Universal three-body recombination via resonant $d$-wave interactions

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 $a\approx -9.73r_{\mathrm{vdW}}$, $r_{\mathrm{vdW}}$ 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 $a\approx 0.956r_{\mathrm{vdW}}$. 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 $a\approx 1.09r_{\mathrm{vdW}}$, or alternatively $a_2=0.902r_{\mathrm{vdW}}$, where $a_2$ 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.

Figure 1

The values of the two-body $s$-wave scattering length $a_l^{*}$ at the point where a $d$-wave ($l=2$) dimer (black curve with squares) and where an $i$-wave ($l=6$) dimer just become bound (red curve with circles) are shown as functions of the number of two-body $s$-wave bound states.

Figure 2

Adiabatic three-body potential curves for $a\approx 0.977r_{\mathrm{vdW}}$. The $2+1$ channels, which corresponding to a dimer plus a free atom at very large distance, are labeled by a combination of a letter and a number. The letter denotes the angular momentum quantum number $l$ of the dimer, and the number labels the channels for the same dimer angular momentum from low to high dimer-binding energies.

Figure 3

Total and partial $J^{\pi }=0^{+}$ three-body recombination rates as functions of the two-body scattering length $a$, shown on a linear scale, near a $d$-wave resonance associated with the $d$-wave dimer formation. The inset shows the same graph with a logarithmic scale for the $y$ axis. The units of $K_3$ are converted to ${\mathrm{cm}}^6/\mathrm{s}$ by using the van der Waals length $r_{\mathrm{vdW}}=101.0$ bohr and mass $132.905429$ amu of ${}^{133}$Cs. The solid vertical line indicates the value of the $s$-wave scattering length at the point where the $d$-wave dimer becomes bound. The two dashed vertical lines indicate the two values of $a$ at which recombination rate is enhanced, denoted A and B, respectively.

Figure 4

(a) $W_{\nu \nu }\left(R\right)$ as a function of hyperradius $R$ at the $K_3$ resonance peak A. (b) $W_{\nu \nu }\left(R\right)$ as a function of hyperradius $R$ around the $K_3$ enhancement peak B. The inset shows the three points to which the potential curves correspond.

Figure 5

Energy of the three-body bound state associated with a $d$-wave dimer as a function of scattering length $a$, both in van der Waals units.

Figure 6

The enhancements for the total three-body recombination rates at about $a=0.995r_{\mathrm{vdW}}$ for Lennard-Jones potential with two and three $s$-wave bound states. $K_3$ is convert to ${\mathrm{cm}}^6/\mathrm{s}$ by using van der Waals length $r_{\mathrm{vdW}}=101.0$ bohr and mass $132.905429$ amu of ${}^{133}$Cs.

Figure 7

The total three-body recombination rates as a function of $r_{\mathrm{vdW}}^5/a_2^5$. The units of $K_3$ are the same as Fig. 6.

Figure 8

Lennard-Jones potential with $\lambda \approx 0.503r_{\mathrm{vdW}}$ (black thicker curve) and $\lambda \approx 0.414r_{\mathrm{vdW}}$ (red thinner curve). The horizontal lines show the bound state spectrum with even angular momentum quantum number $l$. The line styles of solid, dashed, dash-dotted, dash-dot-dotted, and short-dashed indicate $l=0,2,4,6,8$, correspondingly.