Dimer of two bosons in a one-dimensional optical lattice

We investigate theoretically the stationary states of two bosons in a one-dimensional optical lattice within the Bose-Hubbard model. Starting from a finite lattice with periodic boundary conditions, we effect a partial separation of the center-of-mass and relative motions of the two-atom lattice dimer in the lattice momentum representation, and carefully analyze the eigenstates of the relative motion. In the limit when the lattice becomes infinitely long, we find closed-form analytic expressions for both the bound state and the dissociated states of the lattice dimer. We outline the corresponding analysis in the position representation. The results are used to discuss three ways to detect the dimer: by measuring the momentum distribution of the atoms, by finding the size of the molecule with measurements of atom number correlations at two lattice sites, and by dissociating a bound state of the lattice dimer with a modulation of the lattice depth.

Figure 1

The function $f(\omega ,L)$ [Eq. (18)] for $L=16$. The horizontal line represents the right-hand side of Eq. (18) for $\mathcal{K}=-\frac{1}{2}$. The dashed vertical lines are the asymptotes of $f(\omega ,L)$ at the values of $\omega =-\cos q$ such that $f(\omega ,L)=\pm \infty$.

Figure 2

The function $h(q,L)$ [Eq. (69)] for $L=16$. The vertical dashed lines represent the asymptotes of $h(q,L)$; the horizontal line stands for $\mathcal{K}=\frac{1}{2}$.

Figure 3

Contour plot of the momentum distribution in the bound state of the lattice dimer, $f(p,P)$ of Eq. (85), as a function of the momenta of the individual atoms $p$ and the center-of-mass momentum $P$, for $U/J=16$. Brighter shades represent larger values.