Why is the density matrix of a system has this block form? In Ficek's paper (http://zon8.physd.amu.edu.pl/~tanas/spis_pub/pdf/04-joptb-S90.pdf), the density matrix of a two two-level atom system has a
block form like this. Why does it make sense to assume this ?

 A: The basis from the paper is
\begin{align}
\vert 1 \rangle = \vert g_1 \rangle \otimes \vert g_2 \rangle,  \\
\vert 2 \rangle = \vert e_1 \rangle \otimes \vert e_2 \rangle,  \\
\vert 3 \rangle = \vert g_1 \rangle \otimes \vert e_2 \rangle,  \\
\vert 4 \rangle = \vert e_1 \rangle \otimes \vert g_2 \rangle.
\end{align}
The statement in the paper is that if the system is in the block form that you give to begin with, it will always remain in this block form. In particular, there is coherence only between the states for which both qubits are flipped. This implies that the unitary  dynamics conserve the total qubit "parity"
\begin{equation}
P = \textrm{sign}\ S^z_1 S^z_2.
\end{equation}
The upper block has $P = 1$ while the lower block has $P = -1$. Indeed, the unitary part of the master equation (23) respects this symmetry. The non-unitary dynamics only move weight between the blocks, but do not induce coherence. You can easily check this by considering the action of the jump term $S^-_i \vert a \rangle \langle b \vert S^+_j$ term by term, i.e. for all $a,b = 1...4$ and $i,j = 1,2$.
