= Solution
{c}
Quantize a single electromagnetic <normal mode> of <angular frequency> $\omega_c$ and represent one isolated atomic transition as a <two-level system> with raising operator $\sigma_+=|e\rangle\langle g|$. The bare <Hamiltonian operator> is
$$
H_0=\hbar\omega_c a^\dagger a+\frac{\hbar\omega_a}{2}\sigma_z.
$$
The first main approximation is the <electric-dipole approximation>: the field varies negligibly across an atom, $ka_{\rm atom}\ll1$, and the leading coupling is $H_d=-\mathbf d\cdot\mathbf E(\mathbf r_0)$. For states of definite opposite parity, the relevant dipole part is off-diagonal. After choosing operator <phases>, this gives $H_d=\hbar g(a+a^\dagger)(\sigma_++\sigma_-)$, with $g$ fixed by the transition dipole and the one-<photon> field <wave amplitude>.
In the <interaction picture>, $a\sigma_+$ and $a^\dagger\sigma_-$ oscillate at the difference $\omega_a-\omega_c$, whereas $a\sigma_-$ and $a^\dagger\sigma_+$ oscillate at the sum $\omega_a+\omega_c$. The second main approximation is the <rotating-wave approximation>: near <resonance> and for $g\sqrt{n+1}\ll\omega_a+\omega_c$, the latter terms average away on the interaction time scale. The resulting <Jaynes-Cummings model> has
$$
\boxed{H_I=\hbar g(a\sigma_++a^\dagger\sigma_-).}
$$
The <two-level system> and single-<normal mode> truncations are additional assumptions: other atomic transitions and optical <normal modes> must be sufficiently far off <resonance> or weakly coupled. Dissipation is neglected over the interaction time. The retained terms exchange one atomic excitation for one <photon>, conserving $a^\dagger a+\sigma_+\sigma_-$.
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