68 research outputs found
Exponential mixing and log h time scales in quantized hyperbolic maps on the torus
We study the behaviour, in the simultaneous limits \hbar going to 0, t going
to \infty, of the Husimi and Wigner distributions of initial coherent states
and position eigenstates, evolved under the quantized hyperbolic toral
automorphisms and the quantized baker map. We show how the exponential mixing
of the underlying dynamics manifests itself in those quantities on time scales
logarithmic in \hbar. The phase space distributions of the coherent states,
evolved under either of those dynamics, are shown to equidistribute on the
torus in the limit \hbar going to 0, for times t between |\log\hbar|/(2\gamma)
and |\log|\hbar|/\gamma, where \gamma is the Lyapounov exponent of the
classical system. For times shorter than |\log\hbar|/(2\gamma), they remain
concentrated on the classical trajectory of the center of the coherent state.
The behaviour of the phase space distributions of evolved position eigenstates,
on the other hand, is not the same for the quantized automorphisms as for the
baker map. In the first case, they equidistribute provided t goes to \infty as
\hbar goes to 0, and as long as t is shorter than |\log\hbar|/\gamma. In the
second case, they remain localized on the evolved initial position at all such
times
Scarred eigenstates for quantum cat maps of minimal periods
In this paper we construct a sequence of eigenfunctions of the ``quantum
Arnold's cat map'' that, in the semiclassical limit, show a strong scarring
phenomenon on the periodic orbits of the dynamics. More precisely, those states
have a semiclassical limit measure that is the sum of 1/2 the normalized
Lebesgue measure on the torus plus 1/2 the normalized Dirac measure
concentrated on any a priori given periodic orbit of the dynamics. It is known
(the Schnirelman theorem) that ``most'' sequences of eigenfunctions
equidistribute on the torus. The sequences we construct therefore provide an
example of an exception to this general rule. Our method of construction and
proof exploits the existence of special values of Planck's constant for which
the quantum period of the map is relatively ``short'', and a sharp control on
the evolution of coherent states up to this time scale. We also provide a
pointwise description of these states in phase space, which uncovers their
``hyperbolic'' structure in the vicinity of the fixed points and yields more
precise localization estimates.Comment: LaTeX, 49 pages, includes 10 figures. I added section 6.6. To be
published in Commun. Math. Phy
Heteroclinic structure of parametric resonance in the nonlinear Schr\"odinger equation
We show that the nonlinear stage of modulational instability induced by
parametric driving in the {\em defocusing} nonlinear Schr\"odinger equation can
be accurately described by combining mode truncation and averaging methods,
valid in the strong driving regime. The resulting integrable oscillator reveals
a complex hidden heteroclinic structure of the instability. A remarkable
consequence, validated by the numerical integration of the original model, is
the existence of breather solutions separating different Fermi-Pasta-Ulam
recurrent regimes. Our theory also shows that optimal parametric amplification
unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues)
arising from the linearised Floquet analysis
Spectral flow and level spacing of edge states for quantum Hall hamiltonians
We consider a non relativistic particle on the surface of a semi-infinite
cylinder of circumference submitted to a perpendicular magnetic field of
strength and to the potential of impurities of maximal amplitude . This
model is of importance in the context of the integer quantum Hall effect. In
the regime of strong magnetic field or weak disorder it is known that
there are chiral edge states, which are localised within a few magnetic lengths
close to, and extended along the boundary of the cylinder, and whose energy
levels lie in the gaps of the bulk system. These energy levels have a spectral
flow, uniform in , as a function of a magnetic flux which threads the
cylinder along its axis. Through a detailed study of this spectral flow we
prove that the spacing between two consecutive levels of edge states is bounded
below by with , independent of , and of the
configuration of impurities. This implies that the level repulsion of the
chiral edge states is much stronger than that of extended states in the usual
Anderson model and their statistics cannot obey one of the Gaussian ensembles.
Our analysis uses the notion of relative index between two projections and
indicates that the level repulsion is connected to topological aspects of
quantum Hall systems.Comment: 22 pages, no figure
Simultaneous quantization of edge and bulk Hall conductivity
The edge Hall conductivity is shown to be an integer multiple of
which is almost surely independent of the choice of the disordered
configuration. Its equality to the bulk Hall conductivity given by the
Kubo-Chern formula follows from K-theoretic arguments. This leads to
quantization of the Hall conductance for any redistribution of the current in
the sample. It is argued that in experiments at most a few percent of the total
current can be carried by edge states.Comment: 6 pages Latex, 1 figur
One-loop approximation of Moller scattering in Krein-space quantization
It has been shown that the negative-norm states necessarily appear in a
covariant quantization of the free minimally coupled scalar field in de Sitter
spacetime [1,2]. In this processes ultraviolet and infrared divergences have
been automatically eliminated [3]. A natural renormalization of the one-loop
interacting quantum field in Minkowski spacetime () has been
achieved through the consideration of the negative-norm states defined in Krein
space. It has been shown that the combination of quantum field theory in Krein
space together with consideration of quantum metric fluctuation, results in
quantum field theory without any divergences [4]. Pursuing this approach, we
express Wick's theorem and calculate M{\o}ller scattering in the one-loop
approximation in Krein space. The mathematical consequence of this method is
the disappearance of the ultraviolet divergence in the one-loop approximation.Comment: 10 page
Properties and Applications of the Kirkwood-Dirac Distribution
The most famous quasi-probability distribution, the Wigner function, has
played a pivotal role in the development of a continuous-variable quantum
theory that has clear analogues of position and momentum. However, the Wigner
function is ill-suited for much modern quantum-information research, which is
focused on finite-dimensional systems and general observables. Instead, recent
years have seen the Kirkwood-Dirac (KD) distribution come to the forefront as a
powerful quasi-probability distribution for analysing quantum mechanics. The KD
distribution allows tools from statistics and probability theory to be applied
to problems in quantum-information processing. A notable difference to the
Wigner function is that the KD distribution can represent a quantum state in
terms of arbitrary observables. This paper reviews the KD distribution, in
three parts. First, we present definitions and basic properties of the KD
distribution and its generalisations. Second, we summarise the KD
distribution's extensive usage in the study or development of measurement
disturbance; quantum metrology; weak values; direct measurements of quantum
states; quantum thermodynamics; quantum scrambling and out-of-time-ordered
correlators; and the foundations of quantum mechanics, including Leggett-Garg
inequalities, the consistent-histories interpretation, and contextuality. We
emphasise connections between operational quantum advantages and negative or
non-real KD quasi-probabilities. Third, we delve into the KD distribution's
mathematical structure. We summarise the current knowledge regarding the
geometry of KD-positive states (the states for which the KD distribution is a
classical probability distribution), describe how to witness and quantify KD
non-positivity, and outline relationships between KD non-positivity and
observables' incompatibility.Comment: 37 pages, 13 figure
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