Three-body problem for ultracold atoms in quasi-one-dimensional traps

We study the three-body problem for both fermionic and bosonic cold-atom gases in a parabolic transverse trap of length scale $a_{\perp }$. For this quasi-one-dimensional (quasi-1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length $a$ and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths $a_{\mathit{ad}}$ and $b_{\mathit{ad}}$. In the tightly bound “dimer limit” $a_{\perp }∕a\to \infty$, we find $b_{\mathit{ad}}=0$ and $a_{\mathit{ad}}$ is linked to the 3D atom-dimer scattering length. In the weakly bound “BCS limit” $a_{\perp }∕a\to -\infty$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal: $a_{\mathit{ad}}$ and $b_{\mathit{ad}}$ depend both on $a_{\perp }∕a$ and on a parameter $R^{*}$ related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.

Figure 1

Scattering length $a_{\mathit{ad}}∕a_{\perp }$ versus dimensionless binding energy ${\Omega }_B$. The solid curve is the numerical solution to Eq. (3.11), and the dotted (dashed) curves represent the analytical results in the dimer (BCS) limit, respectively. In the inset, $b_{\mathit{ad}}∕a_{\perp }$ is plotted versus ${\Omega }_B$. The solid curve gives $b_{\mathit{ad}}$ from the numerical solution of Eq. (3.11), and the dashed line represents $a_{\mathit{ad}}$ for comparison.

Figure 2

Solutions $w_0\left(r\right)$ to Eq. (4.8) and $w_1\left(r\right)$ to Eq. (4.11).

Figure 3

Solution $w_1\left(0\right)$ (solid line) to Eq. (5.19) as a function of $R^{*}∕a$. The dotted curve gives the asymptotic behavior $w_1\left(0\right)=(8∕9)\sqrt{a∕R^{*}}$ for large $R^{*}∕a$.

Figure 4

Plot of $a_{\mathit{ad}}$ and $b_{\mathit{ad}}$ (solid lines) as function of the dimensionless binding energy ${\Omega }_B$ for the three-boson problem at $R^{*}=0$. The asymptotic behavior is shown (i) for $b_{\mathit{ad}}$ in the BCS limit, $-1∕\sqrt{{\Omega }_B}$, and (ii) for $a_{\mathit{ad}}$ in the dimer limit, $(9∕64)\sqrt{{\Omega }_B}+0.6638∕\sqrt{{\Omega }_B}$.

Figure 5

Parameter $u_0$ appearing in the trimer binding energy, Eq. (5.24), as a function of ${\Omega }_B$ for various values of $R^{*}∕a_{\perp }$. The dotted lines correspond to the asymptotic behavior in the dimer limit, Eq. (5.27).

Figure 6

Reduced energy $ϵ\equiv (E_T∕\hslash {\omega }_{\perp }){(a_{\perp }∕a)}^2$ for the CIT in the dimer limit as a function of $R^{*}∕a$.