The prospects of forming ultracold molecules in states
by magnetoassociation of alkali-metal atoms with Yb
We explore the feasibility of producing ultracold diatomic molecules with nonzero electric and magnetic dipole moments by magnetically associating two atoms, one with zero electron spin and one with nonzero spin. Feshbach resonances arise through the dependence of the hyperfine coupling on internuclear distance. We survey the Feshbach resonances in diatomic systems combining the nine stable alkali-metal isotopes with those of Yb, focussing on the illustrative examples of RbYb and CsYb. We show that the resonance widths may expressed as a product of physically comprehensible terms in the framework of Fermi’s Golden Rule. The resonance widths depend strongly on the background scattering length, which may be adjusted by selecting the Yb isotope, and on the hyperfine coupling constant and the magnetic field. In favorable cases the resonances may be over 100 mG wide.
The successes of cooling gases of atoms to ultracold temperatures have led to great interest in producing molecules at similar temperatures. Because molecules have a richer internal structure and more complex interactions than atoms, ultracold (K) molecules offer the possibility of exploring a wide range of new research areas, including high-precision measurement Hudson et al. (2006); Kawall et al. (2004); Hudson et al. (2002), quantum information DeMille (2002); Jaksch et al. (1999) and quantum simulation Micheli et al. (2006).
Molecules may be formed in ultracold atomic gases either by photoassociation Jones et al. (2006) or by magnetoassociation Köhler et al. (2006). In the latter, cold atomic clouds are subjected to time-dependent magnetic fields that convert atom pairs into molecules by adiabatic passage across zero-energy Feshbach resonances Chin et al. (2010). Recent years have seen substantial progress in producing ultracold molecules made up of pairs of alkali-metal atoms Herbig et al. (2003); Regal et al. (2003); Strecker et al. (2003); Cubizolles et al. (2003); Jochim et al. (2003); Ni et al. (2008); Danzl et al. (2010); Lang et al. (2008); Heo et al. (2012). The molecules are left in high vibrational states and are susceptible to collisional trap loss. For KRb Ni et al. (2008), Cs Danzl et al. (2010), and triplet Rb Lang et al. (2008), it has been possible to transfer the molecules to the absolute ground state by Stimulated Raman Adiabatic Passage (STIRAP).
There is now great interest in the formation of cold molecules that have both electric and magnetic dipole moments Żuchowski et al. (2010); Ivanov et al. (2011); Hansen et al. (2011); Hara et al. (2011); Nemitz et al. (2009); Brue and Hutson (2012). Such molecules offer additional possibilities for manipulation, trapping, and control because they can be influenced by both electric and magnetic fields. In the present paper we investigate the prospects for magnetoassociation of alkali-metal atoms (Alk) with S atoms (specifically Yb) to form heteronuclear diatoms with electron spin .
Ytterbium is an excellent candidate for pairing with the alkali metals. It has 7 stable isotopes (5 zero-spin bosons, 2 fermions), and a closed-shell, singlet-spin electronic structure. Both bosonic Takasu et al. (2003); Fukuhara et al. (2007a, 2009); Sugawa et al. (2011) and fermionic Fukuhara et al. (2007b, c) isotopes have been cooled to quantum degeneracy. Different isotopic combinations have different scattering lengths, and produce molecules with different binding energies; they thus have Feshbach resonances at different magnetic fields.
The existence of magnetically tunable Feshbach resonances requires coupling between a continuum scattering state of the atomic pair and a molecular state that crosses it as a function of magnetic field. For pairs of alkali-metal atoms, this coupling is provided by the difference between the singlet and triplet potential curves and by the magnetic dipolar interaction between the electron spins. However, neither of these effects exists in systems of the type considered here. Instead, the most significant coupling between the atomic and molecular states is provided by the -dependence in the hyperfine coupling constant of the alkali-metal atom Żuchowski et al. (2010). Such -dependences exist in alkali dimers Strauss et al. (2010), but in that case they merely produce small shifts in bound state energies and resonance positions, rather than driving new resonances. If the closed-shell atom has non-zero nuclear spin, it can also couple to the unpaired electron spin. For the case of LiYb Brue and Hutson (2012), this coupling has been found to be much stronger than that due to the Li nucleus. However, this latter effect is less important for the heavier alkali-metal atoms considered here, where the coupling to the alkali-metal nucleus itself is stronger.
