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Physical Review A

Determination of the scattering lengths of 39K from 1u photoassociation line shapes

Carl J. Williams, Eite Tiesinga, Paul S. Julienne, H. Wang, William C. Stwalley, P. L. Gould

Published Dec 1, 1999• 34 citations
Publisher

Abstract

We report threshold scattering properties for ${}^{39}\mathrm{K}(f=1)$ atoms obtained from analysis of experimental photoassociation spectra of the purely long-range ${}^{39}{\mathrm{K}}_{2} {1}_{u}$ state. The existence of hyperfine structure in the excited ${\mathrm{K}}_{2} {1}_{u}$ state and a knowledge of the Hamiltonian structure for two colliding ground state alkali atoms enables us to obtain information on the sign and magnitudes of certain scattering lengths for ground state collisions. Analysis of the experimental photoassociation line shapes shows that the ${}^{39}\mathrm{K}$ scattering lengths are between $\ensuremath{-}3.2$ and $+0.8\mathrm{nm}(\ensuremath{-}{60a}_{0}$ and ${+15a}_{0},$ where ${1a}_{0}=1\mathrm{a}.\mathrm{u}.=0.0529177\mathrm{nm})$ for the $a{}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ molecular state, between +4.8 and +12 nm ${(+90a}_{0}$ and ${+230a}_{0})$ for the ${X}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ state, and between $\ensuremath{-}7.9$ and $+1.1\mathrm{nm} (\ensuremath{-}{150a}_{0}$ and ${+20a}_{0})$ for collisions of two $f=1,m=\ensuremath{-}1$ atoms.

DOI
10.1103/physreva.60.4427

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We report threshold scattering properties for 39K(f=1){}^{39}\mathrm{K}(f=1)39K(f=1) atoms obtained from analysis of experimental photoassociation spectra of the purely long-range 39K21u{}^{39}{\mathrm{K}}_{2} {1}_{u}39K2​1u​ state. The existence of hyperfine structure in the excited K21u{\mathrm{K}}_{2} {1}_{u}K2​1u​ state and a knowledge of the Hamiltonian structure for two colliding ground state alkali atoms enables us to obtain information on the sign and magnitudes of certain scattering lengths for ground state collisions. Analysis of the experimental photoassociation line shapes shows that the 39K{}^{39}\mathrm{K}39K scattering lengths are between \ensuremath−3.2\ensuremath{-}3.2\ensuremath−3.2 and +0.8nm(\ensuremath−60a0+0.8\mathrm{nm}(\ensuremath{-}{60a}_{0}+0.8nm(\ensuremath−60a0​ and +15a0,{+15a}_{0},+15a0​, where 1a0=1a.u.=0.0529177nm){1a}_{0}=1\mathrm{a}.\mathrm{u}.=0.0529177\mathrm{nm})1a0​=1a.u.=0.0529177nm) for the a3\ensuremathΣu+a{}^{3}{\ensuremath{\Sigma}}_{u}^{+}a3\ensuremathΣu+​ molecular state, between +4.8 and +12 nm (+90a0{(+90a}_{0}(+90a0​ and +230a0){+230a}_{0})+230a0​) for the X1\ensuremathΣg+{X}^{1}{\ensuremath{\Sigma}}_{g}^{+}X1\ensuremathΣg+​ state, and between \ensuremath−7.9\ensuremath{-}7.9\ensuremath−7.9 and +1.1nm(\ensuremath−150a0+1.1\mathrm{nm} (\ensuremath{-}{150a}_{0}+1.1nm(\ensuremath−150a0​ and +20a0){+20a}_{0})+20a0​) for collisions of two f=1,m=\ensuremath−1f=1,m=\ensuremath{-}1f=1,m=\ensuremath−1 atoms.
Threshold scattering properties of 39K{}^{39}\mathrm{K}39K atoms are extracted from a line-shape analysis of photoassociation data. Specifically, we consider the rovibrational levels ${(v}^{\ensuremath{'}}=0--6)$ populated in the photoassociation process $\mathrm{K}(4S)+\mathrm{K}(4S)+\ensuremath{\Elzxh}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}{\mathrm{K}}_{2}^{*}[{0}_{g}^{\ensuremath{-}}{(v}^{\ensuremath{'}}{,J}^{\ensuremath{'}})].$ The measured spectra exhibit several general trends, including large ${J}^{\ensuremath{'}}=2$ linewidths, large relative ${J}^{\ensuremath{'}}=4$ peak intensities, and odd rotational lines which are weak or missing. The even ${J}^{\ensuremath{'}}$ features signal the presence of a d-wave shape resonance in the ground-state collisions. The combination of spectral features allows us to place the following limits on the ground-state singlet and triplet scattering lengths, ${a}_{s}{=140}_{\ensuremath{-}9}^{+6}$ a.u. and ${a}_{t}=\ensuremath{-}17\ensuremath{-}{0.045(C}_{6}\ensuremath{-}{C}_{6})\ifmmode\pm\else\textpm\fi{}25$ a.u., C6=3800{C}_{6}=3800C6​=3800 a.u.
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.
High resolution photoassociation spectra of colliding ultracold trapped atoms contain detailed information about ground and excited state interactions. We calculate the asymmetric line shapes for ultracold Na2{\mathrm{Na}}_{2}Na2​ absorption using a resonance scattering expression, proportional to a free-bound Franck-Condon factor, that is justified by full quantum scattering calculations of a collision in a radiation field. The line shapes illustrate Wigner threshold law behavior, which is characteristic of the quantum limit as $T\ensuremath{\rightarrow}0$. Using an adiabatic hyperfine analysis, we calculate and compare a model spectra for J=1,2,3,and4J=1,2,3, \mathrm{and} 4J=1,2,3,and4 features of the Na2{\mathrm{Na}}_{2}Na2​ $^{1}\ensuremath{\Pi}_{g}(\ensuremath{\nu}=48)$ state to recent high resolution experimental data.
We observe the formation of long-range Cs2 Rydberg molecules consisting of a Rydberg and a ground-state atom by photoassociation spectroscopy in an ultracold Cs gas near 6s1/2(F=3,4)→np3/2 resonances (n=26-34). The spectra reveal two types of molecular states recently predicted by D. A. Anderson, S. A. Miller, and G. Raithel [Phys. Rev. A 90, 062518 (2014)]: states bound purely by triplet s-wave scattering with binding energies ranging from 400 MHz at n=26 to 80 MHz at n=34, and states bound by mixed singlet-triplet s-wave scattering with smaller and F-dependent binding energies. The experimental observations are accounted for by an effective Hamiltonian including s-wave scattering pseudopotentials, the hyperfine interaction of the ground-state atom, and the spin-orbit interaction of the Rydberg atom. The analysis enables the characterization of the role of singlet scattering in the formation of long-range Rydberg molecules and the determination of an effective singlet s-wave scattering length for low-energy-electron-Cs collisions.
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.