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arXiv:1110.1204 · Physical Review A

Supercircle description of universal three-body states in two dimensions

F. F. Bellotti, T. Frederico, Marcelo Takeshi Yamashita, D. V. Fedorov, A. S. Jensen, Nikolaj Thomas Zinner

Published Feb 2, 2012• 19 citations
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Abstract

Bound states of asymmetric three-body systems confined to two dimensions are currently unknown. In the universal regime, two energy ratios and two mass ratios provide complete knowledge of the three-body energy measured in units of one two-body energy. We compute the three-body energy for general systems using numerical momentum-space techniques. The lowest number of stable bound states is produced when one mass is larger than two similar masses. We focus on selected asymmetric systems of interest in cold atom physics. The scaled three-body energy and the two scaled two-body energies are related through an equation for a supercircle whose radius increases almost linearly with three-body energy. The exponents exhibit an increasing behavior with three-body energy. The mass dependence is highly nontrivial. Based on our numerical findings, we give a simple relation that predicts the universal three-body energy.

DOI
10.1103/physreva.85.025601

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Bound states of asymmetric three-body systems confined to two dimensions are currently unknown. In the universal regime, two energy ratios and two mass ratios provide complete knowledge of the three-body energy measured in units of one two-body energy. We compute the three-body energy for general systems using numerical momentum-space techniques. The lowest number of stable bound states is produced when one mass is larger than two similar masses. We focus on selected asymmetric systems of interest in cold atom physics. The scaled three-body energy and the two scaled two-body energies are related through an equation for a supercircle whose radius increases almost linearly with three-body energy. The exponents exhibit an increasing behavior with three-body energy. The mass dependence is highly nontrivial. Based on our numerical findings, we give a simple relation that predicts the universal three-body energy.
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We found universal laws for the spectrum of two-dimensional three-body systems, composed by two identical particles and a distinct one (AAB). These universal laws appear when the potential range (r_0) is much smaller than the size of the system. In two dimensions this condition is formulated as (E2 is the two-body energy and ? is the reduced mass). The zero range model, which is very appropriated to establish the universal laws, is introduced through the ?-Dirac potential. We derive the corresponding two-dimensional Faddeev equations for the three-body system and solve them numerically in momentum space. Our results showed that the three-body binding energy monotonically increases with the two-body binding energy, and such dependence is more pronounced than the mass variations. We found that the three-body energy depends logarithmic on the two-body energy for large values. Furthermore, been m=mB/mA the ratio between the masses of the B and A particles, the three-body energy is mass-independent for m ? ? and increase without bounds for m ?0. The limit of two non-interacting identical particles is also studied in the AAB system. We found that the two-dimensional three-body system always support at least two bound states and more bound states appear for m<0.22. Finally, we analyze the particular limit of m ?0 using the adiabatic approximation. This approximation can be used to study the three-body system in two-dimensions with an accuracy better than 10% compared to the solutions of the Faddeev equations, for m ? 0.01.
We calculate root-mean-square radii for a three-body system confined to two spatial dimensions and consisting of two identical bosons (AAA) and one distinguishable particle (BBB). We use zero-range two-body interactions between each of the pairs, and focus thereby directly on universal properties. We solve the Faddeev equations in momentum space and express the mean-square radii in terms of first-order derivatives of the Fourier transforms of densities. The strengths of the interactions are adjusted for each set of masses to produce equal two-body bound-state energies between different pairs. The mass ratio, A=mB/mA\mathcal{A}={m}_{B}/{m}_{A}A=mB​/mA​, between particles BBB and AAA are varied from 0.01 to 100, providing a number of bound states decreasing from 8 to 2. Energies and mean-square radii of these states are analyzed for small A\mathcal{A}A by use of the Born-Oppenheimer potential between the two heavy AAA particles. For large A\mathcal{A}A the radii of the two bound states are consistent with a slightly asymmetric three-body structure. When A\mathcal{A}A approaches thresholds for binding of the three-body excited states, the corresponding mean-square radii diverge inversely proportional to the deviation of the three-body energy from the two-body thresholds. The structures at these three-body thresholds correspond to bound ABABAB dimers and one loosely bound AAA particle.
We consider three-body systems in two dimensions with zero-range interactions for general masses and interaction strengths. The momentum-space Schrödinger equation is solved numerically and in the Born-Oppenheimer (BO) approximation. The BO expression is derived using separable potentials and yields a concise adiabatic potential between the two heavy particles. The BO potential is Coulomb-like and exponentially decreasing at small and large distances, respectively. While we find similar qualitative features to previous studies, we find important quantitative differences. Our results demonstrate that mass-imbalanced systems that are accessible in the field of ultracold atomic gases can have a rich three-body bound state spectrum in two dimensional geometries. Small light-heavy mass ratios increase the number of bound states. For 87Rb-87Rb-6Li and 133Cs-133Cs-6Li we find respectively 3 and 4 bound states.
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.