(9630) Bose Hubbard for two sites - spin language

Consider the Hamiltonian of a 2-site Bose-Hubbard model, where the on-site interaction is \( U \), the hopping frequency between the sites is \( K \), and the potential difference between the sites is \( V \).

(1) Write the Hamiltonian using \( a_j, a_j^{\dagger} \) coordinates, where \(j=1,2\) is the site index, and derive the equations of motion. Clarify what is the solution for \( a_j(t) \) if \( K=0 \).

(2) Write the Hamiltonian using \( S_{x,y,z} \) coordinates, and derive the so-called Bloch equations of motion. Clarify what is the solution if \( K=0 \), and what is the solution if \( U=0 \). Find the critical value of \( U \) above which a separatrix appears.

(3) Write the Hamiltonian using canonical \( (n,\varphi) \) coordinates, and derive the associated equations of motion. Clarify the conditions for getting the equations of a mathematical pendulum. In the latter case clarify what is the ground state energy \( E_{0} \) and what is the separatrix energy \( E_{x} \).