Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.04580

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

Onyango Stephen Okeyo

quant-phSubmitted Sep 4, 2026• Announced Sep 7, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

Comments
15 pages, 7 figures. Submitted to Quantum Inf. Process

Related papers

Anti-Jaynes-Cummings interaction of a two-level atom with squeezed light: A comparison with the Jaynes-Cummings interaction

Christopher Mayero, Joseph Akeyo Omolo

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.

arXiv:2211.13149•Nov 23, 2022•0 citations

Collective photon echoes in the Tavis-Cummings model: distribution independence, two detuning regimes, and the Dicke ladder

Michael Tavis

An ensemble of $N$ two-level molecules prepared in its ground state and sharing a lossless cavity with a weak field does not simply absorb: the intensity collapses and then recurs, in a train of collective photon echoes. Working from the exact solution of the Tavis-Cummings model in the few-photon regime $\bar n<N$, we confirm the echo time $τ_E=4π\sqrt{N+Δ}/g$ numerically at resonance from $N=5$ to $400$, the small-$N$ end discriminating this form from the alternative $4π\sqrt{N-\bar n+Δ}/g$ in its favor. The initial state requires no preparation, being the ground state. The echo time is independent of the initial photon distribution: coherent, thermal, squeezed and oscillatory-squeezed distributions spanning variances from 2 to 34 all recur together, as does a controlled pair with identical mean and variance differing only in the shape of $ρ_{nn}$. The echo amplitude is not: it varies at the tens-of-percent level at fixed mean, including a factor of 1.8 with the squeezing phase at fixed squeezing strength. Detuning organizes the dynamics into two clean regimes separated by a fragmented crossover, the dispersive-branch echo time approaching one-half the resonant one, and sufficient detuning removes the dependence on the initial Dicke state. For arbitrary initial Dicke state, emission replaces absorption at $m\simeq-N/2+\bar n$, and the echo envelope acquires one component per step up the ladder: a single-component echo occurs only at the ground state. A feasibility analysis against a five-qubit superconducting device, with Lindblad simulations of cavity decay and dephasing and full-Hilbert-space disorder simulations, shows the first echoes observable at $N\sim5$-$20$ on existing hardware: the echo survives the dominant loss channel with contrast $\sim e^{-κτ_E/2}$, photons being shielded from cavity decay while resident in the emitters.

quant-pharXiv:2608.21442•Aug 20, 2026•0 citations

Higher-order squeezing oscillations in Jaynes–Cummings model of a pair of cold atoms

N. A. Enaki, V. I. Koroli, Sergiu Bazgan +2 more

We study interaction between a pair of indistinguishable two-level atoms and the single-mode cavity field. It is supposed that the pair of two-level atoms is laser cooled and trapped into the ground vibrational state, in which vibrational quantum number $$\langle n_v\rangle =0$$ . Two Jaynes–Cummings models are investigated. One is the Jaynes–Cummings model with intensity-dependent coupling and the another is the two-photon Jaynes–Cummings model of a pair of indistinguishable two-level atoms. It should be noted that in the present model, at initial moment $$t=0$$ , in intensity-dependent Jaynes–Cummings model the cavity field is prepared in Holstein–Primakoff SU(1,1) CS, while in two-photon Jaynes–Cummings model it is prepared in the squeezed vacuum state. Moreover, at initial moment $$t=0$$ , pair of atoms is supposed to be in the first excited state $$\vert e_1\rangle $$ in both models. By using exact analytical solutions for state-vectors of the coupled atom-field systems amplitude-squared squeezing of the quantized cavity field is examined as a function of the $$\vert \xi \vert $$ parameter. In this situation, in both models higher-order squeezing has the tendency towards oscillations, but exact periodicity of these oscillations is violated by the analogy with the second-order squeezing.

