Two-site Bose-Hubbard hopping and Schr\"odinger cat states

The Bose-Hubbard Hamiltonian can be simplified to have only two lattice sites, in which case the system being described is referred to as a dimer. Due to its structure, the hopping term of the dimer Hamiltonian enjoys invariance in a family of subspaces indexed by a whole number $k$, each subspace corresponding to a system of only $k$ particles. We have invented an inductive argument using the bosonic canonical commutation relations to find the eigenvalues and eigenvectors of the dimer hopping Hamiltonian in its $k$-particle subspaces. In particular, this Hamiltonian, when restricted to one of the $k$-particle subspaces, is exactly the spin projection operator along the $x$-axis, where the number of particles $k$ in the dimer system yields the projection matrix for spin quantum number $s=k/2$. Thus, a new method for computing the eigenvalues and eigenvectors of the $x$-axis spin projector has been unearthed. We use the explicit construction to study the dynamics of coherent states induced by the square of the dimer hopping hamiltonian. We find that it generates Schr\"{o}dinger cat states in the two-site setting.