6,777 research outputs found
Quantum anomalies and linear response theory
The analysis of diffusive energy spreading in quantized chaotic driven
systems, leads to a universal paradigm for the emergence of a quantum anomaly.
In the classical approximation a driven chaotic system exhibits stochastic-like
diffusion in energy space with a coefficient that is proportional to the
intensity of the driving. In the corresponding quantized problem
the coherent transitions are characterized by a generalized Wigner time
, and a self-generated (intrinsic) dephasing process leads to
non-linear dependence of on .Comment: 8 pages, 2 figures, textual improvements (as in published version
Anatomy of quantum chaotic eigenstates
The eigenfunctions of quantized chaotic systems cannot be described by
explicit formulas, even approximate ones. This survey summarizes (selected)
analytical approaches used to describe these eigenstates, in the semiclassical
limit. The levels of description are macroscopic (one wants to understand the
quantum averages of smooth observables), and microscopic (one wants
informations on maxima of eigenfunctions, "scars" of periodic orbits, structure
of the nodal sets and domains, local correlations), and often focusses on
statistical results. Various models of "random wavefunctions" have been
introduced to understand these statistical properties, with usually good
agreement with the numerical data. We also discuss some specific systems (like
arithmetic ones) which depart from these random models.Comment: Corrected typos, added a few references and updated some result
Husimi-Wigner representation of chaotic eigenstates
Just as a coherent state may be considered as a quantum point, its
restriction to a factor space of the full Hilbert space can be interpreted as a
quantum plane. The overlap of such a factor coherent state with a full pure
state is akin to a quantum section. It defines a reduced pure state in the
cofactor Hilbert space. The collection of all the Wigner functions
corresponding to a full set of parallel quantum sections defines the
Husimi-Wigner reresentation. It occupies an intermediate ground between drastic
suppression of nonclassical features, characteristic of Husimi functions, and
the daunting complexity of higher dimensional Wigner functions. After analysing
these features for simpler states, we exploit this new representation as a
probe of numerically computed eigenstates of chaotic Hamiltonians. The
individual two-dimensional Wigner functions resemble those of semiclassically
quantized states, but the regular ring pattern is broken by dislocations.Comment: 21 pages, 7 figures (6 color figures), submitted to Proc. R. Soc.
Regular and Irregular States in Generic Systems
In this work we present the results of a numerical and semiclassical analysis
of high lying states in a Hamiltonian system, whose classical mechanics is of a
generic, mixed type, where the energy surface is split into regions of regular
and chaotic motion. As predicted by the principle of uniform semiclassical
condensation (PUSC), when the effective tends to 0, each state can be
classified as regular or irregular. We were able to semiclassically reproduce
individual regular states by the EBK torus quantization, for which we devise a
new approach, while for the irregular ones we found the semiclassical
prediction of their autocorrelation function, in a good agreement with
numerics. We also looked at the low lying states to better understand the onset
of semiclassical behaviour.Comment: 25 pages, 14 figures (as .GIF files), high quality figures available
upon reques
Quantized chaotic dynamics and non-commutative KS entropy
We study the quantization of two examples of classically chaotic dynamics,
the Anosov dynamics of "cat maps" on a two dimensional torus, and the dynamics
of baker's maps. Each of these dynamics is implemented as a discrete group of
automorphisms of a von Neumann algebra of functions on a quantized torus. We
compute the non- commutative generalization of the Kolmogorov-Sinai entropy,
namely the Connes-Stormer entropy, of the generator of this group, and find
that its value is equal to the classical value. This can be interpreted as a
sign of persistence of chaotic behavior in a dynamical system under
quantization.Comment: a number of misprints corrected, new references and a new section
added. 21 pages, plain Te
Regular-to-Chaotic Tunneling Rates: From the Quantum to the Semiclassical Regime
We derive a prediction of dynamical tunneling rates from regular to chaotic
phase-space regions combining the direct regular-to-chaotic tunneling mechanism
in the quantum regime with an improved resonance-assisted tunneling theory in
the semiclassical regime. We give a qualitative recipe for identifying the
relevance of nonlinear resonances in a given -regime. For systems with
one or multiple dominant resonances we find excellent agreement to numerics.Comment: 4 pages, 3 figures, reference added, small text change
Non-perturbative response: chaos versus disorder
Quantized chaotic systems are generically characterized by two energy scales:
the mean level spacing , and the bandwidth . This
implies that with respect to driving such systems have an adiabatic, a
perturbative, and a non-perturbative regimes. A "strong" quantal
non-perturbative response effect is found for {\em disordered} systems that are
described by random matrix theory models. Is there a similar effect for
quantized {\em chaotic} systems? Theoretical arguments cannot exclude the
existence of a "weak" non-perturbative response effect, but our numerics
demonstrate an unexpected degree of semiclassical correspondence.Comment: 8 pages, 2 figures, final version to be published in JP
Fractal Weyl Law for Open Chaotic Maps
This contribution summarizes our work with M.Zworski on open quantum open
chaoticmaps (math-ph/0505034). For a simple chaotic scattering system (the open
quantum baker's map), we compute the "long-living resonances" in the
semiclassical r\'{e}gime, and show that they satisfy a fractal Weyl law. We can
prove this fractal law in the case of a modified model.Comment: Contribution to the Proceedings of the conference QMath9,
Mathematical Physics of Quantum Mechanics, September 12th-16th 2004, Giens,
Franc
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