Observation of interspecies Feshbach resonances in an ultracold {}^{39}K-{}^{133}Cs mixture and refinement of interaction potentials

Observation of interspecies Feshbach resonances in an ultracold K-Cs mixture
and refinement of interaction potentials

Michael Gröbner Institut für Experimentalphysik und Zentrum für Quantenphysik, Universität Innsbruck, 6020 Innsbruck, Austria    Philipp Weinmann Institut für Quantenoptik und Quanteninformation (IQOQI), Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria    Emil Kirilov Institut für Experimentalphysik und Zentrum für Quantenphysik, Universität Innsbruck, 6020 Innsbruck, Austria    Hanns-Christoph Nägerl Institut für Experimentalphysik und Zentrum für Quantenphysik, Universität Innsbruck, 6020 Innsbruck, Austria    Paul S. Julienne Joint Quantum Institute, University of Maryland and National Institute for Standards and Technology, College Park, Maryland 20742, United States of America    C. Ruth Le Sueur Joint Quantum Centre (JQC) Durham-Newcastle, Department of Chemistry, Durham University, South Road, Durham, DH1 3LE, United Kingdom    Jeremy M. Hutson Joint Quantum Centre (JQC) Durham-Newcastle, Department of Chemistry, Durham University, South Road, Durham, DH1 3LE, United Kingdom

We observe interspecies Feshbach resonances due to s-wave bound states in ultracold K-Cs scattering for three different spin mixtures. The resonances are observed as joint atom loss and heating of the K sample. We perform least-squares fits to obtain improved K-Cs interaction potentials that reproduce the observed resonances, and carry out coupled-channel calculations to characterize the scattering and bound-state properties for K-Cs, K-Cs and K-Cs. Our results open up the possibilities of tuning interactions in K-Cs atomic mixtures and of producing ultracold KCs molecules.

34.50.Cx, 34.20.Cf, 67.60.Bc

I Introduction

The possibility of controlling collisional interactions in ultracold atomic samples to very high precision through Feshbach resonances Chin et al. (2010) is the foundation of many different cold atom experiments. Control of interactions by Feshbach tuning has enabled experiments on tunable quantum gases Cornish et al. (2000); Donley et al. (2001); Weber et al. (2003), the creation of ultracold Feshbach molecules Donley et al. (2002); Regal et al. (2003); Herbig et al. (2003), the formation of Bose-Einstein condensates (BECs) of molecules Jochim et al. (2003); Greiner et al. (2003) and the observation of the BEC-BCS crossover Regal et al. (2004); Bartenstein et al. (2004); Zwierlein et al. (2004), few-body and Efimov physics Kraemer et al. (2006), polaron physics Schirotzek et al. (2009); Koschorreck et al. (2012), and novel states, phase transitions, and dynamics in one-dimensional gases Haller et al. (2009, 2010); Meinert et al. (2016). In these experiments, intraspecies interactions were tuned, in some cases between different spin states of the same species. In recent years, interest has turned to mixtures of quantum gases and the tuning of interspecies interactions. This interest is motivated by the study of exotic phases such as supersolids Titvinidze et al. (2008), the heteronuclear Efimov scenario for a three-body system Wacker et al. (2016); Ulmanis et al. (2016); Barontini et al. (2009), boson-mediated superfluids Efremov and Viverit (2002), quantum phases that involve composite fermions Lewenstein et al. (2004), mixtures under simultaneous superfluidity Ferrier-Barbut et al. (2014), and the possibility of forming samples of ultracold polar ground-state molecules Ni et al. (2008). In particular, the electric dipole moment of heteronuclear molecules gives rise to anisotropic, long-range dipole-dipole interactions that contrast with the isotropic, short-range interaction in atomic experiments Baranov et al. (2012); Lahaye et al. (2009). Combining long-range interactions with optical lattice potentials allows the study of exotic quantum phases such as pair superfluids and the implementation of quantum simulation and quantum information processing Capogrosso-Sansone et al. (2010); Micheli et al. (2007); Trefzger et al. (2011, 2009).

Experimentally, systems of ultracold ground-state molecules are produced in a two-step procedure: First, atoms in nearly quantum-degenerate atomic mixtures are magneto-associated using a Feshbach resonance to form weakly bound molecules. Second, these molecules are optically transferred into the rovibrational ground state by stimulated Raman adiabatic passage (STIRAP) Bergmann et al. (1998); Danzl et al. (2008); Ni et al. (2008). This procedure, which requires precise knowledge of the inter- and intraspecies scattering properties, has recently led to the production of ultracold and dense samples of heteronuclear molecules such as fermionic KRb Ni et al. (2008) and NaK Park et al. (2015) and bosonic RbCs Takekoshi et al. (2014); Molony et al. (2014); Reichsöllner et al. (2017) and NaRb Guo et al. (2016) in their rovibrational ground states. The present paper is aimed towards the goal of producing ultracold KCs molecules by similar methods. Ground-state KCs molecules are of particular interest because of their large electric dipole moment (1.92 D) Aymar and Dulieu (2005) and their stability under two-body molecular collisions Żuchowski and Hutson (2010), which makes KCs the only chemically stable fermionic alkali-metal dimer apart from NaK. Additionally, the two available bosonic isotopes K increase the flexibility in mixing and dimer association with Cs.

In most magneto-association experiments so far, molecules were produced in three-dimensional (3D) bulk atom mixtures Ni et al. (2008); Takekoshi et al. (2014); Molony et al. (2014); Park et al. (2015); Guo et al. (2016). Only a comparatively small fraction of atoms could be converted to heteronuclear dimers, because in bulk samples the process is limited by atomic three-body recombination and vibrational relaxation in atom-molecule and molecule-molecule collisions. Such losses can be suppressed if the two atomic samples are overlapped in an optical lattice, creating either a Bose (Fermi) Mott insulator (band insulator) Moses et al. (2015) or a Bose-Bose double-species Mott insulator Reichsöllner et al. (2017). In both cases reported so far, a Feshbach resonance was exploited in two different ways. First, it was used to null the interspecies interaction at the zero-crossing of the resonance to achieve efficient sample mixing. Subsequently, the resonance was used to form the molecules from atom pairs. Lattice filling fractions of 30% and above have been achieved. Since we aim at a similar strategy for KCs, precise knowledge of the Feshbach resonance positions and widths is crucial.

