix element has been separated into a radialcomponent (cid:104)· · · (cid:105) R , and a spin component (cid:104)· · · (cid:105) spin .The separation of the two components of the matrixelement allows a clear interpretation of the factors thatinfluence the resonance widths. The spin component (cid:104) α (cid:48) | ˆΩ x | α (cid:105) , which was denoted I m f,a ( B ) for mechanism Iin Ref. [34], describes how the coupling strength dependson the spin states that are coupled and how it varies withmagnetic field. The radial component (cid:104) n | ω x ( R ) | k (cid:105) takesaccount of the binding energy of the bound state and thebackground scattering length in the incoming channel.Near threshold, (cid:104) n | ω x ( R ) | k (cid:105) is proportional to k / , sothat ¯∆ is independent of energy to first order.The golden rule approach can be used as an approxi-mate method of calculating widths, but in this paper weuse it only as an interpretative tool. All widths presentedare from coupled-channel calculations. III. COUPLING MECHANISMS
In this section, we explore the resonances caused bythe three principal coupling mechanisms described in Sec.II A. We focus on the general patterns of the resonancepositions and widths, rather than the specific predictions,which are given in Sec. IV. We also consider inelasticdecay.We take Cs+
Yb as our example system in this sec-tion, although the analysis is relevant to other isotopologsand other systems formed from an alkali-metal atomand a closed-shell atom, such as Rb+Sr. The scatteringlength for V elec ( R ) is very small for Cs+ Yb, so that a bg can vary substantially between resonances, and it isimportant to use the normalized width ¯∆ (Eq. 6) ratherthan ∆ itself as the measure of resonance strength. A. Mechanism I
Resonances caused by mechanism I have been investi-gated by Brue and Hutson [34]. However, at that time thebinding energies and scattering lengths for Cs+Yb wereunknown, so they could study only the general proper-ties.The operator ˆΩ I = ˆ i Cs · ˆ s responsible for mechanism Iproduces couplings with selection rule ∆ m f = 0, wherethe notation ∆ x = x bound − x scat indicates the changein quantum number x between the incoming scatteringstate and the resonant bound state. Since there is onlyone atomic state of each m f for each f , mechanism I cou-ples molecular bound states to atomic scattering statesonly if they have different values of f . Each bound state isessentially parallel to the atomic threshold that supportsit, and Fig. 1(b) shows the resulting crossing diagram.The molecular states that produce Feshbach resonancesby mechanism I correspond to f = 4, so at the energy ofthe f = 3 thresholds they are bound by approximatelythe Cs hyperfine splitting. The bound states are thereforesparsely distributed in energy and the corresponding res-onances are sparsely distributed in magnetic field. Thematrix element (cid:104) f m f | ˆΩ I | f (cid:48) m f (cid:105) goes linearly to zero as B →
0, so the resulting resonance widths ¯∆ are propor-tional to B at low fields [34]. Resonances with usefullylarge widths thus exist at accessible magnetic fields onlyif a bound state for f = 4 accidentally falls close to the f = 3 thresholds.Resonances caused by mechanism I are present for bothbosonic and fermionic isotopes of Yb. For bosonic iso-topes, Yb hyperfine couplings are absent. Since there areno significant anisotropic couplings in this case, we usecalculations with L max = 0. However, for fermionic iso-topes ( Yb and
Yb) with nonzero nuclear spin, thehyperfine coupling terms corresponding to mechanismsII and III can alter the resonance widths produced bymechanism I alone and in some cases introduce inelas-tic decay. These effects are discussed in the followingsubsections.
B. Mechanism II
Mechanism II is due to the scalar hyperfine couplingbetween the electron spin and the nuclear spin of afermionic isotopes of Yb, given by the second term inEq. (4). We can separate the corresponding operator E n e r g y ( G H z ) B (Gauss) -6.5-6.0-5.5-5.0-4.5 E n e r g y ( G H z ) f = 3 Cs+ Yb f = 3 n = − f = 3 n = − f = 4 n = − (b)(a) n = − f = 4 f = 4 n = − f = 4 FIG. 2. (Color online) Level-crossing diagram with Fesh-bach resonance positions from mechanism II for Cs +
Yb.The atomic thresholds (thick black lines) are from the upper( f = 4) and lower ( f = 3) hyperfine manifolds in (a) and(b), respectively. The quantum numbers ( f , n ) are given onthe left-hand side for each manifold of molecular levels (thincolored lines). The solid squares, circles and triangles showthe positions of Feshbach resonances caused by mechanism II,with ∆ m f = +1, 0 and −
1, respectively. ˆΩ II = ˆ i Yb · ˆ s into three components,ˆΩ II = ˆΩ + ˆΩ +1II + ˆΩ − = ˆ i Yb ,z ˆ s z + 12ˆ i Yb − ˆ s + + 12ˆ i Yb+ ˆ s − , (11)where ˆ s ± and ˆ i Yb ± are raising and lowering operators.The superscripts on the components ˆΩ x II correspond tothe selection rule ∆ m f = 0, ±
1, and we will similarly re-fer to mechanisms II , II +1 and II − . These calculationsuse L max = 2 in order to take account of inelastic decayas discussed in Sec. III D.The selection rule on ∆ m f is less restrictive for mech-anism II than for mechanism I, and allows Feshbach res-onances with ∆ f = 0 as well as ∆ f = 1. Figure 2 showshow the bound states cross the f = 3 and f = 4 scat-tering thresholds for Cs+ Yb. We consider resonancesthat arise at crossings where there are direct couplingsdue to mechanism II, which are shown as circles, squaresand triangles for mechanisms II , II +1 and II − , respec-tively.Many more resonances arise than for mechanism I. Inparticular, there is a set of resonances at low field, wherethe thresholds are crossed by the least-bound state ( n = −
1) with the same f but ∆ m f = − f = 3 or ∆ m f =+1 for f = 4. The corresponding resonance positions areapproximately B res ( n = −
