A model study on a pair of trapped particles interacting with an arbitrary effective range

TL;DR

This paper investigates how the effective range of interaction influences the eigenvalues and eigenstates of two particles in various harmonic traps, revealing conditions under which finite-range effects mimic or differ from zero-range contact interactions, especially in 1D and quasi-1D regimes.

Contribution

It provides a comprehensive analysis of finite-range interactions in trapped two-particle systems, highlighting differences from contact potential models and exploring the approach to 1D behavior with varying trap aspect ratios.

Findings

Eigenvalues approach noninteracting values when the range exceeds trap length scales.

Zero-range contact potential results are recovered in 3D as the range goes to zero.

In 1D, zero-range limit does not reproduce contact potential results, showing fundamental differences.

Abstract

We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s-wave interactions over a wide range of s-wave scattering length $a_s$ including the unitarity limits $a_s \rightarrow \pm\infty$. Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch {\it et al.} [Foundations of Physics, {\bf28}, 549 (1998)] are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in 1D harmonic trap taking one-dimensional finite- range model potential. We have also calculated bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio $\eta$ of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for $\eta \ge 10000$. However, for large range, one can nearly get 1D results for smaller values of $\eta$. This study will be important for exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.