17,647 research outputs found
Universal and Near-Universal Cycles of Set Partitions
We study universal cycles of the set of -partitions of the
set and prove that the transition digraph associated
with is Eulerian. But this does not imply that universal cycles
(or ucycles) exist, since vertices represent equivalence classes of partitions!
We use this result to prove, however, that ucycles of exist for
all when . We reprove that they exist for odd when and that they do not exist for even when . An infinite family
of for which ucycles do not exist is shown to be those pairs for which
is odd (). We also show that there exist
universal cycles of partitions of into subsets of distinct sizes when
is sufficiently smaller than , and therefore that there exist universal
packings of the partitions in . An analogous result for
coverings completes the investigation.Comment: 22 page
Multiparticle correlations in mesoscopic scattering: boson sampling, birthday paradox, and Hong-Ou-Mandel profiles
The interplay between single-particle interference and quantum
indistinguishability leads to signature correlations in many-body scattering.
We uncover these with a semiclassical calculation of the transmission
probabilities through mesoscopic cavities for systems of non-interacting
particles. For chaotic cavities we provide the universal form of the first two
moments of the transmission probabilities over ensembles of random unitary
matrices, including weak localization and dephasing effects. If the incoming
many-body state consists of two macroscopically occupied wavepackets, their
time delay drives a quantum-classical transition along a boundary determined by
the bosonic birthday paradox. Mesoscopic chaotic scattering of Bose-Einstein
condensates is then a realistic candidate to build a boson sampler and to
observe the macroscopic Hong-Ou-Mandel effect.Comment: 6+11 pages, 3+3 figure
Periodic-Orbit Theory of Universality in Quantum Chaos
We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory,
that full classical chaos is paralleled by quantum energy spectra with
universal spectral statistics, in agreement with random-matrix theory. For
dynamics from all three Wigner-Dyson symmetry classes, we calculate the
small-time spectral form factor as power series in the time .
Each term of that series is provided by specific families of pairs of
periodic orbits. The contributing pairs are classified in terms of close
self-encounters in phase space. The frequency of occurrence of self-encounters
is calculated by invoking ergodicity. Combinatorial rules for building pairs
involve non-trivial properties of permutations. We show our series to be
equivalent to perturbative implementations of the non-linear sigma models for
the Wigner-Dyson ensembles of random matrices and for disordered systems; our
families of orbit pairs are one-to-one with Feynman diagrams known from the
sigma model.Comment: 31 pages, 17 figure
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