1) = (2 i Cs + 1) | ∆ m f | g s µ B E b ( n = −
1) (12)where E b ( n = −
1) is the binding energy of the least-bound state at B = 0. For Cs+ Yb, Eq. (12) gives B res ( n = −
1) = 163 G, consistent with the crossingsshown in Fig. 2. The deviations from Eq. (12) are at mosta few G and arise principally from the non-linearity of theatomic Zeeman effect. The resonance position from theleast-bound state with ∆ f = 0 is approximately the samein the f = 3 and f = 4 manifolds. Even for a systemwhere the binding energy is unknown, the least-boundstate is always within 36 (cid:126) / (2 µ ¯ a ) of threshold [68], andresonances of this type exist provided f remains a nearlygood quantum number at fields up to B res ( n = − n = − f . These start around B = 1200 G, but are much more spread out in field thanthose for n = − m i, Yb , as shown schematically in Fig. 3. Resonances fordifferent m i, Yb have different widths, as discussed be-low. The selection rule on the nuclear spin projectionis ∆ m i, Yb = − ∆ m f ; thus Feshbach resonances occur atdifferent crossing points in the pattern for mechanismsII , II +1 and II − , indicated by circles, squares and tri-angles respectively. The splitting of the threshold levelsis determined solely by the Yb nuclear Zeeman term inEq. (3), while the splitting of the molecular levels hasan additional contribution from the diagonal matrix ele-ments associated with mechanism II. Without this addi-tional contribution, all the resonances for the same valueof ∆ m f would occur at the same field, but its presenceseparates the resonances for different m i, Yb .General properties of the widths of the resonances canbe inferred from Fermi’s golden rule. By contrast withmechanism I, the spin factor in the resonance widths, (cid:104) α (cid:48) | ˆΩ II | α (cid:105) , does not fall to zero as B →
0. This mightseem to suggest usefully large widths for the ∆ f = 0 res-onances that are guaranteed to exist at low field. How-ever, the radial contribution to the resonance widths, (cid:104) n | ω II ( R ) | k (cid:105) , is proportional to E / [34] where E b isthe binding energy of the resonant state below the thresh-old that supports it; through Eq. (12), the width is thusproportional to B / . Thus, although low-field ∆ f = 0 Magnetic fi eld (arb. units) E n e r g y ( a r b . un i t s ) − / − / − / / m i, Yb3 / / m i, Yb = 5 / / / − / − / − / FIG. 3. (Color online) Schematic diagram demonstrating thesplitting pattern for a set of resonances arising from a singlecrossing in Fig. 2, caused by mechanism II for Cs+
Yb. Theatomic thresholds (thick black lines) and molecular boundstates (thin brown lines) are labeled by m i, Yb . The solidsquares, circles and triangles indicate the resonance positionsfor sets with ∆ m f = +1, 0 and −
1, respectively; only oneset appears at each crossing in Fig. 2. The dotted lines showthe bound states without shifts due to mechanism II. Theenergy spacings are typically less that 1 MHz and the set ofresonances typically spans less than 1 G. resonances arising from mechanism II are guaranteed toexist, their widths are also somewhat suppressed, albeitmore weakly and for different reasons than for those aris-ing from mechanism I.There are also ∆ f = 1 resonances from mechanism IIat the f = 3 thresholds. At each threshold, these occurin three sets, corresponding to the three allowed valuesof ∆ m f . As for the resonances arising from mechanismI, which also have ∆ f = 1, these resonances exist at lowfields only if the binding energies are favorable. As shownin Fig. 2, they exist for Cs Yb at the lowest ( m f = 3)threshold at fields from about 600 G upwards, and fromprogressively higher fields at excited thresholds.The four sets of resonances at the m f = 3 thresholdsare examined in Fig. 4; three sets have ∆ f = 1 and onehas ∆ f = 0. The normalized resonance widths are shownas a function of their resonance positions in Fig. 4(a) andas a function of m i, Yb in Fig. 4(b) and (c). For the II ± resonances shown in Fig. 4(b), the resonance widths areproportional to [ i Yb ( i Yb + 1) − m i, Yb ( m i, Yb ∓ i Yb ∓ in ˆΩ ± [33]. For the II resonances, thepattern of widths is more complicated because the atomicscattering and molecular bound states are coupled byboth mechanism I and II. The resonance widths as a func-tion of m i, Yb are shown in Fig. 4(c) for each mechanismseparately and for the combination. For mechanism Ialone, the Yb nuclear spin is not involved, so the width B (Gauss) -5 -4 -3 -2 -1 ¯ ∆ ( m G ) -5/2 -3/2 -1/2 1/2 3/2 5/2 m i, Yb ¯ ∆ ( m G ) -5/2 -3/2 -1/2 1/2 3/2 5/2 m i, Yb ¯ ∆ ( m G ) ¯∆ (I+II) ¯∆ (I) ¯∆ (II)Fermi’s golden rule II +1 (a) (c) II − II II − (b) ¯∆ (II +1 ) ∆ f = 1 × ¯∆ (II − )( ∆ f = 1) II ( ∆ f = 1) × ¯∆ (II − )( ∆ f = 0) ∆ f = 0 ∆ f = 1 FIG. 4. (Color online) Widths of the resonances caused bymechanism II for Cs+
Yb. (a) Overview of the sets ofresonances arising from the four crossings at the f = 3, m f = 3 threshold in Fig. 2. (b) Widths within each set with∆ m f = ± m i, Yb . The lines connecting thepoints are parabolas as expected from Fermi’s golden rule. (c)Widths within the set with ∆ m f = 0, showing separate con-tributions from mechanisms I and II and their combination.Symbols correspond to those in Fig. 2. is constant at ¯∆ =0.04 mG. For mechanism II alone, thewidth is proportional to m i, Yb due to the operator ˆ i Yb,z in ˆΩ . However, the actual resonance width ¯∆(I + II)is proportional to the square of the sum of the couplingmatrix elements. This increases the widths for negative m i, Yb and reduces those for positive m i, Yb .The relative strengths of different sets depend stronglyon the electron-spin components of the states that arecoupled. For example, for the II − set near 200 G, thespin-dependent matrix element (cid:104) α (cid:48) | ˆ i Yb · ˆ s | α (cid:105) is shown as afunction