Bose-Einstein Condensation of Efimovian Triples in the Unitary Bose Gas

In an atomic Bose-Einstein condensate quenched to the unitary regime, we predict the sequential formation of a significant fraction of condensed pairs and triples. At short distances, we demonstrate the two-body and Efimovian character of the condensed pairs and triples, respectively. As the system evolves, their size becomes comparable to the interparticle distance, such that many-body effects become significant. The structure of the condensed triples depends on the size of Efimov states compared with density scales. Unexpectedly, we find universal condensed triples in the limit where these scales are well separated. Our findings provide a new framework for understanding dynamics in the unitary regime as the Bose-Einstein condensation of few-body composites.

Figure 1

Atomic (blue), pair (orange), and triple (green) condensate fraction dynamics in the (a) trimer, (b) crossover, and (c) universal density regimes. The gray dot-dashed lines indicate universal results in the doublet (Hartree-Fock Bogoliubov [80]) model. The comparison of Efimovian (${\kappa }_{*}$) and Fermi ($k_n$) scales strongly determines the relative populations of the condensed few-body composites.

Figure 2

Macroscopic (a) pair and (b) triple order parameter dynamics over a range of densities within the triplet model. Black solid line: universal results within the doublet model. Inset: residual exponents ${\mathrm{\Psi }}_0^{(p)}\propto {\mathrm{\Lambda }}^{{\gamma }_p}$ evaluated at fixed $t=0.05t_n$ converge as expected to 0 as the system becomes increasingly dilute with respect to the range of the interaction ($\mathrm{\Lambda }/k_n\to \infty$).

Figure 3

Internal structures of the normalized (a) pair and (b) triple condensate wave functions as the system evolves for densities within the crossover (orange) and universal (light blue) regimes. The filled circles in (a) indicate the three-body parameter. Insets: normalized pair and triple condensate wave functions in the universal regime at $t=t_n$ compared with the vacuum ground-state Efimov trimer $|{\mathrm{\Psi }}_{3b}^{(0)}(K)|$ (black-dotted) for density $k_n/{\kappa }_{*}=0.35$.

Figure 4

Evolution of the overlap [Eq. (17)] between ground-state Efimov trimer and triple condensate wave function for varying densities, with $P_{3b}^{(0)}(t=0)$ undefined. The numerical upper limit is comparable to the total probability ($\sim 82\%$) projected onto the Efimov channel in the quenched three-body problem with characteristic time scale $t_{3b}^{(0)}=m/\hslash {\kappa }_{*}^2$ (filled circles) [85]. The illustrations indicate the presence or absence of a trimer character in the triple condensate.