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Annals of Physics

A generalized unitary transformation for solving time-independent and time-dependent Jaynes–Cummings models under resonant and selected off-resonance conditions

ANTONINO MESSINA, Hiromichi Nakazato, Antonio S. M. de Castro, Jonathan Alexandre Moraz Almeida Rocha, A. B. Klimov, Roberto Grimaudo, Agostino Migliore

Published Jul 7, 2026• 0 citations

Abstract

Jaynes–Cummings (JC) models are by far the most widely used to study the dynamics of two-level systems, such as atoms or molecules under suitable conditions, coupled to quantized electromagnetic fields. Nowadays, analytical solutions of JC models are more useful than ever for understanding and controlling radiation-matter interactions relevant to quantum information processing, quantum computing, and other quantum technologies. This study presents the analysis and generalization of a method for the solution of the dynamical problem in the time-independent JC model (Carbonaro et al., 1979) that enables an exact description of the atom-field dynamics in the resonant time-dependent JC model and in a nontrivial class of off-resonance time-dependent JC models. Our analysis produces the ready-to-use forms of the evolution operator in the resonant and non-resonant cases, which are then used to derive the evolution of the atomic population inversion for some types of atom-field coupling. The central idea behind the proposed approach offers opportunities for its extension to other time-dependent cases and its generalization to JC-related models.

DOI: 10.1016/j.aop.2026.170605

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Jaynes–Cummings (JC) models are by far the most widely used to study the dynamics of two-level systems, such as atoms or molecules under suitable conditions, coupled to quantized electromagnetic fields. Nowadays, analytical solutions of JC models are more useful than ever for understanding and controlling radiation-matter interactions relevant to quantum information processing, quantum computing, and other quantum technologies. This study presents the analysis and generalization of a method for the solution of the dynamical problem in the time-independent JC model (Carbonaro et al., 1979) that enables an exact description of the atom-field dynamics in the resonant time-dependent JC model and in a nontrivial class of off-resonance time-dependent JC models. Our analysis produces the ready-to-use forms of the evolution operator in the resonant and non-resonant cases, which are then used to derive the evolution of the atomic population inversion for some types of atom-field coupling. The central idea behind the proposed approach offers opportunities for its extension to other time-dependent cases and its generalization to JC-related models.

The Jaynes-Cummings (JC) model is one of the most fundamental models for the interaction between light and matter. In this paper, we first derive the explicit expression for the time evolution operator of the JC model when the light field resonates with a two-level atom. Based on this, we calculate the analytical expressions for the system state and the population of each atomic energy level as a function of time when the two-level atom is initially in a superposition of two energy eigenstates and the light field is in a superposition of three photon number states and a coherent state, respectively. Finally, this paper introduces QuTiP, a useful extension library in Python for quantum mechanics simulation and numerical solution, and uses it to numerically solve the time evolution of this model, obtaining results consistent with the analytical solutions. These computational processes are of great significance in scientific research and university physics teaching.

The unitary transformation method is utilized for the closed form solution of the nonlinear generalization of the Jaynes–Cummings (JC) model, which includes arbitrary multiphoton and intensity-dependent interactions between radiation and matter. Specifically, the unitary transformation leading to the diagonalization of the relevant Hamiltonian is established and energy levels and eigenstates are derived. It is also shown that the results for the JC model can be extended to the anti-Jaynes–Cummings (AJC) model through the map between the two models.

Abstract This study explores the dynamics of a Driven Time‐Dependent Jaynes–Cummings (DTDJC) model, in which the atom and the field are influenced by an external classical field with time‐varying amplitude modulation. By applying a series of unitary transformations and establishing suitable conditions for the time‐dependent parameters of the system, the Hamiltonian operator is transformed into one representing a system not driven by an external field, namely a Free TDJC (FTDJC) system. This mapping allows us to determine, in both the time‐independent and time‐dependent cases, forms of the pumping modulating the field and the atomic population that are amenable to exact analytical solution. The effects of an off‐resonance parameter emerging from the analytical solution of the model and the intensity of the classical pumping field on the dynamics of the atomic population are investigated. The exact solution allows us to reveal a rich dynamics of the system (including long‐time super‐revivals) that could be of use in future quantum technologies.

We present a general procedure to describe the dynamics of N degenerate quantized fields interacting resonantly with a two–level atom, all coupled with the same strength, within the rotating–wave approximation. Starting from the analysis of the two and three field cases, we generalize the method by identifying dynamical invariants that lead to a factorized form of the time–evolution operator. A unitary transformation reduces the problem to an effective Jaynes–Cummings Hamiltonian, where only one field interacts with the atom and the remaining modes contribute as free fields. Assuming initially coherent fields and an atomic superposition, we compute the atomic inversion and the mean photon number, revealing vacuum Rabi oscillations with a frequency determined by an effective coupling constant that exceeds the individual atom–field coupling, as well as the characteristic collapse-revival behavior.

The problem of the interaction of a single two-level atom with a single mode of the quantized electromagnetic field in a coherent state in an ideal resonator in the resonance case is considered. The evolution in time of the atomic inversion, represented by the Jaynes-Cummings sum, is studied. On the basis of the application of the theorem on the approximation of a trigonometric sum by a shorter one (ATS), a new efficient method for approximating the Jaynes-Cummings sum is constructed. New asymptotic formulas for the atomic inversion are found, which approximate it on various time intervals, defined by relation between the atom-field coupling constant and the average photon number in the resonator field before the interaction of the field with the atom. The asymptotics that we obtain give the possibility to predetermine the details of the process of the inversion in the Jaynes-Cummings model depending on the field characteristics.

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