In previous work, we extracted resonance positions and widths for RbSr Żuchowski et al. (2010) and LiYb Brue and Hutson (2012) from coupled-channel quantum scattering calculations. In the present paper, we extend these studies to a range of heavier systems and show how the widths may be broken down into their contributing factors within the framework of Fermi’s Golden Rule.
The theoretical development presented here is applicable to any system made up of an alkali-metal atom paired with a closed-shell atom. In the present study, we have considered the whole range of Alk-Yb systems, but we focus our presentation on the illustrative examples of Rb-Yb, for which the scattering lengths are approximately known, and Cs-Yb, for which they are as yet unknown. In section II we describe the theoretical methods used. In section III we present our results, with discussion of system characteristics that lead to Feshbach resonances suitable for molecule formation.
ii.1 Collisions between alkali-metal and closed-shell atoms
The Hamiltonian for an alkali-metal atom in a S state, interacting with a closed-shell atom in a S state, is
where is the two-atom rotational angular momentum operator and is the interaction operator. and are the single-atom hamiltonians,
where , and are the electron and nuclear spin operators, , and are the electronic and nuclear factors, and and are the Bohr and nuclear magnetons. is the hyperfine coupling constant for the alkali-metal atom and is the external magnetic field, whose direction defines the axis. In the present work we use lower-case angular momentum operators and quantum numbers for individual atoms and upper-case for the corresponding molecular quantities.
The interaction of a S atom with a S atom produces only one molecular electronic state, of symmetry. However, the hyperfine coupling constant of the alkali-metal atom is modified by the presence of the closed-shell atom Żuchowski et al. (2010), and if then there is also hyperfine coupling involving the nucleus of atom Brue and Hutson (2012),
The interaction operator is thus
where is the electronic interaction potential. Most of the theory presented here remains applicable when atom is a non-alkali-metal atom in a multiplet-S state.
Figure 1 shows the energy levels of the Cs atom, with (black solid lines). At zero field the levels may be labeled by quantum numbers , where , whereas at high field the nearly good quantum numbers are and . In the present paper we indicate the lower and upper states for each as and respectively.
The Hamiltonian (1) may be written as the sum of a zeroth-order term and a perturbation ,
The zeroth-order Hamiltonian is separable, and its eigenfunctions are products of atomic functions and radial functions . The latter are eigenfunctions of the 1-dimensional Hamiltonian
with eigenvalues . The eigenvalues of are , where and are the eigenvalues of and . By contrast with the alkali-metal dimers, the molecular states thus lie almost parallel to the atomic states as a function of magnetic field. They also have almost exactly the same spin character. The only terms in the Hamiltonian (1) that couple and are the weak couplings involving and . The former couples states with and , while the latter couples states with and .
The Hamiltonian (1) is entirely diagonal in , so resonances in s-wave scattering can be caused only by bound states. The only interactions off-diagonal in are spin-rotation and nuclear quadrupole interactions, which are neglected in the present work. This again contrasts with the alkali-metal dimers, where the magnetic dipolar interaction between the electron spins and second-order spin-orbit coupling provide relatively strong interactions that produce resonances from bound states with in s-wave scattering.
Figure 1 shows the highest few vibrational states for CsYb with spin character and vibrational quantum numbers , (with respect to threshold) as dashed blue lines, calculated for a potential with an s-wave scattering length bohr. The couplings involving give rise to Feshbach resonances at fields where bound states cross thresholds , shown as solid circles in Figure 1. In the present work we neglect couplings due to and set for all isotopes. This will give accurate results for resonances with but will suppress resonances with , which actually exist for Yb and Yb.
ii.2 Electronic Structure Calculations
ii.2.1 Potential Energy Curves
We have constructed the electronic potential energy curves by carrying out electronic structure calculations at short and medium range and switching to a form incorporating dispersion interactions at long range.
