1,666 research outputs found
The triangle map: a model of quantum chaos
We study an area preserving parabolic map which emerges from the Poincar\' e
map of a billiard particle inside an elongated triangle. We provide numerical
evidence that the motion is ergodic and mixing. Moreover, when considered on
the cylinder, the motion appear to follow a gaussian diffusive process.Comment: 4 pages in RevTeX with 4 figures (in 6 eps-files
Dynamical Localization: Hydrogen Atoms in Magnetic and Microwave fields
We show that dynamical localization for excited hydrogen atoms in magnetic
and microwave fields takes place at quite low microwave frequency much lower
than the Kepler frequency. The estimates of localization length are given for
different parameter regimes, showing that the quantum delocalization border
drops significantly as compared to the case of zero magnetic field. This opens
up broad possibilities for laboratory investigations.Comment: revtex, 11 pages, 3 figures, to appear in Phys. Rev. A, Feb (1997
-Kicked Quantum Rotors: Localization and `Critical' Statistics
The quantum dynamics of atoms subjected to pairs of closely-spaced
-kicks from optical potentials are shown to be quite different from the
well-known paradigm of quantum chaos, the singly--kicked system. We
find the unitary matrix has a new oscillating band structure corresponding to a
cellular structure of phase-space and observe a spectral signature of a
localization-delocalization transition from one cell to several. We find that
the eigenstates have localization lengths which scale with a fractional power
and obtain a regime of near-linear spectral variances
which approximate the `critical statistics' relation , where is related to the fractal
classical phase-space structure. The origin of the exponent
is analyzed.Comment: 4 pages, 3 fig
Quantum localization and cantori in chaotic billiards
We study the quantum behaviour of the stadium billiard. We discuss how the
interplay between quantum localization and the rich structure of the classical
phase space influences the quantum dynamics. The analysis of this model leads
to new insight in the understanding of quantum properties of classically
chaotic systems.Comment: 4 pages in RevTex with 4 eps figures include
The Sato Grassmannian and the CH hierarchy
We discuss how the Camassa-Holm hierarchy can be framed within the geometry
of the Sato Grassmannian.Comment: 10 pages, no figure
Quantum Poincare Recurrences for Hydrogen Atom in a Microwave Field
We study the time dependence of the ionization probability of Rydberg atoms
driven by a microwave field, both in classical and in quantum mechanics. The
quantum survival probability follows the classical one up to the Heisenberg
time and then decays algebraically as P(t) ~ 1/t. This decay law derives from
the exponentially long times required to escape from some region of the phase
space, due to tunneling and localization effects. We also provide parameter
values which should allow to observe such decay in laboratory experiments.Comment: revtex, 4 pages, 4 figure
Parametric Evolution for a Deformed Cavity
We consider a classically chaotic system that is described by a Hamiltonian
H(Q,P;x), where (Q,P) describes a particle moving inside a cavity, and x
controls a deformation of the boundary. The quantum-eigenstates of the system
are |n(x)>. We describe how the parametric kernel P(n|m) = , also
known as the local density of states, evolves as a function of x-x0. We
illuminate the non-unitary nature of this parametric evolution, the emergence
of non-perturbative features, the final non-universal saturation, and the
limitations of random-wave considerations. The parametric evolution is
demonstrated numerically for two distinct representative deformation processes.Comment: 13 pages, 8 figures, improved introduction, to be published in Phys.
Rev.
Entanglement between two subsystems, the Wigner semicircle and extreme value statistics
The entanglement between two arbitrary subsystems of random pure states is
studied via properties of the density matrix's partial transpose,
. The density of states of is close to the
semicircle law when both subsystems have dimensions which are not too small and
are of the same order. A simple random matrix model for the partial transpose
is found to capture the entanglement properties well, including a transition
across a critical dimension. Log-negativity is used to quantify entanglement
between subsystems and analytic formulas for this are derived based on the
simple model. The skewness of the eigenvalue density of is
derived analytically, using the average of the third moment over the ensemble
of random pure states. The third moment after partial transpose is also shown
to be related to a generalization of the Kempe invariant. The smallest
eigenvalue after partial transpose is found to follow the extreme value
statistics of random matrices, namely the Tracy-Widom distribution. This
distribution, with relevant parameters obtained from the model, is found to be
useful in calculating the fraction of entangled states at critical dimensions.
These results are tested in a quantum dynamical system of three coupled
standard maps, where one finds that if the parameters represent a strongly
chaotic system, the results are close to those of random states, although there
are some systematic deviations at critical dimensions.Comment: Substantially improved version (now 43 pages, 10 figures) that is
accepted for publication in Phys. Rev.
Entanglement and dynamics of spin-chains in periodically-pulsed magnetic fields: accelerator modes
We study the dynamics of a single excitation in a Heisenberg spin-chain
subjected to a sequence of periodic pulses from an external, parabolic,
magnetic field. We show that, for experimentally reasonable parameters, a pair
of counter-propagating coherent states are ejected from the centre of the
chain. We find an illuminating correspondence with the quantum time evolution
of the well-known paradigm of quantum chaos, the Quantum Kicked Rotor (QKR).
From this we can analyse the entanglement production and interpret the
ejected coherent states as a manifestation of so-called `accelerator modes' of
a classically chaotic system.Comment: 5 pages, 2 figures; minor corrections, tidied presentatio
Classical diffusion in double-delta-kicked particles
We investigate the classical chaotic diffusion of atoms subjected to {\em
pairs} of closely spaced pulses (`kicks) from standing waves of light (the
-KP). Recent experimental studies with cold atoms implied an
underlying classical diffusion of type very different from the well-known
paradigm of Hamiltonian chaos, the Standard Map.
The kicks in each pair are separated by a small time interval , which together with the kick strength , characterizes the transport.
Phase space for the -KP is partitioned into momentum `cells' partially
separated by momentum-trapping regions where diffusion is slow. We present here
an analytical derivation of the classical diffusion for a -KP
including all important correlations which were used to analyze the
experimental data.
We find a new asymptotic () regime of `hindered' diffusion:
while for the Standard Map the diffusion rate, for , oscillates about the uncorrelated, rate , we find
analytically, that the -KP can equal, but never diffuses faster than,
a random walk rate.
We argue this is due to the destruction of the important classical
`accelerator modes' of the Standard Map.
We analyze the experimental regime , where
quantum localisation lengths are affected by fractal
cell boundaries. We find an approximate asymptotic diffusion rate , in correspondence to a regime in the Standard Map
associated with 'golden-ratio' cantori.Comment: 14 pages, 10 figures, error in equation in appendix correcte
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