kicked rotor

kicked rotor
Darth Rhombus · CC BY-SA 3.0 · source
Name	Kicked rotor
Type	Dynamical system
Introduced	1979
Notable	Chaos, dynamical localization, quantum chaos

kicked rotor

The kicked rotor is a paradigmatic time-periodic mechanical model studied in nonlinear dynamics, classical chaos, and quantum chaos. It appears in theoretical investigations by connecting maps such as the standard map to physical systems probed by researchers from institutions including Princeton University, Massachusetts Institute of Technology, California Institute of Technology, and laboratories associated with Bell Labs and Los Alamos National Laboratory. Its relevance spans topics addressed at conferences like Dynamical Systems Conference and in journals edited by American Physical Society and Institute of Physics publishers.

Introduction

The model idealizes a rigid rotor subject to periodic impulsive torques, producing a discrete-time mapping closely related to the Chirikov standard map and to studies by Boris Chirikov, F. Job, and collaborators at Kurchatov Institute. It serves as a bridge between classical models analyzed in works by Henri Poincaré and modern quantum investigations inspired by researchers such as Casati, Berry, and Izrailev. The kicked rotor informed experimental programs at institutions like Stanford University and Imperial College London and has influenced mathematical treatments found in texts associated with Cambridge University Press and Springer.

Classical dynamics

Classically, the rotor’s phase space evolution is described by a stroboscopic area-preserving map equivalent to the Chirikov standard map studied by Boris Chirikov and elaborated in analyses by Jürgen Moser and Lyndon B. Johnson-era applied mathematicians. The map exhibits transition to widespread chaos via overlap of nonlinear resonances originally analyzed in the context of Kolmogorov–Arnold–Moser (KAM) theory developed by Andrey Kolmogorov, Vladimir Arnold, and Jürgen Moser. Studies connect the rotor to transport mechanisms explored in the work of Pierre-Gilles de Gennes and to diffusion phenomena considered by Albert Einstein and Norbert Wiener in stochastic theory. Classical observables such as rotor momentum follow diffusive scaling until constrained by phase-space structures like cantori described in investigations by John Mather and Robert MacKay.

Quantum kicked rotor

The quantum version replaces classical phase-space variables with operators and exhibits interference phenomena tied to foundational results by Paul Dirac, Werner Heisenberg, and Erwin Schrödinger. The model was quantized and analyzed by Casati, Chirikov, and B. V. Chirikov collaborators, linking to research on level statistics associated with Freeman Dyson ensembles and semiclassical techniques advanced by Michael Berry and Mark Gutzwiller. Quantum resonances and quasienergy spectra refer to Floquet theory developed in contexts involving G. Floquet and studied with methods used by L. Landau and Rudolf Peierls in time-periodic quantum systems. Notable theoretical results relate the rotor’s behavior to Anderson-type localization mechanisms articulated originally by Philip W. Anderson.

Experimental realizations

Physical implementations have been realized using cold-atom platforms pioneered by experimentalists at University of California, Berkeley, NIST, Max Planck Institute, and Ecole Normale Supérieure. Experiments use pulsed optical lattices and standing-wave lasers following techniques from groups led by Mark Raizen, Gustav Weidemueller, and Ian Walmsley, connecting to laser cooling and trapping methods developed by Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips. Other realizations employ microwave-driven superconducting circuits studied at Yale University and Oxford University, and photonic waveguide arrays fabricated in facilities associated with Bell Labs and Riken. Experimental signatures are measured with detection methods used in collaborations with CERN-affiliated teams and national laboratories such as Lawrence Berkeley National Laboratory.

Chaos and dynamical localization

The model provided the first clear demonstration that classically diffusive momentum growth can be halted in the quantum regime by interference, a phenomenon termed dynamical localization that reflects concepts introduced by Philip W. Anderson in condensed matter physics. The connection motivated cross-disciplinary work linking quantum chaos studies by Casati and Berry to localization theory developed in condensed-matter communities around P. A. Lee and T. V. Ramakrishnan. Analytical and numerical characterizations use techniques from random-matrix theory advanced by Eugene Wigner and Mehta and from spectral theory associated with Mark Kac and Peter Lax. Studies of stability, correlation decay, and eigenstate structure reference rigorous results stemming from research programs at Institute for Advanced Study and mathematical physics groups at ETH Zurich.

Extensions and generalizations

Extensions include quasi-periodic driving investigated by groups at Princeton University and ETH Zurich, many-body variants connected to research at Perimeter Institute and Institute for Quantum Information and Matter, and kicked-top analogues studied by Alexey Kitaev-inspired communities. Generalizations explore interactions, open-system coupling addressed by John Preskill-affiliated teams, and topological phenomena analyzed in the context of work by David Thouless and Charles Kane. The kicked rotor continues to inform areas ranging from quantum information experiments at IBM Research to astrophysical applications considered by researchers at Jet Propulsion Laboratory and theoretical studies at Harvard University.

Category:Dynamical systems Category:Quantum chaos