Pseudopotential Method for Higher Partial Wave Scattering

We present a zero-range pseudopotential applicable for all partial wave interactions between neutral atoms. For $p$ and $d$ waves, we derive effective pseudopotentials, which are useful for problems involving anisotropic external potentials. Finally, we consider two nontrivial applications of the $p$-wave pseudopotential: we solve analytically the problem of two interacting spin-polarized fermions confined in a harmonic trap, and we analyze the scattering of $p$-wave interacting particles in a quasi-two-dimensional system.

Figure 1

(a),(b) Energy spectrum of the two fermions interacting in the $p$ wave as a function of the scattering volume $a_p^3/d^3$, with $d=\sqrt{\hslash /\mu {\omega }_z}$. The two particles are confined in an axially symmetric harmonic trap with (a) $\eta ={\omega }_{\perp }/{\omega }_z=0.1$ and (b) $\eta =10$. (c),(d) Energy spectrum of two atoms interacting via a square-well model potential as a function of the well depth $U$. The atoms are confined in an isotropic harmonic potential and interact in (c) the $p$ wave and (d) the $s$ wave.

Figure 2

Two-dimensional differential cross section in forward direction $|f(0){|}^2$ in the scattering of two fermions in a quasi-two-dimensional system as a function of $p$-wave scattering volume $a_p^3$, for different relative energies $E$ of the particles. The inset shows the value of the critical volume $V_c$, at which the system exhibits the confinement-induced resonance.