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arXiv:2302.05646 · New Journal of Physics

Low-energy scatterings and pseudopotential of polarized quadrupoles

Fulin Deng, Wenxian Zhang, Su Yi

Published Jan 25, 2023• 0 citations
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Abstract

Abstract We investigate the low-energy scattering properties of two identical particles interacting via the polarized quadrupolar interaction. It is shown that a series of s- and p-wave resonances appear for identical bosons and fermions, respectively, as the strength of the quadrupolar interaction increases. Interestingly, scattering resonances also appear on the generalized scattering length corresponding to the coupling between the s and d waves. This observation inspires us to propose a new pseudopotential for the quadrupolar interaction. We also explore the bound-state properties of two particles in both free space and harmonic traps.

DOI
10.1088/1367-2630/acb60a

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Abstract We investigate the low-energy scattering properties of two identical particles interacting via the polarized quadrupolar interaction. It is shown that a series of s- and p-wave resonances appear for identical bosons and fermions, respectively, as the strength of the quadrupolar interaction increases. Interestingly, scattering resonances also appear on the generalized scattering length corresponding to the coupling between the s and d waves. This observation inspires us to propose a new pseudopotential for the quadrupolar interaction. We also explore the bound-state properties of two particles in both free space and harmonic traps.
The low-energy scattering properties of two aligned identical bosonic and identical fermionic dipoles are analyzed. Generalized scattering lengths are determined as functions of the dipole moment and the scattering energy. Near resonance, where a new bound state is being pulled in, all nonvanishing generalized scattering lengths diverge, with the a00{a}_{00}a00​ and a11{a}_{11}a11​ scattering lengths being dominant for identical bosons and identical fermions, respectively. Implications for the energy spectrum and the eigenfunctions of trapped two-dipole systems and for pseudopotential treatments are discussed.
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.
An anisotropic pseudopotential arising in the context of collisions of two particles polarized by an external field is rigorously derived and its properties are investigated. Such a low-energy pseudopotential may be useful in describing collective properties of dilute quantum gases, such as molecules polarized by an electric field or metastable 3P2{}^{3}{P}_{2}3P2​ atoms polarized by a magnetic field. The pseudopotential is expressed in terms of the reactance (K) matrix and derivatives of the Dirac $\ensuremath{\delta}$ function. In most applications, it may be represented as a sum of a traditional spherically symmetric contact term and an anisotropic part. The former contribution may be parametrized by a generalized scattering length. The anisotropic part of the pseudopotential may be characterized by the off-diagonal scattering length for dipolar interactions and off-diagonal scattering volume for quadrupolar interactions. The two-body matrix element of the pseudopotential in a basis of plane waves is also derived.
We derive analytical solutions for the system of two ultracold spin-polarized fermions interacting in ppp wave and confined in an axially symmetric harmonic trap. To this end we utilize ppp-wave pseudopotential with an energy-dependent scattering volume. This allows to describe the scattering in tight trapping potentials in the presence of scattering resonances. We verify predictions of the pseudopotential treatment for some model interaction potential obtaining an excellent agreement with exact energy levels. Then we turn to the experimentally relevant case of neutral atom interactions in the vicinity of a ppp-wave Feshbach resonance. In the framework of the multichannel quantum-defect theory we derive relatively simple formula for an energy-dependent scattering volume, and later we apply it to investigate the energy spectrum of trapped atoms close to the ppp-wave Feshbach resonance.
We present exact solutions of a two-body problem for spin-1/21/21/2 fermions with isotropic spin-orbit (SO) coupling and interacting with an arbitrary short-range potential. We find that in each partial-wave scattering channel, the parametrization of a two-body wave function at short interparticle distance depends on the scattering amplitudes of all channels. This reveals the mixed-partial-wave scattering induced by SO couplings. By comparing with results from a square-well potential, we investigate the validity of original pseudopotential models in the presence of SO coupling. We find the sss-wave pseudopotential provides a good approximation for low-energy solutions near sss-wave resonances, given the length scale of SO coupling much longer than the potential range. However, near ppp-wave resonance the ppp-wave pseudopotential gives low-energy solutions that are qualitatively different from exact ones, based on which we conclude that the ppp-wave model can not be applied to the fermion system if the SO coupling strength is larger or comparable to the Fermi momentum.
We present a zero-range pseudopotential applicable for all partial wave interactions between neutral atoms. For p and d waves, we derive effective pseudopotentials, which are useful for problems involving anisotropic external potentials. Finally, we consider two nontrivial applications of the p-wave pseudopotential: we solve analytically the problem of two interacting spin-polarized fermions confined in a harmonic trap, and we analyze the scattering of p-wave interacting particles in a quasi-two-dimensional system.