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arXiv:1401.3314 · Physical Review A

Born-Oppenheimer study of two-component few-particle systems under one-dimensional confinement

Nirav P. Mehta

Published May 13, 2014• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.89.052706

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The energy spectrum, atom-dimer scattering length, and atom-trimer scattering length for systems of three and four ultracold atoms with \ensuremathδ\ensuremath{\delta}\ensuremathδ-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.
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 ∞.
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 consider a few-boson system confined to one dimension with a single distinguishable particle of lesser mass. All particle interactions are modeled with $\ensuremath{\delta}$ functions, but due to the mass imbalance the problem is nonintegrable. Universal few-body binding energies, atom-dimer and atom-trimer scattering lengths, are all calculated in terms of two parameters, namely the mass ratio mL/mH{m}_{\text{L}}/{m}_{\text{H}}mL​/mH​, and ratio gHH/gHL{g}_{\text{HH}}/{g}_{\text{HL}}gHH​/gHL​ of the $\ensuremath{\delta}$-function couplings. We specifically identify the values of these ratios for which the atom-dimer or atom-trimer scattering lengths vanish or diverge. We identify regions in this parameter space in which various few-body inelastic processes become energetically allowed. In the Tonks-Girardeau limit (${g}_{\text{HH}}\ensuremath{\rightarrow}\ensuremath{\infty}$), our results are relevant to experiments involving trapped fermions with an impurity atom.
We study few-body problems in mixed dimensions where two or three heavy atoms are trapped individually in parallel one-dimensional tubes or two-dimensional disks and a single light atom travels freely in three dimensions. Using the Born-Oppenheimer approximation, we find three- and four-body bound states for a broad parameter region. Specifically, the existence of trimer and tetramer states persists to the negative scattering length regime, where no two-body bound state is present. As pointed out by Y. Nishida in an earlier work [Phys. Rev. A 82, 011605(R) (2010)], these few-body bound states are stable against three-body recombination due to geometric separation. In addition, we find that the binding energy of the ground trimer and tetramer state reaches its maximum value when the scattering lengths are comparable to the separation between the low-dimensional traps.
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.
The universal three-body dynamics in ultracold binary gases confined to one-dimensional motion is studied. The three-body binding energies and the (2 + 1)-scattering lengths are calculated for two identical particles of mass m and a different particle of mass m 1, whose interaction is described in the low-energy limit by zero-range potentials. The critical values of the mass ratio m/m 1 at which three-body states occur and the (2 + 1)-scattering length vanishes are determined for both zero and infinite interaction strength λ1 of the identical particles. A number of exact results are listed and asymptotic dependences for both m/m 1 → ∞ and λ1 → −∞ are derived. Combining the numerical and analytic results, we deduce a schematic diagram showing the number of three-body bound states and the sign of the (2 + 1)-scattering length in the plane of the mass ratio and the interaction-strength ratio. The results provide a description of the homogeneous and mixed phases of atoms and molecules in dilute binary quantum gases.