# Atom-field interaction for two level system: decomposition of the dipole moment on $|0\rangle$ and $|1\rangle$

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?

• 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 ? Oct 1, 2018 at 13:55
• 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. Oct 1, 2018 at 14:42