of magnetic field in Fig. 5(a). For this resonance,the dominant electron-spin component is | m s = − / (cid:105) in both the scattering and bound states, but the reso-nance coupling is actually between | m s = 1 / (cid:105) scat and B (Gauss) -1-0.500.51 h α ′ | ˆ i Y b · ˆ s | α i B (Gauss) | h α ′ | ˆ i Y b · ˆ s | α i | m i. Yb = − / m i. Yb = − / m i. Yb = 1 / m i. Yb = − / m i. Yb = − / m i. Yb = − / B (Gauss) | h α ′ | ˆ i Y b · ˆ s | α i | m i, Yb = − / m i, Yb = − / m i, Yb = − / m i Yb = 1 / m i Yb = 3 / (d) m i, Yb − / / I m f =3 II ( ∆ f = 1) (b)(a) (c) II − ( ∆ f = 1)II +1 ( ∆ f = 1)II − ( ∆ f = 0) FIG. 5. (Color online) Spin components (cid:104) α (cid:48) | ˆ i Yb · ˆ s | α (cid:105) of cou-pling matrix elements for mechanism II. (a) Absolute valuesof matrix elements for resonances due to bound states with f = 3, m f = 2 at the f = 3, m f = 3 threshold, as a functionof field, for different m i, Yb . (b) Absolute values of matrix ele-ments for resonances due to bound states with f = 4, m f = 2or 4 at the f = 3, m f = 3 threshold. (c) Matrix elementsfor resonances due to bound states with f = 4, m f = 3 atthe f = 3, m f = 3 threshold. The matrix element for mech-anism I is shown as a thick black line; it adds constructivelyfor m i, Yb < m i, Yb > | m s = − / (cid:105) bound . The small proportion of m s = 1 / (cid:104) α (cid:48) | ˆ i Yb · ˆ s | α (cid:105) in Fig. 5(b) approachzero.A similar argument applies to the sets of resonanceswith ∆ f = 1 and ∆ m f = ±
1. The corresponding spin-dependent matrix elements are shown in Fig. 5(b). Forboth these sets, the dominant electron-spin componentis | m s = 1 / (cid:105) in the scattering state and | m s = − / (cid:105) B (Gauss) -6.5-6.0-5.5-5.0-4.5 E n e r g y ( G H z ) Cs+ Yb f = 3 f = 3 n = − f = 3 n = − f = 4 n = − FIG. 6. (Color online) Level-crossing diagram with Fesh-bach resonances from mechanism III for Cs +
Yb. Thethreshold levels shown (heavy black lines) are from the lowerhyperfine manifolds with f = 3. The quantum numbers ( f , n )are labeled on the left-hand side for each manifold of molec-ular levels (thin colored lines with solid lines for L = 2 anddashed lines for L = 0). The solid squares, circles and trian-gles indicate the positions of Feshbach resonance caused bymechanism III, with ∆ m f = +1, 0 and −
1, respectively. in the bound state. Mechanism II +1 couples these twodominant spin components, but II − couples the smallercomponents that vanish at high field. Consequently, theII +1 resonances have much larger widths than the II − resonances, as shown in Fig. 4(b). C. Mechanism III
Mechanism III is due to the tensor, or anisotropic,hyperfine coupling on the Yb nucleus, described by thethird term in Eq. (4). Like mechanism II, it exists onlyfor the fermionic isotopes of Yb. Unlike mechanisms Iand II, this anisotropic coupling can change the rota-tion of the molecule, with selection rule ∆ L = 2; thereare also ∆ L = 0 terms due to this term, but not for L = 0. Resonances in s-wave scattering arising from di-rect coupling due to mechanism III must therefore comefrom states with L = 2. The other selection rules are∆ m f = 0 , ± m i, Yb = 0 , ±
1. By contrast withmechanism II, a change in ∆ m f + ∆ m i, Yb may be com-pensated by ∆ M L (cid:54) = 0 to conserve M tot . The explicitform of the spin-coupling operator ˆΩ III is more compli-cated than for mechanism II, but it can still be separatedby analogy with Eq. (11) into terms proportional to ˆ s z ,ˆ s + and ˆ s − . We thus subdivide mechanism III into mech-anisms III , III +1 and III − , respectively. These calcula-tions use L max = 4 in order to take account of inelasticdecay as discussed in Sec. III D.Figure 6 shows the L = 2 bound states and the result- ing Feshbach resonances arising from direct coupling dueto mechanism III at the f = 3 thresholds for Cs+ Yb.The L = 0 bound states are shown as dashed lines forcomparison and are identical to those in Fig. 2(b). Each L = 2 state is immediately above the associated L = 0state, with a spacing proportional to an effective rota-tional constant, which varies strongly with the bindingenergy of the state [69]. This produces a pattern of reso-nances very similar to that for mechanism II, but shiftedto somewhat lower field and with additional splittings.Because of the similar separation of the operator intoterms proportional to ˆ s z , ˆ s + and ˆ s − , the general conclu-sions about resonance widths for mechanism II hold formechanism III as well.The most significant difference between mechanisms IIand III is in the internal structure of the sets of reso-nances. Since the bound states for mechanism III have L = 2, there are 5 times as many states, correspondingto different values of M L . Because of the larger num-ber of states, more individual crossings within a set cancause Feshbach resonances, and there can be multiple res-onances at each threshold. Within the set of bound statesfor each M tot , the states with different M L and m i, Yb are mixed by coupling due to mechanism III and the Ybquadrupole term. There are additional small effects dueto spin-rotation and Cs quadrupole coupling. However,the nuclear Zeeman effect and the diagonal matrix ele-ments due to mechanism II separate the states accordingto m i, Yb , and these splittings are generally larger thanthe couplings between them; accordingly, m i, Yb and M L remain useful labels, even though they are not fully con-served.Figure 7 shows the crossing diagrams and the widthsof the resonances in each set as a function of positionfor three typical examples. The first example is the setof resonances due to bound states with f = 3, m f = 0and n = − f = 3, m f = 1 threshold.In this case, the splitting between the bound states issimilar to that between the thresholds. This is becausethe diagonal matrix elements of mechanism II and III areboth