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Physical Review A

Potassium scattering lengths and prospects for Bose-Einstein condensation and sympathetic cooling

Robin Côté, A. Dalgarno, H. Wang, William C. Stwalley

Published Jun 1, 1998• 54 citations
Publisher

Abstract

We have determined the scattering lengths for collisions between the different isotopes of two potassium atoms in singlet and triplet molecular states of ${\mathrm{K}}_{2}$. We show that for the two bosonic species ${}^{39}\mathrm{K}$ and ${}^{41}\mathrm{K}$ the scattering lengths are positive, hence, leading to stable condensates. The fermionic isotope ${}^{40}\mathrm{K}$ also has positive scattering lengths, but the triplet value is nearly zero, hence, a quasi-non-interacting fermionic gas. We also give results for the interspecies collisions. We conclude that double condensates of ${}^{39}\mathrm{K}$ and ${}^{41}\mathrm{K}$ are possible for purely spin-polarized atoms, and sympathetic cooling between the bosonic ${}^{39}\mathrm{K}$ and the fermionic ${}^{40}\mathrm{K}$ will be efficient due to low spin-flip rates and a sizable interspecies scattering length.

DOI
10.1103/physreva.57.r4118

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We have determined the scattering lengths for collisions between the different isotopes of two potassium atoms in singlet and triplet molecular states of K2{\mathrm{K}}_{2}K2​. We show that for the two bosonic species 39K{}^{39}\mathrm{K}39K and 41K{}^{41}\mathrm{K}41K the scattering lengths are positive, hence, leading to stable condensates. The fermionic isotope 40K{}^{40}\mathrm{K}40K also has positive scattering lengths, but the triplet value is nearly zero, hence, a quasi-non-interacting fermionic gas. We also give results for the interspecies collisions. We conclude that double condensates of 39K{}^{39}\mathrm{K}39K and 41K{}^{41}\mathrm{K}41K are possible for purely spin-polarized atoms, and sympathetic cooling between the bosonic 39K{}^{39}\mathrm{K}39K and the fermionic 40K{}^{40}\mathrm{K}40K will be efficient due to low spin-flip rates and a sizable interspecies scattering length.
We determine the interspecies sss-wave triplet scattering length a3{a}_{3}a3​ for all K-Rb isotopic mixtures by measuring the elastic cross section for collisions between 41K^{\mathrm{41}}\mathrm{K}41K and 87Rb^{87}\mathrm{R}\mathrm{b}87Rb in different temperature regimes. The positive value ${a}_{3}=+{163}_{\ensuremath{-}12}^{+57}{a}_{0}$ ensures the stability against collapse of binary $^{41}\mathrm{K}\ensuremath{-}^{87}\mathrm{R}\mathrm{b}$ Bose-Einstein condensates. For the fermion-boson mixture $^{40}\mathrm{K}\ensuremath{-}^{87}\mathrm{R}\mathrm{b}$, we obtain a large and negative scattering length which implies an efficient sympathetic cooling of the fermionic species down to the degenerate regime.
The hyperfine-state-selected scattering properties of potassium atoms at ultralow temperatures are calculated using interaction potentials gleaned from an analysis of recent photoassociation data. We predict that the small, probably negative value of the 39K{}^{39}\mathrm{K}39K triplet scattering length will hamper efforts to produce a Bose-Einstein condensate, unless experiments utilize a broad, accessible magnetic Feshbach resonance. The large positive value calculated for the 41K{}^{41}\mathrm{K}41K triplet scattering length makes it a better candidate for condensation at zero magnetic field. The fermionic isotope 40K{}^{40}\mathrm{K}40K is also predicted to have a large, positive scattering length for elastic collisions between spin states of experimental interest, implying that it can be efficiently evaporatively cooled to the quantum degenerate regime. In addition, certain spin states possess Feshbach resonances that may enable tuning of its interatomic interactions, possibly leading to the formation of Cooper pairs.
