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Bound states of a three-body quantum system in two dimensions in the Born-Oppenheimer approximation

D. S. Rosa

Published Jul 29, 2016• 0 citations
OpenAlex

Abstract

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.

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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 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.
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.
We study the Born–Oppenheimer approximation of a mass-imbalanced three-body system made of two heavy particles of mass M and one light particle of mass m for arbitrary angular momentum. In this system, heavy-light pairs interact via a zero-range force. We construct the light-particle Hamiltonian using self-adjoint extensions of the two-center point interaction and show that the corresponding effective potential is regular at the coincidence point of the heavy particles. Consequently, this model presents an alternative method to finite-range, cutoff, or short-distance heavy-heavy regularizations: the necessary three-body input is encoded in the self-adjoint realization of the light-particle Hamiltonian, while the heavy-light interactions remain point-like. In the unitary limit, after fixing the characteristic length scale, we derive an explicit Efimov spectrum. Our results recover the zero-angular-momentum case of [] and provide a sufficient condition ensuring the absence of non-Efimov bound states. Away from unitarity, we show that the spatial size of the shallowest trimer near the threshold is approximately 2.8 times the heavy-light scattering length, in contrast with the common assumption that these two length scales coincide. We also derive a Bargmann-type bound on the number of three-body bound states and obtain an estimate sharper than previous results. Finally, we illustrate the method with numerical results for selected alkali mixtures.
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 employ the Born-Oppenheimer approximation to find the effective potential in a three-body system consisting of a light particle and two heavy ones when the heavy-light short-range interaction potential has a resonance corresponding to a nonzero orbital angular momentum. In the case of an exact resonance in the p-wave scattering amplitude, the effective potential is attractive and long range; namely, it decreases as the third power of the interatomic distance. Moreover, we show that the range and power of the potential, as well as the number of bound states, are determined by the mass ratio of the particles and the parameters of the heavy-light short-range potential.
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.