# Is possible to obtain a master equation from Rabi's coupled differential equation?

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?

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"