Indian Journal of Physics•Feb 12, 2015•1 citation

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

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.

quant-pharXiv:2609.14563•Sep 15, 2026•0 citations

The anti-Jaynes-Cummings model is solvable : quantum Rabi model in rotating and counter-rotating frames ; following the experiments

Joseph Akeyo Omolo

Abstract This article is a response to the continued assumption, cited even in reports and reviews of recent experimental breakthroughs and advances in theoretical methods, that the antiJaynes-Cummings (AJC)interaction is an intractable energy non-conserving component of the quantum Rabi model (QRM). We present three key features of QRM dynamics : (a) the AJC interaction component has a conserved excitation number operator and is exactly solvable (b) QRM dynamical space consists of a rotating frame (RF) dominated by an exactly solved Jaynes-Cummings (JC) interaction specied by a conserved JC excitation number operator which generates the U(1) symmetry of RF and a correlated counter-rotating frame (CRF) dominated by an exactly solved antiJaynes-Cummings (AJC) interaction specied by a conserved AJC excitation number operator which generates the U(1) symmetry of CRF (c) for QRM dynamical evolution in RF, the initial atom-eld state je0i is an eigenstate of the effective AJC Hamiltonian HAJC, while the effective JC Hamiltonian HJC drives this initial state je0i into a time evolving entangled state, and, in a corresponding process for QRM dynamical evolution in CRF, the initial atom-eld state jg0i is an eigenstate of the effective JC Hamiltonian, while the effective AJC Hamiltonian drives this initial state jg0i into a time evolving entangled state, thus addressing one of the long-standing challenges of theoretical and experimental QRM dynamics; consistent generalizations of the initial states je0i , jg0i to corresponding n 0 entangled eigenstates j+en i , j 􀀀g ni of the AJC in RF and JC in CRF, respectively, provides general dynamical evolution of QRM characterized by collapses and revivals in the time evolution of the atomic, eld mode, JC and AJC excitation numbers for large initial photon numbers ; the JC and AJC excitation numbers are conserved in the respective frames RF, CRF, but each evolves with time in the alternate frame.

arXiv:2103.09546•Apr 8, 2021•3 citations

Phase Dynamics in the Generalized Jaynes-Cummings Model with Intensity-Dependent Coupling Interacting with a Squeezed Vacuum

Fan Au, Xin Fan

WT5BZ]We consider the generalized Jaynes-Cummings model with intensity - dependent coupling interacting with a squeezed vacuum. From the point of view of the new phase formalism of Pegg and Barnett, we have not only found the exact general expressions of the phase distribution and the associated expectation value and variance of the phase of the squeezed vacuum field and their evolution illustrated graphically, but also derived approximately analytic formulas of that for the weakly and strongly squeezed vacuum. The relation between squeezing and the phase properties of the squeezed vacuum is discussed briefly.