We obtained ground-state potential curves for NaYb, KYb, RbYb and CsYb from CCSD(T) calculations (coupled-cluster with single, double, and non-iterative triple excitations) using the Molpro package Werner et al. (2006). For Yb, we used the quasi-relativistic effective core potential (ECP) of Dolg et al. and its corresponding basis set Dolg et al. (1989), with 60 electrons in the inner 4 shells represented by the ECP and the remaining 10 electrons () treated explicitly. ECPs Leininger et al. (1996) and their corresponding basis sets Stoll () were also used for K, Rb, and Cs. An ECP was not used for Na; all 11 electrons were represented with the cc-pvqz basis set of Prascher et al. Prascher et al. (2011). For each system, CCSD(T) calculations were carried out at a series of points from 2 to 40 bohr and the potentials were then interpolated using the reproducing kernel Hilbert space (RKHS) method Ho and Rabitz (1995). The resulting potential curves are shown in Figure 2, together with the LiYb curve of Zhang et al. Zhang et al. (2010), obtained using similar methods but with a fully relativistic ECP for Yb. The well depths and equilibrium distances are given in Table 1.
At long range the potential curves were represented as
The coefficients used for the long-range potential were obtained from Tang’s combination rule Tang (1969) based on the Slater-Kirkwood formula,
using the homonuclear coefficients for Alk-Alk Derevianko et al. (2010) and Yb-Yb Kitagawa et al. (2008) and the static polarizabilities for the alkali-metal atoms Derevianko et al. (2010) and Yb Zhang and Dalgarno (2007). Equation (11) gives coefficients well within 1% of the values of ref. Derevianko et al. (2010) for all the mixed alkali-metal pairs. The results for the Alk-Yb systems are included in Table 1. The and terms were omitted except when fitting to the experimental spectra for RbYb as described in section III.1 below. The short-range and long-range regions of the potential were joined using the switching function of Janssen et al. Janssen et al. (2009) between the distances 28 and 38 bohr.
ii.2.2 Hyperfine Coupling
The hyperfine coupling constant of an atom is a measure of the interaction between its nuclear spin and the electron spin density at the nucleus, which in the case of an alkali metal comes principally from the single valence electron. Approach of another atom perturbs the electronic wavefunction and alters the spin density at the nucleus, so that the coupling between the electron and nuclear spins becomes a function of internuclear distance .
We have calculated the hyperfine coupling constants for the Alk-Yb systems, using density-functional theory with the KT2 functional Keal and Tozer (2003), as implemented in the ADF suite of programs ADF (2007). We fitted these results to a variety of functional forms and found that, in the range of for which the vibrational wavefunctions are non-zero, a Gaussian function gave an adequate fit to the DFT results. These functions are shown in Figure 3 for each of the Alk-Yb systems and the parameters are given in Table 2.
|Li||0.0535111The value of for Li was reported incorrectly in ref. Brue and Hutson (2012).||4.92|
ii.3 Resonance widths from coupled-channel calculations
Near resonance, the s-wave scattering length as a function of magnetic field behaves as Moerdijk et al. (1995)
where is the resonance position and is the background scattering length. The magnitude of the resonance width, , is critical for determining whether magnetoassociation is experimentally feasible. Defining as the field where near resonance, Eq. (12) implies .
In the present work, we obtained scattering lengths principally from coupled-channel calculations. The coupled equations for each field were constructed in an uncoupled basis set and solved using the MOLSCAT package Hutson and Green (1994); González-Martínez and Hutson (2007). The s-wave scattering length was then obtained from the identity Hutson (2007) , where is the diagonal S-matrix element in the incoming channel, , and the collision energy was taken to be 1 nK . MOLSCAT has an option to converge numerically on the fields corresponding to both poles and zeroes in , allowing the extraction of .
It should be noted that Eq. (12) characterizes the scattering length near resonance only for purely elastic scattering. If there exist lower-energy channels that allow decay, then has a non-zero imaginary component and does not follow the simple pole formula (12) Hutson (2007). For , the Hamiltonian (1) allows only elastic scattering even when the alkali-metal atom is in a magnetically excited state. However, when , couplings involving can change , and for alkali-metal atoms in magnetically excited states this provides additional couplings to lower-lying thresholds. We have previously described the behavior of for resonances in such states for the LiYb systems Brue and Hutson (2012).
ii.4 Resonance widths from Golden Rule
Coupled-channel calculations of resonance widths are straightforward but provide relatively little insight into the factors that affect resonance widths. We therefore develop here an alternative approach based on Fermi’s Golden Rule that allows us to understand the factors that determine the widths.
