Beyond Born-Oppenheimer approximation in ultracold atomic collisions

We report on deviations beyond the Born-Oppenheimer approximation in the potassium interatomic potentials. Identifying three up-to-now unknown d-wave Feshbach resonances, we significantly improve the understanding of the K inter-atomic potentials. Combining these observations with the most recent data on known interand intra-isotope Feshbach resonances, we show that Born-Oppenheimer corrections can be determined from atomic collisional properties alone and that significant differences between the homoand heteronuclear case appear.


I. INTRODUCTION
In quantum chemistry and molecular physics, the assumption that the electronic and nuclear motions can be separately treated is well justified by the three orders of magnitude separating the proton and the electron mass. The nuclei are considered as fixed objects at relative distance R when solving the eigenvalue problem of the electron motion resulting in a R-dependent electronic energy which is taken as the potential for the nuclear motion. This approximation leads to large simplifications when solving the Schrödinger equation for molecules and is named Born-Oppenheimer Approximation (BOA) [1]. The BOA is extremely powerful in matching theoretical predictions and spectroscopic results, in particular concerning the understanding of diatomic molecules. One major aspect within the BOA is that the same interatomic potential (BO-potential) is used for different isotopes by simply rescaling the nuclear motion according to the reduced molecular mass. Deviations from this assumption lead to perturbative corrections to the BOA on the order of the electron-to-proton mass ratio. The isotopic dependence of corrections has been discussed in many papers, see for example [2][3][4]. These deviations from the BO-potential approach correspond to shifts in energy levels on the order of ∆E/E ≈ 10 −4 or less and they have been observed in spectroscopy experiments like [5][6][7] and in the dissociation energy of different isotope combinations of hydrogen diatomic molecules [8]. Effects of the corrections to the BOA are much weaker at the collisional threshold of atom pairs as the long range behaviour of the inter-atomic potential is weakly affected by short range variations. Recent developments in molecule cooling and molecule association from ultracold atoms have considerably increased the experimental resolution giving access to study effects beyond the BOA. Corrections to the triplet and singlet scattering length are indeed predicted to be on the order of few * Electronic address: tiemann@iqo.uni-hannover.de † Electronic address: silke.ospelkaus@iqo.uni-hannover.de tenths of the Bohr radius a 0 [7] and are practically undetectable. However, Feshbach resonances are an effective passkey as they carry important information of the short range potential to the atomic threshold [9]. A particular interesting case is given by the collisional properties of ultracold potassium atoms. Potassium features two stable bosonic isotopes ( 39 K and 41 K) and a very long living fermionic one ( 40 K). All these isotopes have been cooled to quantum degeneracy both in single and in two isotope experiments and collisional data for five ( 39 K-39 K, 40 K-40 K, 41 K-41 K, 39 K-41 K, 40 K-41 K) of the six possible combinations are available in literature [10][11][12][13][14]. The comparison of Feshbach resonance positions for different isotope combinations is a promising way to reveal corrections to the BOA. First hints of such corrections were obtained by Falke et al. [7] studying the two cases 39 K-39 K, and 40 K-40 K available at that time.
In this paper we present the experimental observation of up-to-now unmeasured d-wave Feshbach resonances of 39 K and how this allows to improve the knowledge on 39 K 2 . We combine this result with the published literature on potassium Feshbach resonances and we determine corrections to the BOA from collisional data alone.
The paper is structured as follows. In Section II we explain how to reveal effects from beyond BOAcorrections in atomic collisional properties. In Section III we present the observation of three new Feshbach resonances for 39 K, which enhances the knowledge on 39 K 2 to use this dimer as reference for the full isotope analysis. In Section IV we quantify the corrections to the BOA thanks to a multi-parameter fit of the new and already known Feshbach resonance positions.

II. THEORY ASPECTS
To treat the collision of an atom pair of alkali atoms at low kinetic energy we set up the Hamiltonian of the coupled system of the two lowest molecular states X 1 Σ + g and a 3 Σ + u , because the product state of two ground state atoms is generally a mixture of singlet and triplet states. The appropriate Hamiltonian is presented in many papers, e.g. [7,15] and will not be repeated here. It contains the hyperfine interaction, also responsible for the singlet-triplet mixing, the atomic Zeeman interaction and the effective spin-spin interaction of the two atoms in their doublet states. The nuclear motion is governed by the molecular potentials of the two interacting molecular states. The potential functions within the Born-Oppenheimer approximation (BO-potentials) are represented in analytic form as described in detail in [7] in three R-sections divided by an inner R in and outer radius R out : In the intermediate range around the minimum it is described by a finite power expansion with a nonlinear variable function ξ of internuclear separation R: In Eq. 1 the {a i } are fitting parameters and b and R m are chosen such that only few parameters a i are needed for describing the steep slope at the short internuclear separation side and the smaller slope at the large R side by the analytic form of Eq. 1. R m is normally close to the value of the equilibrium separation.
