25 research outputs found
Quantum chaos in the spectrum of operators used in Shor's algorithm
We provide compelling evidence for the presence of quantum chaos in the
unitary part of Shor's factoring algorithm. In particular we analyze the
spectrum of this part after proper desymmetrization and show that the
fluctuations of the eigenangles as well as the distribution of the eigenvector
components follow the CUE ensemble of random matrices, of relevance to
quantized chaotic systems that violate time-reversal symmetry. However, as the
algorithm tracks the evolution of a single state, it is possible to employ
other operators, in particular it is possible that the generic quantum chaos
found above becomes of a nongeneric kind such as is found in the quantum cat
maps, and in toy models of the quantum bakers map.Comment: Title and paper modified to include interesting additional
possibilities. Principal results unaffected. Accepted for publication in
Phys. Rev. E as Rapid Com
Eigenfunction statistics for a point scatterer on a three-dimensional torus
In this paper we study eigenfunction statistics for a point scatterer (the
Laplacian perturbed by a delta-potential) on a three-dimensional flat torus.
The eigenfunctions of this operator are the eigenfunctions of the Laplacian
which vanish at the scatterer, together with a set of new eigenfunctions
(perturbed eigenfunctions). We first show that for a point scatterer on the
standard torus all of the perturbed eigenfunctions are uniformly distributed in
configuration space. Then we investigate the same problem for a point scatterer
on a flat torus with some irrationality conditions, and show uniform
distribution in configuration space for almost all of the perturbed
eigenfunctions.Comment: Revised according to referee's comments. Accepted for publication in
Annales Henri Poincar
Hyperbolic Scar Patterns in Phase Space
We develop a semiclassical approximation for the spectral Wigner and Husimi
functions in the neighbourhood of a classically unstable periodic orbit of
chaotic two dimensional maps. The prediction of hyperbolic fringes for the
Wigner function, asymptotic to the stable and unstable manifolds, is verified
computationally for a (linear) cat map, after the theory is adapted to a
discrete phase space appropriate to a quantized torus. The characteristic
fringe patterns can be distinguished even for quasi-energies where the fixed
point is not Bohr-quantized. The corresponding Husimi function dampens these
fringes with a Gaussian envelope centered on the periodic point. Even though
the hyperbolic structure is then barely perceptible, more periodic points stand
out due to the weakened interference.Comment: 12 pages, 10 figures, Submited to Phys. Rev.
Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map
We rationalize the somewhat surprising efficacy of the Hadamard transform in
simplifying the eigenstates of the quantum baker's map, a paradigmatic model of
quantum chaos. This allows us to construct closely related, but new, transforms
that do significantly better, thus nearly solving for many states of the
quantum baker's map. These new transforms, which combine the standard Fourier
and Hadamard transforms in an interesting manner, are constructed from
eigenvectors of the shift permutation operator that are also simultaneous
eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal)
symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title;
corrected minor error
Semiclassical measures and the Schroedinger flow on Riemannian manifolds
In this article we study limits of Wigner distributions (the so-called
semiclassical measures) corresponding to sequences of solutions to the
semiclassical Schroedinger equation at times scales tending to
infinity as the semiclassical parameter tends to zero (when this is equivalent to consider solutions to the non-semiclassical
Schreodinger equation). Some general results are presented, among which a weak
version of Egorov's theorem that holds in this setting. A complete
characterization is given for the Euclidean space and Zoll manifolds (that is,
manifolds with periodic geodesic flow) via averaging formulae relating the
semiclassical measures corresponding to the evolution to those of the initial
states. The case of the flat torus is also addressed; it is shown that
non-classical behavior may occur when energy concentrates on resonant
frequencies. Moreover, we present an example showing that the semiclassical
measures associated to a sequence of states no longer determines those of their
evolutions. Finally, some results concerning the equation with a potential are
presented.Comment: 18 pages; Theorems 1,2 extendend to deal with arbitrary time-scales;
references adde
Distribution of Husimi Zeroes in Polygonal Billiards
The zeroes of the Husimi function provide a minimal description of individual
quantum eigenstates and their distribution is of considerable interest. We
provide here a numerical study for pseudo- integrable billiards which suggests
