18,413 research outputs found
Geometric phases and anholonomy for a class of chaotic classical systems
Berry's phase may be viewed as arising from the parallel transport of a
quantal state around a loop in parameter space. In this Letter, the classical
limit of this transport is obtained for a particular class of chaotic systems.
It is shown that this ``classical parallel transport'' is anholonomic ---
transport around a closed curve in parameter space does not bring a point in
phase space back to itself --- and is intimately related to the Robbins-Berry
classical two-form.Comment: Revtex, 11 pages, no figures
Dynamical diffraction in sinusoidal potentials: uniform approximations for Mathieu functions
Eigenvalues and eigenfunctions of Mathieu's equation are found in the short
wavelength limit using a uniform approximation (method of comparison with a
`known' equation having the same classical turning point structure) applied in
Fourier space. The uniform approximation used here relies upon the fact that by
passing into Fourier space the Mathieu equation can be mapped onto the simpler
problem of a double well potential. The resulting eigenfunctions (Bloch waves),
which are uniformly valid for all angles, are then used to describe the
semiclassical scattering of waves by potentials varying sinusoidally in one
direction. In such situations, for instance in the diffraction of atoms by
gratings made of light, it is common to make the Raman-Nath approximation which
ignores the motion of the atoms inside the grating. When using the
eigenfunctions no such approximation is made so that the dynamical diffraction
regime (long interaction time) can be explored.Comment: 36 pages, 16 figures. This updated version includes important
references to existing work on uniform approximations, such as Olver's method
applied to the modified Mathieu equation. It is emphasised that the paper
presented here pertains to Fourier space uniform approximation
Observation of a Chiral State in a Microwave Cavity
A microwave experiment has been realized to measure the phase difference of
the oscillating electric field at two points inside the cavity. The technique
has been applied to a dissipative resonator which exhibits a singularity --
called exceptional point -- in its eigenvalue and eigenvector spectrum. At the
singularity, two modes coalesce with a phase difference of We
conclude that the state excited at the singularity has a definitiv chirality.Comment: RevTex 4, 5 figure
Statistical Properties of Many Particle Eigenfunctions
Wavefunction correlations and density matrices for few or many particles are
derived from the properties of semiclassical energy Green functions. Universal
features of fixed energy (microcanonical) random wavefunction correlation
functions appear which reflect the emergence of the canonical ensemble as the
number of particles approaches infinity. This arises through a little known
asymptotic limit of Bessel functions. Constraints due to symmetries,
boundaries, and collisions between particles can be included.Comment: 13 pages, 4 figure
Nonperiodic Orbit Sums in Weyl's Expansion for Billiards
Weyl's expansion for the asymptotic mode density of billiards consists of the
area, length, curvature and corner terms. The area term has been associated
with the so-called zero-length orbits. Here closed nonperiodic paths
corresponding to the length and corner terms are constructed.Comment: 8 pages, 2 figure
Decimation and Harmonic Inversion of Periodic Orbit Signals
We present and compare three generically applicable signal processing methods
for periodic orbit quantization via harmonic inversion of semiclassical
recurrence functions. In a first step of each method, a band-limited decimated
periodic orbit signal is obtained by analytical frequency windowing of the
periodic orbit sum. In a second step, the frequencies and amplitudes of the
decimated signal are determined by either Decimated Linear Predictor, Decimated
Pade Approximant, or Decimated Signal Diagonalization. These techniques, which
would have been numerically unstable without the windowing, provide numerically
more accurate semiclassical spectra than does the filter-diagonalization
method.Comment: 22 pages, 3 figures, submitted to J. Phys.
Deviations from Berry--Robnik Distribution Caused by Spectral Accumulation
By extending the Berry--Robnik approach for the nearly integrable quantum
systems,\cite{[1]} we propose one possible scenario of the energy level spacing
distribution that deviates from the Berry--Robnik distribution. The result
described in this paper implies that deviations from the Berry--Robnik
distribution would arise when energy level components show strong accumulation,
and otherwise, the level spacing distribution agrees with the Berry--Robnik
distribution.Comment: 4 page
Berry phase, topology, and diabolicity in quantum nano-magnets
A topological theory of the diabolical points (degeneracies) of quantum
magnets is presented. Diabolical points are characterized by their diabolicity
index, for which topological sum rules are derived. The paradox of the the
missing diabolical points for Fe8 molecular magnets is clarified. A new method
is also developed to provide a simple interpretation, in terms of destructive
interferences due to the Berry phase, of the complete set of diabolical points
found in biaxial systems such as Fe8.Comment: 4 pages, 3 figure
The three-body problem and the Hannay angle
The Hannay angle has been previously studied for a celestial circular
restricted three-body system by means of an adiabatic approach. In the present
work, three main results are obtained. Firstly, a formal connection between
perturbation theory and the Hamiltonian adiabatic approach shows that both lead
to the Hannay angle; it is thus emphasised that this effect is already
contained in classical celestial mechanics, although not yet defined nor
evaluated separately. Secondly, a more general expression of the Hannay angle,
valid for an action-dependent potential is given; such a generalised expression
takes into account that the restricted three-body problem is a time-dependent,
two degrees of freedom problem even when restricted to the circular motion of
the test body. Consequently, (some of) the eccentricity terms cannot be
neglected {\it a priori}. Thirdly, we present a new numerical estimate for the
Earth adiabatically driven by Jupiter. We also point out errors in a previous
derivation of the Hannay angle for the circular restricted three-body problem,
with an action-independent potential.Comment: 11 pages. Accepted by Nonlinearit
Persistent Currents in Quantum Chaotic Systems
The persistent current of ballistic chaotic billiards is considered with the
help of the Gutzwiller trace formula. We derive the semiclassical formula of a
typical persistent current for a single billiard and an average
persistent current for an ensemble of billiards at finite temperature.
These formulas are used to show that the persistent current for chaotic
billiards is much smaller than that for integrable ones. The persistent
currents in the ballistic regime therefore become an experimental tool to
search for the quantum signature of classical chaotic and regular dynamics.Comment: 4 pages (RevTex), to appear in Phys. Rev. B, No.59, 12256-12259
(1999
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