Dynamics and symmetries of a repulsively bound atom pair in an infinite optical lattice

We investigate the dynamics of two bosons trapped in an infinite one-dimensional optical lattice potential within the framework of the Bose-Hubbard model and derive an exact expression for the wavefunction at finite time. As initial condition we chose localized atoms that are separated by a distance of $d$ lattice sites and carry a center of mass quasi-momentum. An initially localized pair ($d=0$) is found to be more stable as quantified by the pair probability (probability to find two atoms at the same lattice site) when the interaction and/or the center of mass quasi-momentum is increased. For initially separated atoms ($d \neq 0$) there exists an optimal interaction strength for pair formation. Simple expressions for the wavefunction, the pair probability and the optimal interaction strength for pair formation are computed in the limit of infinite time. Whereas the time-dependent wavefunction differs for values of the interaction strength that differ only by the sign, important observables like the density and the pair probability do not. With a symmetry analysis this behavior is shown to extend to the $N$-particle level and to fermionic systems. Our results provide a complementary understanding of the recently observed [Winkler \textit{et al.}, Nature (London) \textbf{441}, 853 (2006)] dynamical stability of atom pairs in a repulsively interacting lattice gas.