Universal ($1+2$)-body bound states in planar atomic waveguides

Shallow heteronuclear trimers are predicted for mixtures of two atomic species strongly trapped in a quasi-two-dimensional (2D) atomic waveguide. The binding energies are functions of the 2D scattering length and of the mass ratio and can be thus tuned by various ways. These universal trimers are composed of two identical noninteracting particles and of a third particle of the other species. Depending on the statistics of the two identical particles, the trimers have an odd (fermions) or even (bosons) internal angular momentum. These results permit one to draw conclusions on the stability issue for the quasi-2D gaseous phase of heteronuclear dimers.

Figure 1

Trimers’ spectrum for two identical bosons of mass $M$ interacting with a third atom of mass $m_i$. The binding energies $E_{\mathrm{trim}}$ are expressed in units of the heteronuclear dimer’s energy $E_{\dim }$ in Eq. (14), as a function of the mass ratio $x=M/m_i$. One can notice that away from threshold, the bound state energy of an $m$ trimer is quasidegenerate with the energy of a trimer with a lower $m$.

Figure 2

Same as Fig. 1 for two identical fermions of mass $M$ interacting with a third atom of mass $m_i$.

Figure 3

Plot of the probability in Eq. (29) $P(k<Q)$ at $Q=q_{\dim }$ as a function of the mass ratio $x$ for the different trimers composed of bosonic atoms of mass $M$ with the binding energies of Fig. 1. Continuous lines, $m=0$ states; dotted lines, $m=2$ states; dashed lines, $m=4$ states.

Figure 4

Same as Fig. 3 for fermionic atoms of mass $M$ with binding energies of Fig. 2. Continuous lines, $m=1$ states; dotted lines, $m=3$ states; dashed lines, $m=5$ states.

Figure 5

Plot of the probability in Eq. (29) $P(k<Q)$ as a function of $1/Q$ in units of $1/q_{\dim }$ for a $m=1$ trimer at threshold ($x=153.9$ and $q-q_{\dim }\ll q_{\dim }$).