proportional to the expectation value (cid:104) m s (cid:105) of theelectron spin projection, and the bound states have m f =0, for which (cid:104) m s (cid:105) = 0 at low field. However, states withthe same m i, Yb but different M L are separated by the Ybquadrupole term. The resulting resonances are separatedinto three subsets corresponding to ∆ m i Yb = +1 , , − g Yb µ B B/ ∆ µ . The patterns of widths fordifferent m i, Yb within these subsets resemble those seenin Fig. 4(b) and (c) for mechanism II, but are distortedby the mixing of the states.The second example is a similar set of resonances with∆ f = 0, but due to f = 3, m f = 2 bound states crossingthe f = 3, m f = 3 threshold. In this case, the expec-tation value (cid:104) m s (cid:105) for the bound states is not near zero,so the molecular states have substantially different split-0 B (Gauss) -4 -2 ¯ ∆ ( µ G ) B (Gauss) -4 -2 ¯ ∆ ( µ G ) B (Gauss) -1 ¯ ∆ ( µ G ) B (Gauss) -5199.40-5199.38-5199.36-5199.34-5199.32-5199.30 E n e r g y ( M H z ) B (Gauss) -5254.40-5254.35-5254.30-5254.25-5254.20-5254.15 E n e r g y ( M H z ) B (Gauss) -5778.8-5778.4-5778.0-5777.6-5777.2 E n e r g y ( M H z ) (a) − / − / − / / / m i, Yb5 / (b) (c)(d) (e) (f) − / m i, Yb5 / Cs- Yb FIG. 7. (Color online) Structure of sets of resonances caused by mechanism III for Cs+
Yb. Panels (a), (b) and (c) showbound-state levels (thin red lines) crossing thresholds (thick black lines) for three representative resonances, with crossings thatcause Feshbach resonances due to direct couplings marked by symbols. Grey squares, white circles, and black triangles showresonances with ∆ m i, Yb = +1, 0, and −
1, respectively. (d), (e) and (f) show the corresponding normalized resonance widths¯∆ as a function of magnetic field. (a) and (d): set of resonances near 78 G at the ( f = 3, m f = 1) threshold; (b) and (e): setof resonances near 79 G at the ( f = 3, m f = 3) threshold; (c) and (f) set of resonances near 552 G at the ( f = 3, m f = 3)threshold. tings to the thresholds. This separates each of the threesubsets (corresponding to ∆ m i, Yb = +1 , , −
1) accordingto m i, Yb , such that the subsets just overlap in field. Thewidths are shown in Fig. 7 and again show the expectedpatterns with respect to m i Yb . However, the resonancesnear the middle of this pattern are the narrowest, so thatin loss spectroscopy the resonances would effectively form two groups in field.The third example is also at the m f = 3 thresh-old, but from a set of resonances with ∆ f = 1. The f = 4 bound states that cause these resonances are moredeeply bound, so the diagonal matrix elements of thespin-dependent terms are much larger. The bound statescan still be labelled by m i, Yb , but the effect of mecha-nism II is so large that the ordering of the bound statesis reversed from that of the thresholds. The splittingsbetween states with the same m i, Yb but different M L ,due to mechanism III and the Yb quadrupole term, arealso much larger, such that the multiplets overlap in somecases. The three subsets with different values of ∆ m i, Yb now completely overlap. As a result, there is no obviousstructure in the pattern of widths as a function of field.These three examples qualitatively explain the pat-terns observed in loss spectroscopy of similar resonancesin Rb+ Sr [41]. Figure 1 of Ref. [41] showed loss pat-terns for three different sets of resonances due to mech-anism III. One of these was for a set of resonances dueto bound states with f = 1, m f = 0 crossing the f = 1, m f = 1 threshold; these bound states have (cid:104) m s (cid:105) ∼
0, soproduce a triple peak as in example 1 above. Another wasfor a set of resonances due to bound states with f = 1, m f = − f = 1, m f = 0 threshold; these bound states have (cid:104) m s (cid:105) (cid:54) = 0, so produce a double peak asin example 2 above. The third was for a set of resonancesdue to bound states with f = 2, m f = − f = 1, m f = − D. Inelastic Decay
Feshbach resonances show signatures of decay whenthe bound state couples to inelastic (open) channels be-low the incoming channel. The primary quantity usedto characterize decay in our calculations is the inelasticdecay width Γ inel B . However, the effect of decay on ex-periments is better quantified by the lifetime τ and theresonant scattering length a res . If the magnitude of a res is too small, the oscillation in the scattering length maynot be sufficient to produce measurable loss in 2-body or3-body loss spectroscopy; at least a res > a is prob-ably necessary to produce measurable loss rates. If thelifetime is too short, molecules formed by magnetoasso-ciation at the resonance will predissociate before furtherexperimental steps; this may pose a problem if the life-times are milliseconds or less.For the bosonic isotopes of Yb, mechanism I couplesthe resonant bound state only to the incoming chan-nel. For collisions at magnetically excited Cs thresholds,the Cs quadrupole and tensor hyperfine couplings can inprinciple cause decay to inelastic channels with L = 2,but these terms are small and the associated decay is very1weak. For example, for the resonance near 654 G for Cs( f = 3, m f = −
3) interacting with
Yb, we calculate a res = 3 . × a , corresponding to τ = 2 . × s. Inthe remainder of this paper, we carry out calculations onbosonic isotopes using L max = 0, which suppresses thisweak decay.For the fermionic isotopes of Yb, any of the couplingoperators in Eq. (4) may cause decay, depending on thecharacter of the resonance. However, there are two sit-uations where there is guaranteed to be very little de-cay. The first is for resonances at the lowest Cs hyperfinethreshold ( f = 3 and m f = 3). These can in principledecay to L = 2 channels with different m i, Yb , but theassociated kinetic energy release is very small and inelas-ticity is strongly suppressed by centrifugal barriers in theoutgoing channels. The second is for II − resonances at f = 3 thresholds. For these, the inelastic channels have∆ m f ≥ m f ≥ f = 4 can always decay to f = 3. This results in signif-icant decay, with a res in the range 19 to 125 a for theresonances due to n = − inel B from −
