One particle in a box: The simplest model for a Fermi gas in the unitary limit

We consider a single quantum particle in a spherical box interacting with a fixed scatterer at the center, to construct a model of a degenerate atomic Fermi gas close to a Feshbach resonance. One of the key predictions of the model is the existence of two branches for the macroscopic state of the gas, as a function of the magnetic field controlling the value of the scattering length. This model is able to draw a qualitative picture of all the different features recently observed in a degenerate atomic Fermi gas close to the resonance, even in the unitary limit.

Figure 1

Model used: a fictitious particle corresponding to the relative motion of a fermion with spin $+1∕2$ and the nearest fermion with spin $-1∕2$ is scattered by the fixed $\delta$ Fermi pseudopotential [22] in the center of a spherical box of radius $R$ with absorbing boundary conditions. The box mimics the interaction effect of the $N∕2-1$ remaining spin $-1∕2$ atoms and the Fermi statistical effect of the remaining $N∕2-1$ spin $+1∕2$ fermions. The coupling constant $g$ is $4\pi {\hslash }^{2}a∕m$, where $a$ is the scattering length of two fermions with opposite spin, $g{\delta }_{\text{reg}}\left(\vec{r}\right)=g\delta \left(\vec{r}\right){\partial }_r(r\bullet )$ is the Fermi pseudopotential which imposes the contact condition (9) on the two-body wave function [21, 22, 23, 24].

Figure 2

Total energy of the gas in units of the Fermi energy, as a function of $-1∕k_{F}a$. Only the first two branches of the model are represented.

Figure 3

Pressure $P$ of the gas in units of the Fermi pressure $P_F={\hslash }^2{k}_F^5∕15{\pi }^{2}m$, as function of $-1∕k_{F}a$. Only the first two branches of the model are represented.