Dipole interaction for a two-level system in a trap potential Consider a two-level system under the following Hamiltonian:
$H=H_{I}+H_{E}+H_{dip}$
where $H_I$ is the Hamiltonian for the two internal states of the two-level system $span \{\left |e \right> {,} \left |g \right> \}$ (e $\rightarrow $ excited state, g $\rightarrow $ ground state), $H_E$ is the trap potential which could be quite general (in fact it could depend also on the internal degrees of freedom, e.g. a potential with different coupling for both states) and $H_{dip}$ is the dipole interaction between a laser and the two-level system, where the laser is treated classically. The trap potential induces a splitting in the energy levels associated within each manifold $\left |e \right>$ and $\left |g \right>$. Thus a possible basis would be $span\{\left |e {,} n_e \right> {,} \left |g {,} n_g\right>\}$ (note that $\left | n_e \right >$ and $\left | n_g \right >$ need not be equal), being this basis the eigenvectors of $H_{E}$ and $H_{I}$. The following paper http://journals.aps.org/pra/pdf/10.1103/PhysRevA.46.2668 proposes what $H_{dip}$ should be for an harmonic trap and running-wave laser; mainly:
$V_{dip}=\frac{\Omega_0}{2} (e^{ik_L\hat{Z}} \left |e \right>\left <g \right|+e^{-ik_L\hat{Z}} \left |g \right>\left <e \right|)$
where $\Omega_0$ and $k_L$ are the Rabi frequency and the laser wavenumber respectively. Similar treatments of what $V_{dip}$ is are found in other references, i.e. they consider the dipole interaction as connecting only internal states. However this is true, I guess (I haven't seen a clear proof of this anywhere), in the case where the trap potential felt by both $\left |e \right>$ and $\left |g \right>$ is the same. My question then, is this also true generally for any trap potential? In other words, why won't light couple say different $n_g$ states within the ground state? What would be a quantum mechanical expression for the dipole moment operator in this case?
 A: The light-matter coupling Hamiltonian you have written is correct in general for a running-wave laser within the dipole approximation (and two-level approximation); it contains no assumption about the trap potential. In fact, it's not true that it connects only internal states. It also connects the motional states due to the $e^{i k \hat{Z}}$ term. This term is actually a displacement operator that shifts the momentum of the atomic centre of mass by an amount $k$ in the $z$ direction. This is exactly what you'd expect from a transition in which the internal state is changed by absorption of energy from a photon. The momentum of the photon also has to go somewhere, and in this case it is transferred to the atomic centre of mass.
A: First of all, the effective coupling strength $\Omega_0$ is determined by two part: the detuning $\delta$ between $\omega_g+\omega_{light}$ and $\omega_e$, and the dipolar overlapping between state $\langle e|\vec{\bf r}|g\rangle$. If the second contribution doesn't vary a lot, then the $\Omega_0\propto\delta^{-1}$. And the separation of different state in frequency is very high, which makes at one particular resonance, unless there are some other reasons giving several quasi-degeneracy state (like hyperfine structure), one should only take one excited state as consideration. 
