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arXiv:1112.1122 · Physical Review A

Mixed-partial-wave scattering with spin-orbit coupling and validity of pseudopotentials

Xiaoling Cui

Published Feb 14, 2012• 58 citations
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Abstract

We present exact solutions of a two-body problem for spin-$1/2$ 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 $s$-wave pseudopotential provides a good approximation for low-energy solutions near $s$-wave resonances, given the length scale of SO coupling much longer than the potential range. However, near $p$-wave resonance the $p$-wave pseudopotential gives low-energy solutions that are qualitatively different from exact ones, based on which we conclude that the $p$-wave model can not be applied to the fermion system if the SO coupling strength is larger or comparable to the Fermi momentum.

DOI
10.1103/physreva.85.022705

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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 study the interplay of spin-orbit coupling (SOC) and strong ppp-wave interactions to the scattering property of spin-12\frac{1}{2}21​ ultracold Fermi gases. Based on a two-channel square-well potential generating ppp-wave resonance, we show that the presence of an isotropic SOC, even for its length being much longer than the potential range, can greatly modify the ppp-wave short-range boundary condition (BC). As a result, the conventional ppp-wave BC cannot predict the induced molecules near ppp-wave resonances, which can be fully destroyed due to strong interference between the sss- and ppp-wave channels. By analyzing the intrinsic reasons for the breakdown of the conventional BC, we propose a ppp-wave BC that can excellently reproduce the exact molecule solutions and also equally apply for a wide class of single-particle potentials besides SOC. This work reveals the significant effect of SOC on both the short- and long-range properties of fermions near ppp-wave resonances, paving the way for future explorations of interesting few- and many-body physics in such systems.
We derive a pseudopotential in two dimensions (2D) with a 2D Rashba spin-orbit coupling (SOC), following in the same spirit of the frame transformation by Guan and Blume [Q. Guan and D. Blume, Phys. Rev. A 95, 020702(R) (2017)]. The frame transformation correctly describes the nontrivial phase accumulation and partial-wave couplings due to the presence of SOC, which modifies the original Fermi pseudopotential in free space, even when the length scale of the SOC is significantly larger than the two-body potential range. As an application, we apply our pseudopotential within the Lippmann-Schwinger equation to obtain an analytical scattering matrix. To demonstrate the validity of our approach, we compare our results with a numerical scattering calculation of the finite-range potential that shows excellent agreement over a wide range of scattering energies and SOC strengths. Our pseudopotential is applicable in the cases of a strong energy-dependent sss-wave scattering length and/or non-negligible ppp-wave interaction, where the original free-space pseudopotential fails. It sets an ideal starting point to explore many-body physics in the presence of synthetic SOC in cold atoms.
Previous work developed a KKK-matrix formalism applicable to positive energies for the scattering between two sss-wave interacting particles with two internal states, isotropic spin-orbit coupling and vanishing center-of-mass momentum [H. Duan, L. You, and B. Gao, Phys. Rev. A 87, 052708 (2013).]. This work extends the formalism to the entire energy regime. Explicit solutions are obtained for the total angular momentum J=0J=0J=0 and 1 channels. The behavior of the partial cross sections in the negative energy regime is analyzed in detail. We find that the leading contributions to the partial cross sections at the negative energy thresholds are governed by the spin-orbit coupling strength kso{k}_{\text{so}}kso​ and the mass ratio. The fact that these contributions are independent of the two-body scattering length as{a}_{s}as​ is a direct consequence of the effective reduction of the dimensionality, and hence of the density of states, near the scattering thresholds due to the single-particle spin-orbit coupling terms. The results are analytically continued to the energy regime where bound states exist. It is shown that our results are consistent with results obtained by alternative approaches. Our formulation, which can be regarded as an extension of the standard textbook partial wave decomposition, can be generalized to two-body systems with other types of spin-orbit coupling, including cases where the center-of-mass momentum does not vanish.
We consider spin-1/2 fermionic atoms with isotropic Rashba spin-orbit coupling in three directions. The interatomic potential is modeled by a square-well potential. We derive the analytic form of the asymptotic wave functions at short range for two fermions in the subspace of zero net momentum and zero total angular momentum. We show that the spin-orbit coupling has perturbative effects on the short-range asymptotic behavior of the wave functions away from resonances. We argue that our conclusion should hold generally.
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.
Synthetic spin-orbit-coupled quantum gases have been widely studied both experimentally and theoretically in the past decade. As shown in previous studies, this modification of single-body dispersion will in general couple different partial waves of the two-body scattering and thus distort the wave function of few-body bound states which determines the short-distance behavior of many-body wave function. In this work, we focus on the two-component Fermi gas with one-dimensional or three-dimensional spin-orbit coupling (SOC) near an sss-wave resonance. Using the method of effective field theory and the operator product expansion, we derive universal relations for both systems, including the adiabatic theorem, viral theorem, and pressure relation, and obtain the momentum distribution matrix $\ensuremath{\langle}{\ensuremath{\psi}}_{a}^{\ifmmode\dagger\else\textdagger\fi{}}(\mathbf{q}){\ensuremath{\psi}}_{b}(\mathbf{q})\ensuremath{\rangle}$ at large q\mathbf{q}q (a,ba,ba,b are spin indices). The momentum distribution matrix shows both spin-dependent and spatial anisotropic features. And the large momentum tail is modified at the subleading order thanks to the SOC. We also discuss the experimental implication of these results depending on the realization of the SOC.