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

arXiv:2011.01707 · Physical Review A

Three-body recombination calculations with a two-body mapped grid method

T. Secker, Jing-Lun Li, P. M. A. Mestrom, S. J. J. M. F. Kokkelmans

Published Mar 12, 2021• 9 citations
arXivHTMLPDF

Abstract

We investigate the prospects of combining a standard momentum space approach for ultracold three-body scattering with efficient coordinate space schemes to solve the underlying two-body problem. In many of those schemes, the two-body problem is numerically restricted up to a finite interparticle distance ${r}_{\mathrm{b}}$. We analyze the effects of this two-body restriction on the two- and three-body level using pairwise square-well potentials that allow for analytic two-body solutions and more realistic Lennard-Jones van der Waals potentials to model atomic interactions. We find that the two-body $t$ operator converges exponentially in ${r}_{\mathrm{b}}$ for the square-well interaction. Setting ${r}_{\mathrm{b}}$ to 2000 times the range of the interaction, the three-body recombination rate can be determined accurately up to a few percent when the magnitude of the scattering length is small compared to ${r}_{\mathrm{b}}$, while the position of the lowest Efimov features is accurate up to the percent level. In addition, we find that with the introduction of a momentum cutoff, it is possible to determine the three-body parameter in good approximation even for deep van der Waals potentials.

DOI
10.1103/physreva.103.032817

Related papers

Short-range physics of the three-body recombination for ultracold helium atoms

Binbin Wang, Yong‐Chang Han

Generally, the three-body recombination (TBR) of the ultracold helium atoms in the zero-collision-energy limit could be described by the original zero-range (Efimov) universal function [E. Braaten, D. Kang, and L. Platter, Phys. Rev. A 75, 052714 (2007)] using the scattering length $a$ of the He-He interaction and a three-body parameter ${a}_{*}$ as input, where ${a}_{*}$ is associated with the positive scattering length at which an interference minimum in the TBR rate occurs at the threshold. This universal property has been reinvestigated by considering various post-Born-Oppenheimer (post-BO) effects on the relevant scattering process. It is found that a linear correction to $1/{a}_{*}$ versus $1/a$ should be introduced to give a better description of these considered TBR cases because of the finite-range effects. Particularly, such a correction may be ascribed not only to the general scattering properties associated with the Efimov channel at the short hyperradial distance, but also to the tunneling probability in the lowest incident channel. In addition, a successful fit of the modified universal function that includes correctly the finite-range effects [E. Garrido, M. Gattobigio, and A. Kievsky, Phys. Rev. A 88, 032701 (2013)] or the zero-range universal function with a linearly variational $1/{a}_{*}$ versus $1/a$ to numerically accurate data also indicates a clear manifestation of the van der Waals universality. Finally, the post-BO effects on the TBR process for the partial wave with the total angular momentum $J>0$ are also discussed.

Physical Review A•Dec 6, 2023•1 citation

Simple model for analyzing Efimov energy and three-body recombination of three identical bosons with van der Waals interactions

Jing-Lun Li, Xue‐Jin Hu, Yong‐Chang Han +1 more

We construct a simple model to calculate the trimer bound state energy ${E}_{T}^{(n)}$ and three-body recombination rate ${K}_{3}^{0}$ of three identical bosons with van der Waals interaction without using any two- or three-body fitting parameter. Using this simple model, we investigate the influence of the van der Waals finite-range effect on ${E}_{T}^{(n)}$ and ${K}_{3}^{0}$. Our calculation shows that the finite-range effect leads to the ground trimer state energy ${E}_{T}^{(0)}$ not crossing the atom-dimer threshold, and the scaled three-body recombination rate ${K}_{3}^{0}/{a}^{4}$ deviating from the universal three-body theory. The results of our simple model agree within a few percent with other theoretical works with van der Waals interaction and also the experimental data.

Physical Review A•Sep 8, 2016•8 citations

Three-body recombination at finite energy within an optical model

Peder K. Sørensen, D. V. Fedorov, A. S. Jensen +1 more

We investigate three-boson recombination of equal mass systems as function of (negative) scattering length, mass, finite energy, and finite temperature. An optical model with an imaginary potential at short distance reproduces experimental recombination data and allows us to provide a simple parametrization of the recombination rate as function of scattering length and energy. Using the two-body van der Waals length as unit we find that the imaginary potential range and also the potential depth agree to within 30% for lithium and cesium atoms. As opposed to recent studies suggesting universality of the threshold for bound-state formation, our results suggest that the recombination process itself could have universal features.

