(9636) Bose-Hubbard model - energy of SF/ST/MI states

(1) Write the Bose-Hubbard Hamiltonian for Bosons in \(L\) site ring using field operators \( a_j \). The on-site interaction is \( U \), the hopping frequency between sites is \( J \), and if the ring is placed in a rotating frame there is also magnetic-like Coriolis force that has flux \( \Phi \).

(2) Write the Hamiltonian using field operators \(b_k \), where \( k \) labels momentum orbitals.

(3) Use creation operators to define self-trapped state (ST) that is formed by condensing \(N\) particles into a single site.

(4) Use creation operators to define Mott insulator state (MI) that is formed placing \( \bar{n}=N/L \) particles in each site. Assume that \( \bar{n} \) is integer.

(5) Use creation operators to define superfluid state (SF) that is formed by condensing \( N \) particles into the zero momentum orbital.

(6) Calculate the energy \( E=\langle H \rangle \) for each of the above states (SF, ST, MI).