I'm reading Environment-assisted quantum transport by Rebentrost at all, where they deal with a system of 2 sites that hosts 1 excitation. They describe this in terms of two states: $\vert 1 \rangle$ denotes an excitation at site 1, $\vert 2 \rangle$ denotes an excitation at site 2. They are coupled, given by the Hamiltonian
\begin{equation} H = \frac{\Delta}{2} \left(\vert1\rangle\langle1\vert\ - \vert2\rangle\langle2\vert\right) + \frac{V}{2}\left( \vert1\rangle\langle2\vert\ + \vert2\rangle\langle1\vert\ \right) \end{equation} where $\Delta$ is the energy mismatch between the sites and $V$ is the coupling strength.
The next ingredient is that both sites couple to an uncorrelated (between the sites) white noise bath, leading to a pure dephasing $\gamma_\phi$ for both states. They write that the master equation is then given by \begin{equation} \dot{\rho} = \gamma_\phi\sum_m\left(A_m \rho A_m^\dagger - \frac{1}{2} A_m A_m^\dagger \rho - \frac{1}{2} \rho A_m A_m^\dagger \right) \end{equation} with $A_m = \vert m \rangle \langle m \vert$. I don't see why this is true. Isn't pure dephasing normally with $\sigma_z$, which here would be $\vert1\rangle\langle1\vert\ - \vert2\rangle\langle2\vert$? The above would miss the minus sign, right?
Assuming I am just not reading the above correctly, this system has a Lindblad term $\sqrt{\gamma_\phi} \left(\vert1\rangle\langle1\vert\ - \vert2\rangle\langle2\vert \right)$. Would it then be the case that, starting from an excitation at site 1, the dephasing leads to population at site 2? I suppose so, because the Hamiltonian is not diagonal; we're not along the z-axis of the Bloch sphere and the dephasing thus leads to rotations between states 1 and 2. Is this correct, and can we quantify it?
What I am then also interested in is a slight change to the above; I figured it is not worth a new question as I think the analysis is very similar. What changes in this picture if only site 1 is coupled to a bath, so that $\gamma_{\phi 2}$ is zero? Would we still end up at site 2, starting from site 1?