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Edit : 2x2 simple system instead, simplification of the question.

I would like to study a system into its diagonal form, but this system is represented by a non-Hermitian Hamiltonian $\tilde{\mathcal{H}}$:

\begin{eqnarray} \tilde{\mathcal{H}} &=& \mathcal{H} +i \gamma \\ &=& \begin{bmatrix} \omega_0 & \Omega \\ \Omega & \omega_1 \end{bmatrix} + i\begin{bmatrix} \gamma_0 & 0 \\ 0 & \gamma_1 \end{bmatrix} \end{eqnarray} where $\mathcal{H}$ represents the unitary part of the Hamiltonian, $\gamma$ the dissipative part. Now, it seems that we have two options, leading to different results:

  • either we diagonalize the full Hamiltonian $\tilde{\mathcal{H}}$: \begin{eqnarray} \tilde{D} &=& \tilde{P}^{-1} \tilde{\mathcal{H}} \tilde{P} \\ &=& \begin{bmatrix} \omega_- + i\gamma_- & 0 \\ 0 & \omega_+ + i\gamma_+ \end{bmatrix} \end{eqnarray} such that the system is now in a fully diagonal form : $\Re(\tilde{D})$ will give the unitary evolution and $\Im(\tilde{D})$ the dissipative part.

  • OR, we only diagonalize $\mathcal{H}$ and we apply the rotation to the dissipative part : \begin{eqnarray} D &=& P^{-1} \mathcal{H}P = \begin{bmatrix} \omega_- & 0 \\ 0 & \omega_+ \end{bmatrix}\\ \gamma_P &=& P^{-1} \gamma P = \begin{bmatrix} \gamma_- & \gamma_\pm \\ \gamma_\mp & \gamma_+ \end{bmatrix} \end{eqnarray} Which leads to a dissipative matrix $\gamma_P$ which is not diagonal.

My questions is : which basis best represents the system? I am in the dark here, any intuitive help would be really appreciated.

Koyot
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  • How do you mean by "valid methods"? All basis changes are valid. What are you going to do and conclude with either? Are you used to the language ? – Cosmas Zachos Feb 10 '21 at 15:44
  • The two basis changes leads to a different qualitative understanding of the problem : either I will look at eigenstates that makes both the unitary and dissipative matrices diagonal, or I will look at eigenstates which are coupled through the non-diagonal terms of $\gamma$ and $\Gamma$. And it troubles me because I don't know which representation I should adopt. – Koyot Feb 10 '21 at 15:50
  • You have not described your application. Have you tried a simplified 2x2 prototype example? – Cosmas Zachos Feb 10 '21 at 16:59
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    Thank you for your suggestions @CosmasZachos. I simplified the question, and put a 2x2 example. About the application : in the end, I want to use the input-output formalism to study the evolution of the system. For example, study the time evolution of the populations of the eigen states (considering an initial distribution of the population). – Koyot Feb 11 '21 at 09:02
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    @Koyot I would put the part about the input-output formalism in the question if I were. In the current version, I would go with the first diagonalization (it's fully diagonal and that is nice). However, in the input-output case, a different basis fixed by the outcoupling matrix could be more useful. Depending on the situation, this could also coincide with the first diagonalization. For example for scattering, this is often the case. Also, I recommend this review for background: https://www.nature.com/articles/s41563-019-0304-9 – Wolpertinger Feb 11 '21 at 15:47

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