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Physical Review A

Role of collisions in creation of overlapping Bose condensates

S. J. J. M. F. Kokkelmans, H. M. J. M. Boesten, BOUDEWIJN J. VERHAAR

Published Mar 1, 1997• 39 citations
Publisher

Abstract

We study the elastic scattering length and inelastic decay rate associated with collisions of $^{23}\mathrm{Na}$ and $^{87}\mathrm{Rb}$ atoms in different hyperfine states: one with atoms in the |f=2,${\mathrm{m}}_{\mathrm{f}}$=2〉 and one with atoms in the |f=1,${\mathrm{m}}_{\mathrm{f}}$=-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 ${\mathrm{a}}_{0}$ 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)${10}^{\mathrm{\ensuremath{-}}11}$ ${\mathrm{cm}}^{3}$ ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$, 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.

DOI
10.1103/physreva.55.r1589

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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 a(22)+(1\ensuremath−1){\mathrm{a}}_{(22)+(1\mathrm{\ensuremath{-}}1)}a(22)+(1\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: G(22)+(1\ensuremath−1){\mathrm{G}}_{(22)+(1\mathrm{\ensuremath{-}}1)}G(22)+(1\ensuremath−1)​=(1.5\ifmmode\pm\else\textpm\fi{}0.7)10\ensuremath−11{10}^{\mathrm{\ensuremath{-}}11}10\ensuremath−11 cm3{\mathrm{cm}}^{3}cm3 s\ensuremath−1{\mathrm{s}}^{\mathrm{\ensuremath{-}}1}s\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 G(22)+(1\ensuremath−1){\mathrm{G}}_{(22)+(1\mathrm{\ensuremath{-}}1)}G(22)+(1\ensuremath−1)​ demonstrated by a recent experiment is shown to be very restrictive for the value of the singlet accumulated phase.
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 $|1,\ensuremath{-}1〉$ 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 $(5.45\ifmmode\pm\else\textpm\fi{}0.26\mathrm{nm})$ scattering lengths for two colliding 87Rb^{87}R\mathrm{b}87Rb atoms in their $|1,\ensuremath{-}1〉$ or ∣2,2〉|2,2〉∣2,2〉 states.
A computation of $^{87}\mathrm{\ensuremath{-}}^{87}$Rb spin-exchange rate constants as functions of the two-body singlet scattering length as{\mathrm{a}}_{\mathrm{s}}as​ demonstrates that the inelastic collision rate is suppressed over a small range of the possible values for as{\mathrm{a}}_{\mathrm{s}}as​. A two-channel model relates this inelastic suppression to an interference phenomenon, manifested in the near coincidence of the singlet and triplet scattering lengths. This mechanism explains the diminished rates measured in recent ``double-trap'' experiments. Combining information extracted from these rates and from previous scattering length measurements allows us to place bounds on the 87Rb^{87}\mathrm{Rb}87Rb singlet scattering length (74--102 a.u.).
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.
A computation of $^{87}\mathrm{\ensuremath{-}}^{87}$Rb spin-exchange rate constants as functions of the two-body singlet scattering length as{\mathrm{a}}_{\mathrm{s}}as​ demonstrates that the inelastic collision rate is suppressed over a small range of the possible values for as{\mathrm{a}}_{\mathrm{s}}as​. A two-channel model relates this inelastic suppression to an interference phenomenon, manifested in the near coincidence of the singlet and triplet scattering lengths. This mechanism explains the diminished rates measured in recent ``double-trap'' experiments. Combining information extracted from these rates and from previous scattering length measurements allows us to place bounds on the 87Rb^{87}\mathrm{Rb}87Rb singlet scattering length (74--102 a.u.).
In this thesis we numerically compute the scattering lengths and bound states for sodium-rubidium collisions at low energy. This work was motivated by experiments which aim to produce Bose-Einstein condensates (BEC) mixtures of sodium-rubidium. Elastic collision properties are important for the rethermalization of the atoms during the evaporative cooling process. Inelastic processes, which we also discuss to some extent, cause trap losses in those experiments. In order to reach the required temperature and density the elastic collision rates should be sufficiently large compared to the inelastic rates. The scattering lengths, which completely specify the elastic collision parameters at low energy, determine the miscibility and phase diagram of the sodium-rubidium condensate mixture. We calculate the scattering lengths approximately and find agreement with previous calculations indicating that miscible phases of sodium and rubidium condensates do not appear to be feasible in the absence of external fields.
We review the information on the interactions between cold ground-state Na atoms following from Na2{\mathrm{Na}}_{2}Na2​ bound states, a Na photoassociation experiment, and a recent observation of field-induced Feshbach resonances in a Na Bose condensate. We obtain a set of Na interaction parameters that is in excellent agreement with all these experimental data. The existing discrepancy between different values for the scattering length ${a}_{1,\ensuremath{-}1}$ of Na atoms in the $|1,\ensuremath{-}1〉$ hyperfine state is resolved. The present interaction parameters enable us to give accurate predictions for the singlet and triplet scattering lengths aS{a}_{S}aS​ and aT,{a}_{T},aT​, ${a}_{1,\ensuremath{-}1},$ scattering lengths for collisions between Na atoms in other hyperfine states, and resonance fields of several yet unobserved Feshbach resonances. We find ${a}_{S}=(19.1\ifmmode\pm\else\textpm\fi{}{2.1)a}_{0},{a}_{T}=(65.3\ifmmode\pm\else\textpm\fi{}{0.9)a}_{0},$ and ${a}_{1,\ensuremath{-}1}(B=0)=(55.4\ifmmode\pm\else\textpm\fi{}{1.2)a}_{0}.$