Dynamical Mean-Field Equations for Strongly Interacting Fermi Gas in a Trap

We derive the time evolution equations at zero temperature for the wavefunctions of the molecular bosons and the fermion pairs in a trapped Fermi gas near a wide Feshbach resonance. The derivation of the equations is based on the variational principle and the BCS-like ansatz state: $|\Phi\rangle=\mathcal{N}e^{\int \phi_b(\mathbf{r})\Psi_b^{\dag}(\mathbf{r})d^3\mathbf{r}} e^ {\int \rho(\mathbf{r},\mathbf{r'})\Psi_{\uparrow}^{\dag}(\mathbf{r}) \Psi_{\downarrow}^{\dag}(\mathbf{r'})d^3\mathbf{r}d^3\mathbf {r'}}|0\rangle$. In deriving the equations, we have assumed that the external trapping potential and the wavefunction of the molecular bosons are spatially slow-varying on the length scale of the size of the fermionic atom pairs, which should be valid over a wide range on the BEC side of resonance, including the resonance point. In the bosonic region ($\mu\le0$, where $\mu$ is the chemical potential), the equations will reduce to one that resembles a Gross- Pitaevskii (GP) equation. We solve the the stationary ground state of the system at different detunings near the crossover region and self-consistently checked our assumptions. The time evolution equations provide macroscopic description for the wavefunctions of the molecular bosons and of the fermion pairs near the interesting BCS-BEC crossover region. In future studies, these equations can be used to analyze the interesting physics of vortices or the excitation spectrum in the Fermi condensate.