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Physical Review A

Simple analytical model for high-partial-wave ultracold molecular collisions

Yan-Peng Bai, Jing-Lun Li, Gao‐Ren Wang, Zhong-Bo Chen, Bo-Wen Si, Shu‐Lin Cong

Published Jun 2, 2020• 6 citations
Publisher

Abstract

The investigation of ultracold atomic or molecular collision is important for us to understand the essence of a chemical reaction. We present a simple analytical model of arbitrary-order partial-wave scatterings for ultracold molecular collision in the threshold regime. The threshold formulas of the transmission and reflection coefficients are given. For high-partial-wave collision, shape resonance will take place because of the existence of a centrifugal barrier. We investigate the effect of shape resonance and short-range interaction on molecular collisions. Moreover, we find a method to fit the loss parameter from the experimental data of ultracold bosonic molecular collision. As an applied example, we calculate the thermally average loss rate for the ultracold $^{87}\mathrm{Rb}\phantom{\rule{0.16em}{0ex}}^{133}\mathrm{Cs}+^{87}\mathrm{Rb}\phantom{\rule{0.16em}{0ex}}^{133}\mathrm{Cs}$ collision, our results are consistent with the recent experimental and theoretical results of Gregory et al. [Nat. Commun. 10, 3104 (2019)].

DOI
10.1103/physreva.101.063605

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The investigation of ultracold atomic or molecular collision is important for us to understand the essence of a chemical reaction. We present a simple analytical model of arbitrary-order partial-wave scatterings for ultracold molecular collision in the threshold regime. The threshold formulas of the transmission and reflection coefficients are given. For high-partial-wave collision, shape resonance will take place because of the existence of a centrifugal barrier. We investigate the effect of shape resonance and short-range interaction on molecular collisions. Moreover, we find a method to fit the loss parameter from the experimental data of ultracold bosonic molecular collision. As an applied example, we calculate the thermally average loss rate for the ultracold 87Rb133Cs+87Rb133Cs^{87}\mathrm{Rb}\phantom{\rule{0.16em}{0ex}}^{133}\mathrm{Cs}+^{87}\mathrm{Rb}\phantom{\rule{0.16em}{0ex}}^{133}\mathrm{Cs}87Rb133Cs+87Rb133Cs collision, our results are consistent with the recent experimental and theoretical results of Gregory et al. [Nat. Commun. 10, 3104 (2019)].
We present a simple model used for investigating the quantum reflection and quantum coherence in ultracold molecular collisions. Analytical expressions of reflection and transmission coefficients, short-range SSS matrix, and transition matrix TTT are derived for the hard-core plus inverse power potentials in the vicinity of the threshold. The effect of the quantum reflection and coherence on the loss rate depends on the scattering length and short-range loss parameter. The quantum coherence results from the phase difference $\ensuremath{\theta}$ between reflection paths. As an applied example, we calculate the loss rate of 23Na87Rb^{23}\mathrm{Na}^{87}\mathrm{Rb}23Na87Rb molecules in the ultracold $^{23}\mathrm{Na}^{87}\mathrm{Rb}\ensuremath{-}^{23}\mathrm{Na}^{87}\mathrm{Rb}$ collision. We can well interpret the experimental results of the ultracold 23Na87Rb^{23}\mathrm{Na}^{87}\mathrm{Rb}23Na87Rb molecular collision [Ye et al., Sci. Adv. 4, eaaq0083 (2018)]. It is found that the quantum coherence has a significant influence on the loss rate of 23Na87Rb^{23}\mathrm{Na}^{87}\mathrm{Rb}23Na87Rb molecules in ultracold molecule-molecule collisions.
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A simple quantum-defect model gives analytic expressions for the complex scattering length and threshold collision rates of ultracold molecules. If the probability of reaction in the short-range part of the collision is high, the model gives universal rate constants for s- and p-wave collisions that are independent of short-range dynamics. This model explains the magnitudes of the recently measured rate constants for collisions of two ultracold 40K87Rb molecules, or an ultracold 40K atom with the 40K87Rb molecule [S. Ospelkaus et al., Science 327, 853 (2010).
We present a unified formalism for describing chemical reaction rates of trapped, ultracold molecules. This formalism reduces the scattering to its essential features, namely, a propagation of the reactant molecules through a gauntlet of long-range forces before they ultimately encounter one another, followed by a probability for the reaction to occur once they do. In this way, the electric-field dependence should be readily parametrized in terms of a pair of fitting parameters (along with a C6{C}_{6}C6​ coefficient) for each asymptotic value of partial-wave quantum numbers $|L,{M}_{L}\ensuremath{\rangle}$. From this, the electric-field dependence of the collision rates follows automatically. We present examples for reactive species, such as KRb, and nonreactive species, such as RbCs.
How does a chemical reaction proceed at ultralow temperatures? Can simple quantum mechanical rules such as quantum statistics, single partial-wave scattering, and quantum threshold laws provide a clear understanding of the molecular reactivity under a vanishing collision energy? Starting with an optically trapped near-quantum-degenerate gas of polar 40K87Rb molecules prepared in their absolute ground state, we report experimental evidence for exothermic atom-exchange chemical reactions. When these fermionic molecules were prepared in a single quantum state at a temperature of a few hundred nanokelvin, we observed p-wave-dominated quantum threshold collisions arising from tunneling through an angular momentum barrier followed by a short-range chemical reaction with a probability near unity. When these molecules were prepared in two different internal states or when molecules and atoms were brought together, the reaction rates were enhanced by a factor of 10 to 100 as a result of s-wave scattering, which does not have a centrifugal barrier. The measured rates agree with predicted universal loss rates related to the two-body van der Waals length.
We investigate the behaviour of single-channel theoretical models of cold and ultracold collisions that take account of inelastic and reactive processes using a single parameter to represent short-range loss. We present plots of the resulting energy-dependence of elastic and inelastic or reactive cross-sections over the full parameter space of loss parameters and short-range phase shifts. We then test the single-channel model by comparing it with the results of coupled-channel calculations of rotationally inelastic collisions between LiH molecules and Li atoms. We find that the range of cross-sections predicted by the single-channel model becomes increasingly accurate as the initial LiH rotational quantum number increases, with a corresponding increase in the number of open loss channels. The results suggest that coupled-channel calculations at very low energy (in the s-wave regime) could in some cases be used to estimate a loss parameter and then to predict the range of possible loss rates at higher energy, without the need for explicit coupled-channel calculations for higher partial waves.