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

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.

Figure 1

Relative energy spectrum as a function of the three-body phase ${\theta }_b$ for three identical bosons in a harmonic trap interacting through zero-range potentials with infinite $s$-wave scattering lengths. The circle, square, and triangle show the ground-state energy for the Gaussian two-body interaction with range $r_0/a_{\mathrm{ho}}=0.06,0.08,$ and $0.1$, respectively. The inset shows the negative energy regime on a log scale. The spacing between the energy levels for fixed ${\theta }_b$ is very close to 515, i.e., very close to the free-space scaling factor.

Figure 2

Hyperradial density $P_{\mathrm{hyper}}\left(R\right)$ for three identical bosons at unitarity. Solid and dotted lines show the PIMC results at $k_{B}T/E_{\mathrm{ho}}=0.4$ for the Gaussian model potential with $r_0/a_{\mathrm{ho}}=0.06$ and $0.1$, respectively (in the main panel, the curves are indistinguishable on the scale shown). The main panel and the inset show the same data but use a different scaling: The main panel uses units derived from the energy of the three-boson system at $T=0$, while the inset employs harmonic oscillator units. For comparison, the dashed line shows the hyperradial density obtained using the zero-range pseudopotential with ${\kappa }_{*}$ determined by the relative energy of the finite-range potential.

Figure 3

Statistical factor $S$ for the $(N-1,1)$ system with interspecies potential $V_{\mathrm{G}}$ with $r_0=0.06a_{\mathrm{ho}}$ and $1/a_s=0$. Squares, crosses, triangles, and circles show the statistical factor $S$ as a function of (a) the temperature $T$ and (b) the inverse temperature $T^{-1}$ for $N=3,4,5$, and 6, respectively.

Figure 4

The lines show (a) the condensate fraction $n_c$ and (b) the superfluid fraction $n_s$ as a function of the temperature $T$ for two Boltzmann particles with zero-range interaction for various $a_s$. The solid, dotted, dashed, dash-dotted, dash-dot-dotted, and dash-dash-dotted lines are for $a_{\mathrm{ho}}/a_s=-\infty ,-2,1,0,1,$ and 2, respectively. In panel (b), the dependence on $a_s$ is small. The insets compare (a) the condensate fraction $n_c$ and (b) the superfluid fraction $n_s$ for two Boltzmann particles (lines; these are the same data as shown in the main parts of the figure) and two identical bosons (squares and circles correspond to $a_{\mathrm{ho}}/a_s=-\infty$ and 0, respectively) as a function of the temperature.

Figure 5

Panels (a) and (b) show radial densities for two identical noninteracting fermions. Solid, dotted, and dashed lines show (a) the scaled radial total density and (b) the scaled radial superfluid density, for $k_{B}T/E_{\mathrm{ho}}=0.5,0.26459,$ and $0.2$, respectively. In panel (a), the dotted line is hardly distinguishable from the dashed line. The solid lines in panels (c)–(e) show (c) the superfluid fraction $n_s$, (d) the classical moment of inertia $I_{\mathrm{c}}$, and (e) the quantum mechanical moment of inertia $I_{\mathrm{q}}$ as a function of the temperature $T$. The diamond, square, and circle mark the temperatures considered in panels (a) and (b).

Figure 6

Energies as a function of the temperature $T$ for three identical bosons at unitarity interacting through $V_{\mathrm{G}}$ with different $r_0$. Circles and squares show the PIMC results for $r_0/a_{\mathrm{ho}}=0.06$ and $0.08$, respectively. For comparison, the solid and dotted lines show the result obtained using the droplet state plus center-of-mass excitations. The dashed line shows the thermally averaged energy for three identical noninteracting bosons. Dash-dot-dotted and dash-dotted lines show results obtained using the simple combined model for $r_0/a_{\mathrm{ho}}=0.06$ and $0.08$ (see the text for discussion).

