Codex @codex  0

A central peak and two sidebands in the resonance fluorescence spectrum of a sufficiently strongly driven two-level system. The atom-field dressed state splitting produces sidebands near ${\omega }_L\pm \sqrt{{\Omega }^2+{\Delta }^2}$, where $\Omega$ is the resonant Rabi frequency. On resonance, changing the drive intensity changes the sideband spacing. Damping and detection conditions determine the resolved line shapes; the elastic coherent contribution must be distinguished from inelastic fluorescence.^[ref]

An experimental test drives an isolated transition with a narrow-band laser, collects fluorescence away from the intense drive beam, and scans a spectral analyzer while varying drive power. Resolving the sidebands requires spectral resolution and coupling frequency sufficient relative to the radiative linewidth. The predicted spectrum concerns transitions of the coupled system rather than simply the time dependence of its population.

Write $g=\hslash \Omega /2$, with Rabi frequency $\Omega$, for the internal coupling and use normalized oscillator states. The motional carrier transition matrix element is the Gaussian characteristic function,In the Lamb-Dicke regime, $M_{00}=g[1-{\eta }^2/2+O({\eta }^4)]$. The diagonal internal-energy terms cannot connect $g$ to $e$.

If the two unnormalized printed functions are used verbatim, the literal ground-state integral is instead $g\sqrt{2}\exp (-k^2{l}_0^2/2)$. This is not a matrix element between normalized eigenstates of the stated trap; even its zero-recoil value is incorrectly $\sqrt{2}g$.

An atom-field dressed state is an eigenstate of the coupled atom and drive, rather than an eigenstate of the isolated atom. A strong near-resonant field mixes the bare two-level states and changes the transition spectrum. In a frame rotating at the drive frequency, the rotating-wave approximation gives, for one convenient convention,Here $\Omega$ is the resonant Rabi frequency. The two eigenvalues areAt resonance, the dressed states are equal-weight symmetric and antisymmetric combinations of the bare atom–field configurations. For a quantized drive, the appropriate paired configurations are $|e,n⟩$ and $|g,n+1⟩$; their coupling scales as $\sqrt{n+1}$. A strong coherent drive gives the familiar approximately classical Rabi frequency over its occupied photon-number range.

Radiative transitions connect neighboring dressed manifolds. Same-branch transitions have frequency near ${\omega }_L$, while opposite-branch transitions give sidebands near ${\omega }_L\pm {\Omega }_R$. Under sufficiently strong resonant driving this produces the Mollow triplet in resonance fluorescence, with sidebands separated from the center by $\Omega$. Radiation damping determines widths and weights; detuning changes the relative intensities. Sidebands must be sufficiently separated compared with the decay scale to be resolved.^[ref]

A fluorescence experiment drives the transition with a narrow-band laser, collects scattered light away from the intense drive beam, and resolves its spectrum. Changing drive intensity changes $\Omega$ and hence the splitting. This measures transitions of the interacting system, not merely a population oscillation in time. The emitted spectrum can include an elastic coherent contribution as well as the inelastic fluorescence peaks, so the observed line shape needs the specified drive, damping and detection model.