Two-body state with $p$-wave interaction in a one-dimensional waveguide under transversely anisotropic confinement

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

Figure 1

The bound-state energy of two ${}^{40}\mathrm{K}$ atoms in the $|F=\frac{9}{2},m_F=-\frac{7}{2}\rangle$ hyperfine state confined in a transversely isotropic waveguide near the $p$-wave Feshbach resonance centered at $B_0=198.8$ G in three dimensions [7, 24]. The dotted horizontal lines are the thresholds where the bound states merge into the continuum as the magnetic field increases. The magnetic field $B$ dependence of the scattering volume $v_1$ as well as the effective range $r_1$ for ${}^{40}\mathrm{K}$ is from Ref. [24]. The inset shows the details of the curves in a smaller magnetic field range for visual clarity.

Figure 2

The same as Fig. 1 but for a transversely anisotropic confinement, i.e., $\eta =2$.

Figure 3

The $p$-wave 1D scattering length $l_{\text{1D}}$ as a function of the magnetic field $B$ in the 1D waveguide with different transverse anisotropy $\eta$. Here, the $p$-wave interaction of two ${}^{40}\mathrm{K}$ atoms in the $|F=\frac{9}{2},m_F=-\frac{7}{2}\rangle$ hyperfine state is tuned by using the $p$-wave Feshbach resonance centered at $B_0=198.8$ G in three dimensions [7, 24].

Figure 4

The position of the $p$-wave confinement-induced resonance in the 1D waveguide denoted by the magnetic field strength $B^{\left(R\right)}$ as a function of the transverse anisotropy $\eta$. The transverse anisotropy $\eta$ can be tuned by increasing ${\omega }_x$ with a fixed ${\omega }_y$. The corresponding parameters are from the ${}^{40}\mathrm{K}$ experiments [7, 24].

Figure 5

The molecular state in the closed-channel Hamiltonian ${\mathcal{H}}_{cl}$ for different transverse anisotropy $\eta$. The horizontal dotted lines denote the continuum threshold of the open channel, while the vertical red dashed lines indicate the position where a zero-energy scattering resonance occurs.