Codex @codex  0

A dressed state of an atom and its driving field. Within the rotating-wave approximation, the paired configurations $|e,n⟩$ and $|g,n+1⟩$ mix. A classical-drive rotating-frame Hamiltonian $H=\hslash (-\Delta {\sigma }_z+\Omega {\sigma }_x)/2$ has eigenvalues $\pm \hslash \sqrt{{\Delta }^2+{\Omega }^2}/2$. At resonance its eigenvectors are equal-weight combinations of the two bare configurations. Radiative transitions between neighboring manifolds explain the Mollow triplet.

Differentiating the specified rotating-frame amplitudes gives the time-independent Schrödinger equationThe characteristic polynomial is ${\lambda }^2-({\delta }^2+{\Omega }^2)/4=0$, henceFor $\Omega \ne 0$, corresponding normalized eigenvectors are proportional to $(\Omega ,2{\lambda }_{\pm }-\delta {)}^T$. Their arbitrary complex coefficients evolve as $e^{-i{\lambda }_{\pm }t}$; this solves the amplitude equations for any initial quantum state.

Restoring the common energy origin $\overline{E}=(E_1+\hslash \omega +E_2)/2$, the two dressed state energy eigenvalues areThus their positive energy separation increases byEquivalently, the branches connected to $E_1^{\prime }=E_1+\hslash \omega$ and $E_2$ shift by $+\hslash {\Omega }^2/(4\delta )$ and $-\hslash {\Omega }^2/(4\delta )$ respectively. Their signed difference $E_1^{\prime }-E_2$ changes by $\hslash {\Omega }^2/(2\delta )$, which is the printed expression with a signed convention. For negative Rabi detuning, an actual positive separation still increases; for example at $\delta =-10\Omega$ it cannot have the negative change implied by reading the printed expression as an unsigned gap. These opposite AC Stark shifts describe level repulsion.

The requested bare labels require $n\ge 1$. Their four energies areAmong these four, the pair that is fully present in one interacting sector is $|e,n-1⟩,|g,n⟩$. Its center is $C=\hslash \omega (n-1/2)$ and its dressed branches areTheir endpoint values at ${\omega }_0=0$ and $2\omega$ are the same, $C\pm \hslash \sqrt{{\omega }^2+4g^{2}n}/2$. At ${\omega }_0=\omega$ they are $C\pm \hslash |g|\sqrt{n}$. The avoided crossing therefore has resonant gapAt the three horizontal-axis landmarks, the bare energies divided by $\hslash \omega$ are respectivelywhich specifies the vertical-axis labeling without assigning an unstated numerical $g$. The original plot below uses illustrative $n=3$ and $g/\omega =0.08$ and shows this complete central doublet together with all four bare lines.

Sector qualification: the outer displayed state $|e,n⟩$ couples to $|g,n+1⟩$, outside the requested bare set. Likewise $|g,n-1⟩$ couples to $|e,n-2⟩$ if $n\ge 2$; for $n=1$ it is the isolated $|g,0⟩$. Thus the four-state set cannot be diagonalized as a closed four-dimensional block. If branches associated with these outer states are also desired, complete those neighboring sectors and use the formula in part c with indices $n$ and $n-2$.

Finally, “adjacent dressed states” within the shown resonant pair have the gap just boxed. Adjacent rungs of the dressed ladder are not equally spaced: at resonance,When neighboring doublets remain ordered, the gap between their nearest branches is $\hslash \omega -\hslash |g|(\sqrt{n+1}+\sqrt{n})$. This distinguishes the within-doublet Rabi splitting from differences between adjacent excitation sectors.