Thermodynamics and coherence of a trapped dipolar Fermi gas

We develop a mean-field treatment of a polarized trapped Fermi gas with dipole-dipole interactions. Our approach is based on self-consistent semiclassical Hartree-Fock theory that accounts for direct and exchange interactions. We discuss our procedure for numerically implementing the calculation. We study the thermodynamic and the first- and second-order correlation properties of the system. We find that the system entropy depends on the trap geometry, allowing the system to be cooled as the trap aspect ratio is increased, and that exchange interactions cause the correlation functions to be anisotropic in the low-temperature regime. We also find that many uniform gas thermodynamic predictions, for which direct interaction effects vanish, are qualitatively unreliable for trapped systems, most notably for oblate traps. We develop a simplified Hartree formalism that is applicable to anisotropic harmonic traps.

Figure 1

Deformation of (a) momentum and (b) position density distributions. Calculation parameters $\lambda =0.1$ [blue (dark gray) lines], $1$ [green (light gray) lines], and 10 [red (gray) lines]. The dimensionless interaction strength is $D_t=1$. Thin curves for $T>0.4T_F^0$ use the variational approach of [28] and crosses at $T=0$ use the variational approach of [17].

Figure 2

Direct energy $E_D$ (solid line) and exchange energy $E_E$ (dashed line) versus temperature for a dipolar Fermi gas. Aspect ratios are $\lambda =0.1$ [blue (dark gray) lines], $1$ [green (light gray) lines], and 10 [red (gray) lines]. The interaction parameter is $D_t=1$. Note that the $E_D$ and $E_E$ results for the $\lambda =1$ case are almost indistinguishable.

Figure 3

Schematic showing how the distribution of dipoles in (a) prolate, (b) spherical, and (c) oblate geometries.

Figure 4

Chemical potential ($\mu /k_B{T}_F^0$) as a function of temperature. Calculation parameters are $\lambda =0.1$ [blue (dark gray) lines]$,1$ [green (light gray) lines], and 10 [red (gray) lines]. The dimensionless interaction strength is $D_t=1$. Values at $T=0.01T_F^0$ are shown with crosses.

Figure 5

Entropy versus temperature for a dipolar Fermi gas. (a) Entropy versus temperature. (b) Difference in entropy from the ideal Fermi gas. The inset shows a magnification of the entropy curves with an adiabatic process from initial condition (i) to final condition (f) indicated. Aspect ratios are $\lambda =0.1$ [blue (dark gray) lines], $1$ [green (light gray) lines], and 10 [red (gray) lines]. The interaction parameters are $D_t=0.5$ (dotted), $D_t=1$ (solid), and $D_t=2$ (dashed).

Figure 6

Heat capacity versus temperature for a dipolar Fermi gas. (a) Heat capacity versus temperature. (b) Change in the interacting heat capacity from the ideal gas heat capacity. Aspect ratios are $\lambda =0.1$ [blue (dark gray) lines], $1$ [green (light gray) lines], and 10 [red (gray) lines]. The interaction parameters are $D_t=0.5$ (dotted), $D_t=1$ (solid), and $D_t=2$ (dashed).

Figure 7

Normalized volume-averaged correlation functions for the dipolar Fermi gas. (a) The first-order correlation function $g^{(1)}(\mathbf{r})$ and high-temperature limit (black curve) $e^{-\pi \left|\mathbf{r}\right|{}^2/{\lambda }_{\mathrm{dB}}^2}$. (b) The second-order correlation function $g^{(2)}(\mathbf{r})$ and high-temperature limit $1-e^{-2\pi \left|\mathbf{r}\right|{}^2/{\lambda }_{\mathrm{dB}}^2}$. Results are given along the $x$ axis (dashed) and along the $z$ axis (dotted) for aspect ratio $\lambda =1$ (with other aspect ratios showing similar behavior) and $D_t=1$. Results are shown for $T=0.01T_F^0$ [blue (gray)] and $T=0.5T_F^0$ [red (light gray)]. $T=0.5T_F^0$ is used for the high-temperature limits. Results for both aspect ratios are indistinguishable from their high-temperature limit at $T=T_F^0$.

Figure 8

Coherence length of a dipolar Fermi gas as a function of temperature: $l_{\mathrm{coh},x}$ (dashed line) and $l_{\mathrm{coh},z}$ (dotted line). Results are given for $D_t=1$ and an aspect ratio of $\lambda =1$. Results for aspect ratios of $\lambda =0.1$ and $\lambda =10$ are similar. The black curve shows ${\lambda }_{\mathrm{dB}}/\sqrt{8\pi }$ for reference.