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arXiv:1908.06918 · Physical Review A

Energetics and structural properties of two- and three-boson systems in the presence of one-dimensional spin-orbit coupling

Qingze Guan, D. Blume

Published Oct 21, 2019• 2 citations
arXivHTMLPDF

Abstract

It was shown recently that the discrete scaling symmetry, which underlies the Efimov effect in the three identical boson system with two-body short-range interactions, survives when single-particle one-dimensional (1D) spin-orbit coupling terms are added to the Hamiltonian. Each three-body energy level in the ordinary Efimov scenario turns into an energy manifold that contains four energy levels in the presence of 1D spin-orbit coupling (equal mixture of Rashba-Dresselhaus coupling). This work provides a detailed characterization of the energy levels in these manifolds. The two-boson energies, which enter into the three-boson scattering threshold, are analyzed in detail. Moreover, the structural properties, e.g., momentum distributions of the two- and three-boson systems, are analyzed for various parameter combinations.

DOI
10.1103/physreva.100.042708

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It was shown recently that the discrete scaling symmetry, which underlies the Efimov effect in the three identical boson system with two-body short-range interactions, survives when single-particle one-dimensional (1D) spin-orbit coupling terms are added to the Hamiltonian. Each three-body energy level in the ordinary Efimov scenario turns into an energy manifold that contains four energy levels in the presence of 1D spin-orbit coupling (equal mixture of Rashba-Dresselhaus coupling). This work provides a detailed characterization of the energy levels in these manifolds. The two-boson energies, which enter into the three-boson scattering threshold, are analyzed in detail. Moreover, the structural properties, e.g., momentum distributions of the two- and three-boson systems, are analyzed for various parameter combinations.
We study the two-body and three-body bound states in ultracold atomic mixtures with one of the atoms subjected to an isotropic spin-orbit (SO) coupling. We consider a system of two identical fermions interacting with one SO-coupled atom. It is found that there can exist two types of three-body bound states, Efimov trimers and universal trimers. The Efimov trimers are energetically less favored by the SO coupling, which will finally merge into the atom-dimer threshold as increasing the SO-coupling strength. Nevertheless, these trimers exhibit a discrete scaling law incorporating the SO-coupling effect. On the other hand, the universal trimers are more favored by the SO coupling. They can be induced at negative sss-wave scattering lengths and with smaller mass ratios than those without SO coupling. These results are obtained by both the Born-Oppenheimer approximation and exact solutions from three-body equations.
A system of bosons in a two-dimensional harmonic trap in the presence of Rashba-type spin-orbit coupling is investigated. An analytic treatment of the ground state of a single atom in the weak-coupling regime is presented and used as a basis for a perturbation theory in the interacting two-boson system. The numerical diagonalization of both the single-particle and the two-boson Hamiltonian matrices allows us to go beyond those approximations and obtain not only the ground state, but also the low-energy spectra and the different energy contributions separately. We show that the expectation value of the spin-orbit term is related to the expectation value of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{z}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{L}}_{z}$ for the eigenstates of the system, regardless of the trapping potential. The low-energy states of the repulsively interacting two-boson system are characterized. In the presence of a sufficiently strong interaction and spin-orbit coupling strength, there is a direct energy-level crossing in the ground state of the system between states of different Jz{J}_{z}Jz​, the third component of the total angular momentum, that changes its structure. This is reflected in a discontinuity in the different energy terms and it is signaled in the spatial density of the system.
We solve the three-body bound-state problem in three dimensions for mass imbalanced systems of two identical bosons and a third particle in the universal limit where the interactions are assumed to be of zero range. The system displays the Efimov effect and we use the momentum-space wave equation to derive formulas for the scaling factor of the Efimov spectrum for any mass ratio assuming either that two or three of the two-body subsystems have a bound state at zero energy. We consider the single-particle momentum distribution analytically and numerically and analyze the tail of the momentum distribution to obtain the three-body contact parameter. Our findings demonstrate that the functional form of the three-body contact term depends on the mass ratio, and we obtain an analytic expression for this behavior. To exemplify our results, we consider mixtures of lithium with either two caesium or rubidium atoms which are systems of current experimental interest.
Near two-body unitarity, the three-boson system is characterized by an approximate discrete scale invariance manifest in a geometric tower of bound states (the Efimov effect). In the absence of a strong four-body force, the four-boson system has two states associated with each Efimov state, one very nearly unstable, the other several times deeper. We study correlations between the excited- and ground-state properties, such as binding energies and radii, up to next-to-leading order in an effective field theory for short-range forces. We obtain the parameters in these correlations from similar correlations arising from existing precise calculations based on short-range potentials. We also derive correlations among excited-state properties that emerge from the proximity of the state to the break-up threshold into a boson and a three-boson bound state, using an effective field theory for ``halo'' states.
In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.