Quasi-one-dimensional scattering with general transverse two-dimensional confinement

Confinement-induced resonances (CIR) in quasi-one-dimensional (quasi-1D) systems have been theoretically predicted and observed in various ultracold atomic gases. Here a regularized local frame transformation method is developed to treat CIR in a quasi-1D system that has an arbitrary transverse trap shape in general. The method is applied to predict the CIR position in a system having a square transverse geometry with hard walls.

Figure 1

This is a sketch of the 2D square-well confinement geometry, and the separation of different length scales in the quasi-1D scattering process.

Figure 2

Plot of quantum numbers $\mathbf{n}=(n_x,n_y)$ that are summed over in the 2D integer plane, shown as smaller circles (pink online). The ring region is defined by the range ${(2n-1)}^2/2\le {|\mathbf{n}+(\frac{1}{2},\frac{1}{2})|}^2\le \frac{{(2n+1)}^2}{2}$, with $n=3$. The large point (blue online) is at $(-\frac{1}{2},-\frac{1}{2})$.

Figure 3

The CIR position obtained from finite-range interaction model calculations is plotted along with the extrapolation to zero range. The square dots (purple online) are from the potential $V=V_0\exp (-\frac{r^2}{2d_0^2})$, while the circle dots (blue online) are from the potential $V=V_0\frac{1}{\cosh {\left(\frac{r}{d_0}\right)}^2}$. The $x$ axis is the effective range of the potential at the position of the CIR. The solid line (green online) is derived from the effective range expansion of the 3D scattering length $|\frac{L_{\perp }}{a_s}|=|\frac{L_{\perp }}{a_s\left(0\right)}|+L_{\perp }^2{E}_0\frac{r_{\mathrm{eff}}}{L_{\perp }}$; $E_0$ is the lowest threshold energy. The single point at $r_{\mathrm{eff}}=0$ (red online) is the predicted CIR position from our regularized frame transformation method. The inset shows the pattern of convergence to the analytical value, which is seen in the numerical calculation near $\frac{r_{\mathrm{eff}}}{L_{\perp }}\sim 0.010$ (the arrows indicate the better converged numerical values of the CIR position). At larger values of the 3D effective range, the numerical calculation begins to deviate from the analytical prediction. This is because the range of interaction $d_0$ in those calculations has become comparable to the CIR scattering length, and hence the “short” -range approximation of the model potential has become less accurate.

Figure 4

Plot of the quasi-1D scattering phase shift in the transverse square-well trapped system, as a function of the 3D scattering length $a_s$. The effective 1D interaction strength in this quasi-1D system is $\frac{g_{1D}}{L_{\perp }{E}_0}=\pi \frac{4a_s/L_{\perp }}{1-{\Lambda }_E\left(\frac{1}{2}{E}_0\right)4a_s/L_{\perp }}$. ${\Lambda }_E\left(\epsilon \right)$ is the regularized Epstein zeta function discussed in the last section. $E_0=\frac{{\hslash }^2}{2\mu }\frac{4{\pi }^2}{L_{\perp }^2}$. The strips of curves clustered together are calculations for a range of energies in different regions of $k$: $k\sim 0.1,0.01,0.001$. $k$ denotes the momentum in the $z$ direction, and the scattering energy $E_z=\frac{k^2}{2\mu }$.