Fermi’s Golden Rule gives an expression for the width of a Feshbach resonance in terms of the off-diagonal matrix element of (Eq. (8)) between the bound state (with vibrational quantum number ) and the continuum state (labeled by wavevector , where ). The Breit-Wigner width in the energy domain, , is
where the continuum function is normalized to a -function of energy and has asymptotic amplitude . At limitingly low collision energy, behaves as Chin et al. (2010),
where is the difference between the magnetic moment of the molecular bound state and that of the free atom pair, which is simply the difference in slope of the crossing lines in Figure 1.
The expression for the magnetic resonance width factorizes into spin-dependent and radial terms,
The quantity is a purely atomic property, which arises because states are eigenfunctions of . Pairs of states with the same are coupled through the operator . At zero field, the states are eigenfunctions of , so that the perturbation has no off-diagonal matrix elements. At sufficiently high field, however, the states are well described by quantum numbers and , such that for a given and ,
The behavior of between these two limits is shown as a function of magnetic field for Cs in Figure 4. For positive the coupling increases monotonically before leveling off to the value (19), while for negative it increases with , peaks, and then declines to the same value. At low fields, the coupling is approximately proportional to , so that the resonance width is proportional to in this region. The range over which this behavior occurs is system-dependent; the coupling elements for lighter alkali metals level off at smaller than for Cs.
The factor in Eq. (16) produces wider resonances when the difference in slope between the bound and continuum states at is small. Particularly shallow crossings and wide resonances can occur when there is a “double crossing” involving a bound state that just dips below the threshold (as a function of ) before rising above it again. The magnetic fields at which this can occur are discussed in Section III.3 below.
The bound and continuum functions, and , are eigenfunctions that correspond to different eigenvalues of the 1-dimensional radial hamiltonian (9). They are thus orthogonal to one another, and the matrix element of Eq. (18) is non-zero only because of the -dependence of .
Figure 5 shows how the integral develops as a function of in a typical case. The upper three panels show , and . Figure 5(d) shows the integrand of Eq. (18), which is the product of the three. For weakly bound states, the bound and continuum functions remain almost in phase with one another across the width of the potential well, so that their product always maintains the same sign. The integral thus accumulates monotonically as shown in the bottom panel. Its value depends principally on between the inner turning point and the potential minimum. Deeply bound states lose phase with the continuum at shorter ranges; in principle this produces some cancelation that reduces the value of the integral, but the effect of this is small for the near-dissociation levels considered here.
Further insight may be gained by considering the integral semiclassically. In the WKB (Wentzel-Kramers-Brillouin) approximation, the bound and continuum wavefunctions both oscillate with amplitudes proportional to in the classically allowed region, where . For very weakly bound states and low collision energies, may be neglected, so
where is the inner classical turning point at . This structure is clearly visible in Figure 5(d).
Near threshold, the WKB approximation gives an incorrect ratio between the short-range and long-range amplitudes of a scattering wavefunction. Quantum Defect Theory (QDT) Mies (1984) corrects for this using an energy-dependent function , which is 1 far from threshold but is given by
at limitingly low energy Chin et al. (2010). The correction amplifies the short-range wavefunction by a factor , which has a minimum value of when but is approximately when .
Combining all these effects gives a semiclassical expression for the Golden Rule width,
where is the normalization integral for the WKB bound-state wavefunction,
which is taken between the classical turning points and at energy . Eq. (22) completely avoids the calculation of any quantal wavefunctions and gives results within 2% of the quantal Golden Rule width (16).
The semiclassical approach may be taken one step further, with a small approximation. For a near-dissociation vibrational state with an interaction potential that varies as at long range, Le Roy and Bernstein Le Roy and Bernstein (1970) have shown that the integral (23) is
where is the gamma function. For the present case, with , is thus proportional to . Deeper bound states thus produce broader resonances, though generally at higher magnetic field. For the bound states of interest here, Eq. (24) is accurate to within 6%.
As described below, different isotopes of Yb offer different values of the scattering length . Eq. (22) shows that large values of may occur when is either very large or very small: is directly proportional to when , and inversely proportional to when .
