From Complexity to Quantum Mechanics: Nonlinearities Underlying Quantum Mechanics

Olivier Brodier, Peter Schlagheck, Denis Ullmo

We present evidence that tunneling processes in near-integrable systems are enhanced due to the manifestation of nonlinear resonances and their respective island chains in phase space. A semiclassical description of this "resonance-assisted" mechanism is given, which is based on a local perturbative description of the dynamics in the vicinity of the resonances. As underlying picture, we obtain that the quantum state is coupled, via a succession of classically forbidden transitions across nonlinear resonances, to high excitations within the well, from where tunneling occurs with a rather large rate. The connection between this description and the complex classical structure of the underlying integrable dynamics is furthermore studied, giving ground to the general coherence of the description as well as guidelines for the identification of the dominant tunneling paths. The validity of this mechanism is demonstrated within the kicked Harper model, where good agreement between quantum and semiclassical (resonance-assisted) tunneling rates is found.

Dieter Schuch

Jincheng Zhang

Adaptive Geometric Topological Fields (AGTF) represents a novel approach to modeling complex systems, particularly within the realms of nonlinear dynamics and quantum information. This research investigates the potential of utilizing quantum entanglement to generate dynamic geometric structures within mathematical fields, enabling the capture of information previously inaccessible through conventional methods. The core mechanism centers around a quantum field description that responds to correlated observations, dynamically altering the field state to reveal underlying complexities. This paper explores the implications of this framework, highlighting its potential for revolutionizing various fields through the integration of quantum mechanics and topology. The research aims to establish the foundational principles and preliminary results supporting the concept and its potential impact.

Gilbert Reinisch

Marek Czachor

Sebastian Deffner, Steve Campbell

Abstract Many complex quantum systems can be described by effectively nonlinear dynamics. While such dynamics have many appealing characteristics, they also make the analysis significantly more involved. This is due to the fact that only a few analytical treatments exist, and that the language of quantum mechanics is built for linear operators. For instance, the very formulation of the quantum adiabatic theorem requires the underlying dynamics to be linear. In this work we show that in a generalized Landau-Zener model, nonlinear dynamics can be leveraged to suppress excitations and coherences of the corresponding linear scenario. To this end, we introduce a generalized “energy spectrum”, which is defined by the expectation values of the energy under the stationary states. As a main result, we show that the nonlinear term in the evolution equation acts like an effective shortcut to adiabaticity for the linear Landau-Zener problem.