The potential is extrapolated for R < R in with: by adjusting the A and B parameters to get a continuous transition at R in ; the final fit uses N s equal to 12 and 6 for X 1 Σ + g and a 3 Σ + u states, respectively, as adequate exponents.
For large internuclear distances (R > R out ) we adopt the standard long range form of molecular potentials: where the exchange contribution is given by and U ∞ is set to zero which fixes the energy reference of the total potential scheme.
The BO-potentials are extended by correction functions U ad (R), which make the full potentials mass dependent. These correction functions [3,4,16] contain matrix elements of the nuclear momentum operators over the electronic wavefunctions of the considered electronic state and other ones with ∆Ω = 0, where Ω is the projection of the total electronic angular momentum onto the molecular axis. U ad (R) is the so called adiabatic correction to the BO-potential function, and it contains the interaction of the considered electronic state with all states according the selection rule ∆Ω = 0 by the nuclear vibrational motion. We do not include non-adiabatic correction with the selection rule ∆Ω = ±1 because it will be negligibly small for collisions with low partial waves as s, p or d.
Watson [3] shows that in the lowest order the mass dependence of these corrections for a molecule AB will be of the form is the atomic mass of atom A(B) and m e the electron mass. For true heteronuclear molecules the coefficients U A (R) and U B (R) will be different, for homonuclear cases in the electronic system such as K 2 both coefficient will be equal and thus the isotope dependence of the correction function will be inversely proportional to the reduced mass µ of the molecule. van Vleck [16] considered the mass dependence of the heteronuclear cases in the hydrogen-deuterium (HD) molecule and found that the corrections should be extended by a term Thus in our case with the nuclei like 39 K-41 K, the representation of the correction functions should read: where µ is the reduced mass for the molecular rovibrational motion. U gen (R) and U asym (R) are functions of the internuclear separation R. The subscripts refer to the general and asymmetric contributions. In our setup, ultracold samples of 39 K atoms are prepared by sympathetic cooling in a bath of evaporatively cooled Na atoms as explained in [17,18]. Compared to the experimental sequence used in our previous works, the mixture here is heavily unbalanced towards 39 K and the Na atom number is practically negligible. During evaporation in a crossed optical dipole trap, the 39 K atoms are initially in the |f = 1, m f = −1 state and are transferred to the target |f = 2, m f = −2 state by rapid adiabatic passage. f is the total angular momentum of the atom and m f its projection. The transfer is based on a 1 ms radio frequency sweep performed at an external magnetic field of about 199 G. At this magnetic field losses are small both in the initial and final state and during the transfer. The sample contains up to 3 × 10 5 atoms at 650 nK in a trap with an average frequency of 2π × 114(5) Hz.
To locate the d-wave resonances, we observe the atom number decreased due to inelastic two-body losses in the proximity of the Feshbach resonance. We ramp the magnetic field strength in 10 ms to the target value. After a fixed holding time, chosen to not lead to complete de- pletion of the atoms at resonance, the magnetic field is ramped back to the magnetic field strength where highfield absorption imaging of the remaining atoms is performed [18]. Figure 1 shows the remaining atom fraction at different values of the magnetic field strength in the vicinity of the predicted d-wave Feshbach resonances. By fitting the loss data with phenomenological Gaussian curves we obtain the following three resonance positions: 125.94(14) G, 188.72(5) G and 227.71(60) G. The predicted width of the resonance at 188.72 G is far below our magnetic field stability of about 30 mG and leads to experimental points not following a Gaussian profile, compare Fig. 1(b). We also measure the remaining atom number at the resonance positions for variable holding time. The data are shown in Fig. 1(d), (e) and (f) for the 125.94 G, 188.72 G and 227.71 G resonance, respectively. The inelastic loss rate coefficients are obtained from a fit to the data according to the two-body loss differential equation including the effects of anti-evaporation heating [19,20] and background lifetime (about 17 s). The loss rate coefficients are summarized in Fig. 1(g) and confirm the expected large difference between fast ( Fig. 1(e)) and slow ( Fig. 1(d) and (f)) losses despite the same d-wave character of the resonances. The values are in good agreement with theoretical predictions using the results of Sec. IV within the error bars, which include statistical uncertainties and the uncertainties on the calibration of temperature, trap frequency and atom number. The values for the 125.94 G ( Fig. 1 (a,d)) and 227.71 G ( Fig. 1 (c,f)) resonances are larger than predicted, probably because of other loss contributions not included in the fit. The measured value for the 188.7 G ( Fig. 1 (b,e)) resonance is instead smaller than expected as the narrow resonance width and the magnetic field jitter do not allow to remain exactly at resonance.

