BCS-BEC crossover of a quasi-two-dimensional Fermi gas: The significance of dressed molecules

We study the crossover of a quasi-two-dimensional Fermi gas trapped in the radial plane from the Bardeen-Cooper-Schrieffer (BCS) regime to the Bose-Einstein condensation (BEC) regime by crossing a wide Feshbach resonance. We consider two effective two-dimensional Hamiltonians within the mean-field level, and calculate the zero-temperature cloud size and number density distribution. For a model 1 Hamiltonian with renormalized atom-atom interaction, we observe a constant cloud size for arbitrary detunings. For a model 2 Hamiltonian with renormalized interactions between atoms and dressed molecules, the cloud size deceases from the BCS to BEC side, which is consistent with the picture of BCS-BEC crossover. This qualitative discrepancy between the two models indicates that the inclusion of dressed molecules is essential for a mean-field description of quasi-two-dimensional Fermi systems, especially on the BEC side of the Feshbach resonance.

Figure 1

The BCS-BEC crossover behavior of a uniform quasi-2D Fermi gas at zero temperature, showing (a) the chemical potential $\mu$, (b) the gap $\Delta$, both in unit of $\hslash {\omega }_z$, and (c) the dressed-molecule fraction $n_b/n$. Notice that the results for ${}^6\text{L}\text{i}$ (solid) and those for ${}^{40}\text{K}$ (dashed) almost coincide as plotted as functions of $a_z/a_s$, indicating a universal behavior around the resonance point. Furthermore, significant difference between model 1 (gray) and model 2 (black) can be observed in (b) and (c), which shows that model 1 is oversimplified at unitarity and on the BEC side of the resonance. The parameters used in these plots are ${\omega }_z=2\pi \times 62\text{kHz}$ and $n{a}_z^2=0.001$.

Figure 2

The Thomas-Fermi cloud size of a quasi-2D Fermi gas of ${}^6\text{L}\text{i}$ over a wide BCS-BEC crossover region. Here, results from model 2 (solid) are compared with those from model 1 (dashed). All curves are normalized to the cloud size of a noninteracting Fermi gas $R_{\text{BCS}}$. Notice that the results of model 2 recover the correct pictures in the BCS and BEC limits, in clear contrast to the model 1 prediction of a flat line. Parameters used for these two plots are ${\omega }_z=2\pi \times 62\text{kHz}$, ${\omega }_{\perp }=2\pi \times 20\text{Hz}$, and the total particle number $N={10}^4$. For reference, the results for an isotropic 3D Fermi gas with the same total particle number is also plotted (dotted), where a single-channel model and a two-channel model are both incorporated to give indistinguishable predictions.

Figure 3

(Color online) The in-trap number density (the solid lines) and dressed-molecule fraction (the dashed lines) distribution along the radial direction of a quasi-2D Fermi gas of ${}^6\text{L}\text{i}$, obtained from model 2 (a–c) and model 1 (d–f). The top panels correspond to the case of $a_z/a_s=-1$ (BCS side), the middle panels to the case of $a_z/a_s=0$ (unitarity), and the bottom panels to the case of $a_z/a_s=1$ (BEC side). The parameters are ${\omega }_z=2\pi \times 62\text{kHz}$, ${\omega }_{\perp }=2\pi \times 20\text{Hz}$, and $N={10}^4$.