arXiv:1307.2854•Physical Review A•Oct 29, 2013•13 citations

Universal three-body recombination via resonant d -wave interactions

Jia Wang, José Paulo D'Incao, Yujun Wang +1 more

For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body $s$-wave dimer, the Efimov effect takes place leading to the formation of an infinite number of three-body (Efimov) states. The lowest Efimov state crosses the three-body break-up threshold when the $s$-wave two-body scattering length is $a\ensuremath{\approx}\ensuremath{-}9.73{r}_{\mathrm{vdW}}$, ${r}_{\mathrm{vdW}}$ being the van der Waals length. This article focuses on a generalized version of this Efimov scenario, where two of the atoms are about to form a two-body $d$-wave dimer, resulting in strong $d$-wave interactions. In his paper [Phys. Rev. A 62, 050702(R) (2000)], Bo Gao predicted that for broad resonances the $d$-wave dimer is always formed when $a\ensuremath{\approx}0.956{r}_{\mathrm{vdW}}$. Here we find that a single universal three-body state associated with the $d$-wave dimer is also formed near the three-body breakup threshold at $a\ensuremath{\approx}1.09{r}_{\mathrm{vdW}}$, or alternatively ${a}_{2}=0.902{r}_{\mathrm{vdW}}$, where ${a}_{2}$ is the two-body $d$-wave scattering length. The signature of such a universal three-body state is signaled experimentally by an enhancement of the three-body recombination rate. The three-body effective potential curves that are crucial for understanding the recombination dynamics are also calculated and analyzed. An improved method to calculate the couplings, effective potential curves, and recombination rate coefficients is presented.

arXiv:1209.4553•Physical Review A•Dec 20, 2012•37 citations

Three-body recombination of two-component cold atomic gases into deep dimers in an optical model

Mathias Mikkelsen, A. S. Jensen, D. V. Fedorov +1 more

Abstract We consider three-body recombination into deep dimers in a mass-imbalanced two-component atomic gas. We use an optical model where a phenomenological imaginary potential is added to the lowest adiabatic hyper-spherical potential. The consequent imaginary part of the energy eigenvalue corresponds to the decay rate or recombination probability of the three-body system. The method is formulated in details and the relevant qualitative features are discussed as functions of scattering lengths and masses. We use zero-range model in analyses of recent recombination data. The dominating scattering length is usually related to the non-equal two-body systems. We account for temperature smearing which tends to wipe out the higher-lying Efimov peaks. The range and the strength of the imaginary potential determine positions and shapes of the Efimov peaks as well as the absolute value of the recombination rate. The Efimov scaling between recombination peaks is calculated and shown to depend on both scattering lengths. Recombination is predicted to be largest for heavy–heavy–light systems. Universal properties of the optical parameters are indicated. We compare to available experiments and find in general very satisfactory agreement.

arXiv:1501.05137•Journal of Physics B Atomic Molecular and Optical Physics•Mar 20, 2015•26 citations

Ultracold three-body recombination in two dimensions

José Paulo D'Incao, Fatima Anis, Brett D. Esry

We study three-body recombination in two dimensions for systems interacting via short-range two-body interactions in the regime of large scattering lengths. Using the adiabatic hyperspherical representation, we derive semianalytical formulas for three-body recombination in both weakly and deeply bound diatom states. Our results demonstrate the importance of long-range corrections to the three-body potentials by showing how they alter the low-energy and scattering-length dependence of the recombination rate for both bosonic and fermionic systems, which exhibit suppressed recombination if compared to the three-dimensional case. We verify these results through numerical calculations of recombination for systems with finite-range interactions and supporting a few two-body bound states. We also study finite-range effects for the energies of the universal three-identical-bosons states and find a slow approach to universal predictions as a function of the scattering length.

