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Starting from a differential equation for $c_i$ such that $c_i c_i^*$ is the probability of being at state $i$, I want to obtain a master equation for $c_i c_i^*\equiv p_i$.

Consider a two-state problem with a sinusoidal oscillating potential. Hamiltonian $$ H_0 = E_1 \vert 1\rangle\langle 1\vert+E_2 \vert 2\rangle\langle 2\vert, $$and a time-dependent potential $$ V(t) = \gamma e^{i\omega t}\vert 1\rangle\langle 2\vert + \gamma e^{-i\omega t}\vert 2\rangle\langle 1\vert. $$ Now, if $c_1$ and $c_2$ are the probabilities of being at state $1$ or $2$, respectively. Standard perturbation theory give us the following differential equation \begin{align*} i \hbar \dot{c}_1 &= \gamma e^{i\omega t}e^{i\omega_{12}t}c_2,\\ i \hbar \dot{c}_2 &= \gamma e^{-i\omega t}e^{-i\omega_{12}t}c_1. \end{align*}

I was trying to obtain a master equation for $p_i$, that is, an expression of the form

$$ \dot{p}_n = \sum_n W_{nn'}p_{n'}-W_{n'n}p_n, $$

for $p_i = c_i c_i^*$. Unfortunately, at this moment I am only able to get this $$ i\hbar \dot{p}_2 = \gamma(e^{-it(\omega+\omega_{12})}c_1 c_2^*-e^{it(\omega+\omega_{12})c_1^* c_2}), $$ not particularly useful.

So, is there any way to obtain $W_{nn'}$ from the given information, I need to keep doing c-numbers algebra?

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The classical master equation you write down applies to classical probabilities. To derive something like it from your quantum system, you need to introduce decoherence effects that wipe out the off-diagonal elements of the density matrix.

If you want to keep quantum effects, you need the Lindblad equation or a "quantum master equation" that governs the evolution of all the entries in the density matrix, and not just the diagonal elements $p_n$. There is some discussion of this on the wipedia page "master equation"

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