Related papers: Around quantum ergodicity
We study a generic model of quantum computer, composed of many qubits coupled by short-range interaction. Above a critical interqubit coupling strength, quantum chaos sets in, leading to quantum ergodicity of the computer eigenstates. In…
With a q-deformed quantum mechanical framework, features of the uncertainty relation and a novel formulation of the Schr\"odinger equation are considered.
The standard generic quantum computer model is studied analytically and numerically and the border for emergence of quantum chaos, induced by imperfections and residual inter-qubit couplings, is determined. This phenomenon appears in an…
After a brief historical review of ergodicity and mixing in dynamics, particularly in quantum dynamics, we introduce definitions of quantum ergodicity and mixing using the structure of the system's energy levels and spacings. Our…
The quantum counterpart of the classically chaotic kicked rotor is investigated using Bohm's appraoch to quantum theory.
Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigenstates are, in average, uniformly distributed on energy surfaces. We show that if N is a hypersurface in the position space satisfying a simple…
Nowadays there is no universally accepted definition of quantum chaos. In this paper we review and critically discuss different approaches to the subject, such as Quantum Chaology and the Random Matrix Theory. Then we analyze the problem of…
Chaos and ergodicity are the cornerstones of statistical physics and thermodynamics. While classically even small systems like a particle in a two-dimensional cavity, can exhibit chaotic behavior and thereby relax to a microcanonical…
The main theorem has been slightly generalized to include a larger class of symbols.
We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in…
This is an expository article for the Encyclopedia of Mathematical Physics on the subject in the title.
The ergodic hypothesis outgrew from the ancient conception of motion as periodic or quasi periodic. It did cause a revision of our views of motion, particularly through Boltzmann and Poincar\'e: we discuss how Boltmann's conception of…
We investigate the equidistribution of the eigenfunctions on quantum graphs in the high-energy limit. Our main result is an estimate of the deviations from equidistribution for large well-connected graphs. We use an exact field-theoretic…
We explore the border between regular and chaotic quantum dynamics, characterized by a power law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. This…
This article is the written version of a talk delivered at the Bexbach Colloquium of Science 2000 and starts with an introduction into quantum chaos and its relationship to classical chaos. The Bohigas-Giannoni-Schmit conjecture is…
In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral…
A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum…
This paper contains a very simple and general proof that eigenfunctions of quantizations of classically ergodic systems become uniformly distributed in phase space. This ergodicity property of eigenfunctions f is shown to follow from a…
We propose a version of the Quantum Ergodicity theorem on large regular graphs of fixed valency. This is a property of delocalization of "most" eigenfunctions. We consider expander graphs with few short cycles (for instance random large…
We prove a Egorov theorem, or quantum-classical correspondence, for the quantised baker's map, valid up to the Ehrenfest time. This yields a logarithmic upper bound for the decay of the quantum variance, and, as a corollary, a quantum…