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arXiv:cond-mat/0607519 · Physical Review A

Energetics and structural properties of three-dimensional bosonic clusters near threshold

Gabriel Hanna, D. Blume

Published Dec 5, 2006• 61 citations
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Abstract

We treat three-dimensional bosonic clusters with up to $N=40$ atoms, interacting additively through two-body van der Waals potentials, in the near-threshold regime. Our study includes super-borromean systems with $N$ atoms for which all subsystems are unbound. We determine the energetics and structural properties such as the expectation value of the interparticle distance as a function of the coupling strength. It has been shown that the coupling strength ${g}_{*}^{(N)}$, for which the $N$-body system becomes unbound, is bounded by the coupling constant ${g}_{*}^{(N\ensuremath{-}1)}$, for which the next smaller system with $N\ensuremath{-}1$ atoms becomes unbound, i.e., ${g}_{*}^{(N)}\ensuremath{\geqslant}(N\ensuremath{-}1){g}_{*}^{(N\ensuremath{-}1)}∕N$. By fitting our numerically determined ground-state energies to a simple functional form with three fitting parameters, we determine the relationship between ${g}_{*}^{(N)}$ and ${g}_{*}^{(N\ensuremath{-}1)}$. Our trimer and tetramer energies fall on the so-called Tjon line, which has been studied in nuclear physics. We confirm the existence of generalized Tjon lines for larger clusters. Signatures of the universal behavior of weakly bound three-dimensional clusters can possibly be observed in ultracold Bose gases.

