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Physical Review Letters

Collisional Stability of Double Bose Condensates

Paul S. Julienne, Frederick H. Mies, Eite Tiesinga, Carl J. Williams

Published Mar 10, 1997• 123 citations
Publisher

Abstract

This paper presents quantitative calculations of collisional loss rates for pure $^{23}N\mathrm{a}$ or $^{87}R\mathrm{b}$ atoms in a binary mixture of hyperfine states $|F,M〉$. We find that the recent observation of a dual Bose condensate consisting of $^{87}R\mathrm{b}$ $|1,\ensuremath{-}1〉$ and $|2,2〉$ atoms is unique and will not occur in Na. We attribute this to the unexpectedly small inelastic spin-exchange rate associated with $^{87}R\mathrm{b}$ s-wave hyperfine collisions. This fortuitous result arises from nearly equal $(5.45\ifmmode\pm\else\textpm\fi{}0.26\mathrm{nm})$ scattering lengths for two colliding $^{87}R\mathrm{b}$ atoms in their $|1,\ensuremath{-}1〉$ or $|2,2〉$ states.

DOI
10.1103/physrevlett.78.1880

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This paper presents quantitative calculations of collisional loss rates for pure 23Na^{23}N\mathrm{a}23Na or 87Rb^{87}R\mathrm{b}87Rb atoms in a binary mixture of hyperfine states ∣F,M〉|F,M〉∣F,M〉. We find that the recent observation of a dual Bose condensate consisting of 87Rb^{87}R\mathrm{b}87Rb ∣1,\ensuremath−1〉|1,\ensuremath{-}1〉∣1,\ensuremath−1〉 and ∣2,2〉|2,2〉∣2,2〉 atoms is unique and will not occur in Na. We attribute this to the unexpectedly small inelastic spin-exchange rate associated with 87Rb^{87}R\mathrm{b}87Rb s-wave hyperfine collisions. This fortuitous result arises from nearly equal (5.45\ifmmode±\else\textpm\fi0.26nm)(5.45\ifmmode\pm\else\textpm\fi{}0.26\mathrm{nm})(5.45\ifmmode±\else\textpm\fi0.26nm) scattering lengths for two colliding 87Rb^{87}R\mathrm{b}87Rb atoms in their ∣1,\ensuremath−1〉|1,\ensuremath{-}1〉∣1,\ensuremath−1〉 or ∣2,2〉|2,2〉∣2,2〉 states.
We study the elastic scattering length and inelastic decay rate associated with collisions of 23Na^{23}\mathrm{Na}23Na and 87Rb^{87}\mathrm{Rb}87Rb atoms in different hyperfine states: one with atoms in the |f=2,mf{\mathrm{m}}_{\mathrm{f}}mf​=2〉 and one with atoms in the |f=1,mf{\mathrm{m}}_{\mathrm{f}}mf​=-1〉 state. For Na the real part of the ${\mathrm{a}}_{(22)+(1\mathrm{\ensuremath{-}}1)}$ scattering length for |22〉+|1-1〉 collisions is predicted to be positive and equal to 65\ifmmode\pm\else\textpm\fi{}5 Bohr length units a0{\mathrm{a}}_{0}a0​ for small B. The zero-temperature low-field decay rate is of the order of a typical exchange rate: ${\mathrm{G}}_{(22)+(1\mathrm{\ensuremath{-}}1)}$=(1.5\ifmmode\pm\else\textpm\fi{}0.7)${10}^{\mathrm{\ensuremath{-}}11}$ cm3{\mathrm{cm}}^{3}cm3 ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$, showing that a two-condensate experiment is not feasible for a Na gas sample in a static magnetic trap. For the case of Rb atoms the present knowledge of the singlet interaction does not allow a similar calculation. The extreme suppression of ${\mathrm{G}}_{(22)+(1\mathrm{\ensuremath{-}}1)}$ demonstrated by a recent experiment is shown to be very restrictive for the value of the singlet accumulated phase.
We studied spin-dependent two-body inelastic collisions in F=2F=2F=2 87Rb^{87}\text{R}\text{b}87Rb Bose-Einstein condensates both experimentally and theoretically. The 87Rb^{87}\text{R}\text{b}87Rb condensates were confined in an optical trap and selectively prepared in various spin states in the F=2F=2F=2 manifold at a magnetic field of 3.0 G. The measured atom loss rates depend on the spin states of colliding atoms. We measured two fundamental loss coefficients for two-body inelastic collisions with total spins of 0 and 2. The loss coefficients determine the loss rates of all the spin pairs. The experimental results for mixtures of all spin combinations are in good agreement with numerical solutions of the Gross-Pitaevskii equations that include the effect of a magnetic field gradient.
Recently, coupled systems of nonlinear Schrödinger equations have been used extensively to describe mixtures Bose–Einstein condensates. In this paper, we study the distribution of two different hyperfine spin states of a binary mixture of three dimensional Bose–Einstein condensates. In a double condensate, an interface may occur due to large intraspecies and interspecies scattering lengths. We prove that there is an asymptotic separation of different phases in the strong coupling (Thomas–Fermi) limit.
A new apparatus featuring a double magneto-optic trap and an Ioffe-type magnetic trap was used to create condensates of $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ atoms in either of the ∣F=2,m=2〉|F\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2,m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2〉∣F=2,m=2〉 or $|F\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1〉$ spin states of 87{}^{87}87Rb. Overlapping condensates of the two states were also created using nearly lossless sympathetic cooling of one state via thermal contact with the other evaporatively cooled state. We observed that (i) the scattering length of the $|1,\ensuremath{-}1〉$ state is positive, (ii) the rate constant for binary inelastic collisions between the two states is $2.2(9)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}{\mathrm{cm}}^{3}/\mathrm{s}$, and (iii) there is a repulsive interaction between the two condensates. Similarities and differences between the behaviors of the two spin states are observed.
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.
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.