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The Journal of Chemical Physics

High-rank separable atom-atom interaction potential used for solving two-body Lippmann-Schwinger and three-body Faddeev equations

Jing-Lun Li, Shu‐Lin Cong

Published Nov 28, 2018• 2 citations
Publisher

Abstract

We derive a high-rank separable potential formula of the atom-atom interaction by using the two-body wave function in the coordinate space as inputs. This high-rank separable potential can be utilized to numerically solve the two-body Lippmann-Schwinger equation and three-body Faddeev equation. By analyzing the convenience and stability of numerical calculations for different kinds of the matrix forms of the Lippmann-Schwinger and Faddeev equations, we can find the optimal forms of the kernal matrices in the two- and three-body scattering equations. We calculate the dimer bound energy, two-body scattering phase shift and off-shell t-matrix, the trimer bound energy, atom-dimer scattering length, and three-body recombination rate using the high-rank separable potentials, taking the identical 4He atoms as an application example. All the calculations converge quickly for the rank number N⩾3. The high-rank separable potential is valid for two-body scattering calculation of 4He atoms, but not accurate enough for reproducing the three-body scattering results by using only two-body s-wave interaction and describing the contributions of two-body high partial-waves to the three-body scattering for the 4He3 system.

DOI
10.1063/1.5045084

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We derive a high-rank separable potential formula of the atom-atom interaction by using the two-body wave function in the coordinate space as inputs. This high-rank separable potential can be utilized to numerically solve the two-body Lippmann-Schwinger equation and three-body Faddeev equation. By analyzing the convenience and stability of numerical calculations for different kinds of the matrix forms of the Lippmann-Schwinger and Faddeev equations, we can find the optimal forms of the kernal matrices in the two- and three-body scattering equations. We calculate the dimer bound energy, two-body scattering phase shift and off-shell t-matrix, the trimer bound energy, atom-dimer scattering length, and three-body recombination rate using the high-rank separable potentials, taking the identical 4He atoms as an application example. All the calculations converge quickly for the rank number N⩾3. The high-rank separable potential is valid for two-body scattering calculation of 4He atoms, but not accurate enough for reproducing the three-body scattering results by using only two-body s-wave interaction and describing the contributions of two-body high partial-waves to the three-body scattering for the 4He3 system.
We study Efimov physics for three identical bosons interacting via a pairwise square-well potential, analyze the validity of the separable approximation as a function of the interaction strength, and investigate what is needed to improve this approximation. We find separable approximations to be accurate for potentials with just one (nearly) bound dimer state. For potentials with more bound or almost bound dimer states, these states need to be included for an accurate determination of the Efimov spectrum and the corresponding three-body observables. We also show that a separable approximation is insufficient to accurately compute the trimer states for energies larger than the finite-range energy even when the two-body TTT matrix is highly separable in this energy regime. Additionally, we have analyzed three distinct expansion methods for the full potential that give exact results and thus improve on the separable approximation. With these methods, we demonstrate the necessity to include higher partial-wave components of the off-shell two-body TTT matrix in the three-body calculations. Moreover, we analyze the behavior of the Efimov states near the atom-dimer threshold and observe the formation of non-Efimovian trimer states as the potential depth is increased. Our results can help to elaborate simpler theoretical models that are capable of reproducing the correct three-body physics in atomic systems.
The two-body scattering length a for 4He{}^{4}\mathrm{He}4He atoms is much larger than their effective range rs.{r}_{s}.rs​. As a consequence, low-energy few-body observables have universal characteristics that are independent of the interaction potential. Universality implies that, up to corrections suppressed by rs/a,{r}_{s}/a,rs​/a, all low-energy three-body observables are determined by a and a three-body parameter ${\ensuremath{\Lambda}}_{*}.$ We give simple expressions in terms of a and ${\ensuremath{\Lambda}}_{*}$ for the trimer binding-energy equation, the atom-dimer scattering phase shifts, and the rate for three-body recombination at threshold. We determine ${\ensuremath{\Lambda}}_{*}$ for several 4He{}^{4}\mathrm{He}4He potentials from the calculated binding energy of the excited state of the trimer and use it to obtain the universality predictions for the other low-energy observables. We also use the calculated values for one potential to estimate the effective range corrections for the other potentials.
Four-body system of 4He{}^4\mathrm{He}4He atoms: Dimer-dimer scattering
The strong short-range repulsion, characteristic to realistic interatomic potentials, complicates the description of weakly-bound few-body systems such as those of \He atoms. The present work proposes an approach for solving this problem and applies it to a realistic system of four 4He{}^4\mathrm{He}4He atoms. The potential is gradually softened such that rigorous four-body equations for bound and scattering tates can be accurately solved in the momentum-space framework, and the results are extrapolated back to the limit of the original potential. Linear correlations between three- and four body quantities are observed, and the accuracy of the procedure is improved by extrapolating in one of the three-body quantities. Results for the 4He{}^4\mathrm{He}4He tetramer ground and excited state binding energies and atom-trimer scattering agree well with at least some of earlier determinations and shed light on the existing disagreements. An additional case of the Phillips correlation line is established for the dimer-dimer scattering length. The trimer production rate via the ultracold two-dimer collisions is estimated, it exhibits significant finite-range effects despite the weak binding of the dimer.
In this work, a system including two neutral atoms confined to an external one-dimensional Morse potential was modelled. The problem can be relevant to cold atom physics, where neutral atoms may be effectively confined in radially tight tubes formed by optical lattices. The atom-atom interaction was considered as a nonlocal separable potential. Analytical expressions for wave-function as well as transition matrix were derived. The contributions of bound states and resonances in the complex energy plane were calculated. For numerical computations, the bound states in a system of argon gas confined in graphite were considered. Since the most important quantity in the low energy quantum scattering problems is “scattering length,” considering various values of Morse parameters, the behavior of this parameter was described versus the reduced energy.
Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the 4He^{4}\mathrm{He}4He atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the sss-wave two-body ttt matrix to a separable form through the unitary-pole approximation and did the same for the sss-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.
We investigate the three-body properties of two identical ↑ fermions and one distinguishable ↓ atom interacting in a strongly confined two-dimensional geometry.We compute exactly the atom-dimer scattering properties and the three-body recombination rate as a function of collision energy and mass ratio m ↑ /m ↓ .We find that the recombination rate for fermions is strongly energy dependent, with significant contributions from higher partial waves at low energies.For m ↑ m ↓ , the s-wave atom-dimer scattering below threshold is completely described by the scattering length.Furthermore, we examine the ↑↑↓ bound states (trimers) appearing at large m ↑ /m ↓ and find that the energy spectrum for the deepest bound trimers resembles that of a hydrogen atom confined to two dimensions.