Occupation numbers of the harmonically trapped few-boson system

TL;DR

This paper analyzes the occupation numbers of harmonically trapped dilute boson systems, comparing perturbative and exact solutions, and explores effects of different Hamiltonians and strong interactions, including Efimov physics.

Contribution

It provides a perturbative framework for calculating occupation numbers in trapped boson systems and compares results with exact and ab initio calculations, highlighting non-universal effects.

Findings

Non-universal corrections affect occupation numbers at subleading order.

Different low-energy Hamiltonians can yield different occupation numbers.

Occupation numbers depend on the Efimov parameter in strongly-interacting three-boson systems.

Abstract

We consider a harmonically trapped dilute $N$-boson system described by a low-energy Hamiltonian with pairwise interactions. We determine the condensate fraction, defined in terms of the largest occupation number, of the weakly-interacting $N$-boson system ($N \ge 2$) by employing a perturbative treatment within the framework of second quantization. The one-body density matrix and the corresponding occupation numbers are compared with those obtained by solving the two-body problem with zero-range interactions exactly. Our expressions are also compared with high precision {\em{ab initio}} calculations for Bose gases with $N=2-4$ that interact through finite-range two-body model potentials. Non-universal corrections are identified to enter at subleading order, confirming that different low-energy Hamiltonians, constructed to yield the same energy, may yield different occupation numbers. Lastly, we consider the strongly-interacting three-boson system under spherically symmetric harmonic confinement and determine its occupation numbers as a function of the three-body "Efimov parameter".