# Eigenvalues of generators [closed]

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

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

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).

• Can you write more about computing L.if we say that $L\rho= \lambda \rho$ is it possible to write all eigenvalues?
– HohO
Commented Oct 7, 2020 at 20:39
• 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? Commented Oct 8, 2020 at 13:12