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Physical Review A

Analysis of threshold effects in ultracold atomic collisions

Frederick H. Mies, Maurice Raoult

Published Jun 14, 2000• 81 citations
Publisher

Abstract

At the ultracold temperatures which occur in cold atom traps and Bose-Einstein condensates, only a few partial waves contribute to the scattering of ground-state alkali-metal atoms, and cross sections are extremely sensitive to threshold effects. We present an analysis of these threshold effects, using a generalized multichannel quantum defect theory (GMQDT) to construct a close-coupled scattering wave function which is analytic in energy across thresholds. We illustrate the theory using the hyperfine transitions in $\mathrm{N}\mathrm{a}+\mathrm{N}\mathrm{a}$ collisions, and show that it gives results completely equivalent to the usual close-coupled cross sections. The virtue of the GMQDT is that it treats both open and closed channels on an equal footing, and all interchannel dynamics is summarized in a single real symmetric matrix $\mathbf{Y}(E)$ which is essentially constant across thresholds (and often over excursions of energy which exceed the hyperfine splittings). The multichannel threshold energy behavior can then be related to calculable properties of the individual channels that are being closed. Many of the smaller spin depolarization cross sections are determined by very long-range ${\ensuremath{\alpha}}^{2}{/R}^{3}$ spin-spin interactions which are not well treated by GMQDT, and we correct these specific elements with a perturbative distorted-wave approximation which yields the observed threshold dependences and brings the GMQDT into perfect agreement with the exact close-coupled results.

