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arXiv:2604.20001

Fractional-Time Jaynes-Cummings Model: Unitary Description of its Quantum Dynamics, Inverse Problem and Photon Statistics

Thiago T. Tsutsui, Danilo Cius, Antonio S. M. de Castro, Fabiano M. Andrade

quant-phSubmitted Apr 21, 2026• Announced Apr 23, 2026• 0 citations
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Abstract

We analyze the quantum dynamics of the fractional-time Jaynes-Cummings model using a recent unitary framework for the fractional-time Schrödinger equation. We examine how the fractional derivative order $α$ influences non-classical features under different initial conditions. For an initial Fock state, fractional evolution introduces transient dynamics and heightened sensitivity to coupling strength. Through an inverse problem approach, we interpret these effects as arising from an effective time-dependent coupling with a strong initial pulse. For an initial coherent state, the fractional order tunes the system between dynamical regimes, with a transition at $α= 0.50 $ where standard collapse-and-revival is replaced by stable, periodic evolution. This regime enhances non-classical field properties, including stronger sub-Poissonian statistics, periodic quadrature squeezing, and the formation of Schrödinger cat states.

Comments
13 pages, 12 figures, comments are welcome

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We analyze the quantum dynamics of the fractional-time Jaynes-Cummings model using a recent unitary framework for the fractional-time Schrödinger equation. We examine how the fractional derivative order ααα influences non-classical features under different initial conditions. For an initial Fock state, fractional evolution introduces transient dynamics and heightened sensitivity to coupling strength. Through an inverse problem approach, we interpret these effects as arising from an effective time-dependent coupling with a strong initial pulse. For an initial coherent state, the fractional order tunes the system between dynamical regimes, with a transition at α=0.50α= 0.50 α=0.50 where standard collapse-and-revival is replaced by stable, periodic evolution. This regime enhances non-classical field properties, including stronger sub-Poissonian statistics, periodic quadrature squeezing, and the formation of Schrödinger cat states.
The time-evolution operator corresponding to the fractional-time Schrödinger equation is nonunitary because it fails to preserve the norm of the vector state in the course of its evolution. However, in the context of the time-dependent non-Hermitian quantum formalism applied to the time-fractional dynamics, it has been demonstrated that a unitary evolution can be achieved for a traceless two-level Hamiltonian. This is accomplished by considering a dynamical Hilbert space embedding a time-dependent metric operator concerning which the system unitarily evolves in time. This allows for a suitable description of a quantum system consistent with the standard quantum mechanical principles. In this work, we investigate the Jaynes-Cummings model in the fractional-time scenario taking into account the fractional-order parameter α and its effect in unitary quantum dynamics. We analyze the well-known dynamical properties, such as the atomic population inversion and the atom-field entanglement, when the atom starts in its excited state and the field in a coherent state.
• A generalized time-dependent Jaynes–Cummings model is investigated within the framework of the time fractional Schr¨odinger equation. • Two distinct fractional formulations are compared, encompassing oscillatory and dissipative non- Markovian dynamics. • Our results shows that fractional order introduces memory effects leading to damped oscillations and asymptotic decay in population inversion and entanglement. • Time-dependent couplings (linear, exponential, and sinusoidal) strongly modulate entanglement generation and decay. • Fractional order acts as a control parameter, preserving non-periodic dynamics under sinusoidal coupling within specific regimes. We investigate the fractional time description of a generalized quantum light-matter system modeled by a time-dependent Jaynes-Cummings (JC) interaction, with different coupling types: constant, linear, exponential, and sinusoidal. Two formulations of the time fractional Schrödinger equation (TFSE) are examined, with a focus on their impact on population inversion and entanglement. Our findings highlight that the introduction of fractional order introduces memory effects, associated with damped oscillations and asymptotic decay. Furthermore, we find that the time-dependent couplings, combined with distinct fractional formulations, influence how these effects occur, ultimately resulting in high or low entanglement. A key finding of our work is that, under sinusoidal coupling, non-periodic dynamics is preserved for both formulations of the TFSE; however, within a certain range, the fractional order can act as a control mechanism for the non-periodic evolution.
One of the most significant physical models for characterizing the dynamics of robust-intensity and strong laser fields, has recently been solved using the so-called multiphoton Jaynes–Cummings model. This generalized model describes the interaction between a two-level atom and a field in the presence of multiple photons simultaneously. In the present study, we consider the generalized form of the equation, known as the fractional equation. An exact fractional solution of the multiphoton Jaynes–Cummings model is derived. Furthermore, the non-Gaussianity of the system is examined using the Wigner and Husimi quasiprobability distributions. Our main result can be summarized as follows: although the initial state is taken as a direct product of the atomic and coherent states, the system remains non-Gaussian when fractional dynamics is taken into account. The results show that the Wigner function consistently exhibits negative regions for small values of the fractional order, corresponding to strong fractional memory. Moreover, it is demonstrated that both the Wigner and Husimi functions are highly sensitive to variations in the parameters of the multiphoton Jaynes–Cummings model. In particular, localization of the quantum state is observed in both functions within a specific region of the phase plane.
We present a two-mode squeezed Jaynes–Cummings model, built upon the formalism of pair coherent states (PCSs), to investigate the dynamics of a two-level atom interacting with a two-mode quantized field. By solving the time-dependent Schrödinger equation under the rotating-wave approximation, we elucidate the system’s quantum evolution, with particular emphasis on how the squeezing degree and photon number difference modulate atomic population inversion and entanglement. We further quantify the nonclassical traits of the two-mode squeezed PCSs via Mandel’s parameter and the violation of the Cauchy–Schwarz inequality, highlighting their sensitivity to model parameters. These findings illuminate the subtle interplay of squeezing, photon statistics, and entanglement in advanced quantum optical systems.
We analyze the time-dependent physical spectrum of a driven Jaynes-Cummings model in which both the two-level system and the quantized cavity mode are subject to coherent classical driving. The time-dependent Hamiltonian is mapped, via well-defined unitary transformations, onto an effective stationary Jaynes-Cummings form. Within this framework, we derive closed-form expressions for the two-time correlation functions of both the atomic and field operators. These correlation functions are subsequently used to evaluate the time-dependent physical spectrum according to the Eberly-Wódkiewicz definition, which properly accounts for finite spectral resolution and transient emission dynamics. We show that the external driving leads to substantial modifications of the atomic spectral response, including controllable frequency shifts and asymmetric line shapes. Importantly, we identify a regime in which the driving parameters are chosen such that the coherent displacement induced in the cavity field exactly cancels out the initial coherent amplitude. In this limit, the system dynamics reduce to that of an effectively vacuum-initialized Jaynes-Cummings model, and the standard vacuum Rabi splitting is recovered. This behavior provides a clear and physically transparent interpretation of the spectral features as arising from coherent field displacement rather than from modifications of the underlying atom-cavity coupling.