(9753) Bosonic bath mode

Consider the following single-mode Hamiltonian for bosons:

\( \mathcal{H} = \omega b^{\dagger} b + f (b^{\dagger}+b) \).
The mode occupation operator is \( n = b^{\dagger} b \).

(1) Define new bosonic operators \( c \) and \( c^{\dagger} \) to get rid of the linear term in \( \mathcal{H} \).

(2) What are the eigen-energies of the Hamiltonian?

(3) Rewrite the Hamiltonian and the transformation using canonical \( (q, p) \) and \( (Q, P) \) coordinates.

(4) The ground state \( |\nu=0 \rangle \) of the Hamiltonian is known as "coherent state". Write explicit expression for it in the old basis \( | n \rangle \). Tip: use translation operator.

(5) Write expression for the excited states \( |\nu=1,2,3,\cdots \rangle \).