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I am trying to diagonalize the Hamiltonian in equation (2) in the paper "Nonlinear Interaction Effects in a Strongly Driven Optomechanical Cavity", by the means of a Bogoliubov transformation, the Hamiltonian is of the form:

$$H_1 = -\Delta d^\dagger d + w_M b^\dagger b + G(d + d^\dagger)(b + b^\dagger)$$

The operators $b$ and $d$ are both bosonic operators. Using the Bogoliubov transformation, I am meeting a dead end where the coefficients $u$ & $v$ in the new defined bosonic operators (e.g. $d = u\alpha^\dagger + v\beta$) have to be set to zero when trying to diagonalize the transformed Hamiltonian. I am wondering whether the Bogoliubov transformation is the one to use for this problem.

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This problem is solvable with the Bogoliubov transformation of a more general form. To diagonalize the hamiltonian, you need to find a solution to the following "eigenproblem": $$ [A, H_1] = \lambda A, $$ where $$ A = u_1 d + u_2 b + v_1 d^\dagger + v_2 b^\dagger. $$ A solution to this problem will allow you to construct creation-annihilation operators of new quasiparticles.

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