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Two-body state with p-wave interaction in one-dimensional waveguides under transversely anisotropic confinement

We theoretically study two atoms with $p$-wave interaction in a one-dimensional waveguide, and investigate how the transverse anisotropy of the confinement affects the two-body state, especially, the properties of the resonance. For bound-state solution, we find there are totally three two-body bound states due to the richness of the orbital magnetic quantum number of $p$-wave interaction, while only one bound state is supported by $s$-wave interaction. Two of them become nondegenerate due to the breaking of the rotation symmetry under transversely anisotropic confinement. For scattering solution, the effective one-dimensional scattering amplitude and scattering length are derived. We find the position of the $p$-wave confinement-induced resonance shifts apparently as the transverse anisotropy increases. In addition, a two-channel mechanism for confinement-induced resonance in a one-dimensional waveguide is generalized to $p$-wave interaction, which was proposed only for $s$-wave interaction before. All our calculations are based on the parameterization of the $^{40}$K atom experiments, and can be confirmed in future experiments.