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Annalen der Physik

Analytical Solutions of the Driven Time‐Dependent Jaynes–Cummings Model

A. Vidiella-Barranco, Antonio S. M. de Castro, Alessandro Sergi, José Antonio Roversi, ANTONINO MESSINA, Agostino Migliore

Published Jul 24, 2025• 5 citations

Abstract

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.

DOI: 10.1002/andp.202500148

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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.

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.

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.

The exact time dependence of the density operator and various physical quantities for the motion-quantized Jaynes-Cummings models (MQJCMs) is given using the operator transformation under which it is shown that the MQJCMs have the same dynamics as the ordinary JCMs in the Kerr medium. The quantum collapses and revivals of the atomic motion due to the quantum properties of light fields and the quantization of atomic position and momentum, are predicted. The effects of the initial atomic momentum distribution and the arbitrary intensity-dependent medium on the collapse and revival phenomena of the population inversion, atomic momentum and radiation are formulated in detail. \textcopyright{} 1996 The American Physical Society.

We study the dynamical features of the Jaynes–Cummings model with the atom driven by an external classical field in the case that the cavity field is initially in a Fock state.We find the dynamical variables by using dressed states in the bare atom representation. We show that the dynamical behavior of the cavity field can be strongly modified by the external classical field. We also find that the cavity field is super-Poissonian after some interaction time if the external field is strong enough.