Temperature dependence of small harmonically trapped atom systems with Bose, Fermi, and Boltzmann statistics

Abstract

While the zero-temperature properties of harmonically trapped cold few-atom systems have been discussed fairly extensively over the past decade, much less is known about the finite-temperature properties. Working in the canonical ensemble, we characterize small harmonically trapped atomic systems as a function of the temperature using analytical and numerical techniques. We present results for the energetics, structural properties, condensate fraction, superfluid fraction, and superfluid density. Our calculations for the two-body system underline that the condensate and superfluid fractions are distinctly different quantities. Our work demonstrates that the path-integral Monte Carlo method yields reliable results for bosonic and fermionic systems over a wide temperature range, including the regime where the de Broglie wavelength is large, i.e., where the statistics plays an important role. The regime where the Fermi sign problem leads to reasonably large signal-to-noise ratios is mapped out for selected parameter combinations. Our calculations for bosons focus on the unitary regime, where the physics is expected to be governed by the three-body parameter. If the three-body parameter is large compared to the inverse of the harmonic oscillator length, we find that the bosons form a droplet at low temperature and behave approximately like a noninteracting Bose and eventually Boltzmann gas at high temperature. The change of the behavior occurs over a fairly narrow temperature range. A simple model that reproduces the key aspects of the phase-transition-like feature, which can potentially be observed in cold atom Bose gas experiments, is presented.

Energy and structural properties ofN-boson clusters attached to three-body Efimov states: Two-body zero-range interactions and the role of the three-body regulator

“…This approach was introduced and benchmarked in Ref. [28]. The basic idea is to place the droplet in a weak external harmonic confinement, whose angular frequency ω is chosen such that the center of mass energy spectrum becomes discretized and the relative motion is unaffected by the trap.…”

“…The diamonds in Fig. 6(a) show the N -boson energy per particle E N /N for the model 2bG as a function of N [21,26,28]. The energy per particle increases approximately linearly with N for N > 6 (for smaller N , some deviations from the linear behavior exist).…”

“…The triangles show the diffusion Monte Carlo (DMC) energies for a Hamiltonian with two-body square well interaction and repulsive three-body hardcore regulator [23]. The diamonds show the energy for the model 2bG [28]. (b) Summary of our PIMC calculations.…”

“…The ground state manifolds of these models, referred to as 2bG, 2bLJ, 2b10-6, and 2b8-6 (see Table I), lack-as we show-a number of key Efimov characteristics. Two-body Gaussian interactions have been employed extensively in the literature [19,21,[26][27][28], sometimes also in combination with a repulsive three-body regulator [23,29].…”

Incorporating exact two-body propagators for zero-range interactions intoN-body Monte Carlo simulations

“…Our simulations pursue an alternative approach, in which the scattering states of the system are discretized in such a way that the relative ground state energy E cluster of the Nbody cluster is much larger than the energy scale introduced by the discretization. We utilize a spherically symmetric harmonic trap and adjust the trapping frequency such that |E cluster | ≫ ω. Simulations are then performed at a temperature where the Bose droplet is in the ground-state dominated liquid-phase [43,56], where the finite temperature introduces center-of-mass excitations but not excitations of the relative degrees of freedom. The temperature T tr at which excitations of the relative degrees of freedom become relevant can be estimated using the "combined model" introduced in Ref.…”