Overall the Golden Rule approximation (22) produces resonance widths that agree within 2% with those from full coupled-channel calculations. It also produces important insights into the origins of the widths, and makes it much easier to select systems and isotopic combinations with experimentally desirable properties.
ii.5 Sensitivity to the interaction potential
The Feshbach resonance positions and widths are strongly dependent on the s-wave scattering length of the system. The background scattering length , the binding energies of high-lying vibrational levels , and the non-integer quantum number at dissociation can all be related to a semiclassical phase integral ,
For a potential with long-range behavior , the scattering length is
where is the mean scattering length of Gribakin and Flambaum Gribakin and Flambaum (1993), which is proportional to . Values for for all the alkali metals with Yb atoms are given for representative isotopes in Table 3. The non-integer quantum number at dissociation is
where the superscript GF distinguishes the Gribakin-Flambaum value from the (less accurate) first-order WKB value (see section III.3). It should be noted that is a single-valued function of the fractional part of and is independent of its integer part.
Potential energy curves from electronic structure calculations for heavy molecules are typically accurate to at best a few percent. For curves that support 35 to 70 bound states, such as those for the systems considered here, this uncertainty is enough to span more than 1 in . It is thus not possible to predict for these systems from electronic structure calculations alone. An experimental measurement is essential to limit the possible range of .
If the uncertainty in is much greater than 1 and we assume that the possible values of (and hence ) are uniformly distributed over such a range of uncertainty in , we find from Eq. (26) that there is a 50% probability that is in the range , and a 70.5% probability that it is in the range .
Different isotopologues of the same molecule have different reduced masses . Since is proportional to , changing between different isotopes of Yb alters , and hence and , in a very well-defined way, which depends only weakly on the potential well depth. For the case of LiYb, changing the heavy-atom isotope has very little effect on the reduced mass and therefore on . For the heavier alkalis, by contrast, changing the Yb isotope allows the scattering length to be tuned over a wide range. Table 4 summarizes the number of bound states and the amount by which it may be tuned for all the alkali-metal + Yb systems.
Iii Results and Discussion
We have previously calculated resonance positions and widths for the LiYb systems Brue and Hutson (2012), using estimates of obtained from thermalization measurements for LiYb Hara et al. (2011); Ivanov et al. (2011). In the following subsections, we present calculations of resonance positions and widths for two cases representative of the heavier alkali metals: RbYb, where the scattering lengths are approximately known, and CsYb, where the scattering lengths have yet to be measured.
Interactions of RbYb mixtures have been studied by Görlitz and coworkers Baumer et al. (2011); Baumer (2010); Münchow et al. (2011); Münchow (2012). Baumer et al. Baumer et al. (2011); Baumer (2010) measured thermalization rates and density profiles for mixtures of Rb with a variety of Yb isotopes, and interpreted the results in terms of background scattering lengths. In particular, RbYb was found to have an extremely large scattering length, which produced phase separation of the atomic clouds, while RbYb was found to have an extremely small one. Münchow et al. Münchow et al. (2011); Münchow (2012) measured 2-photon photoassociation spectra of high-lying vibrational states of the electronic ground state: for RbYb, 6 states were observed with binding energies between about 300 MHz and 60 GHz Münchow et al. (2011); Münchow (2012), whereas for each of Yb, Yb and Yb, two states were observed with binding energies between 100 and 1500 MHz. Münchow Münchow (2012) fitted the binding energies to a Lennard-Jones potential model and inferred from the mass scaling that the potential supports about 66 bound states for RbYb and RbYb, with one fewer state for lighter Yb isotopes. The presence of a bound state very close to dissociation in RbYb produces its large positive scattering length.
The Lennard-Jones potential reproduces the experimental spectra satisfactorily, but the mass scaling determines only the number of bound states and there is no reason to expect the potential to have the correct well depth, equilibrium distance, or inner turning point. These features are however important in the calculation of resonance widths. We have therefore refitted the binding energies measured by Münchow Münchow (2012), together with the scattering length for RbYb, to obtain a new potential curve based on our CCSD(T) results described above. Our best fit was obtained by multiplying the CCSD(T) potential by a scaling factor and adjusting to 2874.7 , producing a potential that supports 66 bound states for RbYb. We also introduced long-range and coefficients related to by a ratio , with an optimum value bohr for the potential above. Since the resulting long-range potential is valid to shorter distances than the pure potential used for the other systems, the switching function Janssen et al. (2009) was applied between 20 and 30 bohr in this case. It should be noted that adequate fits could also be obtained with one or two additional (or fewer) bound states: increasing by 0.036 and by produces a potential with one extra bound state at the bottom of the well but the high-lying states almost unchanged.
