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.
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.
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.
The low-energy spectrum of N-boson clusters with pairwise zero-range interactions is believed to be governed by a three-body parameter. We study the ground state of N-boson clusters with infinite two-body s-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 N-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.
Since the 4He dimer supports only one weakly bound state with an average interatomic distance much larger than the van der Waals length and no deeply bound states, 4HeN clusters with N>2 are a paradigmatic model system with which to explore foundational concepts such as large s-wave scattering length universality, van der Waals universality, Efimov physics, and effective field theories. This work presents structural properties such as the pair and triple distribution functions, the hyper-radial density, the probability to find the Nth particle at a given distance from the center of mass of the other $N\ensuremath{-}1$ atoms, and selected contacts. The kinetic energy release, which can be measured via Coulomb explosion in dedicated size-selected molecular beam experiments---at least for small N -- is also presented. The structural properties are determined for three different realistic $^{4}\mathrm{He}\ensuremath{-}^{4}\mathrm{He}$ interaction potentials and contrasted with those for an effective low-energy potential model from the literature that reproduces the energies of 4HeN clusters in the ground state for N=2 to $N=\ensuremath{\infty}$ at the $\ensuremath{\gtrsim}$95% level with just four input parameters. The study is extended to unitarity (infinite s-wave scattering length) by artificially weakening the interaction potentials. In addition to contributing to the characterization of small bosonic helium quantum droplets, our study provides insights into the effective low-energy theory's predictability of various structural properties.
We study the wave function φ(3) of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length a. We derive asymptotic expansions of φ(3) in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as ln3(B/a) at large hyperradius B=(s12+s22+s32)/2. At order B−2ln−3(B/a), we identify a three-body parameter D with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, D acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of D in terms of φ(3) and the changes of the two-body and three-body potentials. We also study the effects of D and φ(3) on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.
In calculating energy spectra, unlimited accuracy is only obtained if N-body wavefunctions contain N-body correlations except in special cases. These wave-functions are however not practical both for analytic and computational considerations. To this extent, the possiblity of utilizing correlations less than N is investigated. These investigations are based on the hypothesis whereby in D-dimensions, correlations greater than (D + 1)-body correlations are considerably less important. To accomplish these investigations, we compare the quality Q of the wavefunctions, pertaining to two-, three- and four-body correlations, in various dimensions. The trial wavefunctions used in our Monte Carlo method are linear combinations of elementary basis functions that consist of non-linear variational parameters. This method entails a linear optimization nested in a non-linear one. Once these trial wavefunctions are optimized we employ them as basis functions in a correlation function Monte Carlo which reduces systematically the variational bias of the energy estimates. This method is used to calculate ground and excited state energies of small bosonic van der Waal clusters using wavefunctions with two-, three- and four-body correlations. In calculating energy spectra in different dimensions, it was observed that for N atoms there exists an inter-dimensional degeneracy in the energy spectrum for (N − 1)-dimensions and (N + 1)-dimensions. The finite difference method is also used to calculate 'exact' energy spectra of two-body clusters, and these results are used as a benchmark to compare results obtained by our Monte Carlo method. We also describe a method of obtaining better statistics from quantum Monte Carlo estimators based on the assumption that these estimators are afflicted with distributions that resemble the lognormal probability distribution.