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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?

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Your first question asks why you cannot pump a two-level system to create a perfect inversion (100% population in the excited state) whereas a Rabi oscillation can bring all of the state amplitude to the excited state. Excited states separated by optical transitions have finite lifetimes (of order 10 MHz for alkali atoms). Spontaneous emission will damp out the Rabi oscillations.

If you solve for the steady state of the two-level system, you find two regimes. In the weak pump regime, the relationship between excitation and perturbation is linear, like a harmonic oscillator. In the strong pump situation, the relationship is not linear anymore, and you saturate the transition. You will find that you need infinite pump power to have equal population in the ground and the excited state.

In your second question, you say "it is crucial that the excitation of the system has to be 'coherent' in order for the inversion to occur". I do not understand what you mean by this, unless you are trying to say that the excitation needs to be coherent for a perfect Rabi oscillation. A pump source for laser gain medium does not need to be coherent.

A coherent light source roughly means that the phase of the excitation (the light) at any time is well-defined. You need a coherent excitation for Rabi oscillation because phase noise in the excitation will disturb the coherence between the ground and the excited state. If you are familiar with the notion of Bloch sphere, this should become clear, as phase noise disturbs the direction of the axis around which the state vector is revolving.

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