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Physical review. E

Multichannel scattering problem with a nonseparable angular part as a boundary-value problem

Shahpoor Saeidian, Vladimir S. Melezhik

Published Nov 3, 2017• 3 citations
Publisher

Abstract

We have developed an efficient computational method for solving the quantum multichannel scattering problem with a nonseparable angular part. The use of the nondirect product discrete-variable representation, suggested and developed by V. Melezhik, gives us an accurate approximation for the angular part of the desired wave function and, eventually, for the scattering parameters. Subsequent reduction of the problem to the boundary-value problem with well-defined block-band matrix of equation coefficients permits us to use efficient standard algorithms for its solution. We demonstrate the numerical efficiency, flexibility, and good convergence of the computational scheme in a quantitative description of the Feshbach resonances in pair collisions occurring in atomic traps and the scattering in strongly anisotropic traps. The method can also be used for the investigation of further actual problems in quantum physics. A natural extension is a description of spin-orbit coupling, intensively investigated in ultracold gases, and dipolar confinement-induced resonances.

DOI
10.1103/physreve.96.053302

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We have developed an efficient computational method for solving the quantum multichannel scattering problem with a nonseparable angular part. The use of the nondirect product discrete-variable representation, suggested and developed by V. Melezhik, gives us an accurate approximation for the angular part of the desired wave function and, eventually, for the scattering parameters. Subsequent reduction of the problem to the boundary-value problem with well-defined block-band matrix of equation coefficients permits us to use efficient standard algorithms for its solution. We demonstrate the numerical efficiency, flexibility, and good convergence of the computational scheme in a quantitative description of the Feshbach resonances in pair collisions occurring in atomic traps and the scattering in strongly anisotropic traps. The method can also be used for the investigation of further actual problems in quantum physics. A natural extension is a description of spin-orbit coupling, intensively investigated in ultracold gases, and dipolar confinement-induced resonances.
We discuss computational aspects of the developed mathematical models for resonant processes in confined geometry of atomic and atom-ion traps. The main attention is paid to formulation in the nondirect product discrete-variable representation (npDVR) of the multichannel scattering problem with nonseparable angular part in confining traps as the boundary-value problem. Computational efficiency of this approach is demonstrated in application to atomic and atom-ion confinement-induced resonances we predicted recently.
We have developed a method for treating different quantum few-body dynamics without traditional partial-wave analysis. The key element of the method is a nondirect product discrete variable representation (npDVR) we have suggested and developed in application to the time-dependent mutidimensional Schrödinger equations. It happened that this approach was very efficient in quantitative analysis of low-dimensional ultracold few-body systems arising in confined geometry of atomic traps. As an example, we consider the numerical analysis of the four-dimensional Schrödinger equation describing the resonant molecule formation in confined two-component atomic mixture with transferring the energy release to the center-ofmass excitation of forming molecules.
We discuss computational aspects of the developed mathematical models for ultracold few-body processes in atomic traps. The key element of the elaborated computational schemes is a nondirect product discrete variable representation (npDVR) we have suggested and applied to the time-dependent and stationary Schrödinger equations with a few spatial variables. It turned out that this approach is very effcient in quantitative analysis of low-dimensional ultracold few-body systems arising in confined geometry of atomic traps. The effciency of the method is demonstrated here on two examples. A brief review is also given of novel results obtained recently.
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 develop an analytical model for ultracold atom–ion collisions using the multichannel quantum-defect formalism. The model is based on analytical solutions of the r −4 long-range potential and on the application of a frame transformation between asymptotic and molecular bases. This approach allows the description of atom–ion interaction in the ultracold domain in terms of only three parameters: the singlet and triplet scattering lengths, assumed to be independent of the relative motion angular momentum, and the lead dispersion coefficient of the asymptotic potential. We also introduce corrections to the scattering lengths that improve the accuracy of our quantum-defect model for higher-order partial waves, a particularly important result for an accurate description of shape and Feshbach resonances at finite temperature. The theory is applied to the system composed of a 40 Ca + ion and a Na atom, and compared with numerical coupled-channel calculations carried out using ab initio potentials. For this particular system, we investigate the spectrum of bound states, the rate of charge-transfer processes and the collision rates in the presence of magnetic Feshbach resonances at zero and finite temperature.
We develop a grid method for multichannel scattering of atoms in a waveguide with harmonic confinement. This approach is employed to extensively analyze the transverse excitations and deexcitations as well as resonant scattering processes. Collisions of identical bosonic and fermionic as well as distinguishable atoms in harmonic traps with a single frequency $\ensuremath{\omega}$ permitting the center-of-mass (c.m.) separation are explored in depth. In the zero-energy limit and single mode regime we reproduce the well-known confinement-induced resonances (CIRs) for bosonic, fermionic, and heteronuclear collisions. In the case of the multimode regime up to four open transverse channels are considered. Previously obtained analytical results are extended significantly here. Series of Feshbach resonances in the transmission behavior are identified and analyzed. The behavior of the transmission with varying energy and scattering lengths is discussed in detail. The dual CIR leading to a complete quantum suppression of atomic scattering is revealed in multichannel scattering processes. Possible applications include, e.g., cold and ultracold atom-atom collisions in atomic waveguides and electron-impurity scattering in quantum wires.