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arXiv:1211.6342 · Journal of Physics B Atomic Molecular and Optical Physics

Mass-imbalanced three-body systems in two dimensions

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

Published Feb 12, 2013• 29 citations
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Abstract

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.

DOI
10.1088/0953-4075/46/5/055301

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In this thesis we study mass-imbalanced three-body system in two dimensions using BornOppenheimer approximation. Considering the heavy-light particle system interacting through zero-range interaction and disregarding interaction between the heavy-heavy, we study effects the angular momentum and mass ratio difference. Decreasing the mass difference between the heavy and the light particles a smaller number of bound states is found. Increasing the angular momentum we add to the system a repulsive potential, making the system weakly bounded, in this case a small number of bound states are expected. Considering a gaussian potential between the heavy-heavy particles it is possible to unbounded the system and observe tunneling phenomena. The considerable change in the tunneling region for the mean square radii indicates an increase in the system size, making this region interesting to be experimentally detected.
We study a three-body system, formed by a light particle and two identical heavy dipoles, in two dimensions in the Born-Oppenheimer approximation. We present the analytic light-particle wave function resulting from an attractive zero-range potential between the light and each of the heavy particles. It expresses the large-distance universal properties which must be reproduced by all realistic short-range interactions. We calculate the three-body spectrum for zero heavy-heavy interaction as a function of light to heavy mass ratio. We discuss the relatively small deviations from Coulomb estimates and the degeneracies related to radial nodes and angular momentum quantum numbers. We include a repulsive dipole-dipole interaction and investigate the three-body solutions as functions of strength and dipole direction. Avoided crossings occur between levels localized in the emerging small and large-distance minima, respectively. The characteristic exchange of properties such as mean-square radii are calculated. Simulation of quantum information transfer is suggested. For large heavy-heavy-particle repulsion all bound states have disappeared into the continuum. The corresponding critical strength is inversely proportional to the square of the mass ratio, far from the linear dependence from the Landau criterion.
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 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.
The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body subsystems are characterized by two-body energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as a function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the three-body system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22, we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio, and we map out an energy–mass phase diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.
The energy spectrum, atom-dimer scattering length, and atom-trimer scattering length for systems of three and four ultracold atoms with $\ensuremath{\delta}$-function interactions in one dimension are presented as a function of the relative mass ratio of the interacting atoms. The Born-Oppenheimer approach is used to treat three-body (``HHL'') systems of one light and two heavy atoms, as well as four-body (``HHHL'') systems of one light and three heavy atoms. Zero-range interactions of arbitrary strength are assumed between different atoms, but the heavy atoms are assumed to be noninteracting among themselves. Fermionic and bosonic heavy atoms with both positive and negative parity are considered.