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arXiv:2609.14563

Exact $1/4$ mean-rung population and $\sqrt{2}$ continuum revival-time ratio in the anti-Jaynes--Cummings vertex at $f_\star=\sqrt{\bar n+1}$

Onyango Stephen Okeyo

quant-phSubmitted Sep 13, 2026• Announced Sep 15, 2026• 0 citations
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Abstract

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.

Comments
18 pages, 8 figures

Related papers

Residual detuning in the laboratory-frame anti-Jaynes--Cummings model with a squeezed vacuum

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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.

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Properties of the population inversion of two-level atomic mixed-state in the multiphoton anti-Jaynes-Cummings model

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When single mode coherent light field interacts with two-level atomic mixed-state,the properties of the atomic population inversion is studied by means of the multi-photon anti-Jaynes-Cummings model.The influence for the parameters of average photon number of the coherent light field,transition photon number and mixing extent of the atomic initial state on the atomic population inversion has been discussed.The results show that the collapse and revival phenomena of periodicity will also appear in the two-level atomic population inversion.With the increasing of the average photon number of the coherent light field and the transition photon number,the revival period and attenuation time of collapse will change.With the increase of the average photon number of the coherent light field,the oscillation frequency of the atomic population inversion increases correspondingly.The revival period and attenuation time of collapse will not change,but the oscillation amplitude of the atomic population inversion will be increase as the decreasing of mixing extent of the atomic initial state.

