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arXiv:2307.16654

Homonuclear ultracold elastic $s$-wave collisions of alkali atoms via multichannel quantum defect theory

Alyson Laskowski, Nirav P. Mehta

Published Jul 31, 2023• 0 citations
arXivHTMLPDF

Abstract

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 $K$-matrix, ${\mathbf K}_{\text{sr}}$, 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 ${\mathbf K}_{\text{sr}}$ 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 $s$-wave magnetic Feshbach resonances, comparing to experiment where possible, and identify the limitations of various approximations.

DOI
10.48550/arxiv.2307.16654

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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.
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) 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.
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 NNN and not to N3{N}^{3}N3.
We develop a method for calculating multichannel wavefunctions in the spirit of quantum defect theory, based on numerically calculated reference functions. We benchmark the method by calculating cold collisional properties of 85Rb and 6Li in the presence of external magnetic fields tuned across specific s-wave Feshbach resonances and thereby reproducing known results. We then apply the method to calculate experimentally observed d-wave Feshbach resonance in 87Rb-85Rb cold collisions. Our numerical results for this d-wave resonance show good agreement with the experimental observations. The method is applicable to arbitrary interaction potentials and to any energy range near the scattering threshold. The implementation of our method to any multichannel two-body scattering problem is straightforward.
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.