The individual two-body interaction properties of K D’Errico et al. (2007) and Cs Leo et al. (2000); Berninger et al. (2013) are well understood. This has allowed the production of Bose-Einstein condensates for each species separately Weber et al. (2003); Roati et al. (2007); Landini et al. (2012); Salomon et al. (2014) and for both species in the same apparatus Gröbner et al. (2016). The singlet and triplet interaction potentials for KCs have been determined from extensive electronic spectroscopy by Ferber et al. Ferber et al. (2009, 2013); in the present work we designate the potentials of Ref. Ferber et al. (2013) the F2013 potentials. Patel et al. Patel et al. (2014) carried out coupled-channel calculations on the F2013 potentials to obtain the positions and widths of Feshbach resonances for all three K isotopes, including both s-wave and d-wave bound states. However, no experiments have yet been carried out to test these predictions. In this article, we report the observation of Feshbach resonances in an ultracold K-Cs mixture. We prepare the samples in different spin states and search for loss features as we scan the homogeneous magnetic field in the range from 0 to 650 G. The observed resonances in the lowest spin state are observed at magnetic fields about 20 G higher than predicted in Refs. Ferber et al. (2013); Patel et al. (2014). We therefore use coupled-channel calculations to assign the resonances and to fit improved interaction potentials, which we designate G2017. We then use the new potentials to make improved predictions of resonance positions and widths for all three isotopologs of KCs.

Ii Experiment

Techniques for the preparation of ultracold Cs Weber et al. (2003) and K Landini et al. (2012); Gröbner et al. (2016) are well established, but mixing the two species is not straightforward. In particular, mixing the samples in the regime of quantum degeneracy is quite involved and we are pursuing a strategy similar to that demonstrated for RbCs in Ref. Reichsöllner et al. (2017). For the present goal of detecting interspecies Feshbach resonances, however, it is sufficient to mix very cold thermal samples, and even that poses some challenges. The different steps in laser cooling lead to a disparity in the sample temperatures (K for K and K for Cs) and densities. The negative background scattering length of K implies the existence of a Ramsauer-Townsend minimum in the elastic cross section at an energy around  K Landini et al. (2012), where the scattering phase shift passes through zero and the contribution from higher partial waves is still small. This minimum in the elastic cross section, the large losses when K atoms overlap with the Cs magneto-optical trap (MOT), and the strong heating when Cs is loaded into a deep dipole trap make a sequential cooling scheme necessary. We achieve this with a translatable and transformable trap. Specifically, we first load the K sample into a very tight optical trap, subsequently translate this sample vertically to allow for Cs loading and cooling, and finally bring the two species together in a relaxed trap with enlarged waist that is suitable for both species.

Figure 1: Experimental sequence to combine K with Cs as shown in absorption images. (a) In situ image of the K sample after dipole trap loading. (b) Vertically displaced K cloud (upper cloud) to avoid collisional loss with Cs atoms (lower cloud) during Cs laser cooling. (c) Merged samples at the end of the Cs MOT stage. The K sample shows up as the dark red disk at the center of the image. (d) Typical K signal in the crossed trap for a K-Cs mixture after merging. The images in (b) and (c) are overlapped absorption images taken at the respective imaging wavelengths.

The experimental sequence starts with the preparation of an ultracold K sample as described in Ref. Gröbner et al. (2016). In short, after standard laser cooling and spin polarizing on the D and D lines, we load up to atoms in the state into a magnetic quadrupole trap. Hyperfine sublevels of each atom are indicated by alphabetic labels a, b, c, etc., in order of increasing energy. To overcome the Ramsauer-Townsend scattering minimum, we superimpose a dipole trap beam with -waist of 26 m at 1064 nm and an initial power of 15 W, and simultaneously increase the quadrupole field within 5 s from 32 to 75 G/cm (along the coil axis). Although this increases the temperature, the higher density ensures efficient loading of the tight tweezer trap. The quadrupole field is shut off and the magnetic offset field is then ramped to  G. At this field the scattering length for atoms in state is around 100  and we can perform efficient forced evaporative cooling. For this, the power of the single-beam dipole trap is decreased exponentially in 1.5 s to 150 mW. During the first 600 ms of this ramp we move the optical trap 1.2 mm upwards [see Figs. 1(a) to 1(b)]. The vertical transport is achieved by moving a lens and a mirror that are mounted on a motorized translation stage and works without any observable atom loss or heating. We note that magnetization effects related to our stainless steel vacuum chamber require a magnetic polarization stage after the quadrupole trap. Polarization is achieved by pulsing several times up to  G for 100 ms. Without this procedure, laser cooling of Cs, as performed subsequently, is not possible without adjustments in the magnetic field.

At this point, the magnetic trap center is free and we can start loading the Cs MOT. For this, we turn the quadrupole field on again (7.5 G/cm along the coil axis). During the first 100 ms we linearly increase the K trap power to 300 mW and turn on a 15-W dipole trap beam with a waist of 250 m at 1070 nm, crossing the center of the Cs MOT. After 5 s of Cs MOT loading, and before increasing the quadrupole field to 20 G/cm to compress the Cs sample, we superimpose the two clouds [see Fig. 1(c)]. This is done by moving the K trap 0.79 mm downwards in 160 ms. At the same time we dynamically increase the waist of the K trap from 26 to 63 m by shrinking the aperture of an iris with a servomotor and increase the power to 1.2 W. After the compression stage the Cs sample is further cooled and spin-polarized by three-dimensional degenerate Raman-sideband cooling (dRSC) Treutlein et al. (2001); Kraemer et al. (2004). The temperature after dRSC is below 1 K when we release the atoms into free space. Here, however, we cool the atoms into a crossed-dipole trap. When we do so, we measure temperatures of about 7 K and observe some significant atom loss. We attribute the temperature increase and atom loss largely to the mismatch of the Cs cloud size after dRSC to the trapping volume of the crossed-dipole trap and possibly to ac-Stark shifts due to the dipole trap that compromise the performance of dRSC. Also, atoms that are cooled away from the center of the crossed-dipole trap convert potential energy into kinetic energy after extinction of the dRSC lattice beams. In any case, after a hold time of 80 ms at 35.5 G in the crossed-dipole trap, the K and Cs clouds [see Fig. 1(d)] are each found to be in thermal equilibrium, but at different temperatures. They see trap depths of about  K and  K. With around K atoms and Cs atoms in the trap, we measure temperatures of  K and  K. We measure trap frequencies of Hz and Hz and deduce atomic peak densities of  cm and  cm. At this stage, the K atoms are fully spin polarized in the third-lowest energy state . The Cs atoms are 80% polarized in the state, with the rest of the Cs sample mainly populating the state.

