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Physica Scripta

Conditions for bound states of the pseudopotential with harmonic confinement in arbitrary dimensions

Lanie G. McKinney, Brett Allen McKinney

Published Nov 28, 2022• 2 citations
Publisher

Abstract

Abstract We determine the conditions for bound states (E < 0) for arbitrary Cartesian dimension d using a shape-independent regularized pseudopotential with scattering length a for two cold particles in a harmonic trap. It is known for d ≤ 3 that the regularized pseudopotential supports one bound state for positive scattering length but does not support bound states for negative scattering length. We find that the usual (d ≤ 3) positive scattering length bound states rule holds for certain higher odd dimensions d = 4n + 3 (n = 0, 1, …), but the existence of pseudopotential bound states at other odd dimensions requires a negative scattering length. Specifically, bound states are allowed in higher dimensions d = 4n + 1 (n = 1, 2, …) but they require a negative scattering length, which suggests a universe in these dimensions might lead to different chemistry than d = 3. We derive analytical approximations for bound state (E < 0) and scattering (E > 0) energies for a harmonic trap perturbed by the pseudopotential in arbitrary dimensions.

DOI
10.1088/1402-4896/aca6b6

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Abstract We determine the conditions for bound states (E < 0) for arbitrary Cartesian dimension d using a shape-independent regularized pseudopotential with scattering length a for two cold particles in a harmonic trap. It is known for d ≤ 3 that the regularized pseudopotential supports one bound state for positive scattering length but does not support bound states for negative scattering length. We find that the usual (d ≤ 3) positive scattering length bound states rule holds for certain higher odd dimensions d = 4n + 3 (n = 0, 1, …), but the existence of pseudopotential bound states at other odd dimensions requires a negative scattering length. Specifically, bound states are allowed in higher dimensions d = 4n + 1 (n = 1, 2, …) but they require a negative scattering length, which suggests a universe in these dimensions might lead to different chemistry than d = 3. We derive analytical approximations for bound state (E < 0) and scattering (E > 0) energies for a harmonic trap perturbed by the pseudopotential in arbitrary dimensions.
Abstract Introduced eighty years ago, the Fermi pseudopotential has been a powerful concept in multiple fields of physics. It replaces the detailed shape of a potential by a delta-function operator multiplied by a parameter giving the strength of the potential. For Cartesian dimensions d > 1, a regularization operator is necessary to remove singularities in the wave function. In this study, we develop a Fermi pseudopotential generalized to d dimensions (including non-integer) and to non-zero wavenumber, k . Our new fractional calculus approach has the advantage of circumventing singularities that occur in the wave function at integer values of d while being valid arbitrarily close to integer d . In the limit of integer dimension, we show that our generalized pseudopotential is equivalent to previously derived s -wave pseudopotentials. Our pseudopotential generalizes the operator to non-integer dimension, includes energy ( k ) dependence, and simplifies the dimension-dependent coupling constant expression derived from a Green’s function approach. We apply this pseudopotential to the problem of two cold atoms ( k → 0) in a harmonic trap and extend the energy expression to arbitrary dimension.
We study confined scattering of a quantum particle by two centers fixed on the longitudinal axis of a harmonic-waveguide-like trap. The conditions of confinement-induced resonances (CIRs) appearing in these systems, when scattering cross section approaches the unitary limit, are derived for a regularized pseudopotential describing particle interaction with scattering centers. In the limit of a single center, the position of CIR for an even state tends to the well-known result obtained by Olshanii [Phys. Rev. Lett. 81, 938 (1998)]. Our result can be applicable to confined atomic scattering by fixed impurities, such as ions or Rydberg atoms, with possible extension to NNN impurities, or by two-atomic molecules.
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.
The theory of zero-range potentials is investigated in an arbitrary number of dimensions. Except for the trivial one-dimensional case the zero-range potentials are described by nonlocal operators called Fermi pseudopotentials. It is shown that in odd dimensions the Fermi pseudopotentials involve a very simple regularization operator. In even dimensions with the help of dimensional regularization, explicit formulas for the Fermi pseudopotentials are derived. The Green's functions, the propagators, and the exact solutions of the Lippmann-Schwinger equations are derived in explicit forms. In odd dimensions d=1 and 3 the Fermi pseudopotentials can be applied to describe multiphoton processes of atoms and molecules with very short-range interactions. In even dimension d=2 the Fermi pseudopotential can be applied to describe tunneling from laser-driven quantum wells. Physical applications involving higher d are also possible.
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.