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Phase Diagrams for Three-Body and Four-Body Systems in One Dimension

Connor Morehead

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

The phase diagrams for three-body and four-body systems with δ-function interactions are presented. The three-body and four body systems considered are the heavy-heavy-light (HHL) and heavy-heavy-heavy-light (HHHL) systems, respectively. The heavy particles are assumed to be identical. Both systems are treated within the Born-Oppenheimer approximation. The results from [16] are extended to arbitrary values of the heavy-heavy (HH) coupling. The phase diagram presented in [8] is reproduced, and the corresponding HHHL phase-diagram is presented. It was found that in both the three and four body systems the scattering lengths (atom-dimer for HHL and atom-trimer for HHHL) and spectra vary smoothly as the heavy-heavy interaction strength λ is varied from 0 to ∞.

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The phase diagrams for three-body and four-body systems with δ-function interactions are presented. The three-body and four body systems considered are the heavy-heavy-light (HHL) and heavy-heavy-heavy-light (HHHL) systems, respectively. The heavy particles are assumed to be identical. Both systems are treated within the Born-Oppenheimer approximation. The results from [16] are extended to arbitrary values of the heavy-heavy (HH) coupling. The phase diagram presented in [8] is reproduced, and the corresponding HHHL phase-diagram is presented. It was found that in both the three and four body systems the scattering lengths (atom-dimer for HHL and atom-trimer for HHHL) and spectra vary smoothly as the heavy-heavy interaction strength λ is varied from 0 to ∞.
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.
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.
Abstract We study a three-body system, formed by two identical heavy bosons and a light particle, in the Born–Oppenheimer approximation for an arbitrary dimension D . We restrict D to the interval 2 < D < 4, and derive the heavy–heavy D -dimensional effective potential proportional to 1/ R 2 ( R is the relative distance between the heavy particles), which is responsible for the Efimov effect. We found that the Efimov states disappear once the critical strength of the heavy–heavy effective potential 1/ R 2 approaches the limit − ( D − 2 ) 2 / 4 . We obtained the scaling function for the 133 Cs– 133 Cs– 6 Li system as the limit cycle of the correlation between the energies of two consecutive Efimov states as a function of D and the heavy-light binding energy E D 2 . In addition, we found that the energy of the ( N + 1)th excited state reaches the two-body continuum independently of the dimension D when E 2 D / E 3 ( N ) = 0.89 , where E 3 ( N ) is the N th excited three-body binding energy.
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 heavy-heavy-light three-body system confined to one space dimension. Both binding energies and corresponding wave functions are obtained for (i) the zero-range and (ii) two finite-range attractive heavy-light interaction potentials. In the case of the zero-range potential, we apply the method of Skorniakov and Ter-Martirosian to explore the accuracy of the Born-Oppenheimer approach. For the finite-range potentials, we solve the Schr\"odinger equation numerically using a pseudospectral method. We demonstrate that when the two-body ground-state energy approaches zero, the three-body bound states display a universal behavior, independent of the shape of the interaction potential.
We investigate one-dimensional three-body systems composed of two identical bosons and one mass-imbalanced atom (impurity) with attractive two-body and three-body zero-range interactions. In the absence of three-body interaction, we give a complete phase diagram of the number of three-body bound states in the whole region of mass ratio and the ratio of intra- and intercomponent interaction strength via direct calculation of Skornyakov-Ter-Martirosyan equations. We demonstrate that other low-lying three-body bound states emerge when the mass of the impurity particle is different from other two identical particles. We obtain the binding energies together with the corresponding wave functions. When the mass of impurity atom is very large, there are at most three three-body bound states. In the presence of three-body zero-range interaction, we reveal that weak three-body interaction will not always induce one more three-body bound state. At some special parameter points, arbitrary small three-body interaction can generate one more three-body bound state. This corresponds to the transition of the number of three-body bound states induced only by two-body attractive interaction.