Figure 7

Phase-transition-like feature for $N$ identical harmonically trapped bosons interacting through $V_{\mathrm{G}}$ with $1/a_s=0$. (a) Circles and squares show the energy obtained by the PIMC approach for $r_0=0.1a_{\mathrm{ho}}$ and $N=3$ and 4, respectively, as a function of the temperature $T$. The dotted, solid, and dashed lines show the energies for $N=3$, 4, and 5 obtained using the simple combined model. (b) The dotted, solid, and dashed lines show the heat capacity $C_v$ for $N=3$, 4, and 5, respectively, as a function of $T$.

Figure 8

Transition temperature $T_{\mathrm{tr}}$ for $N$ identical bosons in a harmonic trap at unitarity as a function of $N$. The transition temperature is calculated using the simple combined model. The circles show $T_{\mathrm{tr}}$ using the droplet energies for the Gaussian two-body interaction model employed in this work. For comparison, the squares show $T_{\mathrm{tr}}$ using the droplet energies for a model Hamiltonian with attractive two-body and repulsive three-body interactions [53] (to obtain the squares, the three-body eigenenergy $E_{\mathrm{droplet}}=E_{\mathrm{trimer}}$ is chosen such that it agrees with that for the Gaussian two-body interaction model, i.e., the circle and the square agree for $N=3$).

Figure 9

Hyperradial density $P_{\mathrm{hyper}}\left(R\right)$ for three identical bosons at unitarity interacting through $V_{\mathrm{G}}$ with $r_0=0.06a_{\mathrm{ho}}$ for various temperatures $T$. Dash-dash-dotted, solid, dotted, dashed, and dash-dotted lines are for $k_{B}T/E_{\mathrm{ho}}=3,4,5,6,$ and 7, respectively. Panel (a) shows the small-$R$ region while panel (b) shows the large-$R$ region. Note that panels (a) and (b) have different scales for the $x$ and the $y$ axes.

Figure 10

Superfluid fraction $n_s$ as a function of the temperature $T$ for $N$ identical bosons at unitarity. The circles and squares show the PIMC results for the Gaussian potential $V_{\mathrm{G}}$ with $r_0=0.1a_{\mathrm{ho}}$ for $N=3$ and 4, respectively. The error bars are smaller than the symbol size. For comparison, the solid line shows the result obtained using a single-particle model (see text for discussion).

Figure 11

Scaled pair distribution functions $r_{j4}^2{P}_{\mathrm{pair}}\left(r_{j4}\right)$ $\left(j<4\right)$ for the $(3,1)$ system with interspecies interaction $V_{\mathrm{G}}$ with $r_0=0.06a_{\mathrm{ho}}$ and diverging interspecies scattering length $a_s$ at temperature (a) $k_{B}T/E_{\mathrm{ho}}=0.6$, (b) $k_{B}T/E_{\mathrm{ho}}=1.2$, (c) $k_{B}T/E_{\mathrm{ho}}=2$, and (d) $k_{B}T/E_{\mathrm{ho}}=3$. Dashed, solid, and dotted lines are for systems with Fermi, Boltzmann, and Bose statistics, respectively. The error bars are comparable to or smaller than the linewidths. In panel (a), the solid and dotted lines are hardly distinguishable. In panel (d), all three lines nearly coincide.

Figure 12

Superfluid fraction $n_s$ as a function of the temperature $T$ for the $(3,1)$ system with interspecies potential $V_{\mathrm{G}}$ with $r_0=0.06a_{\mathrm{ho}}$ and $1/a_s=0$. The circles, crosses, and squares are obtained from the PIMC simulations with Bose, Boltzmann, and Fermi statistics, respectively. The error bars are only shown when they are larger than the symbol size. For comparison, dotted, solid, and dashed lines show the superfluid fraction for the noninteracting $(3,1)$ systems with Bose, Boltzmann, and Fermi statistics, respectively.