Generalized symmetries, invariant solutions and conservation laws in the Jaynes-Cummings model

arXiv:2606.15538v2 Announce Type: replace-cross Abstract: In this study, we investigate the Jaynes--Cummings model (JCM) using Lie symmetry analysis and conservation-law theory. The dynamics is formulated as a system of partial differential equations by projecting the von Neumann equation onto the atomic degrees of freedom and representing the field mode through its characteristic function. We determined the admitted point and generalized symmetries and constructed invariant solutions by imposing quantum-mechanical constraints. The conventional dressed-state dynamics is recovered, while a second class of solutions with radial dependence expressed through Heun polynomials is obtained for coupled atom--field configurations. We also applied a generating-function methodology to derive the local conservation laws of the JCM differential system in Fourier phase space. In addition to recovering the standard excitation-number constant of motion, we obtained additional conserved currents involving atomic populations, coherence, reduced-state purity, and derivatives of the field characteristic functions. One of these currents yields, at the Fourier-space origin, a balance equation for a combination of atomic purity and coherence whose evolution is driven by atom--field coupling and joint atom--field moments. For globally pure states, the purity contribution provides an indirect connection with the development of atom--field entanglement. The symmetry structure generates generalized symmetries and an infinite hierarchy of conservation laws.

Abstract:In this study, we investigate the Jaynes--Cummings model (JCM) using Lie symmetry analysis and conservation-law theory. The dynamics is formulated as a system of partial differential equations by projecting the von Neumann equation onto the atomic degrees of freedom and representing the field mode through its characteristic function. We determined the admitted point and generalized symmetries and constructed invariant solutions by imposing quantum-mechanical constraints. The conventional dressed-state dynamics is recovered, while a second class of solutions with radial dependence expressed through Heun polynomials is obtained for coupled atom--field configurations. We also applied a generating-function methodology to derive the local conservation laws of the JCM differential system in Fourier phase space. In addition to recovering the standard excitation-number constant of motion, we obtained additional conserved currents involving atomic populations, coherence, reduced-state purity, and derivatives of the field characteristic functions. One of these currents yields, at the Fourier-space origin, a balance equation for a combination of atomic purity and coherence whose evolution is driven by atom--field coupling and joint atom--field moments. For globally pure states, the purity contribution provides an indirect connection with the development of atom--field entanglement. The symmetry structure generates generalized symmetries and an infinite hierarchy of conservation laws.

arXiv:2606.15538 [math-ph]

(or arXiv:2606.15538v2 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2606.15538 Focus to learn more arXiv-issued DOI via DataCite

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Generalized symmetries, invariant solutions and conservation laws in the Jaynes-Cummings model

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