(9750) Bose-Hubbard model - excitations

(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 \( -J \).

(2) Write the Hamiltonian using field operators \(b_k \), where \( k \) labels momentum orbitals. In the following assume that only the 3 lowest orbitals are likely to be populated.

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

(4) Calculate the energy \( E_0=\langle H \rangle_0 \) of the SF state. We regard this as a good approximation for the ground state. Below we would like to obtain an expression for the energies \( E_{\nu,\pm} \) of the excited states, where \( \nu=1,2,\cdots \).

(5) Write a Bogolyubov approximation for the Hamiltonian, considering only the 3 lowest orbitals. Provide explicit expressions for the parameters \( (\varepsilon, \Delta, \text{Const}) \) that appear in

\( \mathcal{H} \ \ \approx \ \ \varepsilon b^{\dagger} b + \varepsilon \tilde{b}^{\dagger} \tilde{b} + \Delta (b^{\dagger} \tilde{b}^{\dagger} + b \tilde{b}) + \text{Const} \)

(6) Write the \( 4 \times 4 \) Hessian matrix \( H \) for the Bogolyubov approximation, and perform symplectic diagonalization to get expressions for the energies \( E_{\nu,\pm} \) of the excited states.

All answers should be expressed using \( (J,U,L,N) \).