Confinement-induced resonances in a cross-dimensional configuration

A lattice-induced opacity is identified in the scattering process of a normally incident matter wave colliding with a two dimensional lattice of atoms. This system can be treated as an analog of a confinement-induced resonance. Specifically, by modifying the $s$-wave scattering length between atoms in the incident matter wave and the lattice-confined atoms, the transmission of the matter wave can be tuned controllably throughout the range from zero to one. The theoretical treatment shows how the mathematical description of confinement-induced resonances can be viewed as an extension of Bragg scattering theory, with alternate opportunities for transmission control.

Figure 1

Sketch of a beam of an atomic matter-wave scattering from an infinite 2D square lattice of fixed atoms. For simplicity, the incoming wave is restricted to be normal to the 2D optical lattice.

Figure 2

This graph plots the quasi-1D transmission probability $|1+f_{0,0}{|}^2$ as a function of the 3D scattering length $a_{sc}$ and the kinetic energy $\epsilon$ of the incoming atom for scattering in the case of a single open channel. There are two regions in this parameter space where the scattering amplitude is close to zero. However, only the one that is associated with a negative 3D scattering length $a_{sc}$ provides a usable lattice-size-induced opacity. At positive values of $a_{sc}$, although there exists a region where $a_{sc}$ and $\epsilon$ cooperatively result in a small transmission amplitude, the tunability there is poorer than the negative-$a_{sc}$ region. This is because in the region $a_{sc}>0$, even when tuning $\epsilon$ and $a_{sc}$ over a large range, does not significantly change the transmission amplitude. The red dot on the $a_{sc}/a$ axes is the resonance position at the lowest threshold.

Figure 3

Sketch of a matter-wave cavity utilizing two parallel 2D optical lattices where the effective 1D interaction strength $g_{\text{1D}}$ for each of them is infinite.

Figure 4

The transmission probability for one open channel, in (from left to right) (i) a two dimensional lattice, (ii) a square-well waveguide, (iii) a two dimensional isotropic harmonic trap. The transmission probabilities for the case of one open channel are plotted. The units in each plot are the relevant system-characteristic energy scale, namely $\frac{{\hslash }^2}{2\mu }{\left(\frac{2\pi }{L}\right)}^2,\frac{{\hslash }^2}{2\mu }{\left(\frac{2\pi }{L_{\perp }}\right)}^2,\hslash {\omega }_{\perp },a_{\perp }=[\hslash /{\left(\mu {\omega }_{\perp })\right]}^{1/2}$. $L$ is the lattice constant, $L_{\perp }$ is the size of the square waveguide, and ${\omega }_{\perp }$ is the transverse trap frequency. Lighter colors correspond to less transmission, and the thin white line locates the positions of confinement-induced opacities. The red dot on the horizontal axes indicates the resonance position at the first threshold energy in these three systems. The white strands correspond to the resonance positions at finite energies. The corresponding resonance positions at the lowest threshold in each system is $a_{sc}/L\approx 0.25,a_{sc}/L_{\perp }\approx 0.17,a_{sc}/a_{\perp }\approx 0.69$. Each scattering-amplitude contour plot has a $g_{\text{1D}}$ plot at threshold energy $\epsilon =0$ as a function of $a_{sc}$.