Feshbach resonances and weakly bound molecular states of boson-boson and boson-fermion NaK pairs

Feshbach resonances and weakly bound molecular states of boson-boson and boson-fermion NaK pairs

Alexandra Viel alexandra.viel@univ-rennes1.fr    Andrea Simoni andrea.simoni@univ-rennes1.fr Institut de Physique de Rennes, UMR 6251, CNRS & Université de Rennes 1, F-35042 Rennes, France
August 20, 2019

We study theoretically magnetically induced Feshbach resonances and near-threshold bound states in isotopic NaK pairs. Our calculations accurately reproduce Feshbach spectroscopy data on NaK and explain the origin of the observed multiplets in the -wave [Phys. Rev. A 85, 051602(R) (2012)]. We apply the model to predict scattering and bound state threshold properties of the boson-boson NaK and NaK systems. We find that the NaK isotopic pair presents broad magnetic Feshbach resonances and favorable ground-state features for producing non-reactive polar molecules by two-photon association. Broad -wave resonances are also predicted for NaK collisions.

I Introduction

Ultracold gases are extraordinary systems to investigate fundamental quantum phenomena in a highly controllable environment leading to a wealth of spectacular experimental and theoretical results. More than a decade ago the experimental production of ultracold mixtures of alkali gases added a new twist to the cold atom field, paving the way towards the study of few- and many-body phenomena absent in a pure homonuclear gas. Few examples include recent experiments with polaronic impurities Schirotzek et al. (2009); Catani et al. (2012); Kohstall et al. (2014), formation of chemically reactive Ni et al. (2008) or non-reactive Takekoshi et al. (2014) ultracold polar molecules, theoretical studies of phase diagrams K. Machida (2006) and of pairing in imbalanced Fermi systems Batrouni et al. (2008); Orso et al. (2010).

In this context, magnetic Feshbach resonances (FR) proved to be a powerful and versatile tool to widely tune few-body interactions Chin et al. (2010), allowing one to explore in a controlled way regimes from the non interacting ideal behavior to strongly interacting systems. A FR also offers the possibility to associate pairs of atoms in weakly bound molecular states using time-dependent magnetic fields Goral et al. (2004). Such molecules have an intrinsic interest due to their long-range nature. Moreover, depending on their spin and spatial structure they can be used as a convenient initial state for producing polar molecules in the ground state via stimulated Raman processes Ni et al. (2008); Takekoshi et al. (2014).

Applications based on resonances require an accurate characterization of the scattering dynamics and of the properties of bound states near the dissociation threshold. Fortunately, theory can predict from a small amount of experimental data the location and the width of magnetic resonances and the relevant molecular state properties. In fact, to date highly quantitative models exist for most alkali isotopic pairs Chin et al. (2010). In this work we focus on NaK mixtures, a system composed of two species that can be individually cooled to ultracold temperatures.

Experiments with Na and the fermionic isotope K are currently being performed at MIT, where heteronuclear FR spectra have been discovered and interpreted based on simplified asymptotic models Park et al. (2012). Magnetic association Wu et al. (2012) and, more recently, transfer to the ro-vibrational ground state of the dimer have also been demonstrated Park et al. (2015a). Accurate Born-Oppenheimer potentials for the ground and the excited states have been built and used to study the adiabatic transfer of a Feshbach molecule to the ground ro-vibrational state Schulze et al. (2013). However, a comprehensive account of scattering and bound state features for this boson-fermion mixture in the electronic ground state is still lacking. In addition, near-threshold properties for the boson-boson pairs NaK and NaK are still unknown. This work aims therefore on one side at giving a more complete picture of the Feshbach physics of NaK, and on the other at providing theoretical predictions for the two purely bosonic pairs, for which experiments are on the way in few groups worldwide. We study both scattering and bound states for an extensive set of hyperfine states, and discuss the experimental implications of our results for interaction control and molecule production.

The paper is organized as follows. Sec. II introduces our theoretical approach and the Born-Oppenheimer potentials optimization procedure based on known experimental data. Sec. III presents results and discussions for the boson-fermion and the boson-boson pairs. Few experimental implications of our results are discussed. A conclusion IV ends this work.

Ii Methods

ii.1 Computational approach

We solve the time-independent Schrödinger equation for bound and scattering states in the well-known framework of the close-coupling approach to molecular dynamics. Briefly, in our approach a basis of Hund’s case (b) molecular states is used to expand the total wavefunction at each value of the interatomic distance . In Hund’s case (b) the spin state of the dimer is represented as with and the electronic and nuclear spin angular momentum, respectively Lefebvre-Brion and Field (2004). The description of the diatomic is completed by assigning the and quantum numbers relative to the orbital angular momentum of the atoms about their center of mass. In this basis the electrostatic Born-Oppenheimer potentials are represented by diagonal matrices with entries the singlet and triplet molecular symmetry adiabatic potential energy curves (see below).

