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arXiv:1206.6072 · Physical Review A

Optimized multichannel quantum defect theory for cold molecular collisions

James F. E. Croft, Jeremy M. Hutson, Paul S. Julienne

Published Aug 20, 2012• 18 citations
arXivHTMLPDF

Abstract

Multichannel quantum defect theory (MQDT) can provide an efficient alternative to full coupled-channel calculations for low-energy molecular collisions. However, the efficiency relies on interpolation of the $\mathbit{Y}$ matrix that encapsulates the short-range dynamics, and there are poles in $\mathbit{Y}$ that may prevent interpolation over the range of energies of interest for cold molecular collisions. We show how the phases of the MQDT reference functions may be chosen so as to remove such poles from the vicinity of a reference energy and dramatically increase the range of interpolation. For the test case of Mg+NH, the resulting optimized $\mathbit{Y}$ matrix may be interpolated smoothly over an energy range of several Kelvin and a magnetic field range of over 1000 gauss. Calculations at additional energies and fields can then be performed at a computational cost that is proportional to the number of channels $N$ and not to ${N}^{3}$.

DOI
10.1103/physreva.86.022711

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Multichannel quantum defect theory (MQDT) can provide an efficient alternative to full coupled-channel calculations for low-energy molecular collisions. However, the efficiency relies on interpolation of the \mathbitY\mathbit{Y}\mathbitY matrix that encapsulates the short-range dynamics, and there are poles in \mathbitY\mathbit{Y}\mathbitY that may prevent interpolation over the range of energies of interest for cold molecular collisions. We show how the phases of the MQDT reference functions may be chosen so as to remove such poles from the vicinity of a reference energy and dramatically increase the range of interpolation. For the test case of Mg+NH, the resulting optimized \mathbitY\mathbit{Y}\mathbitY matrix may be interpolated smoothly over an energy range of several Kelvin and a magnetic field range of over 1000 gauss. Calculations at additional energies and fields can then be performed at a computational cost that is proportional to the number of channels NNN and not to N3{N}^{3}N3.
Multichannel Quantum Defect Theory (MQDT) is shown to be capable of producing quantitatively accurate results for low-energy atom-molecule scattering calculations. With a suitable choice of reference potential and short-range matching distance, it is possible to define a matrix that encapsulates the short-range collision dynamics. Multichannel quantum defect theory can provide an efficient alternative to full coupled-channel calculations for low-energy molecular collisions. However, the efficiency relies on interpolation of the Y matrix that encapsulates the short-range dynamics. It is shown how the phases of the MQDT reference functions may be chosen so as to remove such poles from the vicinity of a reference energy and dramatically increase the range of interpolation. For the test cases of Mg+NH and Li+NH, the resulting optimized Y matrix may be interpolated smoothly over an energy range of several Kelvin and a magnetic field range of over 1000G. Calculations at additional energies and fields can then be performed at a computational cost that is proportional to the number of channels N and not to N cubed. MQDT thus provides a promising method for carrying out low-energy molecular scattering calculations on systems where full exploration of the energy and the field dependence is currently impractical.
Multichannel quantum defect theory (MQDT) is shown to be capable of producing quantitatively accurate results for low-energy atom-molecule scattering calculations. With a suitable choice of reference potential and short-range matching distance, it is possible to define a matrix that encapsulates the short-range collision dynamics and is only weakly dependent on energy and magnetic field. Once this has been produced, calculations at additional energies and fields can be performed at a computational cost that is proportional to the number of channels NNN and not to N3{N}^{3}N3. MQDT thus provides a promising method for carrying out low-energy molecular scattering calculations on systems where full exploration of the energy dependence and the field dependence is currently impractical.
We show that multichannel quantum defect theory (MQDT) can be applied successfully as an efficient computational method for cold molecular collisions in Li+NH\text{Li}+\text{NH}Li+NH, which has a deep and strongly anisotropic interaction potential. In this strongly coupled system, closed-channel poles restrict the range over which the MQDT $\mathbit{Y}$ can be interpolated. We present an improved procedure to transform the MQDT reference functions so that the poles are removed from the energy range of interest. Effects due to very-long-range spin dipolar couplings are outside the scope of MQDT, but can be added perturbatively. This procedure makes it possible to calculate the elastic and inelastic cross sections, over the entire range of energies and fields needed to evaluate the feasibility of sympathetic cooling of NH by Li, using coupled-channel calculations at only five combinations of energy and field.
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.
Multichannel quantum defect theory (MQDT) has been widely applied to resonant and nonresonant scattering in a variety of atomic collision processes. In recent years, the method has been applied to cold collisions with considerable success, and it has proven to be a computationally viable alternative to full close-coupling (CC) calculations when spin, hyperfine, and external field effects are included. In this paper, we describe a hybrid approach for molecule-molecule scattering that includes the simplicity of MQDT while treating the short-range interaction explicitly using CC calculations. This hybrid approach, demonstrated for ${\mathrm{H}}_{2}\text{\ensuremath{-}}{\mathrm{H}}_{2}$ collisions in full dimensionality, is shown to adequately reproduce cross sections for quasiresonant rotational and vibrational transitions in the ultracold $(1\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{K})$ and $\ensuremath{\sim}1\ensuremath{-}10\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ regime spanning seven orders of magnitude. It is further shown that an energy-independent short-range KKK matrix evaluated in the ultracold regime $(1 \ensuremath{\mu}\mathrm{K})$ can adequately characterize cross sections in the mK-K regime when no shape resonances are present. The hybrid CC-MQDT formalism provides an alternative approach to full CC calculations at considerably less computational expense for cold and ultracold molecular scattering.
Homonuclear ultracold elastic sss-wave collisions of alkali atoms via multichannel quantum defect theory
Multichannel quantum defect theory (MQDT) provides a powerful toolkit for describing and understanding collisions of cold alkali 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, high-energy 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 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 to experiment where possible, and identify the limitations of various approximations.