Figure 2: Typical Feshbach resonance signatures in a - mixture. (a) Normalized atom number and (b) temperature for a K sample mixed with Cs after a 900-ms hold time. (a) The remaining fraction of K atoms normalized to a sample without Cs at constant trap depth after a hold time of 100 ms. Each point is an average of at least two measurements and the solid line is a Lorentzian fit to the results. The temperatures in (b) are deduced from five time-of-flight images with different expansion times and the error bars represent statistical errors from the temperature fits. Larger data samples were not possible due to drifts of the vacuum chamber magnetization as discussed in the text.

The dipole trap now allows us to prepare K-Cs mixtures in any desired hyperfine state combination. Here, to provide sufficient input to theoretical modeling, we are mainly interested in combining with , , and . We fully spin-polarize the Cs sample into the state by using a microwave pulse in combination with resonant light to clean out the population. During the pulse, which lasts 6 ms and is resonant with the transition from to , we sweep from 1 to 1.05 G to address magnetic field inhomogeneities, and apply laser light on the S P transition.

Experiment Theory (F2013 potentials)
Spin states
(G) (G) (G) (G)
+ 361.1(1) 3.2(4) 341.89 4.7
442.59(1) 0.28(3) 421.37 0.38
+ 419.3(1) 3.0(5) 399.93 4.3
513.12(1) 0.16(6) 491.39 0.55
+ 491.5(1) 2.1(4) 471.97 3.8
599.32(3) 0.5(1) 575.67 0.44
Table 1: Overview of interspecies Feshbach resonances for mixtures -, -, and -. Experimentally we deduce the positions and FWHM by fitting Lorentzian functions to the loss features. The uncertainties are the statistical errors from the Lorentzian fits. Note that the Lorentzian width is not the same physical quantity as the theoretical width . We note that drifts of the chamber magnetization result in a systematic error of up to  G for .

First, we perform Feshbach spectroscopy on a - mixture. For this, we linearly ramp the magnetic offset field within 10 ms to any desired value in the range from to  G and hold it there for 900 to 1300 ms. During this hold time we exponentially decrease the power of the transformable beam to 520 mW to enhance the loss of K atoms from the crossed trap. In the vicinity of an interspecies Feshbach resonance, the K sample undergoes enhanced trap loss through three-body recombination and heating from the interaction with the hotter Cs sample. To detect the remaining fraction of K atoms we ramp within 10 ms to 0.1 G before applying standard absorption imaging. For this particular spin mixture we scan from 0 to 650 G in steps of 1 G and observe two loss features, one broad and one narrow, located around 491.5 and 599.3 G, respectively. We scan the loss features with finer resolution in . The loss occurs over a range of 0.1 to several G, depending on the resonance and the specific experimental conditions. The results around 491.5 G are shown in Fig. 2(a). The K atom number shows a clear loss maximum. The loss minimum that appears around 495.5 G may be the result of the zero-crossing of the scattering length on the high-field side of the resonance. We fit Lorentzian functions to the loss features to obtain the positions of maximum loss and the full widths at half maximum (FWHM) .

Figure 3: Overview of K-Cs interspecies Feshbach resonances. (a) Loss features for with (black), [red (light gray)], and [blue (dark gray)]. The data are normalized to the atom number away from resonance. (b) Calculated interspecies scattering length for the three hyperfine state combinations as a function of magnetic field , using the G2017 interaction potentials fitted in Sec. III.3.

We also carry out time-of-flight measurements to determine the temperature of the K sample. The results are shown in Fig. 2(b); we observe an increase in temperature from 3.0 to 4.5 K at the same location as the loss is maximal. This temperature increase vanishes when the Cs sample is absent. We attribute the increase in temperature to partial thermalization with the hotter Cs sample. Higher temperatures are probably counteracted by evaporation due to the finite trap depth. As will be seen in Sec. III.4 below, the background scattering length for K-Cs is around 70 , and this relatively small value explains the absence of observed thermalization away from resonance.

For Feshbach spectroscopy with K atoms in and we transfer the K atoms by radio-frequency adiabatic passage from to and, subsequently, to . Each step takes place at  G within 25 ms with an efficiency close to unity. For each spin mixture we again detect a pair of resonances, one broader and the other narrower. All measured Feshbach resonances are shown in Fig. 3(a) and their parameters are summarized in Table 1. We obtain the magnetic field by measuring Cs microwave frequencies at the fields where interspecies loss features are observed. The experimental errors given in Table 1 are statistical errors from the Lorentzian fits. A drift of the magnetization of the stainless steel chamber, which depends on the offset field strength, gives rise to a systematic error on the order of 0.3 G.

Iii Theory

iii.1 Computational methods for bound states and scattering

For the scattering and near-threshold bound states, we solve the Schrödinger equation by coupled-channel methods, using a basis set for the electron and nuclear spins in a fully decoupled representation,


The matrix elements of the different terms in the Hamiltonian in this basis set are given in the Appendix of Ref. Hutson et al. (2008). The calculations in this paper used basis sets with all possible values of and for both atoms, subject to conservation of and parity . For s-wave scattering at a particular threshold, is set by the states of the incoming atoms, , and only channels with even contribute.

Scattering calculations are carried out using the MOLSCAT package Hutson and Green (1994), as modified to handle collisions in magnetic fields González-Martínez and Hutson (2007). At each magnetic field , the wave-function log-derivative matrix at collision energy is propagated from   to 15  using the propagator of Manolopoulos Manolopoulos (1986) with a fixed step size of 0.001 , and from 15 to 3,000  using the Airy propagator Alexander and Manolopoulos (1987) with a variable step size controlled by the parameter Alexander (1984). Scattering boundary conditions Johnson (1973) are applied at   to obtain the scattering S-matrix. The energy-dependent s-wave scattering length is then obtained from the diagonal S-matrix element in the incoming channel using the identity Hutson (2007)


where and is the reduced mass. This reduces to the standard zero-energy scattering length in the low-energy limit.

Weakly bound levels for Feshbach molecules are obtained using the propagation method described in Refs. Hutson (1994); Hutson et al. (2008), using the same step size as for MOLSCAT with a reduced propagation range of   to  . Levels are located either as bound-state energies at a fixed value of the magnetic field using the BOUND package Hutson (1993) or as bound-state fields at a fixed value of the binding energy using the FIELD package Hutson (2011). BOUND and FIELD converge to values of the energy (or field) where the log-derivative matching matrix Hutson (1994) has a zero eigenvalue. Both programs implement a node-count algorithm Hutson (1994) which makes it straightforward to ensure that all bound states that exist in a particular range of energy or field are located.

Zero-energy Feshbach resonances can be located as fields at which the scattering length passes through a pole,


MOLSCAT has the capability to converge on such poles to provide resonance widths and background scattering lengths as well as resonance positions . However, when only resonance positions are required, the FIELD package provides a much cleaner approach: simply running FIELD at zero energy provides a complete list of the energies at which bound states cross threshold, and thus a complete list of resonance positions. The widths and background scattering lengths may then be obtained if required, using scattering calculations with MOLSCAT around the field concerned.

