Resonant scattering properties close to a $p$-wave Feshbach resonance

We present a semianalytical treatment of both the elastic and inelastic collisional properties near a $p$-wave Feshbach resonance. Our model is based on a simple three-channel system that reproduces more elaborate coupled-channel calculations. We stress the main differences between $s$-wave and $p$-wave scattering. We show in particular that, for elastic and inelastic scattering close to a $p$-wave Feshbach resonance, resonant processes dominate over the low-energy behavior.

Figure 1

The $p$-wave model: we consider three internal states, labeled ǀ1⟩, ǀ2⟩, and ǀ3⟩. States ǀ1⟩ and ǀ2⟩ are two open channels corresponding, respectively, to the incoming and decay channels. The released energy in an inelastic collision bringing an atom initially in ǀ1⟩ to ǀ2⟩ is denoted $\Delta$. State ǀ3⟩ is a closed channel that possesses a $p$-wave bound state of energy $\delta$ nearly resonant with state ǀ1⟩. Finally, we assume that these three channels interact through a potential $\widehat{V}$ acting only on the internal states and coupling the two open channels to the closed channel.

Figure 2

Diagrammatic expansion of the $T$ matrix. The full (dashed) lines represent free atoms (molecules). $\mid \alpha ,\mathbf{k}⟩$ is the scattering state of the two particles, in the internal state $\alpha =1$,2. $\mid 3,0_{\mathbf{k}}⟩$ represents the state of a $p$-wave molecule with orbital angular momentum component zero on the $\mathbf{k}$ direction.

Figure 3

Effect of the centrifugal barrier on the bound state in $p$-wave Feshbach resonances. (a) Case of $s$-wave scattering: the bare molecular state goes from $\delta <0$ (full line) to $\delta >0$ (dashed line). In the process, the molecular state becomes unstable and the wave function becomes unbounded. (b) In the case of a $p$-wave bound state, the presence of the centrifugal barrier smooths the transition from $\delta >0$ to $\delta <0$. Even for $\delta >0$, the wave function stays located close to the bottom of the well.

Figure 4

Energy dependence of the elastic cross section. Dots: numerical closed-channel calculation. The left peak corresponds to $m_l=0$ and the right peak to $m_l=\pm 1$. Full line: Fits using Eq. (11).

Figure 5

Full line: numerical calculation of the loss rate for $T=10\mu \mathrm{K}$. Dotted line: asymptotic expansion (12).