Physical origin of the universal three-body parameter in atomic Efimov physics

We address the microscopic origin of the universal three-body parameter that fixes the spectrum of three-atom systems in the Efimov regime. We identify it with the van der Waals two-body correlation, which causes the three-atom system to deform when the three atoms come within the distance of the van der Waals length, effectively preventing them from coming closer due to the kinetic-energy cost associated with three-body deformation. This deformation mechanism explains the universal ratio of the scattering length at the triatomic resonance to the van der Waals length observed in several experiments and confirmed by numerical calculations.

Figure 1

Three-body potentials $U_0\left(R\right)$ and $U_1\left(R\right)$ for different pairwise interactions at unitarity: soft-core van der Waals potential (blue) with $n_b=1–10$ two-body bound states, Lennard-Jones potential (green) with $n_b=1–10$ two-body bound states, and helium potential (red) rescaled to reach unitarity with $n_b=1$ two-body bound state. Note that only the case of the soft-core van der Waals potential with one bound state is significantly different from the other cases. The dashed curve shows the asymptotic Efimov attraction.

Figure 2

Three-dimensional contour plots of the probability distribution in Eq. (11) of finding a particle for a given separation of the two other particles (which are indicated by a pair of small gray balls connected by a black line). For clarity, we only show the probability density behind a plane containing the two particles and shade the contours with an opacity increasing with probability density: the darker, the higher the probability of finding the third particle. The top figures correspond to a separation of $6.0r_{\mathrm{vdW}}$, while the bottom ones correspond to a separation of $1.4r_{\mathrm{vdW}}$. To appreciate the change in configuration between the figures, a typical location of the third particle is indicated by a small green ball connected to the other two particles by green lines. The left figures were computed from the zero-range Efimov theory at unitarity; they show the invariance of the Efimov configuration distribution with respect to the size of the system. The right figures were computed for a Lennard-Jones pairwise potential at unitarity supporting four two-body bound states. At large separations, the probability distribution is consistent with the Efimov configuration distribution, but around each of the two particles there is a noticeable sphere of radius $\sim r_{\mathrm{vdW}}$ in which the probability is significantly suppressed. This suppression leads to an abrupt change in configuration probability when the particles come close.

Figure 3

Zero-energy two-body probability density distribution ${\left|\phi \right|}^2$ (normalized asymptotically to unity) as a function of interparticle distance for different two-body potentials: soft-core van der Waals potential (blue) with $n_b=1–8$ bound states, Lennard-Jones potential (green) with $n_b=1–8$ bound states, and helium potential (red) rescaled to reach unitarity with one bound state. The corresponding potentials are shown in faded colors. The probability density corresponding to the universal van der Waals correlation given in Eq. (13) is shown by the dashed black curve.

Figure 4

Regions of three-body configurations plotted against the hyperradius and one hyperangle, where the molecules illustrate the corresponding configurations. The region in solid red corresponds to the region trivially excluded when at least two particles are within the suppression distance $r_{\mathrm{vdW}}$. The hatched region corresponds to the region excluded by the nonadiabaticity of the deformation occurring when the hyperradius is smaller than $\sim 2r_{\mathrm{vdW}}$. The shades of blue represent the integrated three-body probability obtained by numerically solving the three-body problem with a separable potential Eq. (15) whose two-body correlation reproduces that of a Lennard-Jones potential at the appearance of the first two-body bound state. It shows that the probability is indeed excluded from the red and hatched regions.

Figure 5

Comparison between the Faddeev three-body calculations (left) and the simple two-body correlation model described in the main text (right). The dashed curves show the nonadiabatic kinetic energy $Q_{00}$ for Lennard-Jones potentials of different depths, corresponding to the unitarity limit with different numbers of two-body bound states ranging from 1 to 5. The solid curves show the full three-body potential $U_0\left(R\right)$ obtained by adding to $Q_{00}$ the adiabatic contribution ${\lambda }_0/R^2$ obtained from Faddeev calculations, which is shown by the dotted curves.

Figure 6

Lowest trimer and dimer energy for the separable potential given by Eq. (15), as a function of its inverse scattering length $1/a$. For comparison, the dotted and dashed curves represent the universal dimer energy $\left(E=-\frac{{\hslash }^2}{m{a}^2}\right)$ and the exact van der Waals dimer energy, respectively. Both the abscissa and ordinate are shown in units of the inverse van der Waals length $r_{\mathrm{vdW}}^{-1}$.

Figure 7

Three-body probability (solid gray curve, in arbitrary units) as a function of hyperradius and corresponding effective three-body potential (solid black curve), obtained from the separable model Eq. (15). For comparison, the Efimov potential (dotted curve) and the low-energy Faddeev three-body potential obtained from Eq. (7) (red dashed curve) are also represented. Note that for $R<1.5r_{\mathrm{vdW}}$ the effective potential becomes strongly oscillatory (as can be seen from the nearly vertical lines) in order to reproduce the wave function. Since the physics in that region is correctly described by coupled potentials, the effective potential is not a meaningful construct in that region and the short-range oscillations have no particular significance.