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Journal of the Physical Society of Japan

Evaluating Wigner and Husimi Functions Using Fractional Multiphoton Jaynes–Cummings Model

Eman G. El-Hadidy, K. El Anouz, NASSER METWALLY

Published Jan 30, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.7566/jpsj.95.024004

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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 investigate a generic tripartite quantum system featuring a single qubit interacting concurrently with two quantized harmonic oscillators via nonlinear multiphoton Jaynes-Cummings (MPJC) interactions. Assuming the qubit is initially prepared in a superposition state and the two oscillators are in arbitrary Fock states, we analytically trace the temporal evolution of this tripartite pure initial state. We identify four broad cases, each further divided into two subcases, and derive exact analytical solutions for most cases. Notably, we obtain perfect transfer of excitations between the oscillators by carefully selecting system parameters. In addition, we extensively examine the manner in which the nonclassicalities of various initial oscillator Fock states, quantified by the volume of negative regions in the associated Wigner functions, evolve under the MPJC Hamiltonian, considering diverse system parameters including environmental effects. Besides producing substantial enhancements in the initial value for higher-photon-number states, our analysis reveals that driven solely by the initial qubit energy, with both oscillators initialized in the vacuum state, the nonlinear MPJC interaction yields a significant amount of nontrivial Wigner negativity in the oscillators. The additional nonlinearity introduced by the multiphoton process plays a pivotal role in surpassing the initial nonclassicalities of the photon-number states.
In this paper, a model in which two moving atoms interact with a two-mode field with nondegenerate two-photon transitions is studied in the presence of intensity-dependent coupling. To discuss the entanglement dynamics between subsystems as well as nonclassicality of the system, the time-dependent form of the state vector of the system is exactly obtained. The dynamics of entanglement between the atoms and the field is evaluated by the von Neumann entropy. To examine the degree of entanglement between the atoms, concurrence and negativity are obtained. Then, in order to understand the nonclassicality feature of the state vector of the system, the two-mode Wigner–Weyl quasi-probability distribution function is precisely derived. It is deduced from the numerical results that the domain and the maximum amount of the degree of entanglement and nonclassicality of the system can be appropriately tuned by suitably adopting the nonlinearity function and the field-mode structure parameters.
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= 0.50 $ 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.
We calculate the cavity-field distribution in the Wigner representation for the two-photon resonance of the weakly driven Jaynes-Cummings (JC) oscillator in its strong-coupling limit. Using an effective four-level system, we analytically demonstrate the presence of steady-state and transient bimodality which breaks azimuthal symmetry in phase space. The two steady-state peaks are located at opposite positions and do not correspond to the two-photon amplitude of the driven transition. The developing bimodality is portrayed in parallel with the evolution of the intensity correlation function for the forward-scattered photons, before being finally contrasted to the few-photon steady-state and transient phase-space profiles for the cavity mode in the JC model driven on resonance.