(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 \( n = b^{\dagger} b \).
The standard basis is \( | n \rangle, \ \ n=0,1,2,... \)
The eigenstates are \( | \nu \rangle, \ \ \nu=0,1,2,... \)
Each eigenstate can be written as \( |\nu \rangle = \sum_n \Psi_n |n \rangle \)

(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 \( E_{\nu} \) 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".
Define a translation operator that relates it to \( |n=0 \rangle \),
and derive from it an explicit expression for the expansion coefficients \( \Psi_n \).

(5) Find expression for the expansion coefficients \( \Psi_n \) for general \( | \nu \rangle \).
Tip: The expression can be written as a sum \( \sum_m Q_m \).
Write what are the \( Q_m \) and what is the range of the \(m\)-summation.