IV. ANALYSIS
We start our analysis from potential functions of the two lowest electronic states X 1 Σ + g and a 3 Σ + u derived from spectroscopic observation which have been described in detail in [7]. We refit the spectroscopic data with a smaller set of potential parameters to reduce the risk of obtaining unphysical tiny oscillatory behavior of the potential function. In [7] state X 1 Σ + g was described by 31 parameters, now 22 are sufficient. For state a 3 Σ + u we use 14 parameters compared to 22 in the previous work. The resulting potentials form the starting point for a fit of 49 Feshbach resonances and a comparison of the experimentally determined Feshbach resonance positions with the ones resulting from the coupled channel calculation. We identify the Feshbach resonance position by the maximum scattering rate coefficient at the kinetic energy given by the experimental conditions. We base our analysis on Feshbach resonance data for the isotope combinations 39 K-39 K from D'Errico et al. [11] and [14,21], 40 K-40 K from Regal et al. [22,25], 39 K-41 K from Tanzi et al. [14], 41 K-41 K from Chen et al. [24] and Tanzi et al. [14], and 40 K-41 K from Wu et al. [12]. We summarize the data in Tab. I with their quantum numbers and the reported experimental uncertainty. As quantum numbers we use the projection M of the total angular momentum onto the field axis, the atom pair labels for dressed states and the interval l min − l max of the partial waves. The labels of the atomic dressed states are given by |f, m f . The column 'type' indicates, that for 'el' the peak of the elastic part of the rate coefficient is taken and for 'in' the sum of the inelastic contributions. The evaluation uses atomic hyperfine and g factors from [26]. We fit the data in Tab. I to the BO-potentials adjusting the branches at small (R < R in ) and large (R > R out ) internuclear separations. After few trials it   (1)). Analyzing the obtained fit for the different isotope combinations (see Tab. II, model (1)) reveals that the main part of the sum of squared weighted deviations stems from the isotope combination 39 K-41 K, resulting in σ = 1.235, whereas the other isotope combinations show values below 0.72. A separate fit to the data of the isotope combination 39 K-41 K only underlines the consistency of these observations with a resulting normalized standard deviation of σ = 0.753 (see Tab. II, model ( 39 K 41 K) a ). We thus started an additional fit in an attempt to optimize the result for the isotope combination 39 K-41 K by applying the potentials from the separate fit of 39 K-41 K above as initial values. Note that non-linear fits regularly give slightly different results depending on initial values since it is hard to find the global minimum of the sum of squared weighted deviations. In this second fit we obtain almost the same overall fit quality with a normalized standard deviation of σ = 0.993 but now the main deviation lies in the 40 K-40 K isotope combination with its individual value σ = 1.566 as given in the third line of Tab. II, model ( 39 K 41 K) b ). Since the reduced mas of 39 K 41 K and 40 K-40 K are almost equal, the different behaviour of these two isotope combinations in the two different fits (model (1) and model ( 39 K 41 K) b ) give a strong hint that mass effects beyond the simple mass scaling of the rovibrational motion should be considered.
In the next step, we thus include beyond BO-corrections proportional to the reduced mass for the general case (i.e. part U gen from Eq. 6). Because we can only study the small variations between the naturally existing isotope combinations, it is of advantage to define one isotope combination as reference. This results in a parametrization of the full potentials with BO-corrections for molecule AB given by where the factor of the electron mass in Eq. 6 is incorporated in the new functions BO gen and BO asym and µ ref is the reduced mass of the reference combination.
