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In Chapter 7 of C. J. Foot's Atomic Physics, Foot discusses the interaction of a two-level atom with radiation. He derives the phenomenon of Rabi flopping from the Schrodinger equation, using perturbation theory and the rotating wave approximation as is standard to do. Then he says this:

The population oscillates between the two levels. When $Ωt = π$ all the population has gone from level 1 into the upper state, $|c_2(t)|^2 = 1$, and when $Ωt = 2π$ the atom has returned to the lower state. This behaviour is completely different from that of a two-level system governed by rate equations where the populations tend to become equal as the excitation rate increases and population inversion cannot occur.

What is the distinction he draws here? How does one reconcile the fact that the use of rate equations generally does not allow for population inversion between the two levels, as he says, but that the Schrodinger equation does? Is there a more subtle issue, such as the assumption of coherence in the case of Rabi flopping, involved here? Is it just that the "rate equation" model is simply wrong?

I'm happy to try to clarify the question if it needs it.

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The distinction is precisely given by the balance between the 'coherent dynamics' (ie., the Rabi flopping), and the rate of decoherence. In particular, the coherent dynamics of Rabi oscillations only holds when there is no dissipation, so the system remains in a pure quantum state.

The rate equations governing the system in the presence of decoherence describe the evolution of the density matrix of the system. The system may begin pure but end up quickly in a statistical mixture of the two states. This is a more complete picture for the dynamics of a real-world quantum system, but this picture always simplifies to the picture of coherent dynamics in the limit of no decoherence.

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The two models are very different. Schroedinger's model is unitary evolution, the system oscillates in a periodic motion and does not systematically acquire or lose energy. This corresponds to reversible dynamics such as pendulum in gravity field.

The rate equations model (the golden rule models) is almost always dissipative, there is some systematic increase of something, such as probability for some state. This corresponds to irreversible dynamics, such as pendulum in gravity field experiencing friction, or to an ensemble of pendula in gravity field without friction, but with random phase fluctuations which makes the average coordinate evolve in time in such a way that it approaches some final value (zero), as if there was a friction.

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