Fractional-Time Jaynes–Cummings Model with Unitary Description: Dynamics for Binomial State of Light

Fractional calculus provides useful extensions to models based on ordinary calculus, enabling the description of physical effects such as dissipation and memory. A notable application of this framework in quantum mechanics is the fractional-time Schrödinger equation (FTSE), in which the standard time derivative is replaced by a Caputo derivative carrying a power-law memory kernel. This modification, however, inherently leads to non-unitary evolution of the quantum state. In this work, we apply the FTSE within the paradigmatic Jaynes–Cummings (JC) model to study the evolution of a binomial state of light interacting with matter. The binomial distribution that characterizes the binomial states possesses both coherent and number states as special cases, while the JC model is a cornerstone for studying quantum light–matter interactions, with experimental validation in cavity quantum electrodynamics and applications in quantum information processing. To restore unitarity in our analysis, we follow a recently introduced technique based on time-dependent Dyson maps -- invertible operators $\eta(t)$ that relate the unitary $\left(\hat{u}(t)\right)$ and non-unitary $\left(\hat{U}(t)\right)$ evolution operators via $\hat{u}(t) = \eta(t)\hat{U}(t)\eta^{-1}(0)$. Three distinct binomial distributions are analyzed with population inversion as the primary figure of merit. We show how different derivative orders $\left(\alpha\right)$ distinctly influence the dynamics: a decreasing number of oscillations $\left(\alpha=0.75\right)$, periodicity $\left(\alpha=0.50\right)$, and aperiodicity $\left(\alpha=0.40\right)$.