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arXiv:1811.10663 · Physical Review Letters

Universal Probability Distributions of Scattering Observables in Ultracold Molecular Collisions

Masato Morita, Roman V. Krems, Timur V. Tscherbul

Published Jul 2, 2019• 24 citations
arXivHTMLPDF

Abstract

Currently, quantum dynamics theory cannot be used for quantitative predictions of molecular scattering observables at low temperatures because of two problems. The first problem is the extreme sensitivity of the low-temperature observables to details of potential energy surfaces (PESs) parametrizing the nuclear Schrödinger equation. The second problem is the large size of the basis sets required for the numerical integration of the Schrödinger equation for strongly interacting molecules in the presence of fields, which precludes the application of rigorous quantum theory to all but a few atom-molecule systems. Here, we show that, if the scattering problem is formulated as a probabilistic prediction, quantum theory can provide reliable results with exponentially reduced numerical effort. Specifically, we show that the probability distributions that an observable is in a certain range of values can be obtained by averaging the results of scattering calculations with much smaller basis sets than required for calculations of individual scattering cross sections. Moreover, we show that such distributions do not rely on the precise knowledge of the PES. This opens the possibility of making probabilistic predictions of experimentally relevant observables for a wide variety of molecular systems, currently considered out of reach of quantum dynamics theory. We demonstrate the approach by computing the probability for elastic scattering of CaH and SrOH molecules by Li atoms and SrF molecules by Rb atoms.

