Nondivergent pseudopotential treatment of spin-polarized fermions under one- and three-dimensional harmonic confinement

Abstract

Atom-atom scattering of bosonic one-dimensional (1D) atoms has been modeled successfully using a zerorange delta-function potential, while that of bosonic 3D atoms has been modeled successfully using FermiHuang's regularized s-wave pseudo-potential. Here, we derive the eigenenergies of two spin-polarized 1D fermions under external harmonic confinement interacting through a zero-range potential, which only acts on odd-parity wave functions, analytically. We also present a divergent-free zero-range potential treatment of two spin-polarized 3D fermions under harmonic confinement. Our pseudo-potential treatments are verified through numerical calculations for short-range model potentials.

Analytical treatment of bosonicd-wave scattering in isotropic harmonic waveguides

“…1(c) the analytic expression of the transmission coefficient T d is depicted according to Eqs. (24) and (27). We observe that the DWR is described qualitatively accurate enough (see also insets of Figs.2(a)-(b)), despite the fact that in the analytical calculations we have neglected the contributions of s-wave scattering, since the transversal confinement is very weak.…”

“…Higher-partial wave interactions represent an intriguing perspective in research on ultracold gases and are expected to provide novel manybody phenomena, such as unconventional superfluidity and superconductivity [21][22][23]. So far, theoretical studies with pseudopotentials for ℓ > 0 have mainly been used to describe fermionic systems, e.g., in quasi-onedimensional [13,15,24] or quasi-two-dimensional [14,17] geometries, which have been studied experimentally in [25]. For the case of bosons an analytical description of collisions in a planar waveguide has been derived, where the cross-section possesses s-and d-wave resonances [17].…”

Pseudopotential Method for Higher Partial Wave Scattering

“…We stress that the presence of the full regularization operator ∂ 3 r r 2 on the r.h.s. of the pseudopotential (6) is important, since for anisotropic external potentials the exact wave functions obtained in the pseudopotential method contains terms behaving like x i /r for small r. Such terms, for instance, are not removed by the operator ∇∂ 2 r r 2 [8]. A similar procedure can be repeated for d-wave interactions.…”

“…Their pseudopotential, however, is incorrect with respect to l > 0 waves, as recently shown in [5]. Several alternatives have been proposed [5,6,7,8,9], which have specific limitations. For instance, [5] entails no regularization and is only applicable to mean-field theories; [6] requires knowledge of the wave function in the inner region of the shell potential, and taking the limit of shell radius going to zero in the final step of the calculation.…”

Optical Feshbach resonances of alkaline-earth-metal atoms in a one- or two-dimensional optical lattice

“…All these works relate the effective 1D or 2D scattering properties to the free-space properties. Also closely related are the constructions of pseudo-potentials which can treat confined systems [21,22,23] and pseudo-potentials in low-dimensional systems [22,24].…”