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arXiv:1608.05753 · Physical Review A

Scattering framework for two particles with isotropic spin-orbit coupling applicable to all energies

Qingze Guan, D. Blume

Published Aug 12, 2016• 15 citations
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Abstract

Previous work developed a $K$-matrix formalism applicable to positive energies for the scattering between two $s$-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 $J=0$ 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 ${k}_{\text{so}}$ and the mass ratio. The fact that these contributions are independent of the two-body scattering length ${a}_{s}$ 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.

DOI
10.1103/physreva.94.022706

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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.
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.
We present a comprehensive derivation of a set of universal relations for spin-orbit-coupled Fermi gases in three or two dimensions, which follow from the short-range behavior of the two-body physics. Besides the adiabatic energy relations, the large-momentum distribution, the grand canonical potential, and pressure relation derived in our previous work for three-dimensional systems [Phys. Rev. Lett. 120, 060408 (2018)], we further derive high-frequency tail of the radio-frequency spectroscopy and the short-range behavior of the pair correlation function. In addition, we also extend the derivation to two-dimensional systems with Rashba-type spin-orbit coupling. To simply demonstrate how the spin-orbit-coupling effect modifies the two-body short-range behavior, we solve the two-body problem in the sub-Hilbert space of zero center-of-mass momentum and zero total angular momentum, and we perturbatively take the spin-orbit-coupling effect into account at short distance, since the strength of the spin-orbit coupling should be much smaller than the corresponding scale of the finite range of interatomic interactions. The universal asymptotic forms of the two-body wave function at short distance are then derived, which are independent on the short-range details of interatomic potentials. We find that new scattering parameters emerge in the universal asymptotic forms, apart from the traditional sss- and ppp-wave scattering length (volume) and effective ranges. This is a general and unique feature for spin-orbit-coupled systems. We show how these two-body parameters characterize the universal relations in the presence of spin-orbit coupling. This work probably sheds light on understanding the profound properties of the many-body quantum systems in the presence of the spin-orbit coupling.
The coupling between the spin degrees of freedom and the orbital angular momentum has a profound effect on the properties of nuclei, atoms, and condensed-matter systems. Recently, synthetic gauge fields have been realized experimentally in neutral cold-atom systems, giving rise to a spin-orbit coupling term with ``strength'' kso{k}_{\text{so}}kso​. This paper investigates the interplay between the single-particle spin-orbit coupling term of Rashba type and the short-range two-body sss-wave interaction for cold atoms under external confinement. Specifically, we consider two different harmonically trapped two-atom systems. The first system consists of an atom with spin-orbit coupling that interacts with a structureless particle through a short-range two-body potential. The second system consists of two atoms that both feel the spin-orbit coupling term and that interact through a short-range two-body potential. Treating the spin-orbit term perturbatively, we determine the correction to the ground-state energy for various generic parameter combinations. Selected excited states are also treated. An important aspect of our study is that the perturbative treatment is not limited to small sss-wave scattering lengths but provides insights into the system behavior over a wide range of scattering lengths, including the strongly interacting unitary regime. We find that the interplay between the spin-orbit coupling term and the sss-wave interaction generically enters, depending on the exact parameter combinations of the sss-wave scattering lengths, at order kso2{k}_{\text{so}}^{2}kso2​ or kso4{k}_{\text{so}}^{4}kso4​ for the ground state and leads to a shift of the energy of either sign. While the absence of a term proportional to kso{k}_{\text{so}}kso​ follows straightforwardly from the functional form of the spin-orbit coupling term, the absence of a term proportional to kso2{k}_{\text{so}}^{2}kso2​ for certain parameter combinations is unexpected. The well-known fact that the spin-orbit coupling term couples the relative and center-of-mass degrees of freedom has interesting consequences for the trapped two-particle systems. For example, we find that, for certain parameter combinations, the spin-orbit coupling term turns sharp crossings into avoided crossings with an energy splitting proportional to kso{k}_{\text{so}}kso​. Our perturbative results are confirmed by numerical calculations that expand the eigenfunctions of the two-particle Hamiltonian in terms of basis functions that contain explicitly correlated Gaussians.
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.
The creation of artificial gauge fields in neutral ultracold atom systems has opened the possibility to study the effects of spin-orbit coupling terms in clean environments. This work considers the multichannel scattering properties of two atoms confined by a waveguide in the presence of spin-orbit coupling terms within a KKK-matrix scattering framework. The tunability of resonances, induced by the interplay of the external waveguide geometry, the interactions, and the spin-orbit coupling terms, is demonstrated. Our results for the KKK-matrix elements as well as partial and total reflection coefficients for two identical fermions interacting through a finite-range interaction potential in the singlet channel only are compared with those obtained for a strictly one-dimensional effective low-energy Hamiltonian, which uses the effective coupling constant derived in Zhang et al. [Sci. Rep. 4, 4992 (2014)] and Zhang et al. [Phys. Rev. A 88, 053605 (2013)] as input. In the regime where the effective Hamiltonian is applicable, good agreement is obtained, provided the energy dependence of the coupling constant is accounted for. Our approach naturally describes the energy regime in which the bands associated with excited transverse modes lie below a subset of the bands associated with the lowest transverse modes. The threshold behavior is discussed and scattering observables are linked to bound state properties.
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