5,726 research outputs found
Perturbations and chaos in quantum maps
The local density of states (LDOS) is a distribution that characterizes the
effect of perturbations on quantum systems. Recently, it was proposed a
semiclassical theory for the LDOS of chaotic billiards and maps. This theory
predicts that the LDOS is a Breit-Wigner distribution independent of the
perturbation strength and also gives a semiclassical expression for the LDOS
witdth. Here, we test the validity of such an approximation in quantum maps
varying the degree of chaoticity, the region in phase space where the
perturbation is applying and the intensity of the perturbation. We show that
for highly chaotic maps or strong perturbations the semiclassical theory of the
LDOS is accurate to describe the quantum distribution. Moreover, the width of
the LDOS is also well represented for its semiclassical expression in the case
of mixed classical dynamics.Comment: 9 pages, 11 figures. Accepted for publication in Phys. Rev.
Hypersensitivity and chaos signatures in the quantum baker's maps
Classical chaotic systems are distinguished by their sensitive dependence on
initial conditions. The absence of this property in quantum systems has lead to
a number of proposals for perturbation-based characterizations of quantum
chaos, including linear growth of entropy, exponential decay of fidelity, and
hypersensitivity to perturbation. All of these accurately predict chaos in the
classical limit, but it is not clear that they behave the same far from the
classical realm. We investigate the dynamics of a family of quantizations of
the baker's map, which range from a highly entangling unitary transformation to
an essentially trivial shift map. Linear entropy growth and fidelity decay are
exhibited by this entire family of maps, but hypersensitivity distinguishes
between the simple dynamics of the trivial shift map and the more complicated
dynamics of the other quantizations. This conclusion is supported by an
analytical argument for short times and numerical evidence at later times.Comment: 32 pages, 6 figure
Lyapunov decay in quantum irreversibility
The Loschmidt echo -- also known as fidelity -- is a very useful tool to
study irreversibility in quantum mechanics due to perturbations or
imperfections. Many different regimes, as a function of time and strength of
the perturbation, have been identified. For chaotic systems, there is a range
of perturbation strengths where the decay of the Loschmidt echo is perturbation
independent, and given by the classical Lyapunov exponent. But observation of
the Lyapunov decay depends strongly on the type of initial state upon which an
average is done. This dependence can be removed by averaging the fidelity over
the Haar measure, and the Lyapunov regime is recovered, as it was shown for
quantum maps. In this work we introduce an analogous quantity for systems with
infinite dimensional Hilbert space, in particular the quantum stadium billiard,
and we show clearly the universality of the Lyapunov regime.Comment: 8 pages, 6 figures. Accepted in Phil. Trans. R. Soc.
Loschmidt Echo and the Local Density of States
Loschmidt echo (LE) is a measure of reversibility and sensitivity to
perturbations of quantum evolutions. For weak perturbations its decay rate is
given by the width of the local density of states (LDOS). When the perturbation
is strong enough, it has been shown in chaotic systems that its decay is
dictated by the classical Lyapunov exponent. However, several recent studies
have shown an unexpected non-uniform decay rate as a function of the
perturbation strength instead of that Lyapunov decay. Here we study the
systematic behavior of this regime in perturbed cat maps. We show that some
perturbations produce coherent oscillations in the width of LDOS that imprint
clear signals of the perturbation in LE decay. We also show that if the
perturbation acts in a small region of phase space (local perturbation) the
effect is magnified and the decay is given by the width of the LDOS.Comment: 8 pages, 8 figure
Experimental Implementation of the Quantum Baker's Map
This paper reports on the experimental implementation of the quantum baker's
map via a three bit nuclear magnetic resonance (NMR) quantum information
processor. The experiments tested the sensitivity of the quantum chaotic map to
perturbations. In the first experiment, the map was iterated forward and then
backwards to provide benchmarks for intrinsic errors and decoherence. In the
second set of experiments, the least significant qubit was perturbed in between
the iterations to test the sensitivity of the quantum chaotic map to applied
perturbations. These experiments are used to investigate previous predicted
properties of quantum chaotic dynamics.Comment: submitted to PR
On the Emergence of Nonextensivity at the Edge of Quantum Chaos
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 region corresponds to the edge of chaos for the classical map
from which the quantum chaotic dynamics is derived and can be characterized via
nonextensive entropy concepts.Comment: Invited paper to appear in "Decoherence and Entropy in Complex
Systems", ed. H.T. Elze, Lecture Notes in Physics (Springer, Heidelberg), in
press. 13 pages including 6 figures and 1 tabl
Time Quasilattices in Dissipative Dynamical Systems
We establish the existence of `time quasilattices' as stable trajectories in
dissipative dynamical systems. These tilings of the time axis, with two unit
cells of different durations, can be generated as cuts through a periodic
lattice spanned by two orthogonal directions of time. We show that there are
precisely two admissible time quasilattices, which we term the infinite Pell
and Clapeyron words, reached by a generalization of the period-doubling
cascade. Finite Pell and Clapeyron words of increasing length provide
systematic periodic approximations to time quasilattices which can be verified
experimentally. The results apply to all systems featuring the universal
sequence of periodic windows. We provide examples of discrete-time maps, and
periodically-driven continuous-time dynamical systems. We identify quantum
many-body systems in which time quasilattices develop rigidity via the
interaction of many degrees of freedom, thus constituting dissipative discrete
`time quasicrystals'.Comment: 38 pages, 14 figures. This version incorporates "Pell and Clapeyron
Words as Stable Trajectories in Dynamical Systems", arXiv:1707.09333.
Submission to SciPos
Contributions of plasma physics to chaos and nonlinear dynamics
This topical review focusses on the contributions of plasma physics to chaos
and nonlinear dynamics bringing new methods which are or can be used in other
scientific domains. It starts with the development of the theory of Hamiltonian
chaos, and then deals with order or quasi order, for instance adiabatic and
soliton theories. It ends with a shorter account of dissipative and high
dimensional Hamiltonian dynamics, and of quantum chaos. Most of these
contributions are a spin-off of the research on thermonuclear fusion by
magnetic confinement, which started in the fifties. Their presentation is both
exhaustive and compact. [15 April 2016
Semiclassical approach to fidelity amplitude
The fidelity amplitude is a quantity of paramount importance in echo type
experiments. We use semiclassical theory to study the average fidelity
amplitude for quantum chaotic systems under external perturbation. We explain
analytically two extreme cases: the random dynamics limit --attained
approximately by strongly chaotic systems-- and the random perturbation limit,
which shows a Lyapunov decay. Numerical simulations help us bridge the gap
between both extreme cases.Comment: 10 pages, 9 figures. Version closest to published versio
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