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Physical Review A

Pseudopotential approximation in a harmonic trap

Martin Block, Martin Holthaus

Published Apr 15, 2002• 50 citations
Publisher

Abstract

We construct the pseudopotential for two particles subjected to an isotropic harmonic trapping potential while interacting through a hard-sphere potential, and demonstrate that this trap pseudopotential reduces to the standard ``zero range'' interaction when the ratio of the hard-sphere diameter and the oscillator length is small. We also show that the standard approximation, even when applied to tight traps, is surprisingly accurate.

DOI
10.1103/physreva.65.052102

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•Jan 1, 2009•0 citations
We construct the pseudopotential for two particles subjected to an isotropic harmonic trapping potential while interacting through a hard-sphere potential, and demonstrate that this trap pseudopotential reduces to the standard ``zero range'' interaction when the ratio of the hard-sphere diameter and the oscillator length is small. We also show that the standard approximation, even when applied to tight traps, is surprisingly accurate.
Starting from two identical, interacting particles in an isotropic harmonic trap, we employ first-order perturbation theory to derive the effective s-wave pseudopotential operator for a weakly anisotropic trap. It is shown that such a confinement gives rise to a correction of the s-wave scattering length which is proportional to the square of the ratio of the bare scattering length and the oscillator length. The trap pseudopotential depends on the quantum number of the trapped state, and the effect of a weak anisotropy is dominated by its nearest neighbors. Some implications for experiments with cold bosons in tightly confining potentials are discussed briefly.
Dipolar Bose and Fermi gases, which are currently being studied extensively experimentally and theoretically, interact through anisotropic, long-range potentials. Here, we replace the long-range potential by a zero-range pseudopotential that simplifies the theoretical treatment of two dipolar particles in a harmonic trap. Our zero-range pseudopotential description reproduces the energy spectrum of two dipoles interacting through a shape-dependent potential under external confinement very well, provided that sufficiently many partial waves are included, and readily leads to a classification scheme of the energy spectrum in terms of approximate angular momentum quantum numbers. The results may be directly relevant to the physics of dipolar gases loaded into optical lattices.
We derive a generalized zero-range pseudopotential applicable to all partial wave solutions to the Schrödinger equation based on a delta-shell potential in the limit that the shell radius approaches zero. This properly models all higher order multipole moments not accounted for with a monopolar delta function at the origin, as used in the familiar Fermi pseudopotential for s-wave scattering. By making the strength of the potential energy dependent, we derive self-consistent solutions for the entire energy spectrum of the realistic potential. We apply this to study two particles in an isotropic harmonic trap, interacting through a central potential, and derive analytic expressions for the energy eigenstates and eigenvalues.
In this paper we calculate the energy levels of two trapped atoms interacting through a spherical two-body potential V(r)V(r)V(r) under external confinement quantum mechanically. We then replace V(r)V(r)V(r) by either a regularized energy-independent or an energy-dependent \ensuremath{\delta}-function potential. A comparison shows that the use of an energy-dependent pseudopotential improves significantly upon the use of an energy-independent pseudopotential. The accuracy of the pseudopotential approximation is approximately determined by the ratio ${\ensuremath{\beta}}_{6}{/a}_{\mathrm{ho}},$ where ${\ensuremath{\beta}}_{6}$ and aho{a}_{\mathrm{ho}}aho​ denote the characteristic length scale of the two-body interaction potential V(r)V(r)V(r) and of the trapping potential, respectively.
Ultracold atomic gases have been of great theoretical and experimental interest in the last two decades. In these systems, the de Broglie wavelength of the particles is much greater than the two body van der Waals length. As a result, the details of the two body interaction potential are irrelevant for a large number of applications and the realistic two body interaction potential can be replaced by a simple finite range or zero range model potential that reproduces the scattering quantities of the full interaction potential. This thesis develops zero range pseudopotentials and applies them to trapped two-particle systems. Ultracold gases loaded into optical lattices can be used to realize two particle systems under approximately harmonic confinement. We use pseudopotentials to obtain the eigenspectrum of two particles under external harmonic confinement semi-analytically. Advancements in trapping technology have resulted in the realization of low dimensional systems. We develop pseudopotentials to treat two body interactions in one and two dimensions. We also elaborate on the physics that is unique to one and two dimensional systems. Feshbach resonances allow for the tunability of the effective two body interaction strength in the presence of a magnetic field. To model Feshbach resonances in two and three dimensions we develop coupled two channel zero range potentials. Dipole-dipole interactions in Chromium and polar molecules have been the subject of a lot of recent research. Unlike the interactions between two alkali atoms, these interactions are long range and anisotropic. We explore the scattering properties of two aligned dipoles using a simple shape dependent model potential. To understand a system two aligned dipoles under confinement, we develop a pseudopotential treatment for cylindrically symmetric interaction potentials under cylindrically symmetric harmonic confinement. This pseudopotential can be used to model any cylindrically symmetric interaction potential. Our zero range treatment describes many features of two aligned dipoles under spherically symmetric harmonic confinement accurately but breaks down when applied to two aligned dipoles under cylindrically symmetric harmonic confinement. The breakdown is unique to the long range nature of the dipole-dipole interaction and expected to be absent for short range interactions.