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When discussing two-level systems, spontaneous emission is often neglected `until later'. However, when discussed later, the two-level system is no longer discussed. For example, see Straten and Metcalf.

Why in simple descriptions of the two-level system can spontaneous emission be safely ignored? I think it is simply assumed that the Rabi frequency is far greater than the spontaneous emission rate? Or, to put another way, the lifetime of the excited state is very long lived.

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    $\begingroup$ I'm a bit confused here: every textbook I had in high school and college did not ignore spontaneous emission. The spont.emission coefficient is critical in calculating the available time for a population inversion to contribute to stimulated emission. Moving on to 4-level systems, the need for extremely "fast" spontaneous emission is critical. $\endgroup$ Nov 22, 2021 at 18:16
  • $\begingroup$ @CarlWitthoft I didn't say they ignored spontaneous emission completely, just when formulating the two level system. $\endgroup$
    – jamie1989
    Nov 22, 2021 at 18:18
  • $\begingroup$ In physics it is often helpful to simplify the model to such a degree that we are able to describe it, and therefore understand it. Hence, all models are wrong, but some are useful. $\endgroup$
    – Semoi
    Nov 22, 2021 at 21:05
  • $\begingroup$ The link takes you to the preface of the book. Can you provide a better link or give page numbers. $\endgroup$
    – ProfRob
    Nov 23, 2021 at 7:20

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This is a good and nontrivial question.

The short flippant answer to why spontaneous emission can be ignored is because we ignore it.

What I mean by this is that there are experimental regimes where we can have what you say, either very high Rabi frequencies or very long lifetimes, which are both conceptual statements of the mathematical condition $\Omega \gg \gamma$. In those conditions the predictions from ignoring spontaneous emission match observation really well.

But in my view, ignoring spontaneous emission is more of a learning exercise: you do it to clear away extraneous detail and get to the heart of how the system behaves, namely Rabi flopping. The effects of spontaneous emission are secondary in some sense: you want to see that a light field can transfer population between two levels, and you want to know how fast. The decay due to spontaneous emission is just clutter. Imagine a spherical cow.

There's also a deeper issue here: you'll never get spontaneous emission out of any theory that treats the EM field classically but the atom quantum-ly. Instead you have to quantize the EM field as well; and then the combined atom+field system is, in a real sense, infinitely far from a two-level system, since the states are now $|g,n+1\rangle$ and $|e,n\rangle$ for any $n \geq 0$ (and also $|g,0\rangle$), where the second label indicates the number of photons in the relevant mode (which I incidentally assumed there was only one of).

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    $\begingroup$ +1 but disagree with the last point: it's pretty common to use the density matrix formalism to treat the EM field classically and the atom quantum-ly. Including phenomenological decay terms in the rate equations is pretty common, without changing to quantums states of light. $\endgroup$ Apr 4 at 14:16
  • $\begingroup$ @StevenSagona Very true; but in that case, my point is that you have to sort of stick those on at the end, rather than coming naturally from the theory. $\endgroup$ Apr 10 at 13:13

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