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arXiv:1707.09460 · Journal of Physics B Atomic Molecular and Optical Physics

Potential scattering on a spherical surface

Jian Zhang, Tin-Lun Ho

Published Apr 6, 2018• 20 citations
arXivHTMLPDF

Abstract

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 m 2 − 1 3 + O r o 2 R 2 , where E R = ℏ 2 / ( 2 MR 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 ∼ 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 .

DOI
10.1088/1361-6455/aabc34

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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 m 2 − 1 3 + O r o 2 R 2 , where E R = ℏ 2 / ( 2 MR 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 ∼ 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 .
We analyze the scattering problem of identical bosonic particles confined on a spherical surface. At low scattering energies and for a radius much larger than the healing length, we express the contact interaction strength in terms of the sss-wave scattering length. Adopting this relation, we are then able to regularize the zero-point energy of the spherical Bose gas and to obtain its equation of state, which includes the corrections due to the finite radius of the sphere and coincides with the flat-case result in the infinite-radius limit. We also provide a microscopic derivation of the superfluid density of the system, reproducing a result postulated in a previous work. Our results are relevant for modeling the ongoing microgravity experiments with two-dimensional bubble-trapped Bose-Einstein condensates.
Nonrelativistic particles that are effectively confined to two dimensions can, in general, move on curved surfaces, allowing dynamical phenomena beyond what can be described with scalar potentials or even vector gauge fields. Here we consider a simple case of piecewise uniform curvature: a particle moves on a plane with a spherical extrusion. Depending on the latitude at which the sphere joins the plane, the extrusion can range from an infinitesimal bump to a nearly full sphere that just touches the plane. Free classical motion on this surface of piecewise uniform curvature follows geodesics that are independent of velocity, while quantum mechanical scattering depends on energy. We compare classical, semiclassical, and fully quantum problems, which are all exactly solvable, and show how semiclassical analysis explains the complex quantum differential cross section in terms of interference between two classical trajectories: the sphere on a plane acts as a kind of double slit.
In recent years, ultracold atomic gases confined in curved geometries have attracted considerable theoretical interest. This is motivated by recent realizations of bubble traps in microgravity conditions, which open the possibility of investigating quantum many-body physics beyond the conventional flat-space paradigm. The theoretical interest up to now was mainly focused on Bose gases and their phenomenology, and the study of Fermi gases has been left behind. In this paper, we investigate a two-component repulsive Fermi gas constrained to the surface of a sphere at finite temperature. We first analyze the noninteracting case, showing how the intrinsic geometrical features of the spherical surface give rise to a shell structure and modify the low-temperature thermodynamics compared to the flat two-dimensional gas. Repulsive interactions are then considered through an effective path-integral approach within a Hartree-Fock mean-field approximation, enabling us to derive the grand canonical potential and to regularize the associated Matsubara summation. We then investigate the stability of the spin-balanced state and obtain the finite-temperature Stoner criterion for fermions on a sphere, highlighting the interplay between the repulsive interactions and shell effects.
Although the general properties of the phase shift of the three-dimensional spherically symmetrical square well potential are described in most textbooks on quantum mechanics, specific graphical representations of this function are rare. In this paper we present a number of phase shifts, as a function of energy, for the various values of the angular momentum quantum number. We discuss the general features of these curves. In particular, we consider what happens to the phase shifts if one varies the depth of the well, not only for near-zero energy, but for the entire range of energies. We describe how the bound levels of a deep well are related to the virtual levels of the shallow well. We find points of triple coincidence for curves belonging to three successive l values for given well depth. A proof of this phenomenon is given.
The concepts of scattering length a and effective range re{r}_{e}re​ previously introduced for low-energy scattering from a potential V(r) in a plane and in higher dimensions are extended to include a 1/r potential (strength parameter \ensuremath{\gamma}). Both a and re{r}_{e}re​ have the physical significance of being equal to the radius of an equivalent hard sphere giving rise to the same O(k0{k}^{0}k0) and O(k2{k}^{2}k2) terms in the expression for the phase shift. The method used is based on the properties of the ``local scattering length'' a(r,\ensuremath{\gamma}) for the potential V(r) cut off at radius r and an ``equivalent hard-sphere radius'' a(r,k,\ensuremath{\gamma}) for wave number k\ensuremath{\ne}0. It is shown that these quantities have a smooth behavior for \ensuremath{\gamma}\ensuremath{\rightarrow}0 and for dimension n\ensuremath{\rightarrow}2.
Scattering processes are a fundamental way of experimentally probing distributions and properties of systems in several areas of physics. Considering two-body scattering at low energies, when the de Broglie wavelength is larger than the range of the potential, partial waves with high angular momentum are typically unimportant. The dominant contribution comes from l=0l=0l=0 partial waves, commonly known as sss-wave scattering. This situation is very relevant in atomic physics, e.g. cold atomic gases, and nuclear physics, e.g. nuclear structure and matter. This manuscript is intended as a pedagogical introduction to the topic while covering a numerical approach to compute the desired quantities. We introduce low-energy scattering with particular attention to the concepts of scattering length and effective range. These two quantities appear in the effective-range approximation, which universally describes low-energy processes. We outline a numerical procedure for calculating the scattering length and effective range of spherically symmetric two-body potentials. As examples, we apply the method to the spherical well, modified Pöschl-Teller, Gaussian, and Lennard-Jones potentials. We hope to provide the tools so students can implement similar calculations and extend them to other potentials.