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arXiv:1406.5546 · Physical Review A

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

Yangqian Yan, D. Blume

Published Jul 21, 2014• 12 citations
arXivHTMLPDF

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.

DOI
10.1103/physreva.90.013620

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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.
Ultracold atomic gases with short-range interactions are characterized by a number of universal species-independent relations. Many of these relations involve the two-body Tan contact. Employing the canonical ensemble, we determine the Tan contact for small harmonically trapped two-component Fermi gases at unitarity over a wide range of temperatures, including the zero and high-temperature regimes. A cluster expansion that describes the properties of the NNN-particle system in terms of those of smaller subsystems is introduced and shown to provide an accurate description of the contact in the high-temperature regime. Finite-range corrections are quantified and the role of the Fermi statistics is elucidated by comparing results for Fermi, Bose, and Boltzmann statistics.
The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the sss-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.
Superfluidity in the cold atomic two-species Fermi gas system in the unitary limit of infinite scattering length remains incompletely understood. In particular, a pseudogap phase has been proposed to exist above the superfluid critical temperature. Here we apply the auxiliary-field quantum Monte Carlo method to perform ab initio calculations of the temperature dependence of three quantities---the energy-staggering pairing gap, the condensate fraction, and the heat capacity---in a trapped finite-size cold atom system in the unitary limit of infinite scattering length. As the calculations of the energy-staggering pairing gap require the use of the canonical ensemble, we employ an efficient algorithm for the stabilization of particle-number projection that is essential for reaching convergence in the size of the model space. We observe clear signatures of the superfluid phase transition in all three quantities, including a signature of the recently measured $\ensuremath{\lambda}$ peak in the heat capacity. We also find that the energy-staggering gap does not lead the condensate fraction as the temperature decreases.
Using the finite-temperature path integral Monte Carlo method, we investigate dilute, trapped Bose gases in a quasi-two-dimensional (quasi-2D) geometry. The quantum particles have short-range, sss-wave interactions described by a hard-sphere potential whose core radius equals its corresponding scattering length. The effect of both the temperature and the interparticle interaction on the equilibrium properties such as the total energy, the density profile, and the superfluid fraction is discussed. We compare our accurate results with both the semiclassical approximation and the exact results of an ideal Bose gas. Our results show that for repulsive interactions (i) the minimum value of the aspect ratio, where the system starts to behave quasi-2D, increases as the two-body interaction strength increases; (ii) the superfluid fraction for a quasi-two-dimensionally Bose gas is distinctly different from that for both a quasi-1D Bose gas and a true 3D system, i.e., the superfluid fraction for a quasi-2D Bose gas decreases faster than that for a quasi-1D system and a true 3D system with increasing temperature, and shows a stronger dependence on the interaction strength; (iii) the superfluid fraction for a quasi-2D Bose gas lies well below the values calculated from the semiclassical approximation; and (iv) the Kosterlitz-Thouless transition temperature decreases as the strength of the interaction increases.
We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
We study thermal properties of a trapped Bose-Bose mixture in a dilute regime using quantum Monte Carlo methods. Our main aim is to investigate the dependence of the superfluid density and the condensate fraction on temperature for the mixed and separated phases. To this end we use the diffusion Monte Carlo method in the zero-temperature limit and the path-integral Monte Carlo method for finite temperatures. The results obtained are compared with solutions of the coupled Gross-Pitaevskii equations for the mixture at zero temperature. We notice the existence of an anisotropic superfluid density in some phase-separated mixtures. Our results also show that the temperature evolution of the superfluid density and condensate fraction is slightly different, showing noteworthy situations where the superfluid fraction is smaller than the condensate fraction.