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arXiv:1404.2623 · Physical Review A

Contrasting the wide Feshbach resonances in Li6 and Li7

Paul S. Julienne, Jeremy M. Hutson

Published May 23, 2014• 55 citations
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Abstract

We compare and contrast the wide Feshbach resonances and the corresponding weakly bound states in the lowest scattering channels of ultracold $^{6}\mathrm{Li}$ and $^{7}\mathrm{Li}$. We use high-precision measurements of binding energies and scattering properties to determine interaction potentials that incorporate non-Born-Oppenheimer terms to account for the failure of mass scaling between $^{6}\mathrm{Li}$ and $^{7}\mathrm{Li}$. Correction terms are needed for both the singlet and the triplet potential curves. The universal formula relating binding energy to scattering length is not accurate for either system. The $^{6}\mathrm{Li}$ resonance is open-channel-dominated and the van der Waals formula of Gao [J. Phys. B 37, 4273 (2004)] gives accurate results for the binding energies across much of the resonance width. The $^{7}\mathrm{Li}$ resonance, by contrast, is weakly closed-channel-dominated and a coupled-channel treatment of the binding energies is required. Plotting the binding energies in universal van der Waals form helps illustrate subtle differences between the experimental results and different theoretical forms near the resonance pole.

DOI
10.1103/physreva.89.052715

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We compare and contrast the wide Feshbach resonances and the corresponding weakly bound states in the lowest scattering channels of ultracold 6Li^{6}\mathrm{Li}6Li and 7Li^{7}\mathrm{Li}7Li. We use high-precision measurements of binding energies and scattering properties to determine interaction potentials that incorporate non-Born-Oppenheimer terms to account for the failure of mass scaling between 6Li^{6}\mathrm{Li}6Li and 7Li^{7}\mathrm{Li}7Li. Correction terms are needed for both the singlet and the triplet potential curves. The universal formula relating binding energy to scattering length is not accurate for either system. The 6Li^{6}\mathrm{Li}6Li resonance is open-channel-dominated and the van der Waals formula of Gao [J. Phys. B 37, 4273 (2004)] gives accurate results for the binding energies across much of the resonance width. The 7Li^{7}\mathrm{Li}7Li resonance, by contrast, is weakly closed-channel-dominated and a coupled-channel treatment of the binding energies is required. Plotting the binding energies in universal van der Waals form helps illustrate subtle differences between the experimental results and different theoretical forms near the resonance pole.
We characterize Feshbach resonances in all isotopologues of the Li−Li\mathrm{Li}{-}\mathrm{Li}Li−Li system with improved interaction potentials. Starting from spectroscopically accurate Morse/long-range (MLR) potential-energy curves for the singlet (X1Σ+X^{1}Σ^{+}X1Σ+) and triplet (a3Σ+a^{3}Σ^{+}a3Σ+) electronic states of Li2\mathrm{Li}_2Li2​, we apply small phenomenological inner-wall adjustments (following Julienne and Hutson, Phys. Rev. A 89, 052715 (2014), arXiv:1404.2623v3) and fit the resulting potentials to threshold measurements for the 6Li−6Li^{6}\mathrm{Li}{-}^{6}\mathrm{Li}6Li−6Li and 7Li−7Li^{7}\mathrm{Li}{-}^{7}\mathrm{Li}7Li−7Li isotopologues, including binding energies, scattering lengths, and Feshbach resonance positions. Using the optimized potentials in coupled-channels scattering calculations, we predict and characterize s-wave Feshbach resonances in the 6Li−7Li^{6}\mathrm{Li}{-}^{7}\mathrm{Li}6Li−7Li isotopologue. In its lowest-energy hyperfine channel, all resonances are narrow (∼0.01−0.1\sim 0.01{-}0.1∼0.01−0.1 G), strongly closed-channel dominated, and predominantly triplet in electronic spin character, in marked contrast to the homonuclear systems. These results provide a foundation for designing Raman optical-transfer pathways to produce ultracold Li2\mathrm{Li}_2Li2​ molecules in deeply bound rovibrational levels of both the X1Σ+X^1Σ^{+}X1Σ+ and a3Σ+a^3Σ^{+}a3Σ+ potentials across all three isotopologues.
We employ radio-frequency spectroscopy on weakly bound (6)Li(2) molecules to precisely determine the molecular binding energies and the energy splittings between molecular states for different magnetic fields. These measurements allow us to extract the interaction parameters of ultracold (6)Li atoms based on a multichannel quantum scattering model. We determine the singlet and triplet scattering lengths to be a(s) = 45.167(8)a(0) and a(t) = -2140(18)a(0) (1a(0) = 0.052 917 7 nm), and the positions of the broad Feshbach resonances in the energetically lowest three s-wave scattering channels to be 83.41(15), 69.04(5), and 81.12(10) mT.
We report a theoretical study of Feshbach resonances in 6Li+23Na^{6}\mathrm{Li}+^{23}\mathrm{Na}6Li+23Na and 7Li+23Na^{7}\mathrm{Li}+^{23}\mathrm{Na}7Li+23Na mixtures at ultracold temperatures using accurate interaction potentials in a full quantum coupled-channel calculation. Feshbach resonances for l=0l=0l=0 in the initial collisional open channel 6Li(f=1∕2,mf=1∕2)+23Na(f=1,mf=1)^{6}\mathrm{Li}(f=1∕2,{m}_{f}=1∕2)+^{23}\mathrm{Na}(f=1,{m}_{f}=1)6Li(f=1∕2,mf​=1∕2)+23Na(f=1,mf​=1) are found to agree with previous measurements, leading to precise values of the singlet and triplet scattering lengths for the 6,7Li+23Na^{6,7}\mathrm{Li}+^{23}\mathrm{Na}6,7Li+23Na pairs. We also predict additional Feshbach resonances within experimentally attainable magnetic fields for other collision channels.
We revisit the basic properties of Fano-Feshbach resonances in two-body systems with van der Waals tail interactions, such as ultracold neutral atoms. Using a two-channel model and two different methods, we investigate the relationship between the width and shift of the resonances and their dependence on the low-energy parameters of the system. Unlike what was previously believed [C. Chin et al., Rev. Mod. Phys. 82, 1225 (2010)] for magnetic resonances, we find that the ratio between the width and the shift of a resonance does not depend only on the background scattering length, but also on a closed-channel scattering length. We obtain different limits corresponding to different cases of optical and magnetic resonances. Although the generalization of the theory to the multichannel case remains to be done, we found that our two-channel predictions are verified for a specific resonance of lithium-6.