The other interactions relevant for the ultracold regime included in this work are the atomic hyperfine interaction , the anisotropic spin-spin coupling , and the Zeeman interaction energy with the external magnetic field . Here and are the electron and nuclear spin of the individual atoms and the respective ground-state atomic hyperfine constant for atoms and , and the electron and nuclear gyromagnetic ratios, and the fine structure constant and the Bohr magneton respectively. Such atomic interactions can be expressed in the molecular Hund’s case (b) computational basis using standard methods of angular momentum algebra (see e.g. Tiesinga et al. (1996)).

Bound state calculations are performed using a variable grid approach allowing one to represent over a sufficiently small number of points the rapid oscillations at short range and the long range tail of the dimer wave function Tiesinga et al. (1998). For scattering calculations we use the variable-step Gordon propagation or the renormalized Numerov algorithm to efficiently solve the coupled-channel Schrödinger equation Mies (1973). Once the solution has been built in the computational basis, a frame transformation is applied to express the solution in an asymptotically diagonal representation before using a standard matching procedure to extract the reactance matrix  Mies (1973).

ii.2 Optimization of the molecular potential

We adopt for this work the and the electronic ground state potential of the NaK molecule proposed in Ref. Gerdes et al. (2008). A minor modification is made to ensure a continuous and continuously differentiable expression by fine tuning the parameters given in Gerdes et al. (2008). First, starting from the asymptotic long range expressions we numerically enforce continuity at the switching points and . In addition, smooth damping functions are preferred to the published abrupt change of the potential between the short range repulsive part, the inner well and the long range part. The resulting continuously differentiable expression is given by


where the parameterized functions , and and the switching points and are taken from the work of Tiemann and coworkers Gerdes et al. (2008). Note there is a typo in the Tab. 1 of Ref. Gerdes et al. (2008) where should be replaced by for the constant value. A value of has been found to be suitable for the control parameter of the damping function for the two electronic states. With an infinite value the original potential curves are recovered.

We are now in the position to perform close-coupling calculations for different initial channels. We will conventionally label each asymptotic channel by specifying the separated-atom NaK state with () to which the latter adiabatically correlate as tends to zero. FR have been experimentally observed in the collision between Na in and K in , with and  Park et al. (2012). We compute the -wave scattering length in the relevant channels and search for resonances as poles of as a function of magnetic field. We also search for -wave resonances by locating maxima in the partial -wave elastic cross sections at a fixed collision energy of K. The observed resonance locations are not reproduced accurately by the original potentials. However, a simple modification consisting in introducing the correction terms


near the bottom of the and electronic potentials enables us to model the experimentally measured spectra. More specifically, using  ,   and   a Levenberg-Marquardt algorithm is applied to determine two optimal parameters


In the fitting procedure, we included three wave resonances at 78.3 G, 88.2 G and 81.6 G and an average position of the -wave resonances appearing as a multiplet around 19.19 G Park et al. (2012). For the -wave calculation we initially neglect the spin-coupling term responsible of the multiplet structure thus avoiding possible incorrect labeling of the closely spaced -wave resonances.

After optimization, the resonances positions are theoretically reproduced with a reduced = 0.57. On a more physical ground, the artificial control parameters are usually translated in corresponding singlet and triplet scattering length (see Tab. 1). Our optimized boson-fermion model can also be used for predicting the properties of boson-boson isotopes. Within the Born-Oppenheimer approximation, which is expected to be accurate for all but for the lightest species Julienne and Hutson (2014), it is sufficient to change the value of the reduced mass in the Hamiltonian to compute . Note however that if the number of bound states in our nominal potentials turned out to be incorrect, the predicted (and hence the results of the coupled model) will be systematically shifted.

NaK 255 -84
NaK 63 -838
NaK -3.65 267
Table 1: Singlet and triplet scattering lengths obtained according to our optimized potentials for different NaK isotopic pairs.

We begin our analysis with a detailed discussion of the resonances for the NaK boson-fermion mixture in the section III.1 here after.

NaK channel
+ 78.3 77.78 72.23 -0.668 0.682 -619.3 -1.2 10 -5.5
88.2 88.68 79.82 137. 9.92 -8.8
+ 81.6 81.42 81.18 -8.39 10 0.0120 -552.7 -1.1 10 -0.23
89.8 89.82 83.61 -48.4 0.517 -6.2
108.6 108.91 96.86 141. 12.3 -16.0
+ 96.5 96.39 95.75 -4.84 10 0.0205 -496.4 -1.0 10 -0.6
106.9 106.54 98.83 -90.9 0.426 -7.5
138 136.82 110.53 142. 14.58 -26.2
+ 116.9 117.19 115.62 -3.46 10 0.0283 -443.1 -8.5 10 -1.2
129.5 130.36 119.85 -120. 0.379 -9.8
175 177.44 135.35 143. 17.3 -41.7
Table 2: List of the -wave FR for NaK. Our theoretical resonance positions are compared with the experimental data from Ref. Park et al. (2012). The locations where the scattering length vanishes , the effective range on resonance , the resonance strength parameter and the fitting parameters , , and defined in Eq. (4) are reported in columns 4-9. See text for more details.

