Onyango Stephen Okeyo

At atomic resonance the Jaynes--Cummings (JC) vertex of a single bosonic mode coupled to a two-level emitter is blind to $f=ω/λ$ at zero detuning and Kerr; the anti-Jaynes--Cummings (AJC) vertex is not. We show that the mean-rung excited population of the AJC block equals exactly $1/4$ at $f_\star=\sqrt{\bar n+1}$, while the JC population at the same point is $1/2$. The Poisson-averaged packet value at $\bar n=16$ is $0.24650$, within $3.5\times 10^{-3}$ of the exact value, with the deviation reproduced by $-\bar n/[16(\bar n+1)^2]$. At $f_\star$ the modulation depth of the AJC inversion is $1/2$ and the relative sensitivity of the continuum revival-time ratio $r=t_R^{\rm AJC}/t_R^{\rm JC}$ is maximal with value $1/(2\sqrt{\bar n+1})$. Full diagonalization on $2(n_{\max}{+}1)$ states confirms the analytic values. The $1/4$ value can be generated by Floquet-engineered counter-rotating coupling provided that the static transverse rate satisfies $g_0\ll g_1$. Parameter translations for an ideal trapped-ion blue sideband and a flux-tunable circuit-QED coupler are reported.

The laboratory-frame anti-Jaynes--Cummings (AJC) interaction retains a residual detuning $2fλ$ that is absent from the rotating-frame model. We map two proposed remedies---a Kerr shift $χ(\hat a^\dagger\hat a)^2$ and collective Dicke coupling of $N$ two-level emitters---for a squeezed vacuum on that ladder ($r=1$, $\langle n\rangle=\sinh^2 r\simeq 1.38$). All quoted contrasts are the amplitude of the first turning point of the atomic ground-state population, which coincides with the two-level formula $\mathcal{C}=1/[1+(2f+χ)^2]$ at $r=0$ to $10^{-9}$. Three results follow. (i)~A single Kerr strength never restores unit contrast at $r=1$; the $r=0$ $n$-dependent shift $χ(2n+1)$ cannot cancel $2fλ$ on every occupied Fock component. (ii)~The exact $r=1$ contrast at $χ=0$ is not reproduced by an incoherent sum $\sum_n P_n(r)\,\mathcal{C}_n$ built from the $r=0$ two-level formula; pointwise deviations are several tenths. (iii)~Collective coupling raises the contrast systematically. At $f=5$ one finds $\mathcal{C}=0.140$ ($N=1$) and $\mathcal{C}=0.575$ ($N=8$), above the unsqueezed value $8/[8+(2f)^2]=0.074$. The $N=16$ point at this $f$ remains truncation-limited and is not quoted to three digits. The same $N\sim(2f)^2$ estimate for $\mathcal{C}=1/2$ places trapped-ion values $f\sim 10^{2}$--$10^{3}$ outside the present construction. The relevant platform is ultrastrong circuit QED with $f\sim 1$--$10$. The calculation is a numerical control landscape, not a new solvable limit.