Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Physica Sinica

Ultracold atom-atom collisions by renormalized Numerov method

Jing Bai, Ting Xie

Published Jan 1, 2022• 0 citations
Publisher

Abstract

The renormalized Numerov algorithm is applied to solving time-independent Schrödinger equation relating to atom-atom collisions at ultralow temperature. The proprieties of Feshbach resonance in 39K-133Cs collisions are investigated as an example. The results show that the renormalized Numerov method can give excellent results for ultracold colliding process. In contrast to improved log derivative method, the renormalized Numerov method displays a certain weakness in computational efficiency under the same condition. However, it is much stable in a wide range of grid step size. Hence a new propagating method is proposed by combining renormalized Numerov and logarithmic derivative method which can save computational time with a better accuracy. This algorithm can be used to solve close-coupling Schrödinger equation at arbitrary temperature for two-body collisions.

DOI
10.7498/aps.71.20211308

Related papers

Comparison of log-derivative and renormalized Numerov methods in low-energy two-body collisions

Ting Xie, Lihang Li, Zhigang Sun

Abstract The log-derivative (LOGD) and renormalized Numerov (RN) methods are popularly used in inelastic and reactive scattering calculations. The computational precision of two methods and their improved versions are compared in two-body diatomic and bimolecular collisions at low temperatures. In diatomic 40K-133Cs collision example the single-channel calculations show the solution-improved RN method has the advantage over others when large fixed propagating interval used. The relative error of local scattering length in different propagating sectors are explored. In multi-channel 40K-133Cs collisions the predicted positions of Feshbach resonances by using mapping grid points indicate strong potential-following character in LOGD methods, which is consistent with the analysis on local scattering length in single-channel calculations. In multiple open-channel calculations the elastic and inelastic scatterings and their dependence on collision energy are compared as well. In bimolecular 23Na87Rb-23Na87Rb long-range reactive scattering example the elastic and reactive rate coefficients are investigated. The original LOGD method displays surprisingly relatively high accuracy with less mapping grid points and a wide range of mapping parameters. By increasing the number of mapping grid points the improved LOGD exhibits definitely superiority on precision with proper mapping parameters. We demonstrate our consequent is applicable in other colliding systems.

Physica Scripta•Nov 16, 2023•2 citations

Comparison of numerical methods for the calculation of cold atom collisions

George H. Rawitscher, Brett D. Esry, Eite Tiesinga +2 more

Comparison between three different numerical techniques for solving a coupled channel Schrödinger equation is presented. The benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same diagonal Lennard-Jones potential, one of positive and the other of negative energy. The coupling potential is of an exponential form. The methods are (i) a recently developed spectral type integral equation method based on Chebyshev expansions, (ii) a finite element expansion, and (iii) a combination of an improved Numerov finite difference method and a Gordon method. The computing time and the accuracy of the resulting phase shift is found to be comparable for methods (i) and (ii), achieving an accuracy of ten significant figures with a double precision calculation. Method (iii) achieves seven significant figures. The scattering length and effective range are also obtained.

arXiv:physics/9909005•The Journal of Chemical Physics•Dec 15, 1999•28 citations

Elastic Scattering of Ultracold 133 Cs and 85 Rb Atoms in Triplet State

Sun Jin-Feng, Hu Qiu-Bo, Zunlue Zhu +2 more

Elastic scattering properties of the ultracold interaction for the triplet state of 133 Cs and 85 Rb atoms are studied using two kinds of potentials by the same phase . One is the interpolation potential, and another is Lennard-Jones potential (LJ12,6). The radial Schrodinger equation is also solved using two computational methods, the semiclassical method (WKB), and the Numerov method. Our results are found to be in an excellent agreement with the more recent theoretical values. It shows that the two potentials and methods are applicable for studying ultracold collisions between the mixing alkali atoms.

Communications in Theoretical Physics•Jun 1, 2007•3 citations

Efficient numerical method for locating Feshbach resonances of ultracold molecules in external fields

Yury V. Suleimanov, Roman V. Krems

Collision properties of atoms and molecules in low temperature gases can be controlled by applying an external magnetic or electric field. The external field shifts the energy levels of the colliding particles, which gives rise to Feshbach resonances modifying the scattering cross sections. The resonances occur at particular magnitudes of the external field, where a bound state of the collision complex is degenerate with a scattering state. The positions of the resonances in the external field are usually identified by computing either the scattering cross sections or the bound states of the collision complex as functions of the external field magnitude. We propose a more efficient method for locating Feshbach resonances that requires neither of these computations. In particular, we show that the positions of Feshbach resonances can be identified by computing the log-derivative of the total wave function in a classically allowed region as a function of the external field strength. This procedure is particularly useful for locating narrow Feshbach resonances that may be hard to identify with the other methods.

The Journal of Chemical Physics•Jan 4, 2011•13 citations

Mapped Grid Methods Applied to the Slow Variable Discretization–Enhanced Renormalized Numerov Approach

Juan Blandón, Gregory A. Parker, C. Madrid

We introduce a hyperspherical coordinate mapping procedure to the slow variable discretization-enhanced renormalized Numerov method that allows for more accurate and cost-effective calculations of cold and ultracold atom-dimer scattering. The mapping procedure allows optimization of the numerical grid point spacing by adjusting to the shape of the interaction potential. We show results for elastic scattering in HeH2 and compare the results to previous MOLSCAT calculations by Forrey et al. [ Phys. Rev. A 1999, 59, 2146 ].