Iii Results

iii.1 NaK

We now perform extensive close-coupling calculations with the optimized potentials described in Sec. II.2. Table 2 summarizes our findings for the -wave for magnetic fields up to 1000 G for different hyperfine levels. We report in the table the positions of the poles observed in the calculated -wave scattering length as well as the nearby zero-crossing field where vanishes. The experimental data of Ref. Park et al. (2012) are also reproduced in the table.

The good quality of the theoretical model after optimization is confirmed by the very good agreement (below  G) illustrated in the table for all narrow features experimentally observed in different hyperfine combinations. A larger discrepancy of G is found on broader resonances which may however be more difficult to locate experimentally with accuracy. No additional -wave features are found with respect to the experiment.

In order to extract the magnetic width , the scattering length obtained for each incoming channel is fitted according to a formula appropriate for overlapping resonances Jachmyski and Julienne (2013),


in which a linear variation of as a function of the magnetic field is assumed. Eq. (4) reduces to the well-known standard expression if resonances are isolated and the background scattering length is locally constant. The fitting of the scattering length with Eq. (4) has been carried out for magnetic fields spanning a  G region around each resonance. For overlapping cases, the largest was taken to define the fitting interval. We only report in the table the corresponding , and parameters for which Eq. (4) reproduces the numerical data to an accuracy better than 5% in either the relative error or in the absolute error measured in units of van der Waals radius


The latter quantity ( for NaK) represents the natural value of for scattering in a van der Waals potential Chin et al. (2010).

In addition, for each resonance we extract the effective range defined through the low energy expansion of the elastic reactance matrix element


computed at the resonance value by a linear fit of as a function of collision energy in an appropriate energy domain. We introduce a corresponding intrinsic resonance length to characterize the resonance strength defined as


In the case of isolated resonances the resonance length can be expressed in terms of scattering background and resonance parameters and of the magnetic moment difference between the open and the closed channel as  Chin et al. (2010). As remarked in the supplemental material of Ref. Roy et al. (2013) the Eq. (7) also holds for overlapping resonances to the extent that they are not directly interacting. According to the relative value of the length being much larger (resp. much smaller) than the van der Waals length, resonances are classified as being open channel (resp. closed channel) dominated. The resonance strength parameter defined as


and listed in Tab. 2 is therefore a useful dimensionless indicator of the resonance character Chin et al. (2010).

It is interesting to compare the present close-coupling data with the results of an asymptotic bound state model used for interpretation in Ref. Park et al. (2012) (not shown in the table). Considering the simplicity of the latter, the agreement is good as far as the resonance position and the width of the largest features is concerned. The most serious discrepancy bears on the width of the narrow -wave resonances, which are underestimated by more than one order of magnitude by the asymptotic model.

Feshbach molecules formed by magnetic association can be a good starting point to form molecules in the absolute ro-vibrational ground state using two-photon transfer schemes. A first constraint to be taken into account to achieve such a transfer is that according to electric dipole selection rules only Feshbach molecular states of significant singlet character can be coupled to ground-state singlet molecules if singlet excited states are used as a bridge. If the initial Feshbach molecule turns out to have mostly triplet character one can use excited electronic states of mixed singlet-triplet character as a bridge, an approach suggested for NaK in Schulze et al. (2013) and successfully recently adopted to form NaK in the absolute ground state by a STIRAP two-photon process Park et al. (2015b, a). Moreover, the radial overlap between the excited intermediate state and both the initial and the target ground state molecule must be significant. To gain more insight into the resonance nature and to get a hint at the expected efficiency of two-photon processes we perform bound state coupled-channel calculations. A detailed analysis having already been performed in Schulze et al. (2013) for NaK, here we just stress the main elements for the sake of comparison with the following analysis of the boson-boson pairs.

Figure 1: (color online) Scattering length as a function of the magnetic field for Na + K -wave collisions (top panel). Corresponding molecular energy levels as a function of are shown in the lower panel. The density code denotes the average spin of the molecule.

We depict for instance in Fig. 1 the scattering length and the evolution of the molecular levels near the broadest resonance in the hyperfine absolute ground state which has been successfully used as starting point for STIRAP association Park et al. (2015a). The corresponding average electronic spin is also shown as a function of internuclear distance as color code in the lower panel. The nearly pure triplet character of this molecular state is in principle unfavorable for the production of ground state singlet molecules through the excited singlet manifold. Inspection of lower panel of Fig. 1 might suggest working closer to resonance to increase the single character as the molecular state mixes with the scattering continuum. However, as already noted in Ref. Schulze et al. (2013) such admixture comes to the price of a delocalization of the wavefunction at larger distances, and thus to a decreased overlap with the intermediate excited state. Such strong triplet character is a common feature of the molecular states associated to all broadest resonances in Tab. 2, which is in fact to a good approximation a common molecular state in the triplet potential with a different projection of the total hyperfine angular momentum . In conclusion, use of bridge spin-orbit coupled states to help enhance the transfer efficiency seems necessary for the boson-fermion pair Schulze et al. (2013). We will show below that the situation is significantly different for the boson-boson mixtures.

