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

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.

arXiv comment7 pages, 4 figures

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• Effective multi-body SU($N$)-symmetric interactions of ultracold fermionic atoms on a 3-D lattice
• Generalization of the Fermi Pseudopotential
• Lattice-enabled detection of spin-dependent three-body interactions
• Quantum halo states in two-dimensional dipolar clusters