Local exchange theory for trapped dipolar gases

We develop a practical Hartree-Fock theory for trapped Bose and Fermi gases that interact with dipole-dipole interactions. This theory is applicable at zero and finite temperature. Our approach is based on the introduction of local momentum distortion fields that characterize the exchange effects in terms of a local effective potential. We validate our theory against existing theories, finding excellent agreement with full Hartree-Fock calculations.

Figure 1

Comparison of energies for pure ($g=0$) dipolar theories showing HF (crosses), HLF (solid curves), Hartree (dotted curves, zero for $E_E$ not shown), for aspect ratio $\lambda =0.1$ (blue, dark gray), 1 (green, light gray), 10 (red, gray). (a) Kinetic energy $E_K$, (b) dipolar exchange energy $E_E$. Main figures are for fermions with $D_t=1$ and insets are for a Bose gas at $T=1.5T_c^0$. Dipole strength is parametrized in terms of $D_t=C_{\mathrm{dd}}{N}^{1/6}/\left(4\pi \hslash \omega a_{\mathrm{ho}}^3\right)$, where $a_{\mathrm{ho}}=\sqrt{\hslash /m\omega }$ and $\omega =\sqrt[3]{{\omega }_{\rho }^2{\omega }_z}$, with $T_c^0=\sqrt[3]{N/\zeta \left(3\right)}\hslash \omega /k_B$ and $T_F^0=\sqrt[3]{6N}\hslash \omega /k_B$ the ideal gas Bose-Einstein condensation and Fermi temperature, respectively. Results in the insets terminate at finite $D_t$ due to approaching instability [20].

Figure 2

Comparison of pure ($g=0$) dipolar theories showing HF (crosses), HLF (solid curves), HGF (dashed lines), and Hartree (dotted curves) in the radial (blue, dark gray) and axial (red, gray) directions. (a) $\delta \left(\mathbf{x}\right)$ (Hartree results are not shown, being all zero) with $\lambda =1$ and $D_t=1$ for bosons at $T=1.2T_c^0$ (upper curves) and fermions at $T=0.1T_F^0$ (lower curves). (b) $n\left(\mathbf{x}\right)$ and (c) momentum density $\tilde{n}\left(\mathbf{k}\right)=\int \frac{d\mathbf{x}}{{\left(2\pi \right)}^3}W(\mathbf{x},\mathbf{k})$ for fermions at $T=0.02T_F^0$ with $\lambda =10$ and $D_t=2$. The HGF agreement in (b) and (c) is reasonable and is not shown.