In the present work, basis sets including only functions were used in most cases, since they make the calculations simpler at the b + a and c + a thresholds, where inelastic decay would otherwise exist. However, calculations with were used for the calculations of scattering and bound states on the fitted potentials in Sec. III.4 below. As will be seen, the observed resonances in the a + a channel shifted by no more than 0.01 G when basis functions were included, which is considerably less than the experimental uncertainties in the resonance positions.

iii.2 Potential curves

The KCs interaction potentials of Ferber et al. Ferber et al. (2013) (F2013) were fitted to extensive Fourier transform spectra of the KCs molecule, including vibrational levels up to for the singlet ground state and for the triplet state (although there is significant mixing of the singlet and triplet states for the highest vibrational levels). Each potential curve is constructed in three segments; the central segment from to , with or 1 for the singlet or triplet state, respectively, is represented as a power-series expansion in the variable , where is chosen to be near the equilibrium distance. At long range (), the potentials are


where the dispersion coefficients Derevianko et al. (2001); Porsev and Derevianko (2003); Ferber et al. (2013) are common to both potentials. The exchange contribution is Smirnov and Chibisov (1965)


and makes an attractive contribution for the singlet and a repulsive contribution for the triplet. The central segment is constrained to match the long-range potential at . The potentials are extended to short range () with simple repulsive terms,


The parameters and are chosen to match the values and derivatives of the mid-range potentials at . The potential matching points for KCs are  Å and  Å for the singlet state and  Å and  Å for the triplet state Ferber et al. (2013).

Threshold Unc.
a + a 361.1 360.74 4.4 0.36 0.4
a + a 442.59 442.43 0.37 0.16 0.3
b + a 419.3 419.73 4.0 0.43 0.4
b + a 513.12 513.73 0.52 0.61 0.3
c + a 491.5 492.24 3.6 0.74 0.4
c + a 599.32 598.76 0.39 0.58 0.3
b + a 334.45 0.025
b + a 563.81 0.074
b + b 319.35 0.046
c + a 398.09 0.023
c + a 467.70 0.006
c + a 618.71 0.094
Table 2: Quality of fit to the observed resonance positions, together with the properties of additional resonances due to s-wave bound states predicted by the G2017 potentials. The uncertainties given here include systematic errors and are those that define the weights used in the least-squares fit.

For coupled-channel calculations of the near-threshold bound states and scattering properties, these potentials are supplemented by a coupling , which at long range has a simple magnetic dipole-dipole form that varies as  Stoof et al. (1988); Moerdijk et al. (1995). However, for heavy atoms, second-order spin-orbit coupling provides an additional contribution that has the same tensor form as the dipole-dipole term. is represented as


where is a unit vector along the internuclear axis and is an -dependent coupling constant,


where is the atomic fine-structure constant, is the Hartree energy and is the electron -factor. The second-order spin-orbit coupling has not been obtained from electronic structure calculations for KCs, so in the present work we retained the estimate used in Ref. Patel et al. (2014), obtained by shifting the RbCs function Takekoshi et al. (2012) inwards by 0.125 , to give the same value at the inner turning point for KCs as for RbCs. This gives and as for RbCs, with and .

Figure 4: Bottom panel: Bound states of KCs (solid lines) with [blue (dark gray)], 3 [red (light gray)] and 4 (black), together with the corresponding thresholds (dashed lines), calculated using the G2017 fitted interaction potentials of Sec. III.3 with functions only and shown with respect to the lowest zero-field threshold. The threshold crossings that produce observed resonances are shown with filled circles and those so far unobserved with open circles. Upper panels: Bound states for , 3 and 4, shown relative to the field-dependent c + a, b + a and a + a thresholds.

iii.3 Least-squares fitting

The resonances observed in the present work are due to s-wave bound states that cross the threshold as a function of magnetic field. These are substantially broader than resonances due to d-wave and higher states Patel et al. (2014), which appear only because of the weak anisotropic term in the Hamiltonian. Figure 4 shows the atomic thresholds for , 3 and 4, together with the s-wave bound states responsible for the resonances observed here, calculated on the G2017 fitted potentials described below. There is a state roughly parallel to each of the a + a, b + a and c + a thresholds, bound by 65 to 70 MHz and with the same () character as the threshold. Crossing these near-threshold states are a set of deeper states, bound by 800 to 1100 MHz at zero field, that are closer to horizontal in the bottom panel of Fig. 4. This second set of states correlates at zero field with atoms with and , with resultant , 2, 3, 4, 5 (though does not appear in Fig. 4 because states with are not shown). The observed resonances occur when the near-threshold states are pushed across threshold by mixing with the near-horizontal states at broad avoided crossings, which are complete to varying extents at threshold. The positions of the resonances are thus principally determined by the binding energies of the near-horizontal states, although the states actually crossing threshold have mixed character. An analogous figure showing the bound states on the F2013 potentials Ferber et al. (2013) is included in the Supplemental Material sup (); the pattern of states is visually very similar, despite that fact that the sign of is reversed on the F2013 potentials.

The right-hand columns of Table 1 give the calculated positions and widths of the observed resonances, obtained using the F2013 potentials. It may be seen that the calculated resonance positions are all about 20 G lower than the experiment. It is therefore desirable to adjust the interaction potentials to reproduce the resonance positions. In doing this, we wish to retain as much as possible of the spectroscopically determined potentials of Ref. Ferber et al. (2013), so that the fit to the Fourier transform spectra is affected as little as possible. We found in the initial fitting that it is possible to reproduce the scattering properties by retaining the central and long-range parts of the spectroscopic potential curves and adjusting only the short-range parts for . Small changes to the potential curves in this region have relatively little effect on levels with inner turning points below  cm for the singlet state and  cm for the triplet state. We explored modifications to the values of , with corresponding changes in and to match the values and derivatives of the power-series expansions at ,

() () ()
G2017 potentials F2013 potentials
KCs 18.37 74.88(9) 82.24
KCs 51.44 71.67(45) 41.28
KCs 72.79 179.06(28) 205.25
Table 3: Calculated singlet and triplet scattering lengths for isotopologs of KCs, with 1- statistical uncertainties for .
Figure 5: Scattering length in the a + a channel and energies of near-threshold bound states for KCs. Note that the top 30 MHz is shown on an expanded scale. Resonance widths greater than 1 G are shown as vertical bars with lengths proportional to G).

