84 research outputs found
A Uniform Approximation for the Fidelity in Chaotic Systems
In quantum/wave systems with chaotic classical analogs, wavefunctions evolve
in highly complex, yet deterministic ways. A slight perturbation of the system,
though, will cause the evolution to diverge from its original behavior
increasingly with time. This divergence can be measured by the fidelity, which
is defined as the squared overlap of the two time evolved states. For chaotic
systems, two main decay regimes of either Gaussian or exponential behavior have
been identified depending on the strength of the perturbation. For perturbation
strengths intermediate between the two regimes, the fidelity displays both
forms of decay. By applying a complementary combination of random matrix and
semiclassical theory, a uniform approximation can be derived that covers the
full range of perturbation strengths. The time dependence is entirely fixed by
the density of states and the so-called transition parameter, which can be
related to the phase space volume of the system and the classical action
diffusion constant, respectively. The accuracy of the approximations are
illustrated with the standard map.Comment: 16 pages, 4 figures, accepted in J. Phys. A, special edition on
Random Matrix Theor
Analyzing intramolecular vibrational energy redistribution via the overlap intensity-level velocity correlator
Numerous experimental and theoretical studies have established that
intramolecular vibrational energy redistribution (IVR) in isolated molecules
has a heirarchical tier structure. The tier structure implies strong
correlations between the energy level motions of a quantum system and its
intensity-weighted spectrum. A measure, which explicitly accounts for this
correaltion, was first introduced by one of us as a sensitive probe of phase
space localization. It correlates eigenlevel velocities with the overlap
intensities between the eigenstates and some localized state of interest. A
semiclassical theory for the correlation is developed for systems that are
classically integrable and complements earlier work focusing exclusively on the
chaotic case. Application to a model two dimensional effective spectroscopic
Hamiltonian shows that the correlation measure can provide information about
the terms in the molecular Hamiltonian which play an important role in an
energy range of interest and the character of the dynamics. Moreover, the
correlation function is capable of highlighting relevant phase space structures
including the local resonance features associated with a specific bright state.
In addition to being ideally suited for multidimensional systems with a large
density of states, the measure can also be used to gain insights into the phase
space transport and localization. It is argued that the overlap intensity-level
velocity correlation function provides a novel way of studying vibrational
energy redistribution in isolated molecules. The correlation function is
ideally suited to analyzing the parametric spectra of molecules in external
fields.Comment: 16 pages, 13 figures (low resolution
Complexity, Tunneling and Geometrical Symmetry
It is demonstrated in the context of the simple one-dimensional example of a
barrier in an infinite well, that highly complex behavior of the time evolution
of a wave function is associated with the almost degeneracy of levels in the
process of tunneling. Degenerate conditions are obtained by shifting the
position of the barrier. The complexity strength depends on the number of
almost degenerate levels which depend on geometrical symmetry. The presence of
complex behavior is studied to establish correlation with spectral degeneracy.Comment: 9 revtex pages, 6 Postscript figures (uuencoded
Quantum fingerprints of classical Ruelle-Pollicot resonances
N-disk microwave billiards, which are representative of open quantum systems,
are studied experimentally. The transmission spectrum yields the quantum
resonances which are consistent with semiclassical calculations. The spectral
autocorrelation of the quantum spectrum is shown to be determined by the
classical Ruelle-Pollicot resonances, arising from the complex eigenvalues of
the Perron-Frobenius operator. This work establishes a fundamental connection
between quantum and classical correlations in open systems.Comment: 6 pages, 2 eps figures included, submitted to PR
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.
Symmetry Decomposition of Potentials with Channels
We discuss the symmetry decomposition of the average density of states for
the two dimensional potential and its three dimensional
generalisation . In both problems, the energetically
accessible phase space is non-compact due to the existence of infinite channels
along the axes. It is known that in two dimensions the phase space volume is
infinite in these channels thus yielding non-standard forms for the average
density of states. Here we show that the channels also result in the symmetry
decomposition having a much stronger effect than in potentials without
channels, leading to terms which are essentially leading order. We verify these
results numerically and also observe a peculiar numerical effect which we
associate with the channels. In three dimensions, the volume of phase space is
finite and the symmetry decomposition follows more closely that for generic
potentials --- however there are still non-generic effects related to some of
the group elements
Signatures of Dynamical Tunneling in the Wave function of a Soft-Walled Open Microwave Billiard
Evidence for dynamical tunneling is observed in studies of the transmission,
and wave functions, of a soft-walled microwave cavity resonator. In contrast to
previous work, we identify the conditions for dynamical tunneling by monitoring
the evolution of the wave function phase as a function of energy, which allows
us to detect the tunneling process even under conditions where its expected
level splitting remains irresolvable.Comment: 5 pages, 5 figure
Multi-Layer Cyber-Physical Security and Resilience for Smart Grid
The smart grid is a large-scale complex system that integrates communication
technologies with the physical layer operation of the energy systems. Security
and resilience mechanisms by design are important to provide guarantee
operations for the system. This chapter provides a layered perspective of the
smart grid security and discusses game and decision theory as a tool to model
the interactions among system components and the interaction between attackers
and the system. We discuss game-theoretic applications and challenges in the
design of cross-layer robust and resilient controller, secure network routing
protocol at the data communication and networking layers, and the challenges of
the information security at the management layer of the grid. The chapter will
discuss the future directions of using game-theoretic tools in addressing
multi-layer security issues in the smart grid.Comment: 16 page
The Inhibition of Mixing in Chaotic Quantum Dynamics
We study the quantum chaotic dynamics of an initially well-localized wave
packet in a cosine potential perturbed by an external time-dependent force. For
our choice of initial condition and with small but finite, we find that
the wave packet behaves classically (meaning that the quantum behavior is
indistinguishable from that of the analogous classical system) as long as the
motion is confined to the interior of the remnant separatrix of the cosine
potential. Once the classical motion becomes unbounded, however, we find that
quantum interference effects dominate. This interference leads to a long-lived
accumulation of quantum amplitude on top of the cosine barrier. This pinning of
the amplitude on the barrier is a dynamic mechanism for the quantum inhibition
of classical mixing.Comment: 20 pages, RevTeX format with 6 Postscript figures appended in
uuencoded tar.Z forma
Scars of Invariant Manifolds in Interacting Chaotic Few-Body Systems
We present a novel extension of the concept of scars for the wave functions
of classically chaotic few-body systems of identical particles with rotation
and permutation symmetry. Generically there exist manifolds in classical phase
space which are invariant under the action of a common subgroup of these two
symmetries. Such manifolds are associated with highly symmetric configurations.
If sufficiently stable, the quantum motion on such manifolds displays a notable
enhancement of the revival in the autocorrelation function which is not
directly associated with individual periodic orbits. Rather, it indicates some
degree of localization around an invariant manifold which has collective
characteristics that should be experimentally observable.Comment: 4 pages, RevTeX, 4 PS/EPS-figures, uses psfig.sty, quantum
computation changed, to be published in Physical Review Letter
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