We study the interspecies scattering properties of ultracold Li–Cs mixtures in their two energetically lowest spin channels in the magnetic field range between 800 and 1000 G. Close to two broad Feshbach resonances (FR) we create weakly bound LiCs dimers by radio-frequency association and measure the dependence of their binding energy on the external magnetic field strength. Based on the binding energies and complementary atom loss spectroscopy of three other Li–Cs s -wave FRs we construct precise molecular singlet and triplet electronic ground state potentials using a coupled-channels calculation. We extract the Li–Cs interspecies scattering length as a function of the external field and obtain almost a ten-fold improvement in the precision of the values for the pole positions and widths of the s -wave FRs as compared to our previous work (Pires et al 2014 Phys. Rev. Lett. 112 250404 ). We discuss implications on the Efimov scenario and the universal geometric scaling for LiCsCs trimers.
The susceptibility of ultracold gases to precise manipulation via external fields makes them ideal systems to study fundamental quantum mechanical phenomena. Various regimes, which are not possible to reach in condensed matter or nuclear physics, can be probed with these quantum gases. Especially the tunability of the interactions and confining potentials make that these ultracold gases offer unique possibilities to explore and study universal behavior. The tunability of the dominant two-body interactions in dilute ultracold atomic gases stems from the presence of Feshbach resonances. Via these resonances pairs of ultracold atoms can be associated to form ultracold molecules. In fact, measurements of Feshbach resonance positions can be used to reveal part of the underlying molecular structure. Vica versa, accurate molecular Born-Oppenheimer potentials can be used as input for full numerical coupled-channels calculations to reliably predict Feshbach resonances. For most of the heteronuclear mixtures currently studied however, the relevant molecular potentials are not accurate enough. As the coupled-channels method is often too elaborate and too time consuming to be used to fit the measured data to enable the construction of more accurate potentials, a need for accurate models which complement this established method is created. In this thesis we have developed a simplified model for resonant scattering of two ultracold atoms which fulfills this need. The asymptotic bound-state model (ABM) is a hierarchical model which can be systematically increased in complexity and accuracy. In its most simple form, the ABM only needs two parameters to predict the position and strength of Feshbach resonances: Two binding energies corresponding to the least bound state of the two molecular potentials involved. We demonstrate that this model, although it is solely based on bound states, can incorporate threshold effects. Additionally, the model can be extended to include: Multiple vibrational levels, overlapping resonances, and magnetic dipole-dipole interactions. This versatile model allows for a description of Feshbach resonances in a large variety of systems without accurate knowledge of the relevant molecular Born-Oppenheimer potentials. Being computationally light, the ABM can be used to analyze measurements of Feshbach resonances and hereby expose the basic underlying molecular structure. Additionally, the ABM is well suited to map out all available Feshbach resonances in a system. This ability is particularly useful when an optimal resonance is required for a certain experiment. For the 6Li-40K mixture we have utilized this ability to experimentally observe a strong Feshbach resonance which offers promising perspectives to reach the strongly interacting universal regime in a mass-imbalanced Fermi gas. By analyzing the various inelastic decay processes and the accuracy of the ab initio Born-Oppenheimer potentials, we predict a strong resonance in a mixture of metastable bosonic and fermionic helium atoms. For the homonuclear (bosonic as well as fermionic) helium gas we do not find strong resonances. For an ultracold gas of 40K atoms, an elaborate combined theoretical and experimental effort is presented to map out the Feshbach resonance structure. On the experimental side we observed 36 Feshbach resonances, 31 of which where observed for the first time. These observations are in excellent agreement with coupled-channels calculations which use unmodified Born-Oppenheimer potentials as input. On the theoretical side, we present a comparison between different simplified models of resonance scattering and the full numerical coupled-channels calculations. We have developed the resonant state model (RSM) to describe the resonant scattering of two ultracold atoms. This model is founded on resonant states, the natural (eigen) frequencies of the two-body system, and their non-trivial properties. Threshold effects are implicitly build into the RSM. The RSM can describe the effect of the continuum of scattering states, states which are neglected in the ABM, on the position and strength of a Feshbach resonance by only a few resonant states. Especially when a low-energy resonance is embedded in the continuum of scattering states, the effect of these states on the Feshbach resonance cannot be neglected. The RSM can be viewed as the natural generalization of the ABM. As the RSM can be based on just a few resonant states, it retains the computational simplicity of the ABM. Using the tunability of two-body interactions induced by a Feshbach resonance, we describe a crossover for a rotating Fermi gas between two strongly-correlated many-body states. We demonstrate that the correlation induced long-range interaction will shift the crossover with respect to the non-rotating case.