We have carried out coupled-channel calculations for the RbYb systems using the fitted potential with 66 bound states. For the fermionic isotopes Yb and Yb, we neglected couplings due to . The crossings responsible for the resonances for RbYb are shown in Figure 6 and the resonance positions and widths are given in Table 5 for all resonances located below 5000 G. The corresponding results for RbYb are given in Figure 7 and Table 6 for resonances located below 1500 G. A full listing of all resonances below 10000 G is provided as Supplemental Material Sup (). The resonance positions are generally within about 50 G of those obtained by Münchow Münchow (2012) with a Lennard-Jones model of the potential.
The pattern of widths for RbYb closely follows expectations from Eq. (22). Only RbYb has a resonance below 1000 G, and that has a very low width (300 G), in part because of dropoff in at low fields. Nevertheless, resonances with calculated widths as narrow as 0.2 G have been observed as 3-body loss features in Na Knoop et al. (2011), and resonances a few mG wide have been observed in LiNa at fields as high as 2050 G Schuster et al. (2012). RbYb has particularly large widths as measured by (up to 30 mG), but this is simply because is small in this case: the quantity , which is a better measure of the suitability of a resonance for magnetoassociation Julienne et al. (2004); Góral et al. (2004), is not particularly large for this isotopologue. By contrast, RbYb and RbYb, which both have , have resonances up to 25 mG wide. Experimentally, RbYb displays phase separation that will inhibit molecule formation even for low-temperature thermal clouds Baumer et al. (2011), but RbYb does not Baumer (2010), and is a good candidate for magnetoassociation if the high fields in Table 5 can be achieved.
For RbYb, there are no resonances with . This arises mostly because of the lower hyperfine coupling constant for Rb, which both reduces the magnitude of and further reduces the widths through the factor of described following Eq. (24) above. However, there are several resonances predicted below 1500 G, as shown in Table 6, and some of the broader ones (still below 1 mG width) may be suitable for molecule formation. In particular, our best-fit potential predicts a pair of resonances for for RbYb, where the atomic and molecular states just intersect and undergo a double crossing as shown in Fig. 7. The precise positions and widths of these resonances are very sensitive to the potential details, and indeed Münchow’s Lennard-Jones model predicted that the atomic and molecular states just miss each other instead of just crossing Münchow (2012).
Tables 5 and 6 include only resonances driven by for Rb, which conserve . If the Yb isotope has nuclear spin, as for fermionic Yb and Yb, additional resonances can occur at crossings with , driven by for Yb Brue and Hutson (2012). In particular, RbYb has a lower-field and therefore potentially more accessible group of resonances near 1210 G, where the molecular states with and (not shown in Fig. 6) cross the thresholds with and .
All the resonances in Tables 5 and 6 are strongly closed-channel-dominated. This may be quantified using the dimensionless resonance parameter , where . It may be seen that is never greater than 0.2, and approaches such values only when is very large. In some cases can be less than .
Molecule formation by magnetoassociation is usually carried out by preparing the atomic mixture close to a resonance, on the side where the atomic state lies below the molecular state, and then ramping the field over the resonance. However, for narrow resonances in Cs (a few mG wide, at low fields), Mark et al. Mark et al. (2007) found it effective simply to hold the field on resonance for a few milliseconds. Nevertheless, the most efficient molecule production occurs with a field ramp that is slow enough to cross the resonance adiabatically Julienne et al. (2004); Góral et al. (2004); Hodby et al. (2005). Small field inhomogeneities are not a big problem, as they will simply cause different parts of the cloud to cross the resonance at slightly different times. However, field noise is potentially a problem, particularly high-frequency noise that causes nonadiabatic crossings through the resonance. It will therefore be important to design a molecule creation experiment with very careful field control. In this context it is worth noting that Zürn et al. Zürn et al. (2013) have recently carried out radiofrequency spectroscopy on Li molecules at fields around 800 G with a field precision of mG, which is close to 1 part in , while Heo et al. Heo et al. (2012) achieved molecule formation in LiNa, using a resonance 10 mG wide at 745 G, with active feedback stabilization of the current to achieve field noise less than 10 mG Heo et al. (2012).
