Figure 1

All possible $s$-wave scattering lengths are shown for the lowest three Zeeman states of Li${}^6$ from Ref. [37]. Each marked region gives a different set of length scale discrepancies. Here $a^{(k)}$ is the scattering length between two atoms in states $|i\rangle$ and $|j\rangle$ with $k$ as the component not involved in the interaction.Reuse & Permissions

Figure 2

(a) For an example system having $a^{(1)}=a^{(3)}$ and $a^{(2)}=1000a^{(1)}$, the first four hyperangular eigenvalues are shown as functions of the hyperradius. The solid black horizontal lines show the expected behavior for three identical resonantly interacting bosons. The dashed line gives the behavior of two identical fermions interacting resonantly with a third distinguishable particle. Dotted lines give the expected universal behavior for a single resonant scattering length. Finally, the dot-dashed line is the lowest expected free space eigenvalue for three distinguishable free particles. (b) The coupling strengths between the third and fourth (purple solid curve), the first and fourth (red dashed curve), and the first and third (black dotted curve) adiabatic potentials are shown as a function of $R$.Reuse & Permissions

Figure 3

A schematic picture of the first four hyperradial potentials in region I. The grey areas, labeled by the appropriate scattering lengths, indicate regions where the potentials are changing from one universal behavior to another. The blue region on the left, labeled by $r_0$, indicates the short-range region where the zero-range pseudopotential can no longer be applied.Reuse & Permissions

Figure 4

(a)–(e) Examples of the hyperangular eigenvalues from each region of magnetic field as a function of the hyperradius in atomic units. (f) The three $s$-wave scattering lengths shown as a reference vs the magnetic field strength. The vertical dotted lines, from right to left, show the magnetic fields at which the hyperangular eigenvalues in (a)–(e) were found, $B=675,695,805,820$, and $845$ G, respectively.Reuse & Permissions

Figure 5

A schematic of the path for three-body recombination in region I. Transition regions are labeled by the appropriate length scale, and the short-range nonuniversal region is labeled by $r_0$.Reuse & Permissions

Figure 6

A schematic of the potentials and the path for three-body recombination in region III. Again the labeled grey regions indicate a transition from one universal potential behavior to another. The lowest (green) line represents the hyperradial potential for a deeply bound dimer state. The blue area on the left labeled by $r_0$ is the short-range region not described by zero-range interactions.Reuse & Permissions

Figure 7

A schematic of the lowest hyperradial potential with the path for three-body recombination to deeply bound states. The lowest (green) line represents the hyperradial potential for a deeply bound dimer state. Labeled grey areas indicate transition regions from one universal behavior to another, and the far-left blue region indicates the short-range regime.Reuse & Permissions

Figure 8

(a) The three-body recombination rate from Eq. (66) for ${}^6\mathrm{Li}$ in arbitrary units as a function of magnetic field with $\eta =0.001$ and the short-range length scale chosen to be approximately the van der Waals length, $r_0=r_d\approx 30$ a.u. The large $y$-axis tick marks indicate orders of magnitude. (b) The three scattering lengths $a^{(1)}$ (solid black curve), $a^{(2)}$ (dashed red curve), and $a^{(3)}$ (dot-dashed blue curve) in atomic units as a function of magnetic field in region V.Reuse & Permissions