Here we apply 39 K-39 K as reference. For this combination we have a large number of s-wave resonances and additionally also d-wave resonances. Both together fix the asymptotic branch of the potentials. This is different for the 39 K-41 K isotope combination where only s-wave resonances have been measured. In principle, BO gen is a function of R, but the present data set is too small to derive such function from a fit with acceptable significance. Thus we simplify the condition by assuming correction functions to be proportional to the BO-potential and Eq. 7 reads now where now BO gen and BO asym are fit parameters for the amplitude of the BO-corrections. A crude justification of this assumption is that the normal mass effect in atomic physics, e.g. the Rydberg constant and its nuclear mass dependence, show a similar form of the correction for the binding energy. Furthermore, a molecular potential describes the variation of the kinetic energy within the nuclear vibrational motion as function of R and is therefore a measure of the coupling to the electron motion. Starting with BO gen for states X 1 Σ + g and a 3 Σ + u we perform a fit of all resonances adding the parameter for both electronic states and obtain a normalized standard deviation of σ = 0.786 (the individual deviations are shown in column 'o-c(2)' in Tab. I). This value should be compared with the one from a fit of the pure BO-potentials σ = 0.977. Including beyond Born-Oppenheimer corrections apparently leads to a significantly better fit. Looking again at details of the fit for the different isotope combinations in Tab. II model (2), we see that the combination 39 K-41 K is described with σ = 0.952 whereas the other three show values below 0.6. Since we removed the isotope combination 40 K-41 K from the evaluation already earlier for other reason (see also [27]), the former one is the only heteronuclear case remaining for which the standard deviation is significantly larger than the value seen  (3)) thus a further improvement compared to 0.786 from model (2). Additionally, all individual standard deviations are almost equal to the values obtained by separated fits. The deviations of observation to calculation from the new fit are shown in column 'o-c(3)' of Tab. I. The sequential improvement of the fit quality including beyond Born-Oppenheimer terms underlines the significance of corrections beyond Born-Oppenheimer for the precise description of molecular potentials and the precision derivation of atomic scattering properties. [28] In Tab. III we give the magnitude of the BO-corrections for the two electronic states X 1 Σ + g and a 3 Σ + u . The uncertainty of the significantly determined parameters are about 20 %. For giving a better insight into the BOcorrection we calculate the highest vibrational levels with the correction and compare them with the level energy setting the correction to zero. For the heaviest isotope 41 K-41 K and thus the largest difference to the reference isotope 39 K-39 K we obtain for the level v = 27, N = 0 of the state a 3 Σ + u a difference of 220 kHz and for the state X 1 Σ + g (v = 87, N = 0) it is effectively zero, because here the influence by BO-correction appears only for heteronuclear isotope combinations.

V. DISCUSSIONS
From the different models we calculated the scattering lengths for the pure singlet and triplet state. The results are summarized in Tab. IV for the different isotope combinations using model (1) (BO-approximation) and (3) (including all beyond BO-corrections). Because we choose the isotope combination 39 K-39 K as reference one might expect no difference for the resulting scattering lengths for this isotope pair when using model (1) or (3) respectively. However, we do observe corrections (see Tab. IV). Equal values for 39 K-39 K would result if the  (3) would only vary the BO-correction parameters and anything else be kept constant. But this will be not the optimal fit strategy, because in case (1), i.e. no BO-corrections, existing significant BO-corrections are distributed over the deviations of the fit over all isotopes and thus also the reference isotope is influenced. One can see such different distribution from the standard deviations of 39 K-39 K given for model (1) and (3) in Tab. II, the former one is larger than the latter one. Because of this influence, we only give error estimates for the complete model including BO-corrections in Tab. IV and the differences between model (1) and (3) do not show the true magnitude of the BO-correction. See also the calculation of the energy shift by the BO-correction as given at the end of section IV.
A complete list of scattering lengths was reported in [7]. The new values show a significant improvement by roughly a factor 5 of the error limit. The values agree in most cases within uncertainty limits despite the fact that the former evaluation could only incorporate Feshbach resonances for 39 K-39 K and 40 K-40 K. The paper stated that a weak indication of BO-corrections could be obtained from the resonances. But we believe the present evaluation shows this clearly. Additionally, we were able to study the difference between the homonuclear and heteronuclear case resulting in the values of BO gen and BO asym .
