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arXiv:1110.5633 · Physical Review A

Scattering lengths of calcium and barium isotopes

Umakanth Dammalapati, Lorenz Willmann, Steven Knoop

Published Nov 21, 2011• 4 citations
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Abstract

We have calculated the $s$-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) in order to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecular potential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopes alternative to the recently Bose condensed ${}^{40}$Ca that suffers strong losses because of a very large scattering length. For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths, opening the possibility of BEC for at least one of the isotopes.

DOI
10.1103/physreva.84.054703

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We have calculated the sss-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) in order to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecular potential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopes alternative to the recently Bose condensed 40{}^{40}40Ca that suffers strong losses because of a very large scattering length. For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths, opening the possibility of BEC for at least one of the isotopes.
The s-wave scattering lengths are obtained for all stable isotope combinations of Ca atoms and those values are very sensitive to small changes of the reduced mass. These results rely on a recent measurement of the 40 Ca– 40 Ca scattering length in a Bose condensate of 40 Ca and making small corrections to a Born–Oppenheimer potential that reproduces all known spectroscopic data. We find that the 42 Ca isotope has a scattering length of 417.5 a 0 and, thus, can be Bose condensed. The large interisotope scattering length between 43 Ca and 42 Ca indicates that sympathetic cooling of 43 Ca by 42 Ca atoms will be effective, and 43 Ca– 42 Ca can also be used to study isotopic Bose–Fermi mixtures. We also predict a d-wave shape resonance for collisions of fermionic 43 Ca atoms in different spin components.
We report on the attainment of Bose-Einstein condensation of 86Sr^{86}\mathrm{Sr}86Sr. This isotope has a scattering length of about +800a0+800{a}_{0}+800a0​ and thus suffers from fast three-body losses. To avoid detrimental atom loss, evaporative cooling is performed at low densities around $3\ifmmode\times\else\texttimes\fi{}{10}^{12}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ in a large volume optical dipole trap. We obtain almost pure condensates of $5\ifmmode\times\else\texttimes\fi{}{10}^{3}$ atoms.
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 have achieved Bose-Einstein condensation of ;{40}Ca, the first for an alkaline earth element. The influence of elastic and inelastic collisions associated with the large ground-state s-wave scattering length of ;{40}Ca was measured. From these findings, an optimized loading and cooling scheme was developed that allowed us to condense about 2 x 10;{4} atoms after laser cooling in a two-stage magneto-optical trap and subsequent forced evaporation in a crossed dipole trap within less than 3 s. The condensation of an alkaline earth element opens novel opportunities for precision measurements on the narrow intercombination lines as well as investigations of molecular states at the ;{1}S-;{3}P asymptotes.
The s-wave scattering length for the Positronium-Positronium interaction is estimated semi-phenomenologically with the long-range van der Waals force and the short-range repulsive potential that represents the hard core between two positronium. The obtained value of the scattering length is a∼0.44nma \sim 0.44{\rm nm}a∼0.44nm, and its stability is also checked for different parametrizations. Using this value, the Gross-Pitaevskii equation can be fixed for the Positronium Bose-Einstein condensates (Ps BEC). The static properties of Ps BEC are studied from the solutions of that equation. The phase-transition temperature shift of the Ps BEC due to the Ps-Ps interaction is also evaluated with the mean-field approximation.
We present a theoretical analysis of excitation modes in Bose-Einstein condensates in ultracold alkali-metal gases for large scattering lengths and momenta where corrections to the mean field approximation become important. We assume that the effective interaction in the metastable, single channel, gaseous phase has a well defined Fourier transform that scales with the scattering length. Based on this we show that for increasing scattering lengths or equivalently increasing densities the system becomes less correlated and that at large values of the scattering length Bragg scattering measures directly the Fourier transform of the effective two-body potential. We construct model potentials which fit the recently measured line shifts in 85^{85}85Rb by Papp et al. (Phys. Rev. Lett. {\bf 101}, 135301 (2008)), and show that they fix the low momentum expansion of the effective range function. We find excellent agreement with the experimental data when the effective range is ≪1\ll 1≪1 and the coefficient of the k4k^4k4-term is −7.5±0.5-7.5 \pm 0.5−7.5±0.5 in scattering length units. The resolution in Bragg scattering experiments so far does not reveal details of the frequency dependence in the dynamic structure function S(k,ω)S(k,ω)S(k,ω) and we show that the Feynman spectrum determines the measured line shifts. We propose the possibility of a transition to a novel density wave state.