In addition to the observed - and -wave features, additional -wave resonances are also predicted by our model. We compute the elastic collision rate up to  G and present a restricted magnetic field range in the upper panel of Fig. 2. The lower panel depicts the energy levels of the molecular states responsible for each resonance. Resonance features are detected by local maxima in the elastic collision rate as well as in the inelastic probabilities. The position of these maxima agree to better than 0.02 G for all except the two features around 21.9 G in the channel for which the differences are 0.05 G. It turns out to be easier to extract the location of the resonances from the inelastic probabilities. These positions are summarized in Tab. 3 together with the position of the local maxima in the elastic collision rate when no inelastic process are present.

Figure 2: (color online) Elastic rate computed at a collision energy of K as a function of the magnetic field for Na + K -wave collisions (top panel). Corresponding molecular energy levels as a function of are shown in the lower panel. Curves are resolved according to the projection of total angular momentum.
NaK channel
-9/2 6.51 6.35
-5/2 6.57 6.41
-7/2 6.63 6.47
-7/2 6.85 6.68
-5/2 6.86
-5/2 7.21
-7/2 18.01
-9/2 18.36
-5/2 19.32 19.12
-9/2 19.39 19.18
-7/2 19.47 19.27
-5/2 20.72
-7/2 20.82
-5/2 22.43
-5/2 7.55
-7/2 7.62
-7/2 7.88 7.32
-5/2 7.90 7.54
-3/2 7.90
-3/2 8.27
-5/2 8.33
-3/2 8.87
-7/2 20.09
-5/2 21.76
-7/2 21.91
-5/2 23.49 23.19
-3/2 23.53
-7/2 23.60 23.29
-3/2 25.33
-5/2 25.61
-3/2 28.01
NaK channel
-5/2 8.87
-5/2 9.34 9.23
-3/2 9.33
-3/2 9.82 9.60
-5/2 9.88
-1/2 9.89
-1/2 10.46
-3/2 10.61
-1/2 11.50
-5/2 24.80
-5/2 27.20
-3/2 27.28
-3/2 29.61 29.19
-1/2 29.88 29.45
-5/2 29.93 29.52
-1/2 32.57
-3/2 33.04
-1/2 36.97
-3/2 11.35
-1/2 11.95
-3/2 12.12
-1/2 13.04 12.51
-3/2 13.20
1/2 13.20 12.68
1/2 14.30
-1/2 14.55
1/2 16.31
-3/2 32.00
-3/2 35.74
-1/2 36.10
-1/2 39.87 39.39
1/2 40.37 39.86
-3/2 40.42
1/2 45.18
-1/2 45.85
1/2 53.21
Table 3: Theoretical magnetic field locations of -wave FR for NaK for projection of total angular momentum in different hyperfine atomic channels. Calculations are performed at collision energy of K. Available experimental values are also reported Park et al. (2012).

The -wave multiplets observed in Ref. Park et al. (2012) and reproduced in Tab. 3 are at first sight surprising since the spin-spin interaction typically gives rise to doublets Ticknor et al. (2004). The nature and multiplicity of such magnetic spectrum can be rationalized starting from a picture where the spin interaction is at first neglected. In this situation the total internal spin projection is an exactly conserved quantum number. Let us consider for definiteness the case of two free atoms with . Let us moreover restrict ourselves to , since the contributions are vanishingly small at the present very low collision energies due to centrifugal repulsion. The projection of the total angular momentum, which is strictly conserved, can then only take values , , and . Within this restricted model and fixing , one can build six molecular states with projections that are degenerate, since both and are strictly conserved in the absence of anisotropic spin-spin and of the Zeeman interaction Tiesinga et al. (1996).

If and the spin-spin interaction does not vanish, conservation of total angular momentum guarantees that the six molecular states will give rise to one triply degenerate level with corresponding to , one doubly degenerate level with and with and one singly degenerate level with and with . The energy differences between the three groups is small due to the weakness of the spin-spin interaction. However the mixing of the different within each block of given can be strong.

Figure 3: (color online) Average value of the projection of the total internal spin as a function of for bound states with total angular momentum projection . The right panel presents a blowup for magnetic fields in the 6.4 to 7.2 G range.

In fact, as shown in Fig. 3, even in zero field the values of , and hence of , is undefined with the exception of the bound level with which being essentially isolated retains to high accuracy its and character. Total rotational invariance and thus degeneracy with respect to is broken by the magnetic field which introduces a coupling between states with different quantum number in subspaces of given . As the magnetic field increases, the computed averaged projection in the direction converges for each of the six molecular states toward the large limit, that is three levels with , two with and one with .