DOI
10.1103/physreva.74.063604

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We treat three-dimensional bosonic clusters with up to N=40N=40N=40 atoms, interacting additively through two-body van der Waals potentials, in the near-threshold regime. Our study includes super-borromean systems with NNN atoms for which all subsystems are unbound. We determine the energetics and structural properties such as the expectation value of the interparticle distance as a function of the coupling strength. It has been shown that the coupling strength g∗(N){g}_{*}^{(N)}g∗(N)​, for which the NNN-body system becomes unbound, is bounded by the coupling constant g∗(N\ensuremath−1){g}_{*}^{(N\ensuremath{-}1)}g∗(N\ensuremath−1)​, for which the next smaller system with N\ensuremath−1N\ensuremath{-}1N\ensuremath−1 atoms becomes unbound, i.e., g∗(N)\ensuremath⩾(N\ensuremath−1)g∗(N\ensuremath−1)∕N{g}_{*}^{(N)}\ensuremath{\geqslant}(N\ensuremath{-}1){g}_{*}^{(N\ensuremath{-}1)}∕Ng∗(N)​\ensuremath⩾(N\ensuremath−1)g∗(N\ensuremath−1)​∕N. By fitting our numerically determined ground-state energies to a simple functional form with three fitting parameters, we determine the relationship between g∗(N){g}_{*}^{(N)}g∗(N)​ and g∗(N\ensuremath−1){g}_{*}^{(N\ensuremath{-}1)}g∗(N\ensuremath−1)​. Our trimer and tetramer energies fall on the so-called Tjon line, which has been studied in nuclear physics. We confirm the existence of generalized Tjon lines for larger clusters. Signatures of the universal behavior of weakly bound three-dimensional clusters can possibly be observed in ultracold Bose gases.
Bosonic two-dimensional self-bound clusters consisting of NNN atoms interacting through additive van der Waals potentials become unbound at a critical mass m∗(N){m}_{*}^{(N)}m∗(N)​; m∗(N){m}_{*}^{(N)}m∗(N)​ has been predicted to be independent of the size of the system. Furthermore, it has been predicted that the ground state energy EN{E}_{N}EN​ of the NNN-atom system varies exponentially as the atomic mass approaches m∗{m}_{*}m∗​. This paper reports accurate numerical many-body calculations that allow these predictions to be tested. We confirm the existence of a universal critical mass m∗{m}_{*}m∗​, and show that the near-threshold behavior can only be described properly if a previously neglected term is included. We comment on the universality of the energy ratio EN+1∕EN{E}_{N+1}∕{E}_{N}EN+1​∕EN​ near threshold.
A correlated two-body basis function is used to describe the three-dimensional bosonic clusters interacting via two-body van der Waals potential. We calculate the ground state and the zero orbital angular momentum excited states for Rb(N) clusters with up to N = 40. We solve the many-particle Schrödinger equation by potential harmonics expansion method, which keeps all possible two-body correlations in the calculation and determines the lowest effective many-body potential. We study energetics and structural properties for such diffuse clusters both at dimer and tuned scattering length. The motivation of the present study is to investigate the possibility of formation of N-body clusters interacting through the van der Waals interaction. We also compare the system with the well studied He, Ne, and Ar clusters. We also calculate correlation properties and observe the generalised Tjon line for large cluster. We test the validity of the shape-independent potential in the calculation of the ground state energy of such diffuse cluster. These are the first such calculations reported for Rb clusters.
Small weakly bound droplets determine a number of properties of ultracold Bose and Fermi gases. For example, Efimov trimers near the atom-atom-atom and atom-dimer thresholds lead to enhanced losses from bosonic clouds. Generalizations to four- and higher-body systems have also been considered. Moreover, Efimov trimers have been predicted to play a role in the Bose polaron with large boson-impurity scattering length. Motivated by these considerations, the present work provides a detailed theoretical analysis of weakly bound NNN-body clusters consisting of $N\ensuremath{-}1$ identical bosons (denoted by ``B'') of mass mmm that interact with a single distinguishable impurity particle (denoted by ``X'') of mass MMM. The system properties are analyzed as a function of the mass ratio $\ensuremath{\kappa}$ (values from $\ensuremath{\kappa}=1$ to 50 are considered), where $\ensuremath{\kappa}$ is equal to m/Mm/Mm/M, and the two-body sss-wave scattering length aBX{a}_{\text{BX}}aBX​ between the bosons and the impurity. To reach the universal Efimov regime in which the size of the BBX trimer as well as those of larger clusters is much larger than the length scales of the underlying interaction model, three different approaches are considered: resonance states are determined in the absence of BB and BBX interactions, bound states are determined in the presence of repulsive three-body boson-boson-impurity interactions, and bound states are determined in the presence of repulsive two-body boson-boson interactions. The universal regime, in which the details of the underlying interaction model become irrelevant, is identified.
The results of accurate variational calculations for the ground-state binding energies of several NNN-boson systems in two and three dimensions are reported; the systems specifically considered are small (N=2,3,4,and5N=2, 3, 4, \mathrm{and} 5N=2,3,4,and5) clusters of 4He^{4}\mathrm{He}4He atoms interacting through pairwise Lennard-Jones potentials. Evidence is found strongly suggesting each set of systems have a common functional relationship linking the binding energies and the De Boer quantum constant near threshold for self-binding. These calculations also confirm the existence of a unique threshold value of the interaction strength for all NNN-boson systems in 2D2D2D. In addition, the linear relation between the binding energies of helium clusters present in 3D3D3D holds ture in 2D2D2D as well.
We present an approach to analyze recent experimental evidences of Efimov resonant states in mixtures of ultracold gases, by considering two-species three-body atomic systems bound in a Borromean configuration, where all the two-body interactions are unbound. For such Borromean three-body systems, it is shown that a continuum three-body s -wave resonance emerges from an Efimov state as a scattering length or a three-body scale is moved. The energy and width of the resonant state are determined from a scaling function with arguments given by dimension-less energy ratios relating the two-body virtual state subsystem energies with the shallowest three-body bound state. The peculiar behavior of such resonances is that their peaks are expected to move to lower values of the scattering length, with increasing width, as one raises the temperature. For Borromean systems, two resonant peaks are expected in ultralow-temperature regimes, which will disappear at higher energies. It is shown how a Borromean-Efimov excited bound state turns out to a resonant state by tuning the virtual two-body subsystem energies or scattering lengths, with all energies written in units of the next deeper shallowest Efimov state energy. The resonance position and width for the decay into the continuum are obtained as universal scaling functions (limit cycle) of the dimensionless ratios of the two and three-body scales, which are calculated numerically within a zero-range renormalized three-body model.
The low-energy spectrum of NNN-boson clusters with pairwise zero-range interactions is believed to be governed by a three-body parameter. We study the ground state of NNN-boson clusters with infinite two-body sss-wave scattering length by performing ab initio Monte Carlo simulations. To prevent Thomas collapse, different finite-range three-body regulators are used. The energy and structural properties for the three-body Hamiltonian with two-body zero-range interactions and three-body regulator are in much better agreement with the ``ideal zero-range Efimov theory'' results than those for Hamiltonian with two-body finite-range interactions. For larger clusters we find that the ground-state energy and structural properties of the Hamiltonian with two-body zero-range interactions and finite-range three-body regulators are not universally determined by the three-body parameter, i.e., dependencies on the specific form of the three-body regulator are observed. For comparison, we consider Hamiltonian with two-body van der Waals interactions and no three-body regulator. For the interactions considered, the ground-state energy of the NNN-body clusters is---if scaled by the three-body ground-state energy---fairly universal, i.e., the dependence on the short-range details of the two-body van der Waals potentials is small. Our results are compared with those in the literature.