40 to − µ G and lifetimes from 3 . f = 3, m f < can decay by mechanisms II +1 and III +1 ,while those due to mechanism II +1 can also decay bymechanisms I, II and III . The resulting decay widthsand lifetimes show considerable variation with m i, Yb , butare generally comparable to those at f = 4 thresholds;however a res is considerably larger because ¯∆ is larger.For resonances due to mechanism III, the general pat-terns of decay are similar to those for resonances due tomechanism II. There are additional decay pathways toopen channels with L = 4, which sometimes contributeup to 80% of the decay widths. IV. PROMISING RESONANCES FOREXPERIMENTAL STUDY
In this section we make specific predictions for Fesh-bach resonances that appear promising for experimentalinvestigation. We consider resonances in collisions in-volving Cs in both its ground state ( f = 3, m f = 3)and magnetically excited states. We highlight the mostpromising resonances for each Yb isotope at magneticfields below 2000 G. Tabulations of resonance parame-ters for all resonances at magnetic fields up to 5000 Gare given in the Supplemental Material.There are two experimental situations of particular in-terest. The first is observation of resonances throughtheir enhancement of collisional processes such as 3-bodyrecombination, 2-body inelastic loss, or interspecies ther-malization; this is commonly known as Feshbach spec-troscopy. The second is magnetoassociation of pairs of atoms to form weakly bound molecules, which may becarried out either in an optical trap or in the cells of anoptical lattice. A. Intraspecies Cs collisions
Any experiment carried out in an optical trap is sub-ject to losses due to intraspecies as well as interspeciescollisions. Even in its ground state ( f = 3, m f = 3), ul-tracold Cs suffers from strong 3-body losses at most mag-netic fields, due to large intraspecies scattering lengths.Similar losses exist in magnetically and hyperfine excitedstates, supplemented by 2-body inelastic losses. Thescattering length a ( B ) was tabulated for the ground stateby Berninger et al. [70]. We have recently carried outscattering calculations for pairs of excited Cs atoms inthe same state ( f , m f ), using the interaction potentialsof Ref. [70], for all f = 3 and f = 4 states [71]. We tabu-lated both the complex scattering length a Cs = α Cs − iβ Cs and the rate coefficient k for intraspecies 2-body loss.We estimate that values of k higher than about 10 − cm s − will obscure losses due to interspecies Feshbachresonances.For experiments in an optical trap, we estimate thatintraspecies scattering lengths larger than about 2000 a will produce 3-body losses dominated by intraspecies col-lisions. Even for scattering lengths at the upper end ofthis range, it will probably be necessary to work withCs densities below 10 cm − to moderate intraspecies3-body losses, and with Yb atoms in large excess so thatCs losses due to resonant interspecies collisions are com-petitive.For each interspecies resonances in the Tables below,we give calculated values of α Cs , and k where it exists,at the resonance position. B. Experimental considerations
1. Experiments in optical traps
Optical traps may be used to trap atoms in any internalstate and allow independent control of the applied mag-netic field. Although the atomic cloud is confined to asmall volume, there is nevertheless always some variationin the magnetic field across the sample. This may arisefrom a magnetic field gradient used to levitate the atoms,or from other sources such as curvature in the bias field.There is also inevitably some time-variation of the field,typically on the order of a few mG. For the narrow reso-nances predicted in Cs+Yb, it is likely that only a part ofthe cloud will be on resonance at any one time. The re-sulting loss signal will then be proportional to the rangeof fields over which | a ( B ) | exceeds a critical value a crit .For resonances in elastic scattering, this range is propor-tional to a bg ∆. As described above, for Cs+Yb we havechosen to tabulate the normalized width ¯∆ = ( a bg / ¯ a )∆,2which retains the dimensions of field. The narrowest res-onance observed in recent experiments on RbSr [41] hada calculated normalized width ¯∆ = 0 . > .
04 mG,except below 200 G, where we tabulate resonances with¯∆ > .
004 mG.Feshbach resonances may also be detected through en-hanced interspecies thermalization [72]. This is partic-ularly attractive for Cs+
Yb, where the backgroundscattering length is very low and there will be very lit-tle interspecies thermalization away from resonance. Therate of interspecies thermalization is also expected to beapproximately proportional to ¯∆.Cs atoms in f = 4 excited states are predicted to decayquickly by 2-body inelastic processes [71]. We thereforefocus on Cs+Yb resonances involving Cs atoms in f = 3states. For bosonic isotopes of Yb at any threshold, andfor fermionic isotopes at the lowest threshold, the in-terspecies resonances are undecayed and the scatteringlength passes through a pole at resonance. However, forfermionic isotopes at thresholds with m f < ± a res /
2. If a res is lessthan about 100 a , an interspecies resonance may notproduce a significant peak in 3-body loss.Interspecies 2-body loss occurs only with fermionic iso-topes of Yb in combination with excited states of Cs. Itis very weak away from resonance, but shows a narrowpeak of height proportional to a res at resonance. Theremay be some resonances and conditions under which in-terspecies 2-body loss is faster than 3-body loss.
