Energetics of three interacting mass-imbalanced bodies in a three-dimensional spherical harmonic trap

TL;DR

This paper analyzes the energy spectra of three interacting particles in a 3D harmonic trap using two methods, revealing their agreement in strong interactions and providing insights for thermodynamics and dynamics of such systems.

Contribution

It introduces and compares two approaches for calculating energy eigenstates of three-body systems with mass imbalance in a harmonic trap, including Efimov states.

Findings

Two methods agree well in the strongly interacting limit.

Infinite sum approach cannot uniquely determine Efimov state energies.

Correspondence between boundary conditions and summation truncation in strong interactions.

Abstract

We consider a system of three particles, either three identical bosons or two identical fermions plus an impurity, within a three-dimensional isotropic trap interacting via a contact interaction. Using two approaches, one using an infinite sum of basis states for the wavefunction and the other a closed form wavefunction, we calculate the allowable energy eigenstates of the system as a function of the interaction strength, including the strongly and weakly interacting limits. For the fermionic case this is done while maintaining generality regarding particle masses. We find that the two methods of calculating the spectrum are in excellent agreement in the strongly interacting limit. However the infinite sum approach is unable to uniquely specify the energy of Efimov states, but in the strongly interacting limit there is, to a high degree of accuracy, a correspondence between the three-body parameter required by the boundary condition of the closed form approach and the summation truncation order required by the summation approach. This specification of the energies and wavefunctions forms the basis with which thermodynamic variables such as the virial coefficients or Tan contacts, or dynamic phenomena like quench dynamics can be calculated.