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Physical Review A

Simple model for predicting and analyzing magnetically induced Feshbach resonances

Jing-Lun Li, Xue‐Jin Hu, Gao‐Ren Wang, Yong‐Chang Han, Shu‐Lin Cong

Published Apr 23, 2015• 5 citations
Publisher

Abstract

We present a simple theoretical model for describing ultracold-atomic collision dynamics and Feshbach resonances (FRs) without onerous numerical computation. An effective quantum-defect phase shift ${\ensuremath{\delta}}_{\mathrm{Y}}^{\mathrm{eff}}$ is derived and the reference potentials are shown acting as an assistant tool in the calculation. With the help of singlet and triplet scattering lengths ${a}^{\mathrm{S}}$ and ${a}^{\mathrm{T}}$, as well as an adjustable parameter ${\ensuremath{\gamma}}_{\mathrm{sr}}$, this model can be used to reliably predict FRs. An application example of $^{6}\mathrm{Li}\ensuremath{-}^{40}\mathrm{K}$ collision shows that the results given by the simple model agree quite well with both the relevant experimental data and the coupled-channel calculations.

DOI
10.1103/physreva.91.042708

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We present a simple theoretical model for describing ultracold-atomic collision dynamics and Feshbach resonances (FRs) without onerous numerical computation. An effective quantum-defect phase shift \ensuremathδYeff{\ensuremath{\delta}}_{\mathrm{Y}}^{\mathrm{eff}}\ensuremathδYeff​ is derived and the reference potentials are shown acting as an assistant tool in the calculation. With the help of singlet and triplet scattering lengths aS{a}^{\mathrm{S}}aS and aT{a}^{\mathrm{T}}aT, as well as an adjustable parameter \ensuremathγsr{\ensuremath{\gamma}}_{\mathrm{sr}}\ensuremathγsr​, this model can be used to reliably predict FRs. An application example of 6Li\ensuremath−40K^{6}\mathrm{Li}\ensuremath{-}^{40}\mathrm{K}6Li\ensuremath−40K collision shows that the results given by the simple model agree quite well with both the relevant experimental data and the coupled-channel calculations.
We present a theoretical model for describing ultracold atomic collision based on the multichannel quantum-defect theory. An analytical expression for the quantum-defect matrix Y is derived for the truncated 1/R61/{R}^{6}1/R6 model potential, and threshold behaviors of the quantum-defect parameters are discussed. The model allows us to predict the Feshbach resonances when the singlet and triplet scattering lengths are known or to fit the scattering lengths when a couple of Feshbach resonances is located experimentally. As an illustrative example, we investigate the 6{}^{6}6Li-40{}^{40}40K scattering. The theoretical results agree quite well with experimental values.
We have developed a model of Feshbach resonances in gases of ultracold alkali-metal atoms using the ideas of multichannel quantum-defect theory. Our model requires just three parameters describing the interactions---the singlet and triplet scattering lengths, and the long-range van der Waals coefficient---in addition to known atomic properties. Without using any further details of the interactions, our approach can accurately predict the locations of resonances. It can also be used to find the singlet and triplet scattering lengths from measured resonance data. We apply our technique to 6Li-40K^{6}\text{L}\text{i-}^{40}\text{K}6Li-40K and $^{40}\text{K}\text{\ensuremath{-}}^{87}\text{R}\text{b}$ scattering, obtaining good agreement with experimental results, and with the more computationally intensive coupled channels technique.
Feshbach resonances in ultracold collisions often result from an interplay between many collision channels. Simple two-channel models can be introduced to capture the basic features, but cannot fully reproduce the situation when several resonances from different closed channels contribute to the scattering process. Using the formalism of multichannel quantum defect theory we develop an analytical model of overlapping Feshbach resonances. We find a general formula for the variation of the scattering length with magnetic field in the vicinity of an arbitrary number of resonances, characterized by simple parameters. Our formula is in excellent agreement with numerical coupled-channel calculations for several cases of overlapping resonances in the collisions of two 7{}^{7}7Li atoms or two Cs atoms.
We develop an analytical model for ultracold atom–ion collisions using the multichannel quantum-defect formalism. The model is based on analytical solutions of the r −4 long-range potential and on the application of a frame transformation between asymptotic and molecular bases. This approach allows the description of atom–ion interaction in the ultracold domain in terms of only three parameters: the singlet and triplet scattering lengths, assumed to be independent of the relative motion angular momentum, and the lead dispersion coefficient of the asymptotic potential. We also introduce corrections to the scattering lengths that improve the accuracy of our quantum-defect model for higher-order partial waves, a particularly important result for an accurate description of shape and Feshbach resonances at finite temperature. The theory is applied to the system composed of a 40 Ca + ion and a Na atom, and compared with numerical coupled-channel calculations carried out using ab initio potentials. For this particular system, we investigate the spectrum of bound states, the rate of charge-transfer processes and the collision rates in the presence of magnetic Feshbach resonances at zero and finite temperature.
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 present an analysis of experimentally accessible magnetic Feshbach resonances in ultracold heteronuclear $^{85}\mathrm{Rb}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ and $^{6}\mathrm{Li}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ mixtures. Using recent experimental measurements of the triplet scattering lengths for $^{6}\mathrm{Li}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ and $^{7}\mathrm{Li}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ mixtures and Feshbach resonances for one combination of atomic states, we create model potential curves and fine-tune them to reproduce the measured resonances and to predict the location of several experimentally relevant resonances in Li-Rb collisions. To model $^{85}\mathrm{Rb}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ collisions, we use accurate Rb2{\mathrm{Rb}}_{2}Rb2​ potentials obtained previously from the analysis of experiments on $^{87}\mathrm{Rb}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ collisions. We find resonances that occur at very low magnetic fields, below 10G10\phantom{\rule{0.3em}{0ex}}\mathrm{G}10G, which may be useful for entanglement generation in optical lattices or atom chip magnetic traps.