2. Experiments in lattices
Experiments in 3D optical lattices have several advan-tages. By loading quantum-degenerate gases into the lat-tice and exploiting the superfluid-to-Mott-insulator tran-sition [73], the number of atoms loaded onto a lattice sitecan be controlled and tunneling suppressed. Under suchconditions, intraspecies losses can be completely elimi-nated. Experiments may thus be performed with anyinternal state and at any magnetic field, without restric-tion on the intraspecies scattering properties; this is par-ticularly beneficial when working with atoms such as Cs,where intraspecies loss may otherwise be a limiting fac-tor. The use of an optical lattice also removes the needfor a field gradient to levitate the cloud against gravity.Experiments in lattices are still subject to interspecies2-body loss when it is present. For fermionic isotopesof Yb, combined with excited states of Cs, it may bepossible to detect resonances by searching for 2-body lossas a function of magnetic field in a lattice.
3. Magnetoassociation
Magnetoassociation may be carried out either in anoptical trap or in a lattice cell containing one atom ofeach type. In a confined system, the scattering contin-uum above threshold is replaced by a series of quantizedtranslational levels. A scattering resonance then appearsas a series of avoided crossings between the molecularstates and these quantized levels. The strengths (en-ergy widths) of the avoided crossings are proportionalto ( a bg ∆) / [54, 55]. In magnetoassociation, the goalis to sweep the magnetic field across the lowest of theavoided crossings slowly enough to achieve adiabatic pas-sage. The maximum sweep speed that achieves this isproportional to the square of the strength and thus to a bg ∆ [54, 55]. Because of this, ¯∆ is an appropriate mea-sure of the resonance width for magnetoassociation aswell as for loss spectroscopy.A lattice cell confines a pair of atoms more tightly thanan optical trap, increasing the strength of the avoidedcrossing available for magnetoassociation. The strengthis proportional to ω / [54, 55], where ω is the har-monic trap frequency [74]; the maximum speed of thefield sweep is thus proportional to ω / .For a broad resonance, it is relatively easy to sweep thefield slowly enough to achieve adiabatic passage. How-ever, for narrow resonances such as those considered here,it is more challenging. Field inhomogeneity results onlyin different parts of the sample crossing the resonance atdifferent times. Field noise, however, may result in re-peated crossing and recrossing at speeds that cause nona-diabatic transitions and loss. Very narrow resonancesthus require very stable fields.
4. Molecular lifetimes
Molecules formed by magnetoassociation at a decayedresonance may themselves decay (predissociate) sponta-neously with lifetime τ , as described in section II C. Inpractical terms, it is necessary to stabilize the magneticfield after magnetoassociation sweep before transferringthe molecules to another state. This is likely to be diffi-cult if the molecular lifetime is less than about 100 µ s. C. Cs + bosonic Yb
For Cs interacting with bosonic isotopes of Yb, thereare only a few resonances located below 2000 G. Theseare all caused by mechanism I. Inelastic decay is negligi-ble for these resonances, even for excited states of Cs, asdiscussed in Section III D. The important properties arethe resonance position and width, as well as the proper-ties relevant to background loss of Cs for experiments inan optical trap. Table III lists all resonances that meetthe width criteria described above, together with someadditional ones that warrant discussion.3
TABLE III. Experimentally promising resonances in ultracoldcollisions between Cs and bosonic isotopes of Yb. B res ¯∆ α Cs k Cs-Yb m f (G) (mG) ( a ) (cm s − )133-168 − − .
074 1 . × . × − − − .
098 3 . × . × − − . × . × − . × . × − − . × . × − . × . × − . × . × . × − . × . × − . × . × . × − − −
35 1 . × . × − − . × . × − The resonances for
Yb are the strongest in TableIII, and also have small two-body loss rates for Cs. Thepair of resonances near 1559 G and 3359 G are from adouble crossing between the atomic and molecular stateswith m f = −
3. The relatively large normalized widths¯∆ for these occur both because the background scatter-ing length is large (798 a ) and because the differencebetween the magnetic moments of the atomic and molec-ular states is small near such a double crossing [34]. Theresonance at 3359 G is included in Table III, despite itshigh field, because it is unusually wide and is also in afield range where 3-body loss of Cs is expected to berelatively slow. These resonances are promising for lossspectroscopy. However, Yb has a small negative in-traspecies scattering length [75], which leads to collapseof its condensates [76], so that a lattice with a high fillingfraction will be hard to produce.The Yb isotopes that are most easily cooled to degen-eracy, and are thus most suitable for formation of Mottinsulators, are
Yb [77] and
Yb [78]. The normalizedwidths of the resonances for these isotopes are smallerthan for
Yb, but magnetoassociation in an optical lat-tice may still be feasible. For
Yb, two-body loss ofCs atoms may prevent observation of the resonances byloss spectroscopy. The three resonances for
Yb appearmore suitable for loss spectroscopy, though 3-body lossesof Cs atoms are expected to be fairly fast.
Yb has a large negative intraspecies scatteringlength [75], and has not been cooled to degeneracy. Itnevertheless has resonances that may be observable byloss spectroscopy. The resonance near 1528 G for
Ybwith Cs ( f = 3, m f = 0) appears particularly suitablefor this because of the relatively small background lossesexpected for Cs atoms. Yb has a very low isotopic abundance and the onlyresonance available below 2000 G is the one near 654 G.This resonance might be observable by loss spectroscopy,but has no obvious advantages over those for more abun- dant isotopes.