This limit is however not fully reached when the bound state energies cross threshold giving rise to resonance. We conclude that for the magnetic field values of relevance for the resonances the spin-spin perturbation is comparable with the Zeeman splitting. A treatment of the spin-spin interaction to first order perturbation theory in subspaces spanned by states of given and is thus not appropriate in the present case. As a consequence, one cannot reach the usual conclusion that states with same are degenerate, such approximate degeneracy being lifted by the perturbing effect of nearby states with different .

Generalizing the argument above to other hyperfine combinations, we expect a multiplet of six resonances at low followed by a multiplet with eight resonances for the channel. The channel present an eight-fold multiplet at lower and a nine-fold multiplet for larger values. The two remaining channels, and present two nine-fold multiplets each. Some resonances are only weakly coupled and do not result in marked peaks in the elastic collision rate. More precisely, two molecular state crossings do not give rise to detectable features in the numerical elastic rate for the channel (at 8.87 G and at 24.80 G). For the channel, five molecular state crossings have no detectable effects on the numerical rate (at 11.35, 11.95, 32.00 35.74 and 36.10 G). These features are however clearly seen in the inelastic probabilties.

We now propose an assignment of -wave resonances in the MIT experiment. First of all, one may notice that the experimental spectrum only entails a subset of the predicted multiplets. Some features observed in the theoretical model (for instance, the pair near 6.85 G or the one near 7.90 G) are strong but nearly overlapping, such that one can reasonably assume that they have not been resolved in the experiment. In such cases, for the assignment we only retain the strongest of the two features in the theoretical spectrum. Next, we affect the strongest elastic theoretical features to the experimental positions under the condition that the resulting splitting agrees with the experimental one. The procedure is succesfull in all cases, with the exception of the low spectrum in the channel. Note however that the error given by MIT is relatively large for the 9.60 G resonance.

Few theoretically weak resonances do not have an experimental counterpart, most likely since the corresponding experimental signature has been missed. The quality of our assignment, yet non univocal, strongly suggests that the dominant anisotropic interaction arises from the electronic spins Mies et al. (1996).

iii.2 NaK

We continue our discussion with the most abundant potassium isotope, K, a species for which cooling and Bose-Einstein condensation has traditionally proved to be difficult, yet finally achieved by different techniques Landini et al. (2012); Salomon et al. (2014). We provide results for a series of hyperfine states. Our data can therefore be useful in order to interpret collision data in a pure spin or in the case of partial polarization of the sample. To this aim, calculations of the -wave scattering length are performed for different values of the conserved projection of total angular momentum, . Tab. 4 summarizes the -wave resonances found for NaK for a magnetic field up to 1000 G. We report the positions of the 21 poles observed in the scattering length, , as well as the 17 zero-crossing field, . Note that no zero-crossing exists for Na + K collisions and that a single one at  G is present for the Na K channel.

NaK channel
442.51 405.02 123. 4.00 -114.8 3.72 10 -36.9
536.00 533.72 -174. 0.316 -2.27
35.16 11.51 62.3 1.18
356.21 355.45 -1.81 10 0.0520
498.23 466.24 118. 3.35
606.51 603.13 -88.2 0.430
33.60 19.39 -1.60 10 0.0586 258. 8.2
107.97 39.55 117. 3.22 -569. -62.
116.91 -969. 0.0912 -10760 0.18
116.98 -3.85 10 0.00264
422.51 421.86 -2.02 10 0.0469
566.06 539.80 111. 2.74
688.63 685.97 -170. 0.320
56.31 54.70 -3.04. 10 0.00334
158.18 141. 14.1
498.48 498.22 -4.93 10 0.0201
648.26 627.58 102. 2.20
2.01 75.71 142. 15.2 -183.7 1.86 10 63.2
241.40 107. 2.47 -62.6 53.0
357.96 357.10 -9.66 10 0.0104
657.18 599.44 132. 6.30
Table 4: Same as Tab. 2 but for the isotopic pair NaK. No experimental data are available for this system.

Since we include only -waves, possible narrow spin-spin resonances due for instance to wave couplings are not reproduced by the model. The incoming state for the collision is systematically taken to be the lowest energy state with the given at magnetic field intensity . Note that in general this state may decay by inelastic spin-spin processes if states were included in the basis. These processes will however tend to be slow except very close to resonance and are neglected for computational simplicity. Figures 4, 5 and 6 provide the scattering length as well as the molecular energies for three of the nine studied channels. Numerical data are available upon request to the authors.

Figure 4: (color online) Scattering length as a function of the magnetic field for Na + K -wave collisions (top panel). Corresponding molecular energy levels as a function of are shown in the lower panel. The density code denotes the average spin of the molecule.

Figure 5: (color online) Same as Fig. 4 but for for Na + K collisions.

Figure 6: (color online) Same as Fig. 4 but for for Na + K collisions. The inset shows the behavior of at low field on a larger vertical scale.

