Fractional Nonlinear Quantum Dynamics

Nick Laskin

We show that a quantum system with long-range interparticle interaction can be described by invoking fractional calculus tools. The system under consideration is nonlocal exciton-phonon quantum dynamics on a 1D lattice. It has been shown that long-range power-law exciton-exciton interaction leads to a nonlocal integral term in the motion equation of an exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction with noninteger power, the nonlocal integral term can be expressed through a spatial derivative of fractional order. Considering exciton-phonon dynamics with long-range exciton-exciton interaction, we have obtained the system of two coupled equations, where one is the quantum fractional differential equation for the exciton subsystem while the other is a standard differential equation for the phonon subsystem. It has been found that the system of two coupled equations can be further simplified to come up with the following fractional nonlinear equations of motion: nonlinear fractional Schrödinger equation, nonlinear Hilbert-Schrödinger equation, fractional generalization of Zakharov system, and fractional Ginzburg-Landau equation. The appearance of fractional differential equations in the continuum limit of lattice dynamics allows us to apply powerful tools of fractional calculus to study nonlocal quantum phenomena.

Longzhao Lu, Xiangyang Yu

Alberto Nardin, Iacopo Carusotto

We report numerical studies of the linear and nonlinear edge dynamics of a non-harmonically-confined macroscopic fractional quantum Hall fluid. In the long-wavelength and weak excitation limit, observable consequences of the fractional transverse conductivity are recovered. The first nonuniversal corrections to the chiral Luttinger liquid theory are then characterized: for a weak excitation in the linear response regime, cubic corrections to the linear wave dispersion and a broadening of the dynamical structure factor of the edge excitations are identified; for stronger excitations, sizable nonlinear effects are found in the dynamics. The numerically observed features are quantitatively captured by a nonlinear chiral Luttinger liquid quantum Hamiltonian that reduces to a driven Korteweg--de Vries equation in the semiclassical limit. Experimental observability of our predictions is finally discussed.

Ashish Kumar Patra, Saikumar Krithivasan

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.

Salman Habib

Vasily Evgenevich Tarasov