On page 145 of Exploring the quantum, atom cavities and photons by Serge Haroche and Jean-Michel Raimond, we have a two level system (an atom) interacting with a classical field. From what I understood, we can see the problem as a nucleus with one electron having two orbitals of eigenstates $|0\rangle$ and $|1\rangle$. We thus write the Hamiltonian:

$$ H = \hbar \omega_0 |1\rangle \langle 1| - \vec{d}\cdot\vec{E} $$

Where $\vec{d}=e \vec{r}$ and $\vec{r}$ is the position of the electron relative to the atom. On this page, the author rewrites $\vec{d}=\langle 0 | \vec{r} | 1 \rangle \ |0\rangle \langle 1 | + \langle 1 | \vec{r} | 0 \rangle \ |1\rangle \langle 0 | $. The diagonals terms of this operator vanish for parity reasons (I'm ok with this as fundamental is even and excited state is odd).

I don't understand how we can assume that with the interaction, we still can assume that our atom still has two eigenstates $|0\rangle$ and $|1\rangle$. The interaction could add energy levels for example.

If it is an approximation, what are the requirements to ensure that it is true?


1 Answer 1


For me the interaction could add energy levels for example.

It can't. The dimension of the state space that describes the atom is fixed, and adding an interaction cannot change it.

On the other hand, your construction does rest on the assumption that it is only those two states that are relevant to the evolution. If that assumption fails (which can be because e.g. there's another state that's closer to resonance with the driver, or which has a much stronger coupling, among a myriad possible reasons) then the other states obviously also need to be included, in both the dipole coupling as well as the atomic hamiltonian.

  • $\begingroup$ I think I see what you mean with a "qualitative view". It is like by definition, if my system is described in a basis of $N$ vectors, it is its intrinsic property so whatever the interaction is, it will always be described in a basis of $N$ vectors. But I think it would be interesting to have a source explaining rigorously how we define a quantum system, it would make things way more clear for me. I don't know if you have one in mind ? Like is a quantum system rigorously defined only by dimension of space + hamiltonian ? $\endgroup$
    – StarBucK
    Oct 1, 2018 at 13:55
  • $\begingroup$ You postulated in the question that "we can see the problem as a nucleus with one electron having two orbitals of eigenstates $|0⟩$ and $|1⟩$", which is a direct statement about what state space is appropriate to model your system in the situation you're interested in. There's nothing "qualitative view" about it: it's working strictly within the confines of the model you yourself defined. $\endgroup$ Oct 1, 2018 at 14:42

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