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I want to describe a some 2-level system which begins in its ground state $|g\rangle$ and then is coupled to a coherent laser beam $|\alpha\rangle$ with the energy $\hbar\omega=E_e-E_g$ and thereby jumps to its excited state $|e\rangle$. I'm looking for the operator description of this event. At first I thought it was (ignoring physical normalization constants): $$(\hat{a}+\hat{a}^{\dagger})|\alpha\rangle\otimes|g\rangle=\alpha|\alpha\rangle|g\rangle+|\alpha\rangle|e\rangle\tag{1}$$ which physically seems to mean the annihilation operator takes a photon from the laser beam and adds it, via the creation operator, to the two-level system. But (and this is assuming I've done the algebra here correctly), I don't understand the meaning of the $\alpha|\alpha\rangle|g\rangle$ term that shows up in equation (1). We are beginning in an eigenstate of energy $|e\rangle$ and going into a superposition of energy eigenstates $c_1|g\rangle + c_2|e\rangle$. By physical intuition, it seems to me that the laser -- given sufficient time -- should make the amplitude for $|e\rangle$ essentially $1$ (once normalization is done).

What am I missing here?

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  • $\begingroup$ "should make the amplitude for $|e⟩$ essentially 1" — actually no, this setup would undergo Rabi cycles instead of approaching a stationary state. $\endgroup$
    – Ruslan
    Jun 18 at 21:55
  • $\begingroup$ I should clarify, I meant that thinking in terms of pure physics, you expect that the amplitude for the two-level system to absorb a photon from the laser quickly goes toward 1, and therefore after some (fairly small) time, you would expect the laser to take the stationary state $|g\rangle$ into the stationary state $|e\rangle$ with near certainty. Whereas my equation (1), as you rightly point out, does not have that trend. $\endgroup$
    – Cody Payne
    Jun 18 at 22:00
  • $\begingroup$ Your equation doesn't make sense to me. Why do you expect that a creation operator acting on EM field would change state of the two-level system (say, electron spin), which is another kind of field? The RHS of your equation shouldn't have $|e\rangle$ at all. $\endgroup$
    – Ruslan
    Jun 18 at 22:09
  • $\begingroup$ By definition of what I mean by $\hat{a}^{\dagger}$, $|g\rangle$, and $|e\rangle$, it is true that $\hat{a}^{\dagger}|g\rangle=|e\rangle$ (again, ignoring ALL normalization constants currently). And all I mean by the tensor product $|\alpha\rangle\otimes\|g\rangle$ is the coupled state of the laser with the two level system. My algebra could be wrong, but I don't think anything about my physics is really inconsistent here. $\endgroup$
    – Cody Payne
    Jun 18 at 22:13
  • $\begingroup$ Ah, so your ladder operators act on the two-level system. Then I think you should have $\hat a|g\rangle=0$ (annihilation of vacuum yields zero), and you get just $|\alpha\rangle|e\rangle$ on the RHS. $\endgroup$
    – Ruslan
    Jun 18 at 22:25

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