Quantitative analysis of $p$-wave three-body losses via cascade process

We describe the three-body loss coefficient of identical fermions with $p$-wave interactions using a set of rate equations in which three-body recombination happens via an indirect process. Our theoretical treatment explains experimental results just above the universal scaling law regime of weak interactions. Furthermore, we theoretically extend and experimentally verify the rate equation model for the case of atoms trapped in two dimensions. Moreover, we find that the three-body loss coefficient in a two-dimensional trap is proportional to $A_{p}^{3}$ in the weakly interacting regime, where $A_{p}$ is the scattering area. Our results are useful in understanding three-body physics with $p$-wave interactions.