Statistical properties of the Hamiltonian generating phase state derived by using the generalized Hellmann-Feynman theorem

We study statistical properties of the Hamiltonian (H=ωa†a+κa†N+1+κN+1a) generating phase state. Using the generalized Hellmann-Feynman theorem for ensemble average, we derive its mean energy and find the ratio of the mean energies contributed from the term a†a to that from a†N+1+κN+1a. The relation on the entropy-variation with respect to the dynamic parameters ω and κ is also examined.