# Solving density matrix in a two-level atom

I'm working through some parts of Stephen C. Rand's Lectures on Light: Nonlinear and Quantum Optics, specifically sections in which he works with density matrices. In several places he makes statements about off-diagonal density matrix elements, such as the following:

Off-diagonal elements of the density matrix describe charge oscillations initiated by applied fields. Hence, the full temporal evolution of their amplitudes reflects transient buildup (or decay) of an oscillation prior to the establishment of any steady-state amplitude. The frequency of the oscillation is $$\omega$$, as determined by the driving field. Hence the general form of the solution is

$$\rho_{ij} = \tilde{\rho}_{ij} e^{i\omega t},$$

where $$i,j$$ specify the initial and final states of the transition closest in frequency to $$\omega$$ and $$\tilde{\rho}_{ij}$$ is the slowly varying amplitude of the polarization.

[Emphasis mine.] What justifies writing this down in general? Why should the oscillations of these off-diagonal matrix elements (for which I have little physical intuition) oscillate at the same frequency as the driving field?

The particular example I have in mind is Rabi flopping in a two-level atom, which I attempted (but failed) to derive with density matrices. The populations oscillate with a frequency related to the Rabi frequency $$\Omega \neq \omega$$. What if the off-diagonal elements of the density matrix do, too? How would one know?