Free Expansion of a Weakly-interacting Dipolar Fermi Gas

We theoretically investigate a polarized dipolar Fermi gas in free expansion. The inter-particle dipolar interaction deforms phase-space distribution in trap and also in the expansion. We exactly predict the minimal quadrupole deformation in the expansion for the high-temperature Maxwell-Boltzmann and zero-temperature Thomas-Fermi gases in the Hartree-Fock and Landau-Vlasov approaches. In conclusion, we provide a proper approach to develop the time-of-flight method for the weakly-interacting dipolar Fermi gas and also reveal a scaling law associated with the Liouville’s theorem in the long-time behaviors of the both gases.

💡 Research Summary

This paper presents a comprehensive theoretical study of the free expansion of a weakly‑interacting, polarized dipolar Fermi gas. The authors consider a gas of fermionic particles whose dipole moments are fully aligned along the laboratory z‑axis by a strong external electric field. The dipole–dipole interaction is given by the anisotropic long‑range potential
(v(\mathbf r)=G_d,(1-3z^2/r^2)/r^3),
where (G_d\propto d^2) and (d) is the electric dipole moment. The system is initially confined in a harmonic trap; at time (t=0) the trap is switched off and the gas expands freely.

To treat the many‑body dynamics the authors employ the time‑dependent Hartree‑Fock approximation (TD‑HF A). The two‑body interaction is replaced by a self‑consistent mean field consisting of a direct (Hartree) term and an exchange (Fock) term. By introducing the Wigner function (f(\mathbf r,\mathbf p)) they derive an exact quantum kinetic equation containing a sine operator (Moyal expansion). In the semiclassical limit they truncate the sine at first order in (\hbar) and obtain the Landau‑Vlasov (LV) equation, which conserves phase‑space volume and captures the essential effect of the momentum‑dependent Fock term.

Two initial equilibrium states are considered: (i) a high‑temperature Maxwell‑Boltzmann (MB) gas and (ii) a zero‑temperature Thomas‑Fermi (TF) gas, both confined in a cylindrically symmetric harmonic trap. Their Wigner functions are respectively Gaussian (MB) and a step function of a quadratic form (TF). The authors introduce a quadrupole deformation parameter in momentum space, (\lambda_p=\ln(T_z/T_0)) with (T_z=\langle p_z^2\rangle) and (T_0=(\langle p_x^2\rangle\langle p_y^2\rangle\langle p_z^2\rangle)^{1/3}), to quantify the anisotropy generated during expansion.

Starting from the LV equation they derive an exact evolution equation for (\lambda_p(t)). Assuming that the deviation from the ballistic solution is small, they propose an ansatz for the time‑dependent Wigner function: (\tilde f(\boldsymbol\xi,\mathbf p)=f_0(\tilde{\boldsymbol\xi},\tilde{\mathbf p})), where the tilde variables are scaled versions of the original coordinates and momenta, containing the additional deformation (\lambda(t)). This ansatz respects Liouville’s theorem and reduces to the ballistic solution when (\lambda=0).

Inserting the ansatz into the evolution equation and expanding to first order in the small parameters (\Lambda_0) (initial trap anisotropy), (\lambda_p) and (\dot\lambda), the authors obtain a remarkably simple result: \