# How does "coherence" contribute to inverting a two-level system?

In laser physics I am told that inverting a two-level system is not possible, since it will become transparent once the inversion reaches 50% and no longer be able to absorb more photons. This makes it impossible to create a laser using two-level systems.

However, a two-level system can be described by the equation for the microscopic polarization $p$ and the density of the the upper state $n$ via: $$\frac{\partial p}{\partial t} = -i\omega p + \frac{i}{\hbar}\vec{\mu}\cdot \vec{E}\left(t\right) \left(1-2n\right)- \frac{p}{T_p}$$ $$\frac{\partial n}{\partial t} = 2 \frac{\vec{\mu}\cdot \vec{E}\left(t\right)}{\hbar}\Im\left(p\right) -\frac{n}{T_n}$$ If the applied field $\vec{E}\left(t\right)$ is strong enough for a long enough period of time, this will cause the density $n$ to reach $1$. This means that the system is fully inverted. After that the density will again decrease to $0$ and up again. This is called a Rabi cycle.

I cannot quite understand why this works. What does the two-level system lack, that causes this to happen?

Furthermore, I am told that it is crucial that the excitation of the system has to be "coherent" in order for the inversion to occur. I cannot understand what "coherent" means in this context. What would happen if the excitation was incoherent? What would $\vec{E}\left(t\right)$ look like?