We carried out least-squares fits of potential parameters to the observed resonance positions using the Interactive Non-Linear Least-Squares (I-NoLLS) package Law and Hutson (1997), which gives the user interactive control over step lengths and assignments as the fit proceeds. The quantity optimized in the least-squares fits was the sum of squares of residuals [(observed calculated)/uncertainty], with the uncertainties listed in Table  2. The resonance positions at the a + a, b + a and c + a thresholds are principally sensitive to the triplet potential, so our initial fits varied only, leaving the singlet potential unchanged. The optimum fit was obtained with and gives the fit to the resonance positions shown in Table 2. The short-range coefficients for the best-fit triplet potential are obtained from Eq. (9); their approximate values are cm and  cm Å.

We subsequently explored two-parameter fits, varying both and . However, these produced no significant improvement in the quality of fit. A single-parameter fit varying only was incapable of reproducing the observed resonance positions. We therefore decided to proceed with the single-parameter fit obtained by varying only the triplet potential, which we designate the G2017 potentials; a more extensive refinement will require additional experimental results.

It may be noted that the present modification shifts the vibrational levels of the state by a maximum of about 0.1 cm; this shift is less than 0.01 cm below and peaks around . These shifts are smaller than the Cs hyperfine splitting, but larger than the typical experimental uncertainties of 0.01  cm in Ref. Ferber et al. (2013). A more complete treatment would require refitting the entire potentials, but is not justified at this stage.

Figure 6: Scattering length in the a + a channel and energies of near-threshold bound states for KCs. Resonance widths greater than 1 G are shown as vertical bars with lengths proportional to G).
Figure 7: Scattering length in the a + a channel and energies of near-threshold bound states for KCs. Resonance widths greater than 1 G are shown as vertical bars with lengths proportional to G).

iii.4 Calculations on optimized potentials

Figures 5, 6, and 7 show the scattering lengths and resonance positions, calculated at the lowest (a + a) threshold for all three isotopic combinations of KCs using the optimized G2017 potentials, together with the near-threshold bound states that produce the resonances. Table 3 gives the predicted singlet and triplet scattering lengths and , together with the statistical uncertainty of in the one-parameter space. However, may change outside these limits when additional parameters are fitted. Complete lists of the resonance parameters (positions, widths and background scattering lengths) are given in the Supplemental Material sup ().

The G2017 potentials obtained here predict scattering and bound-state properties that differ in some important ways from those of Ref. Patel et al. (2014). In particular, the triplet scattering lengths all shift to smaller (or more negative) values, and the corresponding near-threshold levels are more deeply bound.

Figure 4 shows that, in addition to the two resonances currently observed in each of the a + a, b + a and c + a channels, there are additional KCs resonances due to s-wave states in the b + a, c + a and b + b channels (where b + b is the lowest threshold at fields between 261 G and 358 G). The positions and widths of these are included in Table 2. They have predicted widths between 6 mG and 0.1 G, and may be useful for molecule formation.

For KCs, Patel et al. Patel et al. (2014) predicted a group of low-field resonances at fields below 50 G, due to a group of d-wave states bound by less than 20 MHz at zero field. The widest of these was a resonance predicted near 50 G with a width of 1 mG, due to a state that was bound by less than 3 MHz at zero field but was nearly parallel to threshold as a function of field. On the G2017 potentials the corresponding states are significantly deeper; the state responsible for the widest resonance is now bound by about 9 MHz at zero field, and the resulting resonance is shifted to 219 G, now with a width of 0.7 mG. Similarly, for KCs, Patel et al. Patel et al. (2014) predicted a group of low-field resonances at fields of 20 to 30 G, again due to a group of very weakly bound d-wave states. These looked promising for molecule formation because of their proximity to the region around 21 G where Cs can be cooled to degeneracy. On the G2017 potentials, however, these states are again significantly deeper, and the resonances are shifted to fields above 70 G.

For fermionic KCs, by contrast, the G2017 potentials appear to offer improved prospects for molecule formation. The older F2013 potentials Ferber et al. (2013) give scattering lengths and . Because these are so similar, even resonances due to s-wave states were predicted in Ref. Patel et al. (2014) to be very narrow and those due to d-wave states even narrower. The G2017 potential, however, has . Because of this, the resonances are shifted to rather higher fields, but they are also considerably broader. For example, the resonance predicted in Ref. Patel et al. (2014) at 264.3 G with a width of  G occurs at 286.0 G on the G2017 potentials with a width of  G. Similarly, the resonance previously predicted at 470 G with a width of  mG is now at 531 G, with a width of  mG. The latter is particularly promising for molecule formation, because it is reasonably close to the region around 556 G where Cs has a moderate scattering length Berninger et al. (2013) and can be cooled efficiently.

Iv Conclusion

We observed Feshbach resonances due to s-wave bound states in ultracold collisions of K and Cs. The resonances occur at magnetic fields about 20 G higher than those predicted in Refs. Ferber et al. (2013); Patel et al. (2014) using interaction potentials fitted to high-resolution Fourier transform spectra. Reproducing the experimental resonance positions requires a significant change to the triplet interaction potential found in Ref. Ferber et al. (2013). We carried out least-squares fits to determine a triplet potential with a modified repulsive wall, which reproduces the Feshbach resonance positions while making only small changes to the deeper vibrational levels.

We used the modified interaction potentials, which we designate G2017, to carry out coupled-channels calculations and make improved predictions of the near-threshold bound states and ultracold scattering properties for all three isotopes of K interacting with Cs. For the case of KCs, the scattering properties are more favorable using the G2017 potentials than was found in Refs. Ferber et al. (2013); Patel et al. (2014). In particular, the G2017 potentials predict a Feshbach resonance that is broad enough to allow tuning of the interactions in a K-Cs Fermi-Bose mixture. The results open up various interesting avenues in cold atom and cold molecule research. These include studies of the dynamics and transport properties of bosonic impurities in low-dimensional Fermi gases, similar to recent experiments where Bloch-type oscillations have been observed for impurity motion in a fermionized one-dimensional Bose gas Meinert et al. (2016). It may also be possible to form fermionic KCs molecules and transfer them to the rovibrational ground state to generate dipolar quantum gases, employing techniques such as those recently demonstrated for fermionic KRb Moses et al. (2015) and for bosonic RbCs Reichsöllner et al. (2017).

V Acknowledgments

We are indebted to R. Grimm for generous support. We thank G. Anich, K. Jag-Lauber, F. Meinert, and G. Unnikrishnan for fruitful discussions. A subroutine to evaluate the potential curves developed in this work, together with sample input and output files for the MOLSCAT, BOUND and FIELD programs and the associated experimental data, has been deposited at doi:10.15128/r1cf95jb44k. We gratefully acknowledge funding by the European Research Council (ERC) under Project No. 278417, by the Austrian Science Foundation (FWF) under Project No. I1789-N20 (joint Austrian-French FWF-ANR project) and under Project No. P29602-N36 and by the UK Engineering and Physical Sciences Research Council under Grant No. EP/N007085/1.


