We derive one-dimensional effective Hamiltonians for spin-orbit-coupled Fermi gases confined in quasi-one-dimensional trapping potentials. For an energy regime around the two-body bound-state energy, the effective Hamiltonian takes a two-channel form, where the populations in transverse excited levels are described by dressed molecules in the closed channel. For an energy regime slightly above the continuum threshold, the effective Hamiltonian takes a single-channel form, where low-energy physics is governed by the one-dimensional interacting strength determined by three-dimensional scattering length and transverse confinement. We further discuss the effect of spin-orbit coupling and effective Zeeman field on the position of confinement-induced resonances and show that these resonances can be understood as Feshbach resonances between the threshold of the transverse ground state and the two-body bound state associated with the transverse excited states. We expect that the shift of confinement-induced resonances can be observed under present experimental technology at attainable temperatures.
We perform an analytical investigation in the framework of generalized
K-matrix theory of the scattering problem in tight isotropic and harmonic waveguides allowing for several open scattering channels. The scattering behavior is explored for identical bosons and fermions, as well as for distinguishable particles, the main aspect being the confinement-induced resonances (CIR) which are attributed to different partial waves. In particular, we present the unitarity bounds which emerge when considering a quasi-one-dimensional system. Unitarity bounds are also given for the transition coefficients, which show the limitations for efficient transversal (de)excitations by means of CIRs. We analyze the CIR for
d waves and find the intriguing phenomenon of a strong transmission suppression in the presence of more than one open channel, which represents an interesting regime to be applied in the corresponding many-particle systems. The corresponding channel threshold singularities are studied and it is shown that these are solely determined by the symmetry class of the partial wave.
We develop a grid method for multichannel scattering of atoms in a waveguide with harmonic confinement. This approach is employed to extensively analyze the transverse excitations and deexcitations as well as resonant scattering processes. Collisions of identical bosonic and fermionic as well as distinguishable atoms in harmonic traps with a single frequency $\ensuremath{\omega}$ permitting the center-of-mass (c.m.) separation are explored in depth. In the zero-energy limit and single mode regime we reproduce the well-known confinement-induced resonances (CIRs) for bosonic, fermionic, and heteronuclear collisions. In the case of the multimode regime up to four open transverse channels are considered. Previously obtained analytical results are extended significantly here. Series of Feshbach resonances in the transmission behavior are identified and analyzed. The behavior of the transmission with varying energy and scattering lengths is discussed in detail. The dual CIR leading to a complete quantum suppression of atomic scattering is revealed in multichannel scattering processes. Possible applications include, e.g., cold and ultracold atom-atom collisions in atomic waveguides and electron-impurity scattering in quantum wires.
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. This paper investigates the interplay between the single-particle spin-orbit coupling term of Rashba type and the short-range two-body
s-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
s-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
s-wave interaction generically enters, depending on the exact parameter combinations of the
s-wave scattering lengths, at order
kso2 or
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 follows straightforwardly from the functional form of the spin-orbit coupling term, the absence of a term proportional to
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. 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.
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
kso and the mass ratio. The fact that these contributions are independent of the two-body scattering length
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.
Low energy scattering amplitudes for two atoms in one- and two-dimensional atomic waveguides are derived for short range isotropic and resonant interactions in high partial wave channels. Taking into account the finite width of the resonance which was neglected in previous works is shown to have important implications in the properties of the confinement induced resonances. For spin polarized fermions in quasi-1D waveguides, it imposes a strong constraint on the atomic density for achieving the Fermi Tonks Girardeau gas. For a planar waveguide, the characteristics of the 2D induced scattering resonances in p- and d-waves are determined as a function of the 3D scattering parameters and of the waveguide frequency.