We calculate binary collision parameters for mixtures of ultracold gases of 85Rb{}^{85}\mathrm{Rb}85Rb and 87Rb{}^{87}\mathrm{Rb}87Rb atoms. We predict a large positive triplet scattering length for mixed-isotope collisions, which implies that a stable mixed-isotope double condensate can be formed by sympathetic evaporative cooling. Moreover, collisions between 85Rb{}^{85}\mathrm{Rb}85Rb and 87Rb{}^{87}\mathrm{Rb}87Rb atoms in the lowest hyperfine manifold exhibit magnetic-field-induced Feshbach resonances which should allow partial control of the interaction between condensates. These considerations make Rb an ideal system for the study of interpenetrating bosonic superfluids.
We use double-resonance photoassociative spectroscopy of ultracold 39K{}^{39}\mathrm{K}39K atoms to precisely determine the triplet $a{}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ scattering length for the various isotopes of potassium. Photoassociation of free 39K{}^{39}\mathrm{K}39K atoms to the pure long-range ${0}_{g}^{\ensuremath{-}}{(v}^{\ensuremath{'}}=0,{J}^{\ensuremath{'}}=2)$ level is followed by stimulated emission to high-lying levels of the $a{}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ potential. The binding energies of levels within 5 GHz below the lowest ground-state 4s1/2(fa=1)+4s1/2(fb=1){4s}_{1/2}{(f}_{a}{=1)+4s}_{1/2}{(f}_{b}=1)4s1/2​(fa​=1)+4s1/2​(fb​=1) hyperfine asymptote are measured by both trap loss and ionization detection. The locations of these near-threshold hyperfine-coupled molecular levels allow us to constrain the triplet potential and thereby determine the triplet scattering length. The result for 39K,{}^{39}\mathrm{K},39K, ${a}_{t}=\ensuremath{-}33\ifmmode\pm\else\textpm\fi{}{5a}_{0}$ (1a0=0.0529177nm),{(1a}_{0}=0.0529177 \mathrm{nm}),(1a0​=0.0529177nm), is a factor of $\ensuremath{\sim}5$ improvement over previous determinations and establishes that a large 39K{}^{39}\mathrm{K}39K Bose-Einstein condensate will not be stable.
We have determined properties of cold 39K^{39}\mathrm{K}39K+39K^{39}\mathrm{K}39K and 41K^{41}\mathrm{K}41K+41K^{41}\mathrm{K}41K collisions by a multichannel inverted perturbation approach applied to spectroscopic data of highly excited 39K^{39}\mathrm{K}39K singlet and triplet bound states. We predict positive scattering lengths for the 39K^{39}\mathrm{K}39K+39K^{39}\mathrm{K}39K and 41K^{41}\mathrm{K}41K+41K^{41}\mathrm{K}41K singlet potentials, as well as for the 41K^{41}\mathrm{K}41K+41K^{41}\mathrm{K}41K triplet potential, and a negative value for 39K^{39}\mathrm{K}39K+39K^{39}\mathrm{K}39K triplet scattering. From a study of the magnetic field dependence we conclude that 41K^{41}\mathrm{K}41K may be a bosonic atom with a Feshbach resonance in the magnetic field range where these atoms can be trapped in the f=1f=1f=1, ${m}_{f}=\ensuremath{-}1$ hyperfine state, thus providing a possibility to switch between stable and instable condensates in a Bose-Einstein condensation experiment.
We report the calculation of the interspecies scattering length for the sodium-rubidium (Na-Rb) system. We present improved hybrid potentials for the singlet ${X}^{1}\ensuremath{\Sigma}{}^{+}$ and triplet ${a}^{3}{\ensuremath{\Sigma}}^{+}$ ground states of the NaRb molecule, and calculate the singlet and triplet scattering lengths as{a}_{s}as​ and at{a}_{t}at​ for the isotopomers 23Na87Rb{}^{23}{\mathrm{Na}}^{87}\mathrm{Rb}23Na87Rb and 23Na85Rb.{}^{23}{\mathrm{Na}}^{85}\mathrm{Rb}.23Na85Rb. Using these values, we assess the prospects for producing a stable two-species Bose-Einstein condensate in the Na-Rb system.