Three-body bound states in a lattice

We pursue three-body bound states in a one-dimensional tight-binding lattice described by the Bose-Hubbard model with strong on-site interaction. Apart from the simple strongly bound “trimer” state corresponding to all three particles occupying the same lattice site, we find two novel kinds of weakly bound trimers with energies below and above the continuum of scattering states of a single-particle (“monomer”) and a bound-particle pair (“dimer”). The corresponding binding mechanism can be inferred from an effective Hamiltonian in the strong-coupling regime which contains an exchange interaction between the monomer and the dimer. In the limit of very strong on-site interaction, the exchange-bound trimers attain a universal value of the binding energy. These phenomena can be observed with cold atoms in optical lattices.

Figure 1

(a) Full three-particle energy spectrum of Hamiltonian (1) with $U=-10J$, versus the total quasimomentum $K$. All bound states are obtained via exact numerical solution of Eq. (4). (a’) Magnified part of the spectrum corresponding to dimer-monomer states. (b) Binding energies $E_B$ for the off-site (weakly bound) trimers at $K=0$ versus the interaction strength $U<0$. (c) Dimer-monomer spectrum of the effective Hamiltonian (5). The two bound states are obtained via numerical solution of Eq. (6).

Figure 2

Top panel: Quasimomentum distributions $|\psi (k_1,k_2,K-k_1-k_2)|{}^2$ of the three-particle bound states for $U=-10J$ and $K=0$; (a1) and (a2) correspond to the off-site (weakly bound) trimers with energies below and above the two body dimer-monomer continuum, while (b) corresponds to the on-site (strongly-bound) trimer. Lower panel: Reduced single-particle momentum distributions ${\rho }_0(k)$ for the cases of (a1), (a2), and (b).