Cesium possesses several properties that make it favorable compared to the other alkali-metal elements for magnetoassociation with Yb. It has the highest mass of the alkali metals, which leads to greater mass scaling through changing the isotope of the closed-shell atom. Its larger mass also provides a higher density of bound states near threshold and thus offers better chances of resonances at low magnetic field. Additionally, its relatively large nuclear spin allows larger off-diagonal elements. Finally, the effects of are larger for Cs than for most of the other alkali metals.
The CsYb potential shown in Fig. 2 supports 70 bound states for all Yb isotopes, and has a background scattering length bohr for CsYb. However, the electronic structure calculations have a degree of inaccuracy, and a plausible change of % in the well depth would produce a change of in . Since the scattering length depends on the fractional part of , it cannot be predicted from these calculations. However, altering the Yb isotopic mass across its possible range from 168 to 176 changes by about 0.70, so that a wide range of background scattering lengths will be accessible by varying the Yb isotope. We have therefore carried out calculations for CsYb as a function of .
The top panel of Figure 8 shows the resonance positions and widths for CsYb as a function of . The plot would be almost identical for any other Yb isotope (though different isotopes will have different scattering lengths). There are multiple resonances for each value of , which occur when bound states cross the scattering threshold . As increases, the binding energies decrease and the most of the crossings (those with positive ) shift to lower magnetic fields. In the case, the position of one resonance changes from G to G as increases from to bohr. When a state becomes too shallow to cross the lower threshold at all, the corresponding resonance line either disappears through or (in the case of a double crossing, as for ) reaches a maximum where the two crossings coalesce.
The middle panel of Figure 8 shows as a function of . The spikes in near occur because of the in the denominator of Eq. (22). However, the strength of the peak in [Eq. (12)] is actually rather than itself, so this situation does not offer particular advantages for molecule formation. For , the widths vary linearly with as described in Section II.4. A particularly interesting feature of this plot is the spike in the widths near bohr, which is physically significant. As noted above, the bound states for this magnetic sublevel experience a double crossing with the lower threshold in this region; as the two crossings approach one another, decreases and increases as given by Eq. (16). A similar spike occurs in the resonance widths near 167 bohr. The inset shows an expanded view of for the range of from to ; as described in section II.5, there is about a 70% probability that lies in this range for any particular isotope.
iii.3 Choosing promising systems
The Fermi Golden Rule treatment developed above shows that the most important properties leading to large resonance widths are a large magnitude of the background scattering length and the occurrence of “double crossings” where the bound and continuum states have similar (small) magnetic moments. It is instructive to consider the conditions where these two enhancements can occur together.
At zero field, the hyperfine splitting of an alkali-metal atom in a S state is . As a function of magnetic field, the splitting between two states with the same value of (neglecting the nuclear Zeeman term) is
For negative values of , this has a minimum value
at a field
The first-order WKB quantisation formula, expressed in terms of the phase integral of Eq. (25), is
Le Roy and Bernstein Le Roy and Bernstein (1970) showed that this implies that, for a long-range potential with , near-dissociation levels exist at energies
However, Eqs. (31) and (32) do not take account of the Gribakin-Flambaum correction Gribakin and Flambaum (1993), which replaces the in Eq. (31) with , where is zero for deeply bound levels but is at dissociation for a long-range potential. This correction may have a significant effect on the energy of the least-bound level Boisseau et al. (1998, 2000), which is responsible for the Feshbach resonances Li-Yb Brue and Hutson (2012) but is small for the slightly deeper levels that are responsible for the resonances in the heavier Alk-Yb systems. As a result, the near-dissociation levels (except ) actually occur at energies close to
when expressed in terms of from Eq. (27).
Very large values of correspond to near-integer values of , so the condition for a very large value of to coexist with a “double crossing” near is that the dimensionless quantity
should be approximately an integer. The quantity may be interpreted as the vibrational quantum number (relative to threshold) that will just give a double crossing between a molecular state associated with the upper hyperfine level and an atomic state at the lower hyperfine threshold with the same . It depends strongly on the alkali-metal isotope through the nuclear spin and hyperfine splitting, but is only very weakly dependent on the Yb isotope chosen. It is proportional to (through ), but is otherwise completely independent of the interaction potential. Values of slightly smaller than than an integer allow a very large value of to coexist with a double crossing further from , or a large negative value of to coexist with a double crossing near . In general, large negative values of may be more favorable for molecule formation than large positive ones, because negative values will not cause phase separation in condensates.