Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 6 / a / Solution

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 ${\sigma }_{+}=|e⟩⟨g|$. The bare Hamiltonian operator isThe first main approximation is the electric-dipole approximation: the field varies negligibly across an atom, $k{a}_{\mathrm{atom}}\ll 1$, 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 $H_d=\hslash g(a+a^{†})({\sigma }_{+}+{\sigma }_{-})$, with $g$ fixed by the transition dipole and the one-photon field wave amplitude.

In the interaction picture, $a{\sigma }_{+}$ and $a^{†}{\sigma }_{-}$ oscillate at the difference ${\omega }_a-{\omega }_c$, whereas $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$ 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 hasThe 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 $a^{†}a+{\sigma }_{+}{\sigma }_{-}$.