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arXiv:1710.03901 · Physical Review A

Ultracold collisions between spin-orbit-coupled dipoles: General formalism and universality

Jia Wang, Christiaan R. Hougaard, Brendan C. Mulkerin, Xia-Ji Liu

Published Apr 23, 2018• 2 citations
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Abstract

A theoretical study of the low-energy scattering properties of two aligned identical bosonic and fermionic dipoles in the presence of isotropic spin-orbit coupling is presented. A general treatment of particles with arbitrary (pseudo)spin is given in the framework of multichannel scattering. At ultracold temperatures and away from shape resonances or closed-channel dominated resonances, the cross section can be well described within the Born approximation to within corrections due to the $s$-wave scattering. We compare our findings with numerical calculations and find excellent agreement.

DOI
10.1103/physreva.97.042709

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A theoretical study of the low-energy scattering properties of two aligned identical bosonic and fermionic dipoles in the presence of isotropic spin-orbit coupling is presented. A general treatment of particles with arbitrary (pseudo)spin is given in the framework of multichannel scattering. At ultracold temperatures and away from shape resonances or closed-channel dominated resonances, the cross section can be well described within the Born approximation to within corrections due to the sss-wave scattering. We compare our findings with numerical calculations and find excellent agreement.
A general treatment of ultracold two-body scattering in the presence of isotropic spin-orbit coupling (SOC) is presented. Owing to the mixing of different partial-wave channels, scattering with SOC is in general a coupled multichannel problem. A systematic method is introduced to analytically solve a class of coupled differential equations by recasting the coupled-channel problem as a simple eigenvalue problem. The exact Green's matrix in the presence of SOC is found, which readily gives the scattering solutions for any two identical particles in any total angular momentum subspace having negligible center-of-mass momentum. Application of this formalism to two spin-1 bosons shows the ubiquitous low-energy threshold behavior for systems with isotropic SOC. A modified threshold behavior shows up, which does not occur for the spin-orbit coupled spin-12\frac{1}{2}21​ system. We also confirm the parity-breaking mechanism for the spontaneous emergence of handedness, which has been proposed by Duan et al. [H. Duan, L. You, and B. Gao, Phys. Rev. A 87, 052708 (2013)]. Additionally, a two-body bound state is found for any arbitrarily small and negative scattering length. Our study sheds light on the few-body side of SOC physics and provides one step towards understanding ultracold scattering in a non-Abelian gauge field.
At the ultracold temperatures which occur in cold atom traps and Bose-Einstein condensates, only a few partial waves contribute to the scattering of ground-state alkali-metal atoms, and cross sections are extremely sensitive to threshold effects. We present an analysis of these threshold effects, using a generalized multichannel quantum defect theory (GMQDT) to construct a close-coupled scattering wave function which is analytic in energy across thresholds. We illustrate the theory using the hyperfine transitions in Na+Na\mathrm{N}\mathrm{a}+\mathrm{N}\mathrm{a}Na+Na collisions, and show that it gives results completely equivalent to the usual close-coupled cross sections. The virtue of the GMQDT is that it treats both open and closed channels on an equal footing, and all interchannel dynamics is summarized in a single real symmetric matrix Y(E)\mathbf{Y}(E)Y(E) which is essentially constant across thresholds (and often over excursions of energy which exceed the hyperfine splittings). The multichannel threshold energy behavior can then be related to calculable properties of the individual channels that are being closed. Many of the smaller spin depolarization cross sections are determined by very long-range ${\ensuremath{\alpha}}^{2}{/R}^{3}$ spin-spin interactions which are not well treated by GMQDT, and we correct these specific elements with a perturbative distorted-wave approximation which yields the observed threshold dependences and brings the GMQDT into perfect agreement with the exact close-coupled results.
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.
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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.
We investigate the scattering cross section of aligned dipolar molecules in low-temperature gases. Over a wide range of collision energies relevant to contemporary experiments, the cross section declines in inverse proportion to the collision speed, and is given nearly exactly by a simple semiclassical formula. At yet lower energies, the cross section becomes independent of energy, and is reproduced within the Born approximation to within corrections due to the s-wave scattering length. While these behaviors are universal for all polar molecules, nevertheless interesting deviations from universality are expected to occur in the intermediate energy range.