arXiv:1411.2321•Physical Review A•Jun 23, 2015•17 citations
We investigate the prospects of combining a standard momentum space approach for ultracold three-body scattering with efficient coordinate space schemes to solve the underlying two-body problem. In many of those schemes, the two-body problem is numerically restricted up to a finite interparticle distance rb{r}_{\mathrm{b}}rb​. We analyze the effects of this two-body restriction on the two- and three-body level using pairwise square-well potentials that allow for analytic two-body solutions and more realistic Lennard-Jones van der Waals potentials to model atomic interactions. We find that the two-body ttt operator converges exponentially in rb{r}_{\mathrm{b}}rb​ for the square-well interaction. Setting rb{r}_{\mathrm{b}}rb​ to 2000 times the range of the interaction, the three-body recombination rate can be determined accurately up to a few percent when the magnitude of the scattering length is small compared to rb{r}_{\mathrm{b}}rb​, while the position of the lowest Efimov features is accurate up to the percent level. In addition, we find that with the introduction of a momentum cutoff, it is possible to determine the three-body parameter in good approximation even for deep van der Waals potentials.
Generally, the three-body recombination (TBR) of the ultracold helium atoms in the zero-collision-energy limit could be described by the original zero-range (Efimov) universal function [E. Braaten, D. Kang, and L. Platter, Phys. Rev. A 75, 052714 (2007)] using the scattering length aaa of the He-He interaction and a three-body parameter a∗{a}_{*}a∗​ as input, where a∗{a}_{*}a∗​ is associated with the positive scattering length at which an interference minimum in the TBR rate occurs at the threshold. This universal property has been reinvestigated by considering various post-Born-Oppenheimer (post-BO) effects on the relevant scattering process. It is found that a linear correction to 1/a∗1/{a}_{*}1/a∗​ versus 1/a1/a1/a should be introduced to give a better description of these considered TBR cases because of the finite-range effects. Particularly, such a correction may be ascribed not only to the general scattering properties associated with the Efimov channel at the short hyperradial distance, but also to the tunneling probability in the lowest incident channel. In addition, a successful fit of the modified universal function that includes correctly the finite-range effects [E. Garrido, M. Gattobigio, and A. Kievsky, Phys. Rev. A 88, 032701 (2013)] or the zero-range universal function with a linearly variational 1/a∗1/{a}_{*}1/a∗​ versus 1/a1/a1/a to numerically accurate data also indicates a clear manifestation of the van der Waals universality. Finally, the post-BO effects on the TBR process for the partial wave with the total angular momentum J>0J>0J>0 are also discussed.
We construct a simple model to calculate the trimer bound state energy ET(n){E}_{T}^{(n)}ET(n)​ and three-body recombination rate K30{K}_{3}^{0}K30​ of three identical bosons with van der Waals interaction without using any two- or three-body fitting parameter. Using this simple model, we investigate the influence of the van der Waals finite-range effect on ET(n){E}_{T}^{(n)}ET(n)​ and K30{K}_{3}^{0}K30​. Our calculation shows that the finite-range effect leads to the ground trimer state energy ET(0){E}_{T}^{(0)}ET(0)​ not crossing the atom-dimer threshold, and the scaled three-body recombination rate K30/a4{K}_{3}^{0}/{a}^{4}K30​/a4 deviating from the universal three-body theory. The results of our simple model agree within a few percent with other theoretical works with van der Waals interaction and also the experimental data.
We investigate three-boson recombination of equal mass systems as function of (negative) scattering length, mass, finite energy, and finite temperature. An optical model with an imaginary potential at short distance reproduces experimental recombination data and allows us to provide a simple parametrization of the recombination rate as function of scattering length and energy. Using the two-body van der Waals length as unit we find that the imaginary potential range and also the potential depth agree to within 30% for lithium and cesium atoms. As opposed to recent studies suggesting universality of the threshold for bound-state formation, our results suggest that the recombination process itself could have universal features.
For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body sss-wave dimer, the Efimov effect takes place leading to the formation of an infinite number of three-body (Efimov) states. The lowest Efimov state crosses the three-body break-up threshold when the sss-wave two-body scattering length is $a\ensuremath{\approx}\ensuremath{-}9.73{r}_{\mathrm{vdW}}$, rvdW{r}_{\mathrm{vdW}}rvdW​ being the van der Waals length. This article focuses on a generalized version of this Efimov scenario, where two of the atoms are about to form a two-body ddd-wave dimer, resulting in strong ddd-wave interactions. In his paper [Phys. Rev. A 62, 050702(R) (2000)], Bo Gao predicted that for broad resonances the ddd-wave dimer is always formed when $a\ensuremath{\approx}0.956{r}_{\mathrm{vdW}}$. Here we find that a single universal three-body state associated with the ddd-wave dimer is also formed near the three-body breakup threshold at $a\ensuremath{\approx}1.09{r}_{\mathrm{vdW}}$, or alternatively a2=0.902rvdW{a}_{2}=0.902{r}_{\mathrm{vdW}}a2​=0.902rvdW​, where a2{a}_{2}a2​ is the two-body ddd-wave scattering length. The signature of such a universal three-body state is signaled experimentally by an enhancement of the three-body recombination rate. The three-body effective potential curves that are crucial for understanding the recombination dynamics are also calculated and analyzed. An improved method to calculate the couplings, effective potential curves, and recombination rate coefficients is presented.
Abstract We consider three-body recombination into deep dimers in a mass-imbalanced two-component atomic gas. We use an optical model where a phenomenological imaginary potential is added to the lowest adiabatic hyper-spherical potential. The consequent imaginary part of the energy eigenvalue corresponds to the decay rate or recombination probability of the three-body system. The method is formulated in details and the relevant qualitative features are discussed as functions of scattering lengths and masses. We use zero-range model in analyses of recent recombination data. The dominating scattering length is usually related to the non-equal two-body systems. We account for temperature smearing which tends to wipe out the higher-lying Efimov peaks. The range and the strength of the imaginary potential determine positions and shapes of the Efimov peaks as well as the absolute value of the recombination rate. The Efimov scaling between recombination peaks is calculated and shown to depend on both scattering lengths. Recombination is predicted to be largest for heavy–heavy–light systems. Universal properties of the optical parameters are indicated. We compare to available experiments and find in general very satisfactory agreement.
We study three-body recombination in two dimensions for systems interacting via short-range two-body interactions in the regime of large scattering lengths. Using the adiabatic hyperspherical representation, we derive semianalytical formulas for three-body recombination in both weakly and deeply bound diatom states. Our results demonstrate the importance of long-range corrections to the three-body potentials by showing how they alter the low-energy and scattering-length dependence of the recombination rate for both bosonic and fermionic systems, which exhibit suppressed recombination if compared to the three-dimensional case. We verify these results through numerical calculations of recombination for systems with finite-range interactions and supporting a few two-body bound states. We also study finite-range effects for the energies of the universal three-identical-bosons states and find a slow approach to universal predictions as a function of the scattering length.