(9751) Bogolyubov transformation for Bosons - one or two sites

Consider the one site Hamiltonian with squeezing term:
\( \mathcal{H} = \varepsilon a^{\dagger} a + \frac{\Delta}{2} (a^{\dagger} a^{\dagger} + a a) \)

(1) Write semiclassical equations of motion for \( z = (a,\bar{a}) \) using matrix notations.
(2) Write expression for the Bogolyubov frequencies.
(3) Find the Bogolyubov transformation that diagonalize the matrix.

Guidance:
See Lectures notes [51.6] and [37.2].
Use the convention \( c^{\dagger} = u a^{\dagger} +v a \).
Write the transformation in matrix notations as \( \tilde{z} = S z \).
Express the results of the diagonalization in terms of \( \theta \) defined via \( \tanh(\theta)=\Delta/\varepsilon \).
Using trigonometric identities derive explicit expression for \(|u|^2\) and \(|v|^2\).

Consider the two site Hamiltonian with pairing interaction:
\( \mathcal{H} = (\varepsilon_1 a_1^{\dagger} a_1 + \varepsilon_2 a_2^{\dagger} a_2) + \Delta (a_1^{\dagger} a_2^{\dagger} + a_1 a_2) \)

(4) Use the same procedure to diagonalize the above two-site Hamiltonian.
Note that the \( 4 \times 4 \) Hessian matrix separates into two \( 2 \times 2 \) blocks.