Window for Efimov physics for few-body systems with finite-range interactions

Abstract

We investigate the two lowest-lying weakly bound states of N ≤ 8 bosons as functions of the strength of two-body Gaussian interactions. We observe the limit for validity of Efimov physics. We calculate energies and second radial moments as functions of scattering length. For identical bosons we find that two (N − 1)-body states appear before the N -body ground states become bound. This pattern ceases to exist for N ≥ 7 where the size of the ground state becomes smaller than the range of the two-body potential. All mean-square-radii for N ≥ 4 remain finite at the threshold of zero binding, where they vary as (N − 1) p with p = −3/2, −3 for ground and excited states, respectively. Decreasing the mass of one particle we find stronger binding and smaller radii. The identical particles form a symmetric system, while the lighter particle is further away in the ground states. In the excited states we find the identical bosons either surrounded or surrounding the light particle for few or many bosons, respectively. We demonstrate that the first excited states for all strengths resemble two-body halos of one particle weakly bound to a dense N -body system for N = 3, 4. This structure ceases to exist for N ≥ 5.

Universality of size-energy ratio in four-body systems

“…The scaled sizes approach constant values in the limit of weak binding for all R . This is in agreement with the study of four-body Brunnian systems 30 and with the recent analysis on the validity limits of Efimov physics 12 . Using the stochastic variational method, both studies concluded that the radius of few-body clusters approaches a constant value for very weak binding.…”

“…The reported size is extrapolated from exact ground-state results and it does not depend on the Lennard-Jones 6–12 potential parameters. In fact, previous variational estimates 12,30 also predicted finite size. This behavior is opposite to the dimer and trimer cases whose scaled size diverges in both cases 3,12,15 .…”

“…The discovery of the Efimov effect 1 naturally opened the question whether the same effect could exist in N -body systems, but it was concluded rather soon 9 that there is no true Efimov effect for four or more identical bosons. Despite of that the prediction of a variety of universal bound states linked to the Efimov trimer motivated new research that was extended 5,10–12 to non-identical particles and ( N > 3)-body systems.…”

Van der Waals five-body size-energy universality

“…In this case, particles are far away and pair potentials barely affect the probability outside the range of the van der Waals potential. That scenario is similar to the one of finite and contact interactions and thus, it is in agreement with previous theoretical findings 10 , 52 that scaled size saturates in the weak binding limit. Weak binding, which does not support smaller clusters, holds pentamers through mediated interactions of additional particles.…”

“…The Efimov effect was very soon observed in other cold atom systems, including those with non-identical particles and -body systems, in which a variety of universal bound states linked to the Efimov trimer was found. 6 – 10 . A further unexpected van der Waals universality appeared for three atoms interacting with potential in the ultracold regime, near Feshbach resonances 6 , 7 , 11 – 13 .…”