Confinement-induced resonance in quasi-one-dimensional systems under transversely anisotropic confinement

We theoretically investigate the confinement-induced resonance for quasi-one-dimensional quantum systems under transversely anisotropic confinement, using a two-body $s$-wave-scattering model in the zero-energy collision limit. We predict a single resonance for any transverse anisotropy, whose position shows a slight downshift with increasing anisotropy. We compare our prediction with the recent experimental result by Haller et al. [Phys. Rev. Lett. 104, 153203 (2010)], in which two resonances are observed in the presence of transverse anisotropy. The discrepancy between theory and experiment remains to be resolved.

Figure 1

The bound states of the total relative Hamiltonian ${\mathrm{Ĥ}}_{\mathrm{rel}}$ (the solid curve) and the excited Hamiltonian ${\mathrm{Ĥ}}_e$ (the dotted curve) with $\eta =1$.

Figure 2

The same as Fig. 1 but for $\eta =2$.

Figure 3

The effective 1D coupling constant, $g_{1D}$ [in units of $\hslash {}^2/({\mathit{ma}}_y)$], as a function of $a_{3D}/a_y$ at different transverse anisotropy $\eta =1$ (solid line), $1.5$ (dashed line), and $3$ (dot-dashed line).

Figure 4

Resonance positions of CIR, $a_{3D}^{(R)}/a_y$, as a function of the transverse anisotropy $\eta ={\omega }_x/{\omega }_y$. For sufficient large or small values of transverse anisotropy, the quasi-1D system crosses over to the quasi-2D regime.

Figure 5

Resonance positions of CIRs, obtained by solving $a_{3D}^{(R)}/a_y=\sqrt{2}/\mathcal{C}$, as a function of the transverse anisotropy $\eta ={\omega }_x/{\omega }_y$. The experimental results (symbols) are compared with our zero-energy $s$-wave-scattering prediction (solid line). In the presence of transverse anisotropy, it is difficult to determine directly the resonance positions from atom loss. Therefore, experimentally the positions were obtained by determining the associated atom number minima and then subtracting a constant offset. In accord with this procedure, we have uniformly offset the experimental data in such a way that the measured resonance position in the symmetric limit ($\eta =1$) is equal to the known theoretical prediction by Olshanii [10]. The dashed and dot-dashed lines are the resonance positions, obtained respectively by solving, $a_{3D}^{(R)}/a_y=\sqrt{2\eta }/\mathcal{C}$ and $a_{3D}^{(R)}/a_y=\sqrt{2/\eta }/\mathcal{C}$. See the text for more details.