Two-site Bose-Hubbard model ($H=-t(a_L^{†}{a}_R+a_R^{†}{a}_L)+\frac{U}{2}{\sum }_{j=L,R}{n}_j(n_j-1)$)

Two localized orbitals coupled by quantum tunnelling and on-site interactions give the smallest Bose-Hubbard model. A symmetric double-well potential with lowest one-particle energies $E_0,E_1$ has $t=(E_1-E_0)/2$ and common diagonal energy $(E_0+E_1)/2$. At fixed total particle number, that common diagonal term is constant. A fixed-number two-mode condensate gives a simple variational method for this model, while stronger interactions can require occupation-correlated trial quantum states.