D. Cs + fermionic Yb
For Cs interacting with fermionic isotopes of Yb, res-onances can be driven by any of the three mechanismsdiscussed in Sec. III. This provides more resonances thanfor bosonic isotopes, particularly at low field. The reso-nances that meet the criteria described above are listedin Table IV for
Yb and Table V for
Yb. Each entryin the Tables represents a set of closely spaced resonancescorresponding to different values of m i, Yb (and M L formechanism III), as described in Sec. III. For each set,only the widest is given. Full tabulations of the reso-nances, including all those in each set and those that areexcluded from Tables IV and V by one or more of thecriteria, are given in the Supplemental Material.The resonances for Yb follow similar patterns tothose for
Yb, discussed in Sec. III. For
Yb, there is agroup of resonances around 74 G caused by bound stateswith n = − f .These are all caused by mechanism II. The correspond-ing resonances from n = − f = 4 crossing f = 3 thresholds, and arise from mech-anisms I and II. The n = − f = 4lies approximately 360 MHz below the f = 3 thresholdat zero field; it causes resonances starting around 150 G.At each threshold, there are resonances of this type with∆ m f = +1, 0 and −
1, at progressively increasing fields,though not all of them meet the criteria for inclusion inTable IV.Most of the resonances for fermionic Yb are subject todecay. Tables IV and V include values of the resonantscattering length a res and the lifetime τ that characterizethis decay [79]. Many of the resonances at f = 4 thresh-olds have a res < a and are likely to be difficult toobserve in loss spectroscopy.Resonances due to mechanism III are included in Ta-bles IV and V. For Yb, only 3 resonances meet thecriteria for inclusion. For
Yb, there are none thatmeet the criteria, so we have included the widest unde-cayed resonance, at 553 G. Resonances due to mecha-nism III at excited thresholds are strongly decayed, with a res < a , as exemplified by the resonance at 113 G forCs (3,2) interacting with Yb. Such resonances are un-likely to be observable in loss spectroscopy because a ( B )deviates so little from its background value.There are several resonances in Tables IV and V forCs ( f = 3, m f = 3) interacting with each of Yb and
Yb. These resonances occur at fields where α Cs islarge, so that experiments in an optical trap are likely tobe hampered by fast intraspecies 3-body losses. However,they would be good candidates for magnetoassociationin an optical lattice. The strongest resonances in thiscategory are those at 148 G for Yb and at 619 G and700 G for
Yb.4
TABLE IV. Experimentally promising resonances in ultracold collisions between Cs and
Yb. Parameters are given for thewidest resonance in each set, and the corresponding value of m i, Yb is given. B res ¯∆ a res τ α Cs k ( f, m f ) m i, Yb (∆ f, ∆ m f ) mechanism (G) (mG) ( a ) (s) ( a ) (cm s − )(3 , − / , −
1) II 75 0.0048 1 . × . × . × (3 , − / ,
1) III 81 0.0063 6 . ×
37 1 . × (3 ,
3) 1 / ,
1) II 149 0.33 1 . × . × . × (3 ,
3) 1 / ,
0) I+II 171 0.065 4 . × . × . × (3 , − / , −
1) II 74 0.0081 1 . × . × . × . × − (3 , − / ,
1) III 113 0.0065 9.6 7 . × − − . × . × − (3 ,
2) 1 / ,
1) II 203 0.34 9 . × . × − . × . × − (3 ,
2) 1 / ,
0) I+II 247 0.18 9 . × . × − . × . × − (3 , − / , −
1) II 315 0.054 6 . ×
87 1 . × . × − (3 , − / , −
1) II 1517 0.057 2 . ×
17 1 . × . × − (3 , − / , −
1) II 74 0.0097 1 . × . × . × . × − (3 , − / ,
1) III 180 0.0074 3.3 3 . × − − . × . × − (3 ,
1) 1 / ,
1) II 315 0.38 6 . × . × − − . × . × − (3 ,
1) 1 / ,
0) I+II 423 0.44 6 . × . × − . × . × − (3 , − / , −
1) II 613 0.14 1 . ×
11 1 . × . × − (3 , − / , −
1) II 1373 0.069 2 . × − . × . × − (3 , − / , −
1) II 73 0.0097 1 . × . × − . × . × − (3 ,
0) 1 / ,
1) II 613 0.46 1 . × . × − − . × . × − (3 ,
0) 1 / ,
0) I+II 934 1.22 9 . × . × − . × . × − (3 , − / , −
1) II 1243 0.067 3 . ×
14 1 . × . × − (3 , − / , −
1) II 1444 0.14 2 . × . × . × − (3 , − − / , −
1) II 73 0.0081 1 . × . × − . × . × − (3 , − − / , −
1) II 1125 0.056 3 . ×
16 3 . × . × − (3 , −
1) 1 / ,
1) II 1444 0.47 3 . × . × − . × . × − (3 , − − / , −
1) II 73 0.0049 1 . × . × . × . × − (4 , −
4) 1 / ,
1) II 72 0.0066 97 3 . × − . × . × − (4 , −
4) 1 / ,
1) II 927 0.042 9 . × . × − . × . × − (4 , −
3) 1 / ,
1) II 73 0.011 91 2 . × − . × . × − (4 , −
3) 1 / ,
1) II 1021 0.075 2 . × . × − . × . × − (4 , −
2) 1 / ,
1) II 73 0.015 78 1 . × − . × . × − (4 , −
2) 1 / ,
1) II 1126 0.098 1 . × . × − . × . × − (4 , −
1) 1 / ,
1) II 74 0.016 64 1 . × − . × . × − (4 , −
1) 1 / ,
1) II 1243 0.11 58 1 . × − . × . × − (4 ,
0) 1 / ,
1) II 74 0.016 50 8 . × − . × . × − (4 ,
0) 1 / ,
1) II 1373 0.11 34 1 . × − . × . × − (4 ,
1) 1 / ,
1) II 74 0.015 37 6 . × − . × . × − (4 ,
1) 1 / ,
1) II 1517 0.098 20 9 . × − . × . × − (4 ,
2) 1 / ,
1) II 75 0.011 24 5 . × − . × . × − (4 ,
2) 1 / ,
1) II 1673 0.075 11 8 . × − . × . × − (4 ,
3) 1 / ,
1) II 75 0.0066 12 5 . × − . × . × − (4 ,
3) 1 / ,
1) II 1840 0.042 4.8 7 . × − . × . × − Cs atoms in magnetically excited states offer additionalpossibilities. Promising candidates for observation in lossspectroscopy include those near 202 G and 423 G for
