Fractional Revival Dynamics in Kerr-Type Systems: Angular Momentum Moments and Classical Analogs

TL;DR

The paper develops observable-based diagnostics for quantum revivals and fractional revivals in Kerr-type nonlinear systems, focusing on angular-momentum moments in addition to standard autocorrelation measures. By deriving explicit time evolutions for moments and demonstrating that higher-order angular-momentum moments selectively reveal fractional revivals, the work provides a practical toolkit for experimental detection. Using the Kerr Hamiltonian $H=\hat{N}(\hat{N}-1)\hbar\chi$ (or $H=a^{\dagger 2}a^2\hbar\chi$) as a paradigmatic model, it analyzes autocorrelation, phase-space dynamics, and quantum carpets to illustrate revival phenomena and their quantum-classical correspondences. The study also draws classical analogs—like the Talbot effect and dancing pendulums—to place revival behavior within a broader dynamical context, and outlines computational approaches (e.g., QuTiP) for visualizing observables. Overall, the results broaden experimental diagnostics of fractional revivals and establish angular-momentum moments as sensitive, interpretable indicators of revival structure with potential applications in quantum optics and spin-based platforms.

Abstract

Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.

Figures (24)

• Figure 1: The absolute squares of the co-efficients of coherent states with for i) $\nu = 1$, ii) $\nu = 10$, iii)$\nu = 100$ for $H = (a^{\dagger}a+\frac{1}{2})\hbar \chi$.
• Figure 2: Absolute square of Auto Correlation Function for i) $\nu = 1$, ii) $\nu = 10$, iii)$\nu = 100$ for $H = (a^{\dagger}a+\frac{1}{2})\hbar \chi$.
• Figure 3: Absolute square of Auto Correlation Function for i) $\nu = 1$, ii) $\nu = \sqrt{10}$, iii) $\nu = 10$ for $H = a^{\dagger 2}a^2 \hbar\chi$
• Figure 4: An Optical Talbot Carpet. https://upload.wikimedia.org/wikipedia/commons/6/69/Optical_Talbot_Carpet.png
• Figure 5: An apparatus of 15 pendulums, with monotonically increasing lengths. The link for this animation is provided in Appendix C