Journal of Atomic and Molecular Physics•Jan 1, 2011•0 citations
Exact 1/41/41/4 mean-rung population and 2\sqrt{2}2​ continuum revival-time ratio in the anti-Jaynes--Cummings vertex at f⋆=nˉ+1f_\star=\sqrt{\bar n+1}f⋆​=nˉ+1​
At atomic resonance the Jaynes--Cummings (JC) vertex of a single bosonic mode coupled to a two-level emitter is blind to f=ω/λf=ω/λf=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 1/41/41/4 at f⋆=nˉ+1f_\star=\sqrt{\bar n+1}f⋆​=nˉ+1​, while the JC population at the same point is 1/21/21/2. The Poisson-averaged packet value at nˉ=16\bar n=16nˉ=16 is 0.246500.246500.24650, within 3.5×10−33.5\times 10^{-3}3.5×10−3 of the exact value, with the deviation reproduced by −nˉ/[16(nˉ+1)2]-\bar n/[16(\bar n+1)^2]−nˉ/[16(nˉ+1)2]. At f⋆f_\starf⋆​ the modulation depth of the AJC inversion is 1/21/21/2 and the relative sensitivity of the continuum revival-time ratio r=tRAJC/tRJCr=t_R^{\rm AJC}/t_R^{\rm JC}r=tRAJC​/tRJC​ is maximal with value 1/(2nˉ+1)1/(2\sqrt{\bar n+1})1/(2nˉ+1​). Full diagonalization on 2(nmax⁡+1)2(n_{\max}{+}1)2(nmax​+1) states confirms the analytic values. The 1/41/41/4 value can be generated by Floquet-engineered counter-rotating coupling provided that the static transverse rate satisfies g0≪g1g_0\ll g_1g0​≪g1​. 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 2fλ2fλ2fλ that is absent from the rotating-frame model. We map two proposed remedies---a Kerr shift χ(a^†a^)2χ(\hat a^\dagger\hat a)^2χ(a^†a^)2 and collective Dicke coupling of NNN two-level emitters---for a squeezed vacuum on that ladder (r=1r=1r=1, ⟨n⟩=sinh⁡2r≃1.38\langle n\rangle=\sinh^2 r\simeq 1.38⟨n⟩=sinh2r≃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 C=1/[1+(2f+χ)2]\mathcal{C}=1/[1+(2f+χ)^2]C=1/[1+(2f+χ)2] at r=0r=0r=0 to 10−910^{-9}10−9. Three results follow. (i)~A single Kerr strength never restores unit contrast at r=1r=1r=1; the r=0r=0r=0 nnn-dependent shift χ(2n+1)χ(2n+1)χ(2n+1) cannot cancel 2fλ2fλ2fλ on every occupied Fock component. (ii)~The exact r=1r=1r=1 contrast at χ=0χ=0χ=0 is not reproduced by an incoherent sum ∑nPn(r) Cn\sum_n P_n(r)\,\mathcal{C}_n∑n​Pn​(r)Cn​ built from the r=0r=0r=0 two-level formula; pointwise deviations are several tenths. (iii)~Collective coupling raises the contrast systematically. At f=5f=5f=5 one finds C=0.140\mathcal{C}=0.140C=0.140 (N=1N=1N=1) and C=0.575\mathcal{C}=0.575C=0.575 (N=8N=8N=8), above the unsqueezed value 8/[8+(2f)2]=0.0748/[8+(2f)^2]=0.0748/[8+(2f)2]=0.074. The N=16N=16N=16 point at this fff remains truncation-limited and is not quoted to three digits. The same N∼(2f)2N\sim(2f)^2N∼(2f)2 estimate for C=1/2\mathcal{C}=1/2C=1/2 places trapped-ion values f∼102f\sim 10^{2}f∼102--10310^{3}103 outside the present construction. The relevant platform is ultrastrong circuit QED with f∼1f\sim 1f∼1--101010. The calculation is a numerical control landscape, not a new solvable limit.
We considered the anti-Jaynes-Cummings (AJC) interaction of a two-level atom in an initial ground state interacting with a field mode in an initial squeezed coherent state at arbitrary values of squeeze parameter r and provided the Jaynes-Cummings (JC) interaction as a comparison. We analysed the degree of entanglement (DEM) measured by the von Neumann entropy and the nature of the field quantified by the Mandel Q parameter in relation to the atomic population inversion during the AJC interaction and separately the corresponding JC interaction. We noted in our examples that at r>1.4, photon statistics evolved to super-Poissonian from sub-Poissonian during the respective AJC, JC interactions. Further, for high values of r, the form of the time evolution of atomic population inversion depicted enhanced ringing revivals at the collapse region in comparison to the case of an initial coherent state. What is more, at higher values of r the time evolution of DEM showed more rapid oscillations and recorded higher values, concurrently, an increase in the degree of mixedness.
A variant Jaynes-Cummings model (JCM) where the two-state atom coupled to the cavity mode is driven by a strong external field reveals many interesting behaviors. On resonance, a large scale oscillatory behavior of the mean cavity photon number is predicted. It varies sinusoidally with the amplitude of 2(/spl epsi//g)/sup 2/, centered at the same value, where /spl epsi/ is the (real) driving field amplitude and g the vacuum Rabi frequency. The period of the oscillation was found to be exactly twice the "revival time" of the atomic inversion, i.e., T/sub 0/=(4/spl pi//g)(/spl epsi//g). Furthermore, the oscillation then shows another kind of "collapse-and-revival" type behavior as time proceeds. It is very similar to that of the atomic inversion, but the size of oscillation and the time scales are fundamentally different. The "revival" time of the oscillation, for instance, is found to be exactly one half of the so-called "fractional revival" time of the atomic inversion. Our current interest is directed to the radiative behavior of the driven JCM in a more general situation where dissipation as well as detuning is introduced.
Starting from the Jaynes-Cummings (J-C) model describing the interaction of a trapped ion with a standing-wave laser field, we derive the interaction Hamiltonian on the nonlinear J-C model of the trapped ion excitated resonantly with 1-quantum vibrational mode. After solving the time-dependent states of this model, we study the evolution of population inversion for the ion numerically. It is shown that the period of the quantum collapse and revival of the population inversionis related with the Lamb-Dick parameter and the position of the ion in the standing-wave laser field. With the increasing of the parameter, it will decrease the period of the collapse and revival. When moving the ion from the node of the standing-wave laser to the antinode, the period of the collapse and revival becomes longer.
When single mode coherent light field interacts with two-level atomic mixed-state,the properties of the atomic population inversion is studied by means of the multi-photon anti-Jaynes-Cummings model.The influence for the parameters of average photon number of the coherent light field,transition photon number and mixing extent of the atomic initial state on the atomic population inversion has been discussed.The results show that the collapse and revival phenomena of periodicity will also appear in the two-level atomic population inversion.With the increasing of the average photon number of the coherent light field and the transition photon number,the revival period and attenuation time of collapse will change.With the increase of the average photon number of the coherent light field,the oscillation frequency of the atomic population inversion increases correspondingly.The revival period and attenuation time of collapse will not change,but the oscillation amplitude of the atomic population inversion will be increase as the decreasing of mixing extent of the atomic initial state.