The Journal of Physical Chemistry A•Jan 21, 2016•3 citations

Surveying ultracold atom-atom binary collisions by utilizing renormalized Numerov method

中国科学院大连化学物理研究所,, 大连职业技术学院,

采用重归一化Numerov算法求解关于超低温双原子碰撞问题的非含时薛定谔方程组。以39K-133Cs碰撞为例，研究了超低温下双原子Feshbach共振的性质。结果表明，重归一化Numerov算法可以很精确地描述超冷条件下碰撞过程。和改进的logarithmic derivative算法相比，同等参数条件下，重归一化Numerov方法在计算效率上虽然有一定劣势，但在大格点步长参数范围内有着更好的稳定性。提出重归一化Numerov和logarithmic derivative算法相结合的传播方法，在保证结果精度的同时大大减少了计算时间。此项算法也可以应用于求解任意温度下的两体碰撞耦合薛定谔方程组。

Acta Physica Sinica•Jan 1, 2021•0 citations
The renormalized Numerov algorithm is applied to solving time-independent Schrödinger equation relating to atom-atom collisions at ultralow temperature. The proprieties of Feshbach resonance in 39K-133Cs collisions are investigated as an example. The results show that the renormalized Numerov method can give excellent results for ultracold colliding process. In contrast to improved log derivative method, the renormalized Numerov method displays a certain weakness in computational efficiency under the same condition. However, it is much stable in a wide range of grid step size. Hence a new propagating method is proposed by combining renormalized Numerov and logarithmic derivative method which can save computational time with a better accuracy. This algorithm can be used to solve close-coupling Schrödinger equation at arbitrary temperature for two-body collisions.
Abstract The log-derivative (LOGD) and renormalized Numerov (RN) methods are popularly used in inelastic and reactive scattering calculations. The computational precision of two methods and their improved versions are compared in two-body diatomic and bimolecular collisions at low temperatures. In diatomic 40K-133Cs collision example the single-channel calculations show the solution-improved RN method has the advantage over others when large fixed propagating interval used. The relative error of local scattering length in different propagating sectors are explored. In multi-channel 40K-133Cs collisions the predicted positions of Feshbach resonances by using mapping grid points indicate strong potential-following character in LOGD methods, which is consistent with the analysis on local scattering length in single-channel calculations. In multiple open-channel calculations the elastic and inelastic scatterings and their dependence on collision energy are compared as well. In bimolecular 23Na87Rb-23Na87Rb long-range reactive scattering example the elastic and reactive rate coefficients are investigated. The original LOGD method displays surprisingly relatively high accuracy with less mapping grid points and a wide range of mapping parameters. By increasing the number of mapping grid points the improved LOGD exhibits definitely superiority on precision with proper mapping parameters. We demonstrate our consequent is applicable in other colliding systems.
Comparison between three different numerical techniques for solving a coupled channel Schrödinger equation is presented. The benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same diagonal Lennard-Jones potential, one of positive and the other of negative energy. The coupling potential is of an exponential form. The methods are (i) a recently developed spectral type integral equation method based on Chebyshev expansions, (ii) a finite element expansion, and (iii) a combination of an improved Numerov finite difference method and a Gordon method. The computing time and the accuracy of the resulting phase shift is found to be comparable for methods (i) and (ii), achieving an accuracy of ten significant figures with a double precision calculation. Method (iii) achieves seven significant figures. The scattering length and effective range are also obtained.
Elastic scattering properties of the ultracold interaction for the triplet state of 133 Cs and 85 Rb atoms are studied using two kinds of potentials by the same phase . One is the interpolation potential, and another is Lennard-Jones potential (LJ12,6). The radial Schrodinger equation is also solved using two computational methods, the semiclassical method (WKB), and the Numerov method. Our results are found to be in an excellent agreement with the more recent theoretical values. It shows that the two potentials and methods are applicable for studying ultracold collisions between the mixing alkali atoms.
Collision properties of atoms and molecules in low temperature gases can be controlled by applying an external magnetic or electric field. The external field shifts the energy levels of the colliding particles, which gives rise to Feshbach resonances modifying the scattering cross sections. The resonances occur at particular magnitudes of the external field, where a bound state of the collision complex is degenerate with a scattering state. The positions of the resonances in the external field are usually identified by computing either the scattering cross sections or the bound states of the collision complex as functions of the external field magnitude. We propose a more efficient method for locating Feshbach resonances that requires neither of these computations. In particular, we show that the positions of Feshbach resonances can be identified by computing the log-derivative of the total wave function in a classically allowed region as a function of the external field strength. This procedure is particularly useful for locating narrow Feshbach resonances that may be hard to identify with the other methods.
We introduce a hyperspherical coordinate mapping procedure to the slow variable discretization-enhanced renormalized Numerov method that allows for more accurate and cost-effective calculations of cold and ultracold atom-dimer scattering. The mapping procedure allows optimization of the numerical grid point spacing by adjusting to the shape of the interaction potential. We show results for elastic scattering in HeH2 and compare the results to previous MOLSCAT calculations by Forrey et al. [ Phys. Rev. A 1999, 59, 2146 ].
采用重归一化Numerov算法求解关于超低温双原子碰撞问题的非含时薛定谔方程组。以39K-133Cs碰撞为例，研究了超低温下双原子Feshbach共振的性质。结果表明，重归一化Numerov算法可以很精确地描述超冷条件下碰撞过程。和改进的logarithmic derivative算法相比，同等参数条件下，重归一化Numerov方法在计算效率上虽然有一定劣势，但在大格点步长参数范围内有着更好的稳定性。提出重归一化Numerov和logarithmic derivative算法相结合的传播方法，在保证结果精度的同时大大减少了计算时间。此项算法也可以应用于求解任意温度下的两体碰撞耦合薛定谔方程组。