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Pseudo-potential of an arbitrary isotropic interaction in two-dimensional systems

Sheng-Ming Shih, Daw-Wei Wang

Published Feb 17, 2009• 0 citations
OpenAlex

Abstract

We analytically derive the general pseudo-potential operator of an arbitrary isotropic interaction for particles confined in two-dimensional (2D) systems, using the same frame work developed by Huang and Yang for 3D scattering [Phys. Rev. {\bf 105}, 767 (1957)]. We also analytically derive the low energy dependence of the phase-shift for an arbitrary potential with a power-law decaying tail, $V_{\rm 2D}(\rho)\propto \rho^{-\alpha}$ (for $\alpha>2$). We apply our results to the 2D dipolar gases ($\alpha=3$) as an example, calculating the momentum and dipole moment dependence of the pseudo-potential for both $s$- and p-wave scattering channels if the two scattering particles are in the same 2D layer. We also calculate the result for the s-wave scattering between particles in two different (parallel) layers. Our analytic and numerical results can be directly applied to the study of many-body properties of magnetic dipolar atoms and/or polar molecules in a general 2D system.

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We analytically derive the general pseudo-potential operator of an arbitrary isotropic interaction for particles confined in two-dimensional (2D) systems, using the same frame work developed by Huang and Yang for 3D scattering [Phys. Rev. {\bf 105}, 767 (1957)]. We also analytically derive the low energy dependence of the phase-shift for an arbitrary potential with a power-law decaying tail, V2D(ρ)∝ρ−αV_{\rm 2D}(\rho)\propto \rho^{-\alpha}V2D​(ρ)∝ρ−α (for α>2\alpha>2α>2). We apply our results to the 2D dipolar gases (α=3\alpha=3α=3) as an example, calculating the momentum and dipole moment dependence of the pseudo-potential for both sss- and p-wave scattering channels if the two scattering particles are in the same 2D layer. We also calculate the result for the s-wave scattering between particles in two different (parallel) layers. Our analytic and numerical results can be directly applied to the study of many-body properties of magnetic dipolar atoms and/or polar molecules in a general 2D system.
We analytically derive the general pseudopotential operator of an arbitrary isotropic interaction for particles confined in two-dimensional (2D) systems using the frame work developed by Huang and Yang for three-dimensional scattering. We also analytically derive the low-energy dependence of the scattering phase shift for an arbitrary interaction with a power-law decaying tail, ${V}_{2\text{D}}(\ensuremath{\rho})\ensuremath{\propto}{\ensuremath{\rho}}^{\ensuremath{-}\ensuremath{\alpha}}$ (for $\ensuremath{\alpha}>2$). We apply our results to the 2D dipolar gases $(\ensuremath{\alpha}=3)$ as an example, calculating the momentum and dipole moment dependences of the pseudopotential for both sss- and ppp-wave scattering channels if the two scattering particles are in the same 2D layer. Results for the sss-wave scattering between particles in two different (parallel) layers are also investigated. Our results can be directly applied to the systems of dipolar atoms and/or polar molecules in a general 2D geometry.
Abstract In this paper, we provide the two-body exact solutions of the two-dimensional (2D) Schrödinger equation with isotropic ± 1 / r 3 interactions. An analytic quantum defect theory is constructed based on these solutions and it is applied to investigate the scattering properties as well as two-body bound states of an ultracold polar molecules confined in a quasi-2D geometry. Interestingly, we find that for the attractive case, the scattering resonance happens simultaneously in all partial waves, which has not been observed in other systems. The effect of this feature on the scattering phase shift across such resonances is also illustrated.
We propose a framework for calculating scattering and bound-state properties in anisotropic two-dimensional potentials. Using our method, we derive systematic approximations of partial wave phase shifts and binding energies. Moreover, the method is suitable for efficient numerical computations. We calculate the sss-wave phase shift and binding energy of polar molecules in two layers polarized by an external field along an arbitrary direction. We find that scattering depends strongly on their polarization direction and that absolute interlayer binding energies are larger than thermal energies at typical ultracold temperatures.
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.
Dipolar quantum gases, encompassing atoms and molecules with significant dipole moments, exhibit unique long-range and anisotropic dipole-dipole interactions (DDI), distinguishing them from systems dominated by short-range contact interactions. This review explores their behavior across dimensions, focusing on magnetic atoms in quasi-2D in comparison to 3D. In 3D, strong DDI leads to phenomena like anisotropic superfluidity, quantum droplets stabilized by Lee-Huang-Yang corrections, and supersolid states with density modulations. In 2D, we discuss a new scenario where DDI induces angle-dependent Berezinskii-Kosterlitz-Thouless transitions and potential supersolidity, as suggested by recent experimental realizations of strongly dipolar systems in quasi-2D geometries. We identify key challenges for future experimental and theoretical work on strongly dipolar 2D systems. The review concludes by highlighting how these unique 2D dipolar systems could advance fundamental research as well as simulate novel physical phenomena.
Abstract In this paper, we have provided exact two-body solutions to the 2D and 3D Schrödinger equations with isotropic van der Waals potentials of the form ± 1 / r 6 . Based on these solutions, we developed an analytical quantum defect theory (QDT) applicable to both quasi-2D and 3D geometries, and applied it to study the scattering properties and bound-state spectra of ultracold polar molecules confined in these geometries. Interestingly, we find that in the attractive (repulsive) van der Waals potential case, the short-range interaction can be effectively modeled by an infinite square barrier (finite square well), which leads to narrow and dense (broad and sparse) resonance structures in the quantum defect parameter. In the quasi-2D attractive case, shape resonances can appear in an ordered fashion across different partial waves, characterized by sharp phase jumps as the scattering energy is varied. Furthermore, the low-energy analytical expansions derived from QDT show excellent agreement with the exact numerical results, validating the accuracy and usefulness of our analytical approach in describing two-body physics governed by long-range van der Waals interactions.