that the zeroes tend to diffuse over phase space in a manner reminiscent of
chaotic systems but nevertheless contain a subtle signature of
pseudo-integrability. We also find that the zeroes depend sensitively on the
position and momentum uncertainties with the classical correspondence best when
the position and momentum uncertainties are equal. Finally, short range
correlations seem to be well described by the Ginibre ensemble of complex
matrices.Comment: includes 13 ps figures; Phys. Rev. E (in press
Crystal properties of eigenstates for quantum cat maps
Using the Bargmann-Husimi representation of quantum mechanics on a torus
phase space, we study analytically eigenstates of quantized cat maps. The
linearity of these maps implies a close relationship between classically
invariant sublattices on the one hand, and the patterns (or `constellations')
of Husimi zeros of certain quantum eigenstates on the other hand. For these
states, the zero patterns are crystals on the torus. As a consequence, we can
compute explicit families of eigenstates for which the zero patterns become
uniformly distributed on the torus phase space in the limit . This
result constitutes a first rigorous example of semi-classical equidistribution
for Husimi zeros of eigenstates in quantized one-dimensional chaotic systems.Comment: 43 pages, LaTeX, including 7 eps figures Some amendments were made in
order to clarify the text, mainly in the 4 first sections. Figures are
unchanged. To be published in: Nonlinearit
How Chaotic is the Stadium Billiard? A Semiclassical Analysis
The impression gained from the literature published to date is that the
spectrum of the stadium billiard can be adequately described, semiclassically,
by the Gutzwiller periodic orbit trace formula together with a modified
treatment of the marginally stable family of bouncing ball orbits. I show that
this belief is erroneous. The Gutzwiller trace formula is not applicable for
the phase space dynamics near the bouncing ball orbits. Unstable periodic
orbits close to the marginally stable family in phase space cannot be treated
as isolated stationary phase points when approximating the trace of the Green
function. Semiclassical contributions to the trace show an - dependent
transition from hard chaos to integrable behavior for trajectories approaching
the bouncing ball orbits. A whole region in phase space surrounding the
marginal stable family acts, semiclassically, like a stable island with
boundaries being explicitly -dependent. The localized bouncing ball
states found in the billiard derive from this semiclassically stable island.
The bouncing ball orbits themselves, however, do not contribute to individual
eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing
ball eigenstates in the stadium can be derived. The stadium billiard is thus an
ideal model for studying the influence of almost regular dynamics near
marginally stable boundaries on quantum mechanics.Comment: 27 pages, 6 figures, submitted to J. Phys.
Scarring Effects on Tunneling in Chaotic Double-Well Potentials
The connection between scarring and tunneling in chaotic double-well
potentials is studied in detail through the distribution of level splittings.
The mean level splitting is found to have oscillations as a function of energy,
as expected if scarring plays a role in determining the size of the splittings,
and the spacing between peaks is observed to be periodic of period
{} in action. Moreover, the size of the oscillations is directly
correlated with the strength of scarring. These results are interpreted within
the theoretical framework of Creagh and Whelan. The semiclassical limit and
finite-{} effects are discussed, and connections are made with reaction
rates and resonance widths in metastable wells.Comment: 22 pages, including 11 figure
Physics of the Riemann Hypothesis
Physicists become acquainted with special functions early in their studies.
Consider our perennial model, the harmonic oscillator, for which we need
Hermite functions, or the Laguerre functions in quantum mechanics. Here we
choose a particular number theoretical function, the Riemann zeta function and
examine its influence in the realm of physics and also how physics may be
suggestive for the resolution of one of mathematics' most famous unconfirmed
conjectures, the Riemann Hypothesis. Does physics hold an essential key to the
solution for this more than hundred-year-old problem? In this work we examine
numerous models from different branches of physics, from classical mechanics to
statistical physics, where this function plays an integral role. We also see
how this function is related to quantum chaos and how its pole-structure
encodes when particles can undergo Bose-Einstein condensation at low
temperature. Throughout these examinations we highlight how physics can perhaps
shed light on the Riemann Hypothesis. Naturally, our aim could not be to be
comprehensive, rather we focus on the major models and aim to give an informed
starting point for the interested Reader.Comment: 27 pages, 9 figure