  • Chin et al. (2010) C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
  • Cornish et al. (2000) S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 85, 1795 (2000).
  • Donley et al. (2001) E. A. Donley, N. R. Clausen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Nature 412, 295 (2001).
  • Weber et al. (2003) T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, Science 299, 232 (2003).
  • Donley et al. (2002) E. A. Donley, N. R. Claussen, S. T. Thompson, and C. E. Wieman, Nature 417, 529 (2002).
  • Regal et al. (2003) C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature 424, 47 (2003).
  • Herbig et al. (2003) J. Herbig, T. Kraemer, M. Mark, T. Weber, C. Chin, H.-C. Nägerl, and R. Grimm, Science 301, 1510 (2003).
  • Jochim et al. (2003) S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, and R. Grimm, Science 302, 2101 (2003).
  • Greiner et al. (2003) M. Greiner, C. A. Regal, and D. S. Jin, Nature 426, 537 (2003).
  • Regal et al. (2004) C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004).
  • Bartenstein et al. (2004) M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, Phys. Rev. Lett. 92, 120401 (2004).
  • Zwierlein et al. (2004) M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle, Phys. Rev. Lett. 92, 120403 (2004).
  • Kraemer et al. (2006) T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl, et al., Nature 440, 315 (2006).
  • Schirotzek et al. (2009) A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwierlein, Phys. Rev. Lett. 102, 230402 (2009).
  • Koschorreck et al. (2012) M. Koschorreck, D. Pertot, E. Vogt, B. Fröhlich, M. Feld, and M. Köhl, Nature 485, 619 (2012).
  • Haller et al. (2009) E. Haller, M. Gustavsson, M. J. Mark, J. G. Danzl, R. Hart, G. Pupillo, and H. C. Nägerl, Science 325, 1224 (2009).
  • Haller et al. (2010) E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, M. Gustavsson, M. Dalmonte, G. Pupillo, and H.-C. Nägerl, Nature 466, 597 (2010).
  • Meinert et al. (2016) F. Meinert, M. Knap, E. Kirilov, K. Jag-Lauber, M. B. Zvonarev, E. Demler, and H.-C. Nägerl, arXiv:1608.08200 (2016).
  • Titvinidze et al. (2008) I. Titvinidze, M. Snoek, and W. Hofstetter, Phys. Rev. Lett. 100, 100401 (2008).
  • Wacker et al. (2016) L. J. Wacker, N. B. Jørgensen, D. Birkmose, N. Winter, M. Mikkelsen, J. Sherson, N. Zinner, and J. J. Arlt, Phys. Rev. Lett. 117, 163201 (2016).
  • Ulmanis et al. (2016) J. Ulmanis, S. Häfner, R. Pires, E. D. Kuhnle, Y. Wang, C. H. Greene, and M. Weidemüller, Phys. Rev. Lett. 117, 153201 (2016).
  • Barontini et al. (2009) G. Barontini, C. Weber, F. Rabatti, J. Catani, G. Thalhammer, M. Inguscio, and F. Minardi, Phys. Rev. Lett. 103, 043201 (2009).
  • Efremov and Viverit (2002) D. V. Efremov and L. Viverit, Phys. Rev. B 65, 134519 (2002).
  • Lewenstein et al. (2004) M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, Phys. Rev. Lett. 92, 050401 (2004).
  • Ferrier-Barbut et al. (2014) I. Ferrier-Barbut, M. Delehaye, S. Laurent, A. T. Grier, M. Pierce, B. S. Rem, F. Chevy, and C. Salomon, Science 345, 1035 (2014).
  • Ni et al. (2008) K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Science 322, 231 (2008).
  • Baranov et al. (2012) M. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller, Chem. Rev. 112, 5012 (2012).
  • Lahaye et al. (2009) T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, Rep. Prog. Phys. 72, 126401 (2009).
  • Capogrosso-Sansone et al. (2010) B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, and G. Pupillo, Phys. Rev. Lett. 104, 125301 (2010).
  • Micheli et al. (2007) A. Micheli, G. Pupillo, H. P. Büchler, and P. Zoller, Phys. Rev. A 76, 043604 (2007).
  • Trefzger et al. (2011) C. Trefzger, C. Menotti, B. Capogrosso-Sansone, and M. Lewenstein, J. Phys. B 44, 193001 (2011).
  • Trefzger et al. (2009) C. Trefzger, C. Menotti, and M. Lewenstein, Phys. Rev. Lett. 103, 035304 (2009).
  • Bergmann et al. (1998) K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998).
  • Danzl et al. (2008) J. G. Danzl, E. Haller, M. Gustavsson, M. J. Mark, R. Hart, N. Bouloufa, O. Dulieu, H. Ritsch, and H.-C. Nägerl, Science 321, 1062 (2008).
  • Park et al. (2015) J. W. Park, S. A. Will, and M. W. Zwierlein, Phys. Rev. Lett. 114, 205302 (2015).
  • Takekoshi et al. (2014) T. Takekoshi, L. Reichsöllner, A. Schindewolf, J. M. Hutson, C. R. Le Sueur, O. Dulieu, F. Ferlaino, R. Grimm, and H.-C. Nägerl, Phys. Rev. Lett. 113, 205301 (2014).