DOI
10.1103/physrevlett.123.013401

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Currently, quantum dynamics theory cannot be used for quantitative predictions of molecular scattering observables at low temperatures because of two problems. The first problem is the extreme sensitivity of the low-temperature observables to details of potential energy surfaces (PESs) parametrizing the nuclear Schrödinger equation. The second problem is the large size of the basis sets required for the numerical integration of the Schrödinger equation for strongly interacting molecules in the presence of fields, which precludes the application of rigorous quantum theory to all but a few atom-molecule systems. Here, we show that, if the scattering problem is formulated as a probabilistic prediction, quantum theory can provide reliable results with exponentially reduced numerical effort. Specifically, we show that the probability distributions that an observable is in a certain range of values can be obtained by averaging the results of scattering calculations with much smaller basis sets than required for calculations of individual scattering cross sections. Moreover, we show that such distributions do not rely on the precise knowledge of the PES. This opens the possibility of making probabilistic predictions of experimentally relevant observables for a wide variety of molecular systems, currently considered out of reach of quantum dynamics theory. We demonstrate the approach by computing the probability for elastic scattering of CaH and SrOH molecules by Li atoms and SrF molecules by Rb atoms.
This thesis describes four types of cold molecular collision systems with increasing complexity: from simple atom-diatomic molecule to complicated polyatomic molecule-polyatomic molecule. The early thesis work is concerned with three open questions pertaining to collision dynamics of cold molecules. We demonstrate the possibility of controlling collisional decoherence of ultracold molecules by tuning an external magnetic field. We then provide insight into the feasibility of evaporative cooling of molecules, and for the first time incorporate the uncertainty analysis of the potential energy surface (PES) into scattering calculations. In addition, we use classical trajectory methods to study the effects of the interaction strength and the geometry of rigid polyatomic molecules on the formation of long-lived collision complexes at low collision energies. The second half of the thesis work is focused on combining statistical methodology and scattering calculations to address two major problems in molecular dynamics calculations: increasing computational complexity and uncertainties due to inaccuracies of PES. Using a small number of scattering calculations, we show that we can build a Gaussian Process (GP) model to statistically approximate collision outcomes for complex molecules, and then perform the uncertainty analysis and the sensitivity analysis. We also demonstrate that trained by a combination of classical and quantum calculations, a GP model can provide an accurate description of the quantum scattering cross sections, even near scattering resonances.
The elastic scattering amplitudes of indistinguishable, bosonic, strongly polar molecules possess universal properties at the coldest temperatures due to wave propagation in the long-range dipole-dipole field. Universal scattering cross sections and anisotropic threshold angular distributions, independent of molecular species, result from careful tuning of the dipole moment with an applied electric field. Three distinct families of threshold resonances also occur for specific field strengths, and can be both qualitatively and quantitatively predicted using elementary adiabatic and semiclassical techniques. The temperatures and densities of heteronuclear molecular gases required to observe these universal characteristics are predicted.
Quantum scattering calculations for strongly interacting molecular systems are computationally demanding due to the large number of molecular states coupled by the anisotropy of atom - molecule interactions. We demonstrate that thermal rate coefficients for total (elastic + inelastic) atom - molecule scattering are insensitive to the interaction anisotropy of the underlying potential energy surface. In particular, we show that the rate coefficients for Rb-H2_22​ and Rb-N2_22​ scattering at room temperature can be computed to 1% accuracy with anisotropy set to zero, reducing the complexity of coupled channel quantum scattering calculations to numerical solutions of a single differential equation. Our numerical calculations and statistical analysis based on Gaussian process regression elucidate the origin and limitations of the invariance of the total scattering rate coefficients to changes in atom - molecule interaction anisotropy.
Collisions between atoms and molecules govern how energy is exchanged and redistributed in the gas phase, underpinning phenomena from spectral line broadening and interstellar molecular excitation to the cooling and trapping of molecules. A quantitative account of these collisions requires connecting the theoretical description of the interaction between the collision partners to measurable quantities such as cross sections. However, scattering theory as presented in quantum mechanics textbooks does not readily translate into numerical methods. Here, we formulate the scattering problem in Dirac notation, as customary in present-day quantum mechanics courses, and show how to build a practical numerical framework from it. We derive the "coupled channel" representation of the scattering problem with correct asymptotic behaviour and introduce Numerov's method to obtain time-independent wavefunctions, from which we calculate cross sections, the observables measured in scattering experiments. Starting from single channel scattering, we build up to the realistic collision of a ground-state atom with a diatomic molecule.
This thesis reports on a variety of calculations on cold and ultracold scattering, with a broad theme of how best to consider and understand complex systems in simple ways. Firstly, we investigate quantum defect theory. We demonstrate that it is not only an excellent model for simple systems, but can also provide simple predictions of the \emph{range} of possible behaviours for complex systems, in particular for a model of collisional losses. These predictions agree well with expensive coupled-channels calculations in cases where the full calculations also predict only the range of possible behaviours. Secondly, we consider effects relating to thermalisation of cold and ultracold gases. We show that considering the correct transport cross section, ση(1)\sigma_\eta^{(1)}ση(1)​, is important for determination of scattering lengths and their signs by interspecies thermalisation. This cross section is also important to the understanding of high-quality simulations of sympathetic cooling in a microwave trap, which suggest Rb is likely to be a good coolant for CaF. We also correct an error in the interpretation of previous results for sympathetic cooling in a magnetic trap, showing this may work from over 100 mK for Li+CaF and many Kelvin when using atomic hydrogen as a coolant. Thirdly, we study quantum chaos in ultracold collisions. We find very clear and strong signs of chaos in Li+CaH. We also show that a more strongly coupled system, Li+CaF, is \emph{not} fully chaotic and that there is unexpected structure in the levels of chaos as the CaF rotational constant is varied. We also show that signatures of chaos can emerge in a very simple atom-atom system, Yb(1S0{}^1S_01S0​)+Yb(3P2{}^3P_23P2​), which interacts on only two Born-Oppenheimer potentials. Finally, we examine the idea that metastable states in 2-body scattering greatly enhance 3-body recombination at ultracold temperatures. We attempt to put it on a more rigorous theoretical grounding by considering Smith's collision lifetime and related quantities, but those are shown to lack clear interpretations in the ultracold regime. We therefore consider 3-body scattering theory and arrive at some general conclusions about how we expect such 2-body features to appear in 3-body scattering and suggest possible ways forward.
Because the de Broglie wavelength of ultracold molecules is very large, the cross sections for collisions of molecules at ultracold temperatures are always computed by the time-independent quantum scattering approach. Here, we report the first accurate time-dependent wave packet dynamics calculation for reactive scattering of ultracold molecules. Wave packet dynamics calculations can be applied to molecular systems with more dimensions and provide real-time information on the process of bond rearrangement and/or energy exchange in molecular collisions. Our work thus makes possible the extension of rigorous quantum calculations of ultracold reaction properties to polyatomic molecules and adds a new powerful tool for the study of ultracold chemistry.