Correct way to write the eigenvector in second quantization

I am studying diagonalization of a quadratic bosonic Hamiltonian of the type:



$$ H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j + \frac{1}{2}\displaystyle\sum_{<i,j>} [B_{ij} a_i^\dagger a_j^\dagger + B_{ij}^* a_j a_i ]

$$



in second quantization.



I can write this in matrix form as

$$ H = \frac{1}{2} \alpha^\dagger M \alpha - \frac{1}{2} tr(A)$$

where $$ \alpha = \begin{pmatrix} a \\ a^\dagger \\ \end{pmatrix} $$

, $$ \alpha^\dagger = \begin{pmatrix}

a & a^\dagger

\end{pmatrix} $$

and M is $$ M = \begin{pmatrix}

A & B\\

B^* & A^* \\

\end{pmatrix} $$



The Hamiltonian is called diagonal when it is expressed as:



$$ H = \beta^\dagger N \beta - \frac{1}{2} tr(A) $$

where $$ \beta = \begin{pmatrix}

b \\

b^\dagger \\

\end{pmatrix} $$

, $$ \beta^\dagger = \begin{pmatrix}

b & b^\dagger

\end{pmatrix} $$

and N is a 2-by-2 matrix .



Question: Can I numerically diagonalize the matrix M to get eigenvalues and eigenvectors of the Hamiltonian?

If yes, then what would be the right way to write the eigenvector in second quantization?



e.g. If for 2$\times$2 matrix M, one of the numerically calculated eigenvectors is $$\begin{pmatrix}

p\\

q \\

\end{pmatrix} $$ , then, should it be written as $$p \,a|0> + q \, a^\dagger|0> $$

(where the column $\alpha$ has been used as the basis)

or

$$p \,a^\dagger|0> + q \, a|0> $$

(where $\alpha^\dagger$ has been used as the basis)?



Note: where |0> is the vacuum state for 'a' type (bosonic) particles.

End of Question.



Note : Consider a simpler Hamiltonian

$$ H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j $$

and note that its eigenvectors are of the form

$$(a_1^\dagger a_2^\dagger ... ) |0> $$