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As I understand from the official QuTiP guidlines, it is only capable of solving "cross-damping off" Master Equation in form:

$\dot{\rho(t)} = -\dfrac{i}{\hbar}[H(t),\rho(t)] + \sum\limits_n\dfrac{1}{2}\left[ C_n\rho(t)C_n^{\dagger} - \rho(t)C_n^{\dagger}C_n - C_n^{\dagger}C_n\rho(t) \right].$

Where $C_n$ are collapse operators. However when we need to consider cross-damping, the ME takes the form like:

$\dot{\rho(t)} = -\dfrac{i}{\hbar}[H(t),\rho(t)] + \sum\limits_{i,j}\dfrac{1}{2}\left[ C_i\rho(t)C_j^{\dagger} - \rho(t)C_i^{\dagger}C_j - C_i^{\dagger}C_j\rho(t) \right].$

Is QuTiP capable of solving this? And how one can manage to code this if it is?

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In QuTiP, one can construct generalized dissipators with the lindblad_dissipator function listed under "Superoperators and Liouvillians" here. You can pass in separate operators to act from the left and right:
\begin{align} D[a,b] = a\rho b^\dagger - \frac{1}{2}a^\dagger b \rho - \frac{1}{2} \rho a b^\dagger \end{align} Then, in the master equation solver mesolve, if you pass in such dissipators in the collapse operator argument, QuTiP will detect superoperator contributions and add them directly to the total Liouvillian.

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