As in the boson-fermion case we choose to parameterize the field-dependent -wave scattering length by the unique expression Eq. (4) over a magnetic field range of around the resonance and compute the resonance length in order to assess the resonance strength. For overlapping resonances a unique and values are given, whereas for isolated resonances, we give a local and , when the latter is non vanishing. We achieve the sufficient required accuracy (below 5% as for the K isotope) for three of the nine initial channels considered. Many combinations are found to be not well described by Eq. (4), in particular in the presence of energetically degenerate channels that give rise to characteristic threshold singularities Newton (1959).

Both open and closed channel dominated resonances are available in suitable hyperfine combinations. A particularly interesting feature is the one near 442 G for collisions in the absolute hyperfine ground state , which is strictly stable under two-body inelastic collisions and open-channel dominated; See also Figure 4. Its large magnetic width  G should allow one to tune to desired values with high accuracy and thus possibly to explore the quantum phases predicted in free space and under optical-lattice confinement for a variety of geometries Duan et al. (2003); Altman et al. (2003); García-March et al. (2014). Also note that for vanishing the scattering length is negative and very large in magnitude , a feature related to the presence of a virtual state with positive energy. Variation with of the position of the virtual state results in the rapid variation of with magnetic field observed for small .

The known FR for Na + Na collisions in the ground state are located at large fields  G Laue et al. (2002) in a region where and present regular non resonant behavior 111For reference, and .. Tuning of the interspecies scattering length can be used to increase the cross section for sympathetic cooling, for instance to cool K by thermal contact with ultracold Na. A comparison of Fig. 4 with the Fig.4 of Ref. D’Errico et al. (2007) shows that at the field  G at which K has been condensed Roati et al. (2007) the is slightly negative. A double BEC of sodium and potassium will thus be miscible and stable against collapse Riboli and Modugno (2002). Moreover, if the double condensate is adiabatically loaded in an optical lattice the attractive character of the NaK effective interaction will favor the loading of Na and K pairs at the lattice cells. This should be an advantageous starting point to associate Feshbach molecules and thus implement STIRAP schemes to form ultracold molecules in the absolute ground state.

Indeed, as compared to the boson-fermion case of Sec. III.1 one can verify from the given by the color code in the lower panel of Fig. 4 that the situation is here favorable, since the molecule presents hyperfine-induced singlet-triplet mixing even far from dissociation. Beyond the average spin character, we also represent in Fig. 7 the detail of the singlet and triplet components of the coupled wave-function, defined as and with the projectors on the and subspaces, respectively.

Figure 7: (color online) The (left panel) and (right panel) electron spin components in arbitraty units of last-below-threshold multichannel wavefunction with as a function of the magnetic field for the NaK dimer.

Interestingly, the amplitude reaches its maximum right before the resonance, at  G. Most importantly, Fig. 7 shows that maintains a short-range character with maximum amplitude for .

Such short-range character is confirmed quantitatively in Fig. 8 by calculations of the average distances .

Figure 8: Average distance and partial singlet/triplet norms (defined in the text) of the last-below-threshold multichannel wavefunction with as a function of the magnetic field for the NaK dimer.

Moreover, the partial norms depicted in the figure show that the singlet admixture is significant in the region of interest. Analyses of the electronic excited state structure of NaK and of the corresponding Frank-Condon factor for transfer of the Feshbach molecule to the excited state is beyond the scope of this work. However Ref. Schulze et al. (2013) finds relatively favorable Frank-Condon factors in the case of Bose-Fermi Feshbach molecules, that present similar spatial extent as the present bosonic ones but with significantly smaller singlet component. We expect therefore that suitable excited states can be found to implement an efficient two-photon transfer in the present case.

Let us now consider collisions for atoms in the first excited hyperfine level reproduced in Fig. 5. Similar to the case of the absolute ground state, a large negative scattering length rapidly varying with is predicted at low magnetic fields. A point of non-analiticity is expected at a magnetic field  G as the and channels become degenerate. It is interesting to observe here that the expected cusp in the elastic scattering matrix element is accompanied by poles in occurring right before (after) the degeneracy point in the (the ) channel a peculiar effect stemming from the interplay beetween channel degeneracy and Feshbach physics; see Fig. 5 and Tab. 4.

The case of collisions is shown in Fig. 6 and is particularly relevant for the applications since is the lowest magnetically trappable atomic state of K and Na at low . Moreover, below 259 G the NaK combination is stable under -wave collisions since it is the lowest hyperfine state with . Note that due to the presence of a Feshbach resonance at very low magnetic fields G the scattering zero-field length is negative and very large in magnitude (see inset of Fig. 6). Bose-Einstein condensation has been achieved in this hyperfine level using magnetic tuning of to suitable values Landini et al. (2012); Salomon et al. (2014). Interestingly, Fig. 5 of Ref. D’Errico et al. (2007) shows that for magnetic field between the two homonuclear K resonances located in at about 33 and 163 G the is positive, thus ensuring the stability of a K condensate. In the same magnetic field region varies from being large and positive to large and negative, allowing one to explore the phase diagram of a quantum degenerate NaK mixture as a function of the mutual interaction strength.