  • Molony et al. (2014) P. K. Molony, P. D. Gregory, Z. Ji, B. Lu, M. P. Köppinger, C. R. Le Sueur, C. L. Blackley, J. M. Hutson, and S. L. Cornish, Phys. Rev. Lett. 113, 255301 (2014).
  • Reichsöllner et al. (2017) L. Reichsöllner, A. Schindewolf, T. Takekoshi, R. Grimm, and H.-C. Nägerl, Phys. Rev. Lett. 118, 073201 (2017).
  • Guo et al. (2016) M. Guo, B. Zhu, B. Lu, X. Ye, F. Wang, R. Vexiau, N. Bouloufa-Maafa, G. Quéméner, O. Dulieu, and D. Wang, Phys. Rev. Lett. 116, 205303 (2016).
  • Aymar and Dulieu (2005) M. Aymar and O. Dulieu, J. Chem. Phys. 122, 204302 (2005).
  • Żuchowski and Hutson (2010) P. S. Żuchowski and J. M. Hutson, Phys. Rev. A 81, 060703 (2010).
  • Moses et al. (2015) S. A. Moses, J. P. Covey, M. T. Miecnikowski, B. Yan, B. Gadway, J. Ye, and D. S. Jin, Science 350, 659 (2015).
  • D’Errico et al. (2007) C. D’Errico, M. Zaccanti, M. Fattori, G. Roati, M. Inguscio, G. Modugno, and A. Simoni, New J. Phys. 9, 223 (2007).
  • Leo et al. (2000) P. J. Leo, C. J. Williams, and P. S. Julienne, Phys. Rev. Lett. 85, 2721 (2000).
  • Berninger et al. (2013) M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. Nägerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M. Hutson, Phys. Rev. A 87, 032517 (2013).
  • Roati et al. (2007) G. Roati, M. Zaccanti, C. D’Errico, J. Catani, M. Modugno, A. Simoni, M. Inguscio, and G. Modugno, Phys. Rev. Lett. 99, 010403 (2007).
  • Landini et al. (2012) M. Landini, S. Roy, G. Roati, A. Simoni, M. Inguscio, G. Modugno, and M. Fattori, Phys. Rev. A 86, 033421 (2012).
  • Salomon et al. (2014) G. Salomon, L. Fouché, S. Lepoutre, A. Aspect, and T. Bourdel, Phys. Rev. A 90, 033405 (2014).
  • Gröbner et al. (2016) M. Gröbner, P. Weinmann, F. Meinert, K. Lauber, E. Kirilov, and H.-C. Nägerl, J. Mod. Optic. 63, 1829 (2016).
  • Ferber et al. (2009) R. Ferber, I. Klincare, O. Nikolayeva, M. Tamanis, H. Knöckel, E. Tiemann, and A. Pashov, Phys. Rev. A 80, 062501 (2009).
  • Ferber et al. (2013) R. Ferber, O. Nikolayeva, M. Tamanis, H. Knöckel, and E. Tiemann, Phys. Rev. A 88, 012516 (2013).
  • Patel et al. (2014) H. J. Patel, C. L. Blackley, S. L. Cornish, and J. M. Hutson, Phys. Rev. A 90, 032716 (2014).
  • Treutlein et al. (2001) P. Treutlein, K. Y. Chung, and S. Chu, Phys. Rev. A 63, 051401 (2001).
  • Kraemer et al. (2004) T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H.-C. Nägerl, and R. Grimm, Appl. Phys. B 79, 1013 (2004).
  • Hutson et al. (2008) J. M. Hutson, E. Tiesinga, and P. S. Julienne, Phys. Rev. A 78, 052703 (2008), note that the matrix element of the dipolar spin-spin operator given in Eq. A2 of this paper omits a factor of .
  • Hutson and Green (1994) J. M. Hutson and S. Green, MOLSCAT computer program, version 14, distributed by Collaborative Computational Project No. 6 of the UK Engineering and Physical Sciences Research Council (1994).
  • González-Martínez and Hutson (2007) M. L. González-Martínez and J. M. Hutson, Phys. Rev. A 75, 022702 (2007).
  • Manolopoulos (1986) D. E. Manolopoulos, J. Chem. Phys. 85, 6425 (1986).
  • Alexander and Manolopoulos (1987) M. H. Alexander and D. E. Manolopoulos, J. Chem. Phys. 86, 2044 (1987).
  • Alexander (1984) M. H. Alexander, J. Chem. Phys. 81, 4510 (1984).
  • Johnson (1973) B. R. Johnson, J. Comput. Phys. 13, 445 (1973).
  • Hutson (2007) J. M. Hutson, New J. Phys. 9, 152 (2007).
  • Hutson (1994) J. M. Hutson, Comput. Phys. Commun. 84, 1 (1994).
  • Hutson (1993) J. M. Hutson, BOUND computer program, version 5, distributed by Collaborative Computational Project No. 6 of the UK Engineering and Physical Sciences Research Council (1993).
  • Hutson (2011) J. M. Hutson, FIELD computer program, version 1 (2011).
  • Derevianko et al. (2001) A. Derevianko, J. F. Babb, and A. Dalgarno, Phys. Rev. A 63, 052704 (2001).
  • Porsev and Derevianko (2003) S. G. Porsev and A. Derevianko, J. Chem. Phys. 119, 844 (2003).
  • Smirnov and Chibisov (1965) B. M. Smirnov and M. I. Chibisov, Sov. Phys. JETP 21, 624 (1965).
  • Stoof et al. (1988) H. T. C. Stoof, J. M. V. A. Koelman, and B. J. Verhaar, Phys. Rev. B 38, 4688 (1988).
  • Moerdijk et al. (1995) A. J. Moerdijk, B. J. Verhaar, and A. Axelsson, Phys. Rev. A 51, 4852 (1995).
  • Takekoshi et al. (2012) T. Takekoshi, M. Debatin, R. Rameshan, F. Ferlaino, R. Grimm, H.-C. Nägerl, C. R. Le Sueur, J. M. Hutson, P. S. Julienne, S. Kotochigova, et al., Phys. Rev. A 85, 032506 (2012).
  • (72) See Supplemental Material at [to be inserted by publisher].
  • Law and Hutson (1997) M. M. Law and J. M. Hutson, Comput. Phys. Commun. 102, 252 (1997).