DOI
10.1103/physreva.62.012708

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At the ultracold temperatures which occur in cold atom traps and Bose-Einstein condensates, only a few partial waves contribute to the scattering of ground-state alkali-metal atoms, and cross sections are extremely sensitive to threshold effects. We present an analysis of these threshold effects, using a generalized multichannel quantum defect theory (GMQDT) to construct a close-coupled scattering wave function which is analytic in energy across thresholds. We illustrate the theory using the hyperfine transitions in Na+Na\mathrm{N}\mathrm{a}+\mathrm{N}\mathrm{a}Na+Na collisions, and show that it gives results completely equivalent to the usual close-coupled cross sections. The virtue of the GMQDT is that it treats both open and closed channels on an equal footing, and all interchannel dynamics is summarized in a single real symmetric matrix Y(E)\mathbf{Y}(E)Y(E) which is essentially constant across thresholds (and often over excursions of energy which exceed the hyperfine splittings). The multichannel threshold energy behavior can then be related to calculable properties of the individual channels that are being closed. Many of the smaller spin depolarization cross sections are determined by very long-range \ensuremathα2/R3{\ensuremath{\alpha}}^{2}{/R}^{3}\ensuremathα2/R3 spin-spin interactions which are not well treated by GMQDT, and we correct these specific elements with a perturbative distorted-wave approximation which yields the observed threshold dependences and brings the GMQDT into perfect agreement with the exact close-coupled results.
We introduce a choice of reference wave functions that allows multichannel quantum defect theory (MQDT) to describe ultracold collisions involving high-partial-wave quantum numbers LLL. This requires a careful standardization of the MQDT reference wave functions at long range to ensure their linear independence. To illustrate the simplicity and accuracy of the resulting theory, we perform a comprehensive calculation of $L\ensuremath{\le}2$ Fano-Feshbach resonances in the range 0−−10000--10000−−1000 G for the scattering of 40K+87Rb{}^{40}\mathrm{K}{+}^{87}\mathrm{Rb}40K+87Rb in their lowest hyperfine states. Moreover, we derive analytic expressions for our universal set of MQDT parameters in the threshold regime.
Ultracold collisions of cold atoms or molecules make the bound states of the collision complex formed from the two colliding species accessible for control and manipulation of the cold species or the complex. Such resonances are best treated by a resonant scattering theory, which in the ultracold domain can take advantage of the properties of the long-range potential and the methods of multichannel quantum defect theory. Coupled channels calculations on the threshold scattering states and bound states of the 40K87Rb molecule illustrate the ideas and methodology of quantum defect theory using the long-range potential and also demonstrate the spin properties of the bound states throughout the spectrum.
Ultracold collisions of cold atoms or molecules make the bound states of the collision complex formed from the two colliding species accessible for control and manipulation of the cold species or the complex. Such resonances are best treated by a resonant scattering theory, which in the ultracold domain can take advantage of the properties of the long-range potential and the methods of multichannel quantum defect theory. Coupled channels calculations on the threshold scattering states and bound states of the 40 K 87 Rb molecule illustrate the ideas and methodology of quantum defect theory using the long-range potential and also demonstrate the spin properties of the bound states throughout the spectrum. 1
The binary collision dynamics of 2Dg,3/2{}^{2}{D}_{g,3/2}2Dg,3/2​ ground state scandium atoms is studied from first principles. We employ 30 coupled diabatic ab initio potentials in a coupled-channels study of the scattering dynamics of cold and ultracold scandium atoms in external magnetic fields. Due to the long-ranged magnetic dipolar interaction, the field dependence of the cross section does not follow the threshold laws derived by Volpi and Bohn [Phys. Rev. A 65, 052712 (2002)]. In the field-free case, the near-threshold cross section is independent of the collision energy, and hence the cross section does not follow the well-established Wigner threshold laws. The observed threshold behavior is explained in the Born approximation. For energies above $1\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{K}$, inelastic collisions are driven by the anisotropic nonrelativistic electronic interaction. For energies below $100\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{K}$, the ratio of elastic-to-inelastic collisions is likely to be favorable for evaporative cooling. Both anisotropy in the long-range interaction and in the short-range potential contribute to large cross sections for inelastic collisions at higher energies and lead to a small ratio of elastic-to-inelastic collisions. This is in agreement with the large rates for Zeeman relaxation of submerged-shell atoms observed experimentally. The effect of the uncertainty in the ab initio potential is sampled by scaling the reduced mass and is found to have little influence on the conclusions drawn from this work.
We analyze, within the generalized multichannel quantum defect theory framework, atomic binary collision cross sections over the $1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{K}$ to 10mK10\phantom{\rule{0.3em}{0ex}}\mathrm{mK}10mK energy range just above the entrance channel threshold, a domain where the Wigner threshold law should apply. By adjusting the strength of a constant external magnetic field a Feshbach resonance is tuned at will over this energy range. In the threshold regime, the quasibound state interacts with an opening continuum whose wave function presents a strong energy dependence which reflects the breakdown of the WKB approximation. The effective discrete-continuum interactions become very sensitive to the energy. The consequences of these variations have been investigated by including threshold effects in the analysis of the Fano configuration-interaction theory [Phys. Rev. 124, 1866 (1961)] in terms of quantum defect theory quantities proposed by [Lecomte J. Phys. B 20, 3645 (1987)]. This analysis shows that the energy variations are in general so important that it becomes meaningless to associate a width with a Feshbach resonance. However, it is still possible to define the resonance energy, as long as the energy variation of the shift of the resonance, induced by the effective discrete-continuum interaction, remains linear over an energy range corresponding to the magnitude of the shift.
Multichannel quantum-defect theory (MQDT) provides a powerful toolkit for describing and understanding collisions of cold alkali-metal atoms. Various MQDT approximations differ primarily in how they characterize the so-called short-ranged KKK matrix Ksr{\mathbf{K}}_{\text{sr}}Ksr​, which encapsulates the short-ranged physics into a handful of low-energy parameters that exhibit simple and smooth dependence on energy and field. Here, we compare three different methods for computing Ksr{\mathbf{K}}_{\text{sr}}Ksr​ for homonuclear collisions of alkali-metal atoms, from lithium to cesium. The MQDT calculations are benchmarked against numerically converged coupled-channels calculations that use a log-derivative propagator out to the asymptotic region. We study how well these approximations reproduce positions of sss-wave magnetic Feshbach resonances, comparing with experiment where possible, and identify the limitations of various approximations.