Yb and those near 165 G, 720 G and 1004 G for
Yb.The resonance near 165 G for m f = 2 has a width similarto that near 168 G for m f = 3, but α Cs is much smaller,corresponding to much slower 3-body loss. The 2-bodyloss rate k is also very small. V. CONCLUSION
We present a comprehensive theoretical study of mag-netically tunable Feshbach resonances in ultracold colli-sions between Cs and Yb atoms. We carry out coupled-channel calculations of the complex scattering length andanalyze the results to obtain resonance positions andwidths. For resonances in collisions of Cs in magneticallyexcited states, we also extract parameters that charac-terize resonance decay and the lifetime of the molecularstates responsible for the resonances.We use an accurate interaction potential recently de-5
TABLE V. Experimentally promising resonances in ultracold collisions between Cs and
Yb. Parameters are given for thewidest resonance in each set, and the corresponding value of m i, Yb is given. B res ¯∆ a res τ α Cs k ( f, m f ) m i, Yb (∆ f, ∆ m f ) mechanism (G) (mG) ( a ) (s) ( a ) (cm s − )(3 , − / , −
1) II 167 0.011 2 . × . × . × (3 , − / ,
1) III 553 0.0053 6 . × . × − . × (3 ,
3) 1 / ,
1) II 620 0.48 3 . ×
29 2 . × (3 , − / ,
0) I+II 700 0.32 1 . × . × . × (3 , − / , −
1) II 165 0.018 1 . × . × . × . × − (3 ,
2) 1 / ,
1) II 804 0.49 7 . × . × − . × . × − (3 , − / ,
0) I+II 933 0.86 6 . × − . × . × − (3 , − / , −
1) II 163 0.022 8 . × . × − . × . × − (3 ,
1) 1 / ,
1) II 1107 0.50 2 . × . × − . × . × − (3 , − / ,
0) I+II 1326 1.80 4 . × . × − − . × . × − (3 , − / , −
1) II 1622 0.048 4 . × . × . × . × − (3 , − / , −
1) II 161 0.022 7 . × . × − . × . × − (3 ,
0) 1 / ,
1) II 1624 0.52 88 1 . × − . × . × − (3 , − / , −
1) II 1815 0.13 3 . × . × . × . × − (3 , − / ,
0) I+II 1983 3.30 5 . × . × − − . × . × − (3 , − − / , −
1) II 159 0.018 8 . × . × . × . × − (3 , − − / , −
1) II 1566 0.10 8 . × . × . × . × − (3 , − − / , −
1) II 157 0.011 1 . × . × . × . × − (3 , − − / , −
1) II 1357 0.058 2 . × . × . × . × − (3 , − − / , −
1) II 1744 -0.30 1 . × . × . × . × − (4 , − − / ,
1) II 157 0.015 37 6 . × − . × . × − (4 , − − / ,
1) II 1188 0.069 36 7 . × − . × . × − (4 , − − / ,
1) II 158 0.026 76 7 . × − . × . × − (4 , − − / ,
1) II 1362 0.13 35 5 . × − . × . × − (4 , − − / ,
1) II 160 0.033 1 . × . × − . × . × − (4 , − − / ,
1) II 1574 0.17 27 4 . × − . × . × − (4 , − − / ,
1) II 162 0.037 1 . × . × − . × . × − (4 , − − / ,
1) II 1828 0.19 20 3 . × − . × . × − (4 , − / ,
1) II 165 0.037 1 . × . × − . × . × − (4 , − / ,
1) II 167 0.033 75 6 . × − . × . × − (4 , − / ,
1) II 169 0.026 47 5 . × − . × . × − (4 , − / ,
1) II 171 0.015 21 4 . × − . × . × − termined from photoassociation spectroscopy [43], whichgives reliable scattering lengths for all isotopic combina-tions of Cs and Yb and gives accurate predictions forthe energies of the molecular states that cause Feshbachresonances.The resonances are driven by couplings due to spin-dependent terms in the Hamiltonian that vary with theinternuclear distance. We carry out electronic structurecalculations of the distance-dependence of all the im-portant spin-dependent interactions, including the scalarhyperfine, tensor hyperfine, nuclear electric quadrupole,and spin-rotation terms. The resulting couplings allowus to make quantitative predictions of resonance widthsand other properties.For bosonic isotopes of Yb, with zero nuclear spin, theresonances are driven almost entirely by the distance-dependence of the scalar hyperfine interaction on Cs. Thegeneral features of the resulting resonances have beenexplored in previous work, but the much improved in-teraction potential used here allows us to make specific predictions of the resonance positions and widths for thefirst time.For fermionic isotopes of Yb, with non-zero nuclearspin, there are several additional terms in the hyper-fine Hamiltonian, including significant anisotropic termsthat couple atomic and molecular states with differentvalues of the partial-wave (or molecular rotation) quan-tum number L . The additional terms cause additionalFeshbach resonances. They also split both the atomicand molecular states: the atomic states are split intoregularly spaced Zeeman components, but the molecularstates are split in more complicated ways, particularlyfor L >
0, and several different spin-dependent termscontribute. Each Feshbach resonance that would exist inthe absence of these terms is split into a closely spacedset of resonances, spread over 1 G or less.A particular feature of the fermionic systems is thatbound states below one Cs f = 3 threshold can causeresonances at another f = 3 threshold with a differentvalue of m f . Because these states can be very weakly6bound, they can cause resonances at relatively low field.We have made a complete set of predictions for allFeshbach resonances below 5000 G for all isotopic com-binations. We have identified resonances that are par-ticularly promising for experimental investigation, bothto detect resonances in an optical trap and to formmolecules by magnetoassociation in an optical lattice. ACKNOWLEDGMENTS
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