Stability of fermionic gases close to a $p$-wave Feshbach resonance

We study the stability of the paired fermionic $p$-wave superfluid made out of identical atoms all in the same hyperfine state close to a $p$-wave Feshbach resonance. First we reproduce known results concerning the lifetime of a three-dimensional superfluid, in particular, we show that it decays at the same rate as its interaction energy, which may preclude its equilibration before it decays. Then we proceed to study its stability in the case when the superfluid is confined to two dimension (2D) by means of an optical harmonic potential. We find that the relative stability is improved in 2D in the BCS regime, such that the decay rate is now parametrically slower than the appropriate interaction energy scale, leading to the possibility of an equilibrated $p$-wave superfluid in 2D.

Figure 1

The renormalized propagator of spin-1 molecules. Here the standard notations for the two-channel model Eq. (9) are used: The thin straight lines are fermionic propagators, the thin wavy lines are bare bosonic propagators, and the thick wavy lines are full bosonic propagators.

Figure 2

The diagram corresponding to the scattering of two atoms.

Figure 3

The diagram corresponding to the wave function of free fermions. The square block represents the atom-molecule $T$-matrix, and the three outgoing fermionic lines represent the three fermions whose wave function we would like to compute.

Figure 4

The series of diagrams which sum to give the scattering $T$-matrix of a spin-$1\text{molecule}$ and a fermionic atom.

Figure 5

(Color online) $T_1\left(p\right)$ and $T_2\left(p\right)$ obtained by solving Eq. (65) for ${\omega }_0=-{10}^{-6}\frac{{\Lambda }^2}{m}$ and $c_2={10}^6$ (blue, solid). The solution uses Gaussian-Legendre quadrature for the numerical integration [43] with 2000 grid points. Also shown are the analytical solutions (78) with parameters chosen as in Eq. (79) (black, dashed).