We calculated, using seven realistic $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials in the literature, the Efimov spectra of the
4He trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr\"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the
4He trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length
rvdW, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius ${R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius ${R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$ makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the
4He trimer with the realistic potentials.
A comparison is made between the number of Efimov States present in
4He trimers as calculated using the Faddeev equations in the Faddeev coordinate-momentum approach with estimates predicted by N=(1/\ensuremath{\pi})ln(\ensuremath{\Vert}a\ensuremath{\Vert}/
r0) using realistic phenomenological He-He potentials. It is found that there is good agreement for systems supporting few states and that agreement remains when the Efimov states are forced to disappear by mathematically altering the potentials.
For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body
s-wave dimer, the Efimov effect takes place leading to the formation of an infinite number of three-body (Efimov) states. The lowest Efimov state crosses the three-body break-up threshold when the
s-wave two-body scattering length is $a\ensuremath{\approx}\ensuremath{-}9.73{r}_{\mathrm{vdW}}$,
rvdW being the van der Waals length. This article focuses on a generalized version of this Efimov scenario, where two of the atoms are about to form a two-body
d-wave dimer, resulting in strong
d-wave interactions. In his paper [Phys. Rev. A 62, 050702(R) (2000)], Bo Gao predicted that for broad resonances the
d-wave dimer is always formed when $a\ensuremath{\approx}0.956{r}_{\mathrm{vdW}}$. Here we find that a single universal three-body state associated with the
d-wave dimer is also formed near the three-body breakup threshold at $a\ensuremath{\approx}1.09{r}_{\mathrm{vdW}}$, or alternatively
a2=0.902rvdW, where
a2 is the two-body
d-wave scattering length. The signature of such a universal three-body state is signaled experimentally by an enhancement of the three-body recombination rate. The three-body effective potential curves that are crucial for understanding the recombination dynamics are also calculated and analyzed. An improved method to calculate the couplings, effective potential curves, and recombination rate coefficients is presented.
We formulate a method to study two-body correlations in a system of N identical bosons interacting via central two-body potentials. We use the adiabatic hyperspherical approach and assume a Faddeev-like decomposition of the wave function. For a fixed hyperradius we derive variationally an optimal integro-differential equation for the hyperangular eigenvalue and wave function. This equation reduces substantially by assuming the interaction range much smaller than the size of the N-body system. At most, one-dimensional integrals then remain. We view a Bose-Einstein condensate pictorially as a structure in the landscape of the potential given as a function of the one-dimensional hyperradial coordinate. The quantum states of the condensate can be located in one of the two potential minima. We derive and discuss properties of the solutions and illustrate with numerical results. The correlations lower the interaction energy substantially. The new multibody Efimov states are solutions independent of details of the two-body potential. We compare with mean-field results and available experimental data.
We present numerical solutions of the Schr\"odinger equation for three identical bosons from very large to essentially infinite two-body
s-wave scattering lengths
a. We have considered scattering lengths up to $2\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ and solved the hyperangular part of the Schr\"odinger equation for distances up to
106a.u. From these, we obtained the three-body effective potentials, hyperspherical channel functions and the asymptotic behavior of the nonadiabatic couplings in order to characterize the main aspects of the Efimov states. These results allow us to test and quantify the assumptions related to the Efimov effect.