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If I have a hamiltonian like $\omega\sigma_z$ and 2 Lindblad operators as $\gamma\sigma_-$ and $\gamma\sigma_+$ how can I find eigenvalues of generators?

I think I should put the general form of $\rho$ and these parameters in the Lindblad equation and find $\dot{\rho}$ and then write it like

$$\dot{\rho}=L[\rho]$$

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If I understood correctly, you are interested in the Liouvillian superoperator $L$, given by: $$ L[\rho]=-i[\omega\sigma_z,\rho]+\gamma\left(\sigma_-\rho\sigma_+-\frac{1}{2}\{\sigma_+\sigma_-,\rho\}+\sigma_+\rho\sigma_--\frac{1}{2}\{\sigma_-\sigma_+,\rho\}\right). $$

For finite-dimensional systems, finding its eigenvalues is quite straightforward. A brute-force, still quite immediate method consists in writing the $2\times 2$ density matrix of the qubit as 4-dimensional vector: if the density matrix reads $$ \begin{pmatrix} \rho_{00}&\rhọ_{01}\\ \rho_{10}&\rhọ_{11}\\ \end{pmatrix}, $$
construct the column vector $|\!\,|\rho\rangle =(\rho_{00},\rho_{01},\rho_{10},\rho_{11})^T$. Then, write the Liouvillian $L$ as a $4\times 4$ matrix acting on the vectorized density matrix $|\!\,|\rho\rangle$ (have a look at this reference for further details, or check this previous question). You can quickly verify that: $$ L=\begin{pmatrix} -\gamma&0&0&\gamma\\ 0&-\gamma+2i\omega&0&0\\ 0&0&-\gamma-2i\omega&0\\ \gamma&0&0&-\gamma\\ \end{pmatrix}. $$

We finally find the eigenvalues: $$ \lambda_1=0,\quad \lambda_2= -\gamma-2i \omega,\quad\lambda_3=-\gamma+2i\omega,\quad\lambda_4=-2\gamma. $$ As we may expect, they are the eigenvalues describing the emission and absorption of a two-level atom in an infinite-temperature thermal bath. For the very simple case of a single qubit that we are considering, they can be also derived through the standard Bloch equations [1].

[1] Breuer & Petruccione, The theory of open quantum systems, Oxford University Press (2002).

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  • $\begingroup$ Can you write more about computing L.if we say that $L\rho= \lambda \rho$ is it possible to write all eigenvalues? $\endgroup$
    – HohO
    Commented Oct 7, 2020 at 20:39
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    $\begingroup$ To compute $L$ either you follow the method described in the links I have written, or you perform standard brute-force representation of an operator as a matrix: you apply $L$ on the vectorized basis $(1,0,0,0)^T$, $(0,1,0,0)^T$, etc. As for the second part of your question, I don't really understand what you are asking: if you write $L\rho=\lambda\rho$ you are saying that $\rho$ is an eigenvector with given eigenvalue $\lambda$. What about all the eigenvalues? $\endgroup$ Commented Oct 8, 2020 at 13:12

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