Potential Scattering on a Spherical Surface

Abstract

The advances in cold atom experiments have allowed construction of confining traps in the form of curved surfaces. This opens up the possibility of studying quantum gases in curved manifolds. On closed surfaces, many fundamental processes are affected by the local and global properties, i.e. the curvature and the topology of the surface. In this paper, we study the problem of potential scattering on a spherical surface and discuss its difference with that on a 2D plane. For bound states with angular momentum $m$, their energies ($E_m$) on a sphere are related to those on a 2D plane ($-|E_{m,o}|$) as $E_m=-|E_{m,o}|+E_R^{}\left[\frac{m^2-1}{3}+O\left(\frac{r_o^2}{R^2}\right)\right]$, where $E_R^{}={\hslash }^2/(2M{R}^2)$, and $R$ is the radius of the sphere. Due to the finite extent of the manifold, the phase shifts on a sphere at energies $E\sim E_R^{}$ differ significantly from those on a 2D plane. As energy $E$ approaches zero, the phase shift in the planar case approaches $0$, whereas in the spherical case it reaches a constant that connects the microscopic length scale to the largest length scale $R$.

5 pages, 2 figures

References in corpus (3)

• Radio-frequency dressed state potentials for neutral atoms
• Manipulation of ultracold atoms in dressed adiabatic radio frequency potentials
• Emergent Gauge Field for a Chiral Bound State on Curved Surface

Cited by in corpus (15)

• Observation of ultracold atomic bubbles in orbital microgravity
• Bose-Einstein Condensation on the Surface of a Sphere
• Quantum Bubbles in Microgravity
• Topological superfluid transition in bubble-trapped condensates
• Low-dimensional quantum gases in curved geometries
• Perspective on Quantum Bubbles in Microgravity
• Superfluid vortex dynamics on a spherical film
• Superfluid vortex dynamics on an ellipsoid and other surfaces of revolution
• Shell-shaped atomic gases
• Scattering theory and equation of state of a spherical two-dimensional Bose gas
• Shell-Shaped Quantum Droplet in a Three-Component Ultracold Bose Gas
• Quantum Scattering of Spinless Particles in Riemannian Manifolds
• Bose-Einstein condensates and the thin-shell limit in anisotropic bubble traps
• Dimer problem on a spherical surface
• Confinement-Induced Resonances in Spherical Shell Traps