Solution of dynamics of density matrix Given the dynamics of the density matrix:
$
\frac{d}{d t}\begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} = \begin{pmatrix} \lambda i(\rho_{10}-\rho_{01})+\lambda^2\rho_{11} & \lambda i(\rho_{11} -\rho_{00})+\lambda^2 \rho_{01} \\
  \lambda i(\rho_{00}-\rho_{11}) +\lambda^2 \rho_{10} & 
  \lambda i(\rho_{01}-\rho_{10}) +\lambda^2 \rho_{11} \end{pmatrix}
$
How can this system of differential equations be solved, since they refer to each other. With initial condition $\rho_{ij}\in \mathbb{R}$.
 A: You don't have a mistake in a11 element in the right matrix, do you ?
Might be that I didn't get your question. But what the problem to place your initial condition into matrix ? One of the ways is expressing second order derivative over time through the first one as well you can introduce a new variable that might be
$\rho_{01}-\rho_{10}$ and $\rho_{11}-\rho_{00}$.
It looks like description of 2 level atomic system, try to find books with "resonances in atom" or "nonlinear resonances in atoms" tag.
A: As Lelesquiz points out, that looks like a somewhat standard matrix differential equation to me. The Wikipedia link does give a solution method for the matrix, but I think that it might be easier to do some remapping:
\begin{align}
\rho_{00}&\to v_1\\
\rho_{01}&\to v_2\\
\rho_{10}&\to v_3\\
\rho_{11}&\to v_4
\end{align}
and write it as (assuming what you wrote is correct and that I did the right mapping, you should double check this)
$$
\frac{d}{dt}\left(\begin{array}{c}v_1\\v_2\\v_3\\v_4\end{array}\right)=\left(\begin{array}{c}\lambda i(v_3-v_2)+\lambda^2v_1 \\ \lambda i(v_4-v_1)+\lambda^2v_2 \\ \lambda i(v_1-v_4)+\lambda^2 v_3 \\ \lambda i(v_2-v_3)+\lambda^2v_4\end{array}\right)
$$
which makes it more clear that this can be solved numerically with Runge-Kutta methods because it is a simple vector with coupled components. You may want to note that, given the complex term, stability is going to be an issue.