We obtained a significant contribution for the triplet state a 3 Σ + u by BO gen for both fit cases but for the singlet state X 1 Σ + g only for the heteronuclear isotope pairs. This is probably related to the fact that the closest singlet state, namely A 1 Σ + u , has u symmetry compared to g symmetry for the singlet ground state. These two can only couple by the symmetry breaking part of the Hamiltonian responsible for the BO-correction [29]. The situation for the triplet state is different, where the energetically closest is b 3 Π u and has u symmetry as the triplet ground state. We should note that the magnitudes of both effects, BO gen and BO asym cannot be directly compared, because the former one is referenced to 39 K-39 K and thus describes only the difference between the isotope pairs whereas the latter indicates the total effect. We evaluated the isotope dependence by using the precise Feshbach spectroscopy and checked finally that the obtained BO-corrections have only little influence in the deep rovibrational levels measured by molecular spectroscopy, e.g. in Ref. [7,30,31], which have an uncertainty in the order of few thousands of cm −1 or about 100 MHz compared to 1 MHz or better for the Feshbach spectroscopy. For this purpose we went back to the full data set from the spectroscopy for iterating the fit for obtaining the consistent description of the complete data set from molecular and Feshbach spectroscopy. The final parameter sets of the potentials are given in the appendix.

VI. CONCLUSION AND OUTLOOK
We use an analysis of the complete set of all known Feshbach resonances in different K isotope combinations to derive potential energy curves for states X 1 Σ + g and a 3 Σ + u and find clear signatures of beyond BO-corrections. We base our work on the discussion of H 2 and HD molecules by van Vleck [16] and find correction terms for the homonuclear and heteronuclear cases when analysing homo-and heteronuclear isotope combinations of K respectively. Unfortunately, our analysis of heteronuclear cases is restricted to the 39 K-41 K isotope combination, although, in principle, more isotope combinations exist. However, available Feshbach resonance data of the 40 K-41 K [12] isotope combination show very large deviations which are beyond a realistic description [32]. We therefore excluded this isotope combination from the analysis given in Sec. IV. To allow for an extended analysis of heteronuclear beyond BO-corrections, it would be very much desirable to revisit observed Feshbach resonances in the |9/2, 9/2 + |1, 1 channel of the 40 K-41 K isotope combination and extend measurements to resonances within other collision channels such as |9/2, -9/2 +|1, 1 . In the same context, the 39 K-40 K isotope is of great interest. Here, it would be particularly favorable to study collisions in the |9/2, -7/2 +|1, 1 and |9/2, -5/2 +|1, 1 channels. In these channels well-separated Feshbach resonances in a magnetic field region below 200 G should be found whereas sharp resonances in the |9/2, -9/2 +|1, 1 channel will be overlapped by a very broad resonance. Furthermore, the above-mentioned channels will show sharp Feshbach resonances in the range of 800 to 850 G. We believe that such studies will settle the discussion of the importance of BO-corrections in cases of homo-and hetero-nuclear pairs of homo-polar molecules. In the same spirit, it would be very interesting to analyze Feshbach resonances in the different isotope combination of the homo-polar molecule Li 2 . There exists a detailed analysis [33] of spectroscopic data of the X 1 Σ + g -A 1 Σ + u transition in the Li 2 considering homonuclear BOcorrections (BO gen (R) from Eq. 7). The study includes data from 7 Li-6 Li isotopologue, however, the authors do not mention any need to distinguish between homo-and hetero-nuclear corrections. The data set for the 7 Li-6 Li molecule is small compared to that of both homonuclear molecules 7 Li-7 Li and 6 Li-6 Li, thus it could be not sufficiently significant for the above mentioned distinction. For Li 2 there exist also measurements of Feshbach resonances for the homonuclear cases, see the latest report by Gerken et al. [34], but nothing on 7 Li-6 Li. Thus studies of Feshbach resonances of Li-Li would be very worth to investigate both homonuclear and heteronuclear beyond BO corrections. We conclude that very interesting Feshbach spectroscopy is ahead of us to work out and highlight the importance of BO-corrections in the understanding of cold collisions.

VII. ACKNOWLEDGEMENT
We gratefully acknowledge financial support from the European Research Council through ERC Starting Grant POLAR and from the Deutsche Forschungsgemeinschaft (DFG) through CRC 1227 (DQ-mat), project A03 and FOR2247, project E5. K.K.V. and P.G. thank the Deutsche Forschungsgemeinschaft for financial support through Research Training Group 1991.

VIII. APPENDIX
Tab. V, VII and VI show the potential parameters (defined in Eqs. 1, 3 and 4) for the two states X 1 Σ + g and a 3 Σ + u , as derived during the evaluation. These results are improved potentials compared to the published ones [7], not only because the Feshbach data were largely extended but also the number of potential parameters is significantly reduced leading to a more stringent potential form with less danger of showing tiny oscillatory unphysical effects.    [2] R. Herman and A. Asgharian, Journal of Molecular Spec-