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Journal of Research of the National Institute of Standards and Technology

Treasure of the past X - A spectroscopic determination of scattering lengths for sodium atom collisions (Reprinted from the Journal of Research of the National Bureau of Standards, vol 101, pg 505-520, 1996

Eite Tiesinga, Carl J. Williams, Paul S. Julienne, K. M. Jones, Paul D. Lett, William D. Phillips

Published Jan 1, 2002• 1 citation
Publisher

Abstract

We report a preliminary value for the zero magnetic field Na (2)S(f = 1, m = - 1) + Na (2)S(f = 1, m = - 1) scattering length, a 1,-1. This parameter describes the low-energy elastic two-body processes in a dilute gas of composite bosons and determines, to a large extent, the macroscopic wavefunction of a Bose condensate in a trap. Our scattering length is obtained from photoassociative spectroscopy with samples of uncondensed atoms. The temperature of the atoms is sufficiently low that contributions from the three lowest partial waves dominate the spectrum. The observed lineshapes for the purely long-range [Formula: see text] molecular state enable us to establish key features of the ground state scattering wavefunction. The fortuitous occurrence of a p-wave node near the deepest point (R e = 72 a 0) of the [Formula: see text] potential curve is instrumental in determining a 1,-1 = (52 ± 5) a 0 and a 2.2 = (85 ± 3) a 0, where the latter is for a collision of two Na (2)S(f = 2, m = 2) atoms.

DOI
10.6028/jres.107.011

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We report a preliminary value for the zero magnetic field Na (2)S(f = 1, m = - 1) + Na (2)S(f = 1, m = - 1) scattering length, a 1,-1. This parameter describes the low-energy elastic two-body processes in a dilute gas of composite bosons and determines, to a large extent, the macroscopic wavefunction of a Bose condensate in a trap. Our scattering length is obtained from photoassociative spectroscopy with samples of uncondensed atoms. The temperature of the atoms is sufficiently low that contributions from the three lowest partial waves dominate the spectrum. The observed lineshapes for the purely long-range [Formula: see text] molecular state enable us to establish key features of the ground state scattering wavefunction. The fortuitous occurrence of a p-wave node near the deepest point (R e = 72 a 0) of the [Formula: see text] potential curve is instrumental in determining a 1,-1 = (52 ± 5) a 0 and a 2.2 = (85 ± 3) a 0, where the latter is for a collision of two Na (2)S(f = 2, m = 2) atoms.
We report a preliminary value for the zero magnetic field Na 2 S(f = 1, m = Ϫ 1) + Na 2 S(f = 1, m = Ϫ 1) scattering length, a1,Ϫ1.This parameter describes the lowenergy elastic two-body processes in a dilute gas of composite bosons and determines, to a large extent, the macroscopic wavefunction of a Bose condensate in a trap.Our scattering length is obtained from photoassociative spectroscopy with samples of uncondensed atoms.The temperature of the atoms is sufficiently low that contributions from the three lowest partial waves dominate the spectrum.The observed lineshapes for the purely long-range 0g Ϫ molecular state enable us to establish key features of the ground state scattering wavefunction.The fortuitous occurrence of a p -wave node near the deepest point (Re = 72 a0) of the 0g Ϫ potential curve is instrumental in determining a1,Ϫ1 = (52 Ϯ 5) a0 and a2,2 = (85 Ϯ 3) a0, where the latter is for a collision of two Na 2 S(f = 2, m = 2) atoms.
The scattering lengths that characterize the low-energy elastic scattering of 23 Na atoms by 39 K, 40 K and 41 K atoms, when they interact via the X 1 Σ + and a 3 Σ + states of the NaK dimer, are calculated. The systematic variation of the short- and long-range parts of the interaction potentials reveals that the calculated scattering lengths are sensitive to uncertainties in the potentials. It is concluded that precise spectroscopic measurements of the least-bound ro-vibrational levels supported by the ground electronic state, which would reduce the uncertainties in both the molecular potentials, are needed for an accurate determination of the scattering lengths.
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 ${a}_{1,\ensuremath{-}1}(B=0)=(55.4\ifmmode\pm\else\textpm\fi{}{1.2)a}_{0}.$
In photoassociation spectroscopy, the line intensities of a given vibrational progression exhibit zero-signal modulation reflecting the node structure of the sss-wave ground state wave function of two free colliding atoms. This leads to the determination of the scattering length. We performed photoassociation of cold Cs atoms polarized in the Zeeman sublevel f=4f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4f=4, mf=4{m}_{f}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4mf​=4. We analyzed the intensities of the lines associated with the Cs2{\mathrm{Cs}}_{2}Cs2​ ${0}_{g}^{\ensuremath{-}}$ state dissociating to the 6s1/2+6p3/2{6s}_{1/2}{+6p}_{3/2}6s1/2​+6p3/2​ asymptote. This yields a value of the Cs triplet state scattering length, ${a}_{T}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}530a}_{0}$, while consistency requirements impose a value of the multipole ground state molecular coefficient, C6=6510{C}_{6}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6510C6​=6510 a.u.
A simple analytical formula, a=a\ifmmode\bar\else\textasciimacron\fi{}[1-tan(\ensuremath{\pi}/n-2)tan{\ensuremath{\Phi}-[\ensuremath{\pi}/2(n-2)]}, is obtained for the scattering length in atomic collisions. Here a\ifmmode\bar\else\textasciimacron\fi{}=cos[\ensuremath{\pi}/(n-2)]{ \ensuremath{\surd}2M\ensuremath{\alpha} /[\ensuremath{\Elzxh}(n-2)]${\mathrm{}}}^{2/(\mathit{n}\mathrm{\ensuremath{-}}2)}$[\ensuremath{\Gamma}(n-3)/(n-2)]/[T(n-1)/(n-2)] is the mean scattering length determined by the asymptotic behavior of the potential U(r)\ensuremath{\sim}-\ensuremath{\alpha}/Rn{\mathit{R}}^{\mathit{n}}Rn, (n=6 for atom-atom scattering or n=4 for ion-atom scattering), M is the reduced mass of the atoms, and \ensuremath{\Phi} is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of a\ifmmode\bar\else\textasciimacron\fi{}, the average scattering length, also determines the slope of the s-wave phase shifts beyond the near-threshold region. The formula is applicable to the collisions of atoms cooled down in traps, where the scattering length determines the character of the atom-atom interaction. Our calculation shows that repulsion between atoms (a>0) is more likely than attraction with a ``probability'' of 75%. For the Cs-Cs scattering in the $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathit{u}}$ state, a\ifmmode\bar\else\textasciimacron\fi{}=95.5aB{\mathit{a}}_{\mathit{B}}aB​ has been obtained, where aB{\mathit{a}}_{\mathit{B}}aB​ is the Bohr radius. The comparison of the calculated cross-section energy dependence with the experimental data gives evidence for a positive value for the Cs-Cs scattering length, which makes cesium Bose gas stable.
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.