Universal and non-universal effective $N$-body interactions for ultracold harmonically-trapped few-atom systems

TL;DR

This paper derives the ground-state energy of ultracold atoms in a harmonic trap using effective quantum field theory, revealing both universal and non-universal three-body interactions and confirming results with numerical calculations.

Contribution

It introduces a systematic EFT approach up to order l^{-4} for ultracold atoms, highlighting the emergence of non-universal three-body effects due to a logarithmic divergence.

Findings

Effective three-body interaction contains a non-universal component.

Numerical calculations confirm EFT predictions.

Explicit calculation of non-universal three-body energy contribution.

Abstract

We derive the ground-state energy for a small number of ultracold atoms in an isotropic harmonic trap using effective quantum field theory (EFT). Atoms are assumed to interact through pairwise energy-independent and energy-dependent delta-function potentials with strengths proportional to the scattering length $a$ and effective range volume $V$, respectively. The calculations are performed systematically up to order $l^{-4}$, where $l$ denotes the harmonic oscillator length. The effective three-body interaction contains a logarithmic divergence in the cutoff energy, giving rise to a non-universal three-body interaction in the EFT. Our EFT results are confirmed by nonperturbative numerical calculations for a Hamiltonian with finite-range two-body Gaussian interactions. For this model Hamiltonian, we explicitly calculate the non-universal effective three-body contribution to the energy.