To conclude our analysis for this isotope, we provide in Tab. 5 the spectrum of -wave resonances, limiting ourselves to the absolute ground state.

NaK channel
1 242.83
1 260.19
1 354.02
1 370.58
1 395.53
1 462.21
1 491.27
1 529.67
NaK channel
2 242.68
2 353.95
2 370.82
2 438.27
2 461.84
2 491.68
3 354.01
3 437.99
3 462.21
Table 5: Theoretical magnetic field locations of -wave FR for NaK for projection of total angular momentum in different hyperfine atomic channels. Calculations are performed at collision energy of K.

As in the case of the boson-fermion mixtures Wu et al. (2012) such resonances can be experimentally observable even at ultracold temperatures. Fig. 9 shows the elastic collision rate for different projections presenting a rich spectrum with nearby peaks of multiplicities three, two and one. Closer inspection shows that triply degenerate peaks are the usual doublets Ticknor et al. (2004), with the peaks arising from spin-spin induced mixing of states being nearly degenerate and slightly shifted with respect to the peak. Larger multiplicities like the ones in Fig. 2 are not observed here since -wave resonances occur at larger magnetic fields; See the discussion in Sec. III.1. According to the value of in Fig. 9, doubly degenerate components arise from coupling to states with or to states with . Finally, all singly degenerate levels in the figure are due to coupling with .

Figure 9: (color online) Elastic collision rate computed at a collision energy of K as a function of the magnetic field for Na + K -wave collisions. The dots materialize the occurence of maxima in the rate for each projection .

iii.3 NaK

We now provide numerical data for the other bosonic pair NaK. Let us first recall that K has been brought to Bose-Einstein condensation using Rb as a coolant or more recently by direct evaporation Kishimoto et al. (2009). Resonances exist for collisions in different hyperfine states with magnetic widths of several Gauss; see Tab. 6.

NaK channel
20.90 20.90 -6.55 10 1.56 10 334.80 -1.03 10 3.57 10
51.23 51.30 -2.76 10 3.52 10 7.10 10
73.35 77.97 85.2 1.60 4.59
470.08 476.41 104. 2.25 6.32
531.59 532.16 -458. 1.68 10 5.63 10
(-235.65) (6.89 10)
33.26 33.26 -6.18 10 1.66 10
35.53 35.53 -3.70 10 2.76 10
66.48 66.61 -1.58 10 5.93 10
87.53 90.94 58.1 1.12
165.58 165.60 -4.72 10 2.16 10
453.37 453.37 -1.03 10 9.88 10
499.41 506.39 108. 2.50
566.30 567.17 -244. 2.60 10
35.05 35.05 -5.67 10 1.80 10 246.1 9.71 10 8.08 10
63.46 63.48 -2.87 10 3.55 10
72.53 72.53 -8.85 10 1.16 10
106.20 107.69 -77.5 4.51 10
183.36 183.36 -1.25 10 8.17 10
370.11 370.11 -1.20 10 8.51 10
481.53 481.53 -1.34 10 7.62 10
531.87 539.42 111. 2.70
604.62 605.36 -309. 2.23 10
66.97 66.98 -8.54 10 1.20 10
129.37 129.60 -1.98 10 4.81 10
149.33 149.32 -2.68 10 3.82 10
156.22 156.22 -1.16 10 8.84 10
209.92 209.93 -5.51 10 1.85 10
391.25 391.25 -1.26 10 8.13 10
512.63 512.63 -3.84 10 2.66 10
567.79 575.74 113. 2.82
137.27 137.27 -4.13 10 2.48 10 212.75 0 -2.43 10
146.65 146.65 -9.73 10 1.05 10
245.19 252.66 91.4 1.77
500.76 500.82 -9.51 10 1.06 10
601.15 606.56 92.9 1.81
Table 6: Same as Tab. 4 but for NaK.

Such broad resonances are essentially open-channel dominated, with resonance strength of . Several closed channel dominated features are also readily available in each hyperfine channel we studied. A distinctive feature of NaK is the large and positive for all the hyperfine combinations. The parameterization Eq. (4) for overlapping resonances is used for the absolute ground state, where it is found to be accurate only if an artificial pole is added in Eq. (4) at negative . Such a pole mimicks the effect of a virtual state, i.e. a quasi-bound state located at positive energy and that would give a resonance at negative values of . The position obtained through the fitting procedure is given in Table 6 in parenthesis to distinguish from physical poles of . The corresponding scattering length is given in the top panel of Fig. 10.

Figure 10: (color online) Scattering length as a function of the magnetic field for Na + K collisions (top panel) in the -wave. Corresponding molecular energy levels as a function of are shown in the lower panel. The density code denotes the average spin of the molecule.