Appendix A Supplemental Material: Observation of interspecies Feshbach resonances in an ultracold K-Cs mixture and refinement of interaction potentials

(G) (G) (G) ()
G2017 F2013 G2017 G2017
potentials potentials potentials potentials
9.335 2.995 63.7 2 2
16.450 5.482 64.3 2 3
16.670 10.784 64.3 2 2
23.293 16.311 64.9 2 2
31.344 18.154 65.5 2 3
219.346 202.900 72.9 2 2
219.972 49.593 72.8 2 4
257.539 240.221 74.2 2 3
274.019 255.726 74.9 2 2
318.249 298.576 79.2 2 3
338.197 318.143 86.0 2 5
344.508 323.897 91.5 2 2
359.899 326.943 0.012 420 2 4
360.745 341.895 4.6 68 0 4
396.419 373.849 63.8 2 3
435.855 412.433 72.0 2 2
442.429 421.364 0.37 68.6 0 4
763.737 697.020 0.031 73.0 2 6
782.083 714.609 73.0 2 5
802.159 375.354 73.2 2 4
824.886 757.456 73.5 2 3
828.375 760.131 0.005 73.4 2 5
842.083 734.709 0.002 73.5 2 4
864.584 778.978 0.002 73.7 2 4
867.490 798.340 73.7 2 3
881.298 813.139 73.9 0 4
889.799 819.977 74.0 2 2
920.147 849.801 74.8 2 3
929.603 860.524 0.045 74.6 0 4
940.700 869.412 74.7 2 2
978.997 907.538 0.019 84.9 2 3
985.676 915.564 1.1 73.1 0 4
999.386 926.772 67.7 2 2
Table 4: Resonance properties predicted from the optimized G2017 potentials for the a + a channel of KCs, with the positions from Ref. Patel et al. (2014) for comparison. Resonances were located using MOLSCAT, with set to 1,500 .
(G) (G) (G) ()
G2017 F2013 G2017 G2017
potentials potentials potentials potentials
72.950 50.848 171.5 2 2
80.281 47.266 174.0 2 3
82.902 23.890 172.5 2 6
87.111 53.035 173.2 2 4
88.120 25.677 171.9 2 5
90.238 39.244 171.7 2 5
90.866 56.608 171.9 2 2
95.460 42.860 173.1 2 4
98.090 55.545 173.5 2 3
100.577 28.413 173.9 2 4
105.056 32.104 174.6 2 3
112.269 68.538 175.6 2 2
114.513 64.875 176.0 2 3
117.826 36.635 176.5 2 2
131.464 90.098 179.1 2 5
137.038 98.542 180.7 2 4
143.299 108.839 188.8 2 3
146.325 98.503 345 2 2
146.492 113.926 86.2 0 4
148.031 87.381 167.8 2 4
149.012 94.276 159.2 2 3
155.171 126.591 173.0 2 2
168.004 120.590 178.1 0 4
169.953 91.559 178.6 2 2
171.823 111.038 179.2 0 4
173.280 90.442 179.1 2 3
208.658 109.860 189.9 2 2
216.655 171.198 195.8 0 4
236.047 168.192 176 0 4
705.820 629.690 184.2 2 2
789.035 737.859 186.4 2 2
794.615 746.956 186.6 2 3
803.232 755.110 186.9 2 2
852.093 806.405 189.8 2 4
862.873 818.335 190.9 2 3
874.210 830.925 192.4 2 2
905.078 861.029 201.7 2 5
920.652 877.705 220.8 2 4
935.245 884.925 183 0 4
935.745 894.133 -909 2 3
950.626 910.602 148.3 2 2
Table 5: Resonance properties predicted from the optimized G2017 potentials for the a + a channel of KCs, with the positions from Ref. Patel et al. (2014) for comparison. Resonances were located using MOLSCAT, with set to 1,500 .
Figure 8: Bottom panel: Bound states of KCs (solid lines) with [blue (dark gray)], 3 [red (light gray)] and 4 (black), together with the corresponding thresholds (dashed lines), calculated using the interaction potentials of Ref. Ferber et al. (2013) with functions only and shown with respect to the lowest zero-field threshold. Upper panels: Bound states for , 3 and 4, shown relative to the field-dependent c + a, b + a and a + a thresholds.
(G) (G) (G) ()
G2017 F2013 G2017 G2017
potentials potentials potentials potentials
31.002 27.103 2 +1/2
32.976 29.035 2 -1/2
35.308 31.293 2 -3/2
35.775 31.611 2 +1/2
38.048 33.935 2 -5/2
38.760 34.382 2 -1/2
41.263 37.034 2 -7/2
42.324 37.679 2 -3/2
42.976 38.160 2 +1/2
46.592 41.636 2 -5/2
47.660 42.366 2 -1/2
51.748 46.435 2 -7/2
53.410 47.546 2 -3/2
54.539 48.410 2 +1/2
60.587 54.030 2 -5/2
62.447 57.590 0 -3/2
62.537 55.439 2 -1/2
69.698 62.295 2 -7/2
72.924 64.609 2 -3/2
75.492 66.582 2 +1/2
75.524 69.853 0 -3/2
86.780 76.911 2 -5/2
91.630 80.518 2 -1/2
96.335 89.010 0 -3/2
105.702 93.892 2 -7/2
114.882 100.745 2 -3/2
124.087 107.659 2 +1/2
133.522 122.768 0 -3/2
149.344 131.536 2 -5/2
165.689 146.133 2 -7/2
169.871 149.152 2 -7/2
171.606 148.042 2 -1/2
173.035 152.632 2 -7/2
176.113 156.289 2 -7/2
176.689 157.129 2 -5/2
181.215 160.074 2 -5/2
185.920 163.642 2 -5/2
190.232 167.934 2 -5/2
190.509 170.875 2 -3/2
194.198 172.611 2 -5/2
195.011 173.418 2 -3/2
199.611 176.821 2 -3/2
(G) (G) (G) ()
G2017 F2013 G2017 G2017
potentials potentials potentials potentials
203.797 181.086 2 -3/2
208.400 185.944 2 -3/2
208.414 188.721 2 -1/2
208.624 180.762 2 -7/2
212.577 190.265 2 -1/2
212.873 192.186 2 -3/2
215.626 196.709 0 -3/2
217.044 192.995 2 -1/2
221.848 196.981 2 -1/2
226.715 202.190 2 -1/2
232.248 211.942 2 +1/2
232.589 208.400 2 -1/2
236.460 213.627 2 +1/2
237.183 215.961 2 -1/2
239.858 214.741 2 +1/2
243.371 216.624 2 +1/2
247.638 221.339 2 +1/2
250.777 230.247 0 -3/2
253.115 227.532 2 +1/2
256.464 234.147 0 -3/2
257.695 235.275 2 +1/2
262.890 239.549 0 -3/2
265.979 224.767 2 -3/2
270.122 246.441 0 -3/2
277.215 254.519 0 -3/2
285.964 264.340 0 -3/2
411.372 318.650 2 +1/2
441.876 379.663 2 -5/2
489.071 466.458 2 +1/2
531.217 470.254 0 -3/2
555.460 531.469 2 -1/2
613.799 588.830 2 -3/2
661.323 627.835 2 +1/2
667.781 642.012 2 -5/2
704.214 677.444 0 -3/2
718.940 692.463 2 -7/2
741.968 706.122 2 -1/2
817.559 779.664 2 -3/2
890.432 850.612 2 -5/2
944.619 902.844 0 -3/2
949.117 892.623 2 +1/2
961.839 920.113 2 -7/2
Table 6: Resonance properties predicted from the optimized potentials for the a + a channel of KCs, with the positions from Ref. Patel et al. (2014) for comparison. Resonances with widths tabulated were located using MOLSCAT, with set to 1,500 ; those without tabulated widths proved impossible to locate using MOLSCAT and must be much narrower than 1 nG. Their tabulated positions are those of the threshold crossings identified by FIELD. Because of mixing between bound states, the mapping between resonances on the G2017 and F2013 potentials is uncertain for some resonances below 200 G.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description