•Jan 1, 2000•0 citations
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.
An ensemble of NNN two-level molecules prepared in its ground state and sharing a lossless cavity with a weak field does not simply absorb: the intensity collapses and then recurs, in a train of collective photon echoes. Working from the exact solution of the Tavis-Cummings model in the few-photon regime nˉ<N\bar n<Nnˉ<N, we confirm the echo time τE=4πN+Δ/gτ_E=4π\sqrt{N+Δ}/gτE​=4πN+Δ​/g numerically at resonance from N=5N=5N=5 to 400400400, the small-NNN end discriminating this form from the alternative 4πN−nˉ+Δ/g4π\sqrt{N-\bar n+Δ}/g4πN−nˉ+Δ​/g in its favor. The initial state requires no preparation, being the ground state. The echo time is independent of the initial photon distribution: coherent, thermal, squeezed and oscillatory-squeezed distributions spanning variances from 2 to 34 all recur together, as does a controlled pair with identical mean and variance differing only in the shape of ρnnρ_{nn}ρnn​. The echo amplitude is not: it varies at the tens-of-percent level at fixed mean, including a factor of 1.8 with the squeezing phase at fixed squeezing strength. Detuning organizes the dynamics into two clean regimes separated by a fragmented crossover, the dispersive-branch echo time approaching one-half the resonant one, and sufficient detuning removes the dependence on the initial Dicke state. For arbitrary initial Dicke state, emission replaces absorption at m≃−N/2+nˉm\simeq-N/2+\bar nm≃−N/2+nˉ, and the echo envelope acquires one component per step up the ladder: a single-component echo occurs only at the ground state. A feasibility analysis against a five-qubit superconducting device, with Lindblad simulations of cavity decay and dephasing and full-Hilbert-space disorder simulations, shows the first echoes observable at N∼5N\sim5N∼5-202020 on existing hardware: the echo survives the dominant loss channel with contrast ∼e−κτE/2\sim e^{-κτ_E/2}∼e−κτE​/2, photons being shielded from cavity decay while resident in the emitters.
We study interaction between a pair of indistinguishable two-level atoms and the single-mode cavity field. It is supposed that the pair of two-level atoms is laser cooled and trapped into the ground vibrational state, in which vibrational quantum number ⟨nv⟩=0\langle n_v\rangle =0⟨nv​⟩=0 . Two Jaynes–Cummings models are investigated. One is the Jaynes–Cummings model with intensity-dependent coupling and the another is the two-photon Jaynes–Cummings model of a pair of indistinguishable two-level atoms. It should be noted that in the present model, at initial moment t=0t=0t=0 , in intensity-dependent Jaynes–Cummings model the cavity field is prepared in Holstein–Primakoff SU(1,1) CS, while in two-photon Jaynes–Cummings model it is prepared in the squeezed vacuum state. Moreover, at initial moment t=0t=0t=0 , pair of atoms is supposed to be in the first excited state ∣e1⟩\vert e_1\rangle ∣e1​⟩ in both models. By using exact analytical solutions for state-vectors of the coupled atom-field systems amplitude-squared squeezing of the quantized cavity field is examined as a function of the ∣ξ∣\vert \xi \vert ∣ξ∣ parameter. In this situation, in both models higher-order squeezing has the tendency towards oscillations, but exact periodicity of these oscillations is violated by the analogy with the second-order squeezing.
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.
Abstract This article is a response to the continued assumption, cited even in reports and reviews of recent experimental breakthroughs and advances in theoretical methods, that the antiJaynes-Cummings (AJC)interaction is an intractable energy non-conserving component of the quantum Rabi model (QRM). We present three key features of QRM dynamics : (a) the AJC interaction component has a conserved excitation number operator and is exactly solvable (b) QRM dynamical space consists of a rotating frame (RF) dominated by an exactly solved Jaynes-Cummings (JC) interaction specied by a conserved JC excitation number operator which generates the U(1) symmetry of RF and a correlated counter-rotating frame (CRF) dominated by an exactly solved antiJaynes-Cummings (AJC) interaction specied by a conserved AJC excitation number operator which generates the U(1) symmetry of CRF (c) for QRM dynamical evolution in RF, the initial atom-eld state je0i is an eigenstate of the effective AJC Hamiltonian HAJC, while the effective JC Hamiltonian HJC drives this initial state je0i into a time evolving entangled state, and, in a corresponding process for QRM dynamical evolution in CRF, the initial atom-eld state jg0i is an eigenstate of the effective JC Hamiltonian, while the effective AJC Hamiltonian drives this initial state jg0i into a time evolving entangled state, thus addressing one of the long-standing challenges of theoretical and experimental QRM dynamics; consistent generalizations of the initial states je0i , jg0i to corresponding n 0 entangled eigenstates j+en i , j 􀀀g ni of the AJC in RF and JC in CRF, respectively, provides general dynamical evolution of QRM characterized by collapses and revivals in the time evolution of the atomic, eld mode, JC and AJC excitation numbers for large initial photon numbers ; the JC and AJC excitation numbers are conserved in the respective frames RF, CRF, but each evolves with time in the alternate frame.
WT5BZ]We consider the generalized Jaynes-Cummings model with intensity - dependent coupling interacting with a squeezed vacuum. From the point of view of the new phase formalism of Pegg and Barnett, we have not only found the exact general expressions of the phase distribution and the associated expectation value and variance of the phase of the squeezed vacuum field and their evolution illustrated graphically, but also derived approximately analytic formulas of that for the weakly and strongly squeezed vacuum. The relation between squeezing and the phase properties of the squeezed vacuum is discussed briefly.