Both K and Na homonuclear resonances in are narrow and quite sparse. Combination of the present and the magnetic spectra in Refs D’Errico et al. (2007) and Laue et al. (2002) for K and Na respectively shows that homonuclear and heteronuclear resonances take place at well separated locations. Note that the large for NaK and the nonresonant values of order for both Na and K imply that two Bose-Einstein condensates will tend to phase separate. However, the heteronuclear resonances can be used to reduce or even change the sign of , such to favor miscibility and eventually the realization of overlapping quantum gases of Na and K in free space or in optical lattices.

Let us now discuss magnetic association of Na and K atoms when they are prepared in the respective ground hyperfine levels. A calculation of the quantum average depicted as density code in the lower panel of Fig. 10 readily shows that resonances arise from states with dominant triplet character. Note that the large background scattering length implies the existence of a molecular level close to the dissociation threshold; See lower panel of Fig. 10. Let us consider performing magnetic association near the two broadest FR. Based on our data, three routes can be envisioned, yet presenting drawbacks.

If molecules are formed at the 73 G FR and molecular curve crossings are swept through diabatically, one stays in the “background” weakliest bound molecular level. Unfortunately, as shown in Fig. 11 and 12, the state has long-range character with .

Figure 11: (color online) The (left panel) and (right panel) electron spin components using the same arbitrary units of fig 7 of the multichannel wavefunction with as a function of the magnetic field for the NaK dimer. The molecule is created at 73.35 G and the molecular state is followed diabatically with decreasing .

Figure 12: Average distance and partial singlet/triplet norms (defined in the text) of the last-below-threshold multichannel wavefunction with as a function of the magnetic field for the NaK dimer. The molecular state is followed diabatically as for Fig. 11.

Therefore, in spite of the sufficient singlet character predicted in Fig. 12 poor overlap is expected with the excited molecular states. Note that since quantum numbers of this background state are essentially atomic ones or Hund’s case (e) the projections and on the Hund’s case (b) spin-coupled basis in Fig. 11 have virtually identical spatial profiles.

An alternative route consists in following adiabatically the entrance state through the first avoided crossing near 50 G. As shown in the lower panel of Fig. 10 this leads however to the formation of a molecule with relatively poor singlet admixture.

If one uses the broad resonance at 470 G as an entrance gate, a long magnetic field sweep down to  G would be needed before a small is attained, as it can be inferred from Fig. 13 and the main panel of Fig. 14.

Figure 13: (color online) The (left panel) and (right panel) electron spin components using the same arbitrary units of fig 7 of the multichannel wavefunction with as a function of the magnetic field for the NaK dimer. The molecule is created at 470 G and the molecular state is followed adiabatically with decreasing .

Figure 14: Average distance and partial singlet/triplet norms (defined in the text) of the last-below-threshold multichannel wavefunction with as a function of the magnetic field for the NaK dimer. The molecular state is followed adiabatically as for Fig. 13.

However, the inset in the latter figure indicates that molecule shrinking also corresponds to a drop in the singlet character, thus requiring a compromise to be found. In order to draw firmer conclusions a detailed analysis of the excited states will be needed.

Finally, as illustrated in Fig. 15 our model predicts a series of -wave Feshbach resonances at both weak and strong magnetic fields in the absolute ground state.

Figure 15: (color online) Elastic rate computed at a collision energy of K as a function of the magnetic field for Na + K -wave collisions. The color dots above each peaks materialize the occurence of maxima in the rate for each projection .

Tab. 7 confirms as expected that multiplet splittings at small are “anomalous” in the same sense as for boson-fermion pair, whereas like in NaK they follow standard patterns at large .

NaK channel
1 17.82
1 20.72
1 31.47
1 40.94
1 133.49
1 377.46
1 386.90
1 418.20
1 426.43
1 438.89
1 466.48
1 478.98
1 495.58
NaK channel
2 17.79
2 20.47
2 41.89
2 377.84
2 417.19
2 427.37
2 453.54
2 465.11
2 480.30
3 17.84
3 418.19
3 452.11
3 466.47
Table 7: Same as Tab. 5 but for NaK.

Experimental observation of the corresponding magnetic spectra would provide a valuable piece of information to confirm the accuracy of our model for collision in this boson-boson mixture.

Iv Conclusions

We have presented an extensive compendium of the ground state scattering properties of isotopic NaK mixtures in an external magnetic field. Our results complement existing theory and experimental data on the boson-fermion pair NaK. The Feshbach resonance locations and strengths we predict for the boson-boson pairs should be of major interest for experiments in which control of the atom-atom interaction is a requirement. Our spin-resolved analysis of Feshbach molecules also provides an important piece of information for designing magnetoassociation and two-photon transfer scheme of Feshbach molecules to the absolute ro-vibrational ground state.


This work is supported by the Agence Nationale de la Recherche (Contract COLORI No. ANR-12-BS04-0020-01).


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