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arXiv:1704.08935

Two bosons in a narrowly resonant trap

Bingwei Long, Jianfeng Wang, Songlin Lyu

Published Apr 28, 2017• 1 citation
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Abstract

It is rare for an $S$-wave resonance to remain narrow while approaching threshold as a result of the interaction parameters being fine tuned. Such an interaction is, however, realized by pion-charmed baryon system $πΣ_c$, which has a narrow resonance $Λ_c^+(2595)$ located a few MeVs above its threshold. We study the consequence of narrowly resonant inter-species interaction in a three-body system composed of two identical light bosons and an extremely heavy particle. We concern ourselves in the present paper with resonances of the three-body system, as opposed to bound states. The focus is on the technical aspects of constructing the integral equation for three-body amplitudes and of analytically continuing the integral equation to locate the "trimer" resonance pole in the unphysical sheet of the center-of-mass energy plane.

DOI
10.48550/arxiv.1704.08935

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It is rare for an SSS-wave resonance to remain narrow while approaching threshold as a result of the interaction parameters being fine tuned. Such an interaction is, however, realized by pion-charmed baryon system πΣcπΣ_cπΣc​, which has a narrow resonance Λc+(2595)Λ_c^+(2595)Λc+​(2595) located a few MeVs above its threshold. We study the consequence of narrowly resonant inter-species interaction in a three-body system composed of two identical light bosons and an extremely heavy particle. We concern ourselves in the present paper with resonances of the three-body system, as opposed to bound states. The focus is on the technical aspects of constructing the integral equation for three-body amplitudes and of analytically continuing the integral equation to locate the "trimer" resonance pole in the unphysical sheet of the center-of-mass energy plane.
We revisit the problem of three identical bosons in free space, which exhibits a universal hierarchy of bound states (Efimov trimers). Modeling a narrow Feshbach resonance within a two-channel description, we map the integral equation for the three-body scattering amplitude to a one-dimensional Schrödinger-type single-particle equation, where an analytical solution of exponential accuracy is obtained. We give exact results for the trimer binding energies, the three-body parameter, the threshold to the three-atom continuum, and the recombination rate.
We investigate the formation of three-body bound states in the continuum by tracing pole trajectories in the complex energy plane under variation of system parameters. Using a one-dimensional model of two identical bosons and a distinguishable particle interacting via Gaussian potentials, we systematically vary the interaction strength, interaction range, and mass ratio. Our results confirm the parametric nature of few-body bound states in a continuum (BIC) and extend this characterization to a broader set of system parameters. Specifically, we find that variations of both interaction parameters and the mass ratio can lead to the formation of at least one three-body BIC. However, the exact shape of trajectories differs, and for the mass ratio variation we find a more regular pattern with multiple BIC locations. These results suggest that the mechanism of few-body BIC formation is more sensitive to the kinematic structure of the problem than to the specific details of the two-body interaction.
We calculate the universal spectrum of trimer and tetramer states in heteronuclear mixtures of ultracold atoms with different masses in the vicinity of the heavy-light dimer threshold. To extract the energies, we solve the three- and four-body problem for simple two- and three-body potentials tuned to the universal region using the Gaussian expansion method. We focus on the case of one light particle of mass mmm and two or three heavy bosons of mass MMM with resonant heavy-light interactions. We find that trimer and tetramer cross into the heavy-light dimer threshold at almost the same point and that as the mass ratio M/mM/mM/m decreases, the distance between the thresholds for trimer and tetramer states becomes smaller. We also comment on the possibility of observing exotic three-body states consisting of a dimer and two atoms in this region and compare with previous work.
Planar three-body bound states induced by ppp-wave interatomic resonance
We consider the bound states of a system consisting of a light particle and two heavy bosonic ones, which are restricted in their quantum mechanical motion to two space dimensions. A ppp-wave resonance in the heavy-light short-range potential establishes the interaction between the two heavy particles. Due to the large ratio of the atomic masses this planar three-body system is perfectly suited for the Born-Oppenheimer approximation which predicts a Coulomb energy spectrum with a Gaussian cut-off.
We investigate the appearance of resonances in three-body systems using pionless effective field theory at leading order with two complementary methods. The Faddeev equation is analytically continued to the unphysical sheet adjacent to the positive real energy axis using a contour rotation. We consider both the three-boson system and the three-neutron system. For the former, we calculate the trajectory of Borromean three-body Efimov states turning into resonances as they cross the three-body threshold. For the latter, we find no sign of three-body resonances or virtual states at leading order. This result is validated by exploring the level structure of three-body states in a finite volume approach.
The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. Two identical fermions of the mass mmm and a particle of the mass m1m_1m1​ with the zero-range two-body interaction in the states of the total angular momentum L=1 are considered. Using the boundary condition model for the s-wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependencies of the three-body binding energies on the mass ratio m/m1m/m_1m/m1​ for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m/m1gelambda1approx8.17260m/m_1 \\ge \\lambda_1 \\approx 8.17260m/m1​gelambda1​approx8.17260 and m/m1gelambda2approx12.91743m/m_1 \\ge \\lambda_2 \\approx 12.91743m/m1​gelambda2​approx12.91743, respectively. For m/m1altlambda1m/m_1 \\alt \\lambda_1m/m1​altlambda1​ and m/m1altlambda2m/m_1 \\alt \\lambda_2m/m1​altlambda2​, the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied and it is shown that a two-hump structure in the mass-ratio dependencies of the cross sections is connected with arising of the bound states.