153 research outputs found
Quantum Optical Random Walk: Quantization Rules and Quantum Simulation of Asymptotics
Rules for quantizing the walker+coin parts of a classical random walk are
provided by treating them as interacting quantum systems. A quantum optical
random walk (QORW), is introduced by means of a new rule that treats quantum or
classical noise affecting the coin's state, as sources of quantization. The
long term asymptotic statistics of QORW walker's position that shows enhanced
diffusion rates as compared to classical case, is exactly solved. A quantum
optical cavity implementation of the walk provides the framework for quantum
simulation of its asymptotic statistics. The simulation utilizes interacting
two-level atoms and/or laser randomly pulsating fields with fluctuating
parameters.Comment: 18 pages, 3 figure
Prime decomposition and correlation measure of finite quantum systems
Under the name prime decomposition (pd), a unique decomposition of an
arbitrary -dimensional density matrix into a sum of seperable density
matrices with dimensions given by the coprime factors of is introduced. For
a class of density matrices a complete tensor product factorization is
achieved. The construction is based on the Chinese Remainder Theorem and the
projective unitary representation of by the discrete Heisenberg group
. The pd isomorphism is unitarily implemented and it is shown to be
coassociative and to act on as comultiplication. Density matrices with
complete pd are interpreted as grouplike elements of . To quantify the
distance of from its pd a trace-norm correlation index is
introduced and its invariance groups are determined.Comment: 9 pages LaTeX. Revised version: changes in the terminology, updates
in ref
q-Symmetries in DNLS-AL chains and exact solutions of quantum dimers
Dynamical symmetries of Hamiltonians quantized models of discrete non-linear
Schroedinger chain (DNLS) and of Ablowitz-Ladik chain (AL) are studied. It is
shown that for -sites the dynamical algebra of DNLS Hamilton operator is
given by the algebra, while the respective symmetry for the AL case is
the quantum algebra su_q(n). The q-deformation of the dynamical symmetry in the
AL model is due to the non-canonical oscillator-like structure of the raising
and lowering operators at each site.
Invariants of motions are found in terms of Casimir central elements of su(n)
and su_q(n) algebra generators, for the DNLS and QAL cases respectively.
Utilizing the representation theory of the symmetry algebras we specialize to
the quantum dimer case and formulate the eigenvalue problem of each dimer
as a non-linear (q)-spin model. Analytic investigations of the ensuing
three-term non-linear recurrence relations are carried out and the respective
orthonormal and complete eigenvector bases are determined.
The quantum manifestation of the classical self-trapping in the QDNLS-dimer
and its absence in the QAL-dimer, is analysed by studying the asymptotic
attraction and repulsion respectively, of the energy levels versus the strength
of non-linearity. Our treatment predicts for the QDNLS-dimer, a
phase-transition like behaviour in the rate of change of the logarithm of
eigenenergy differences, for values of the non-linearity parameter near the
classical bifurcation point.Comment: Latex, 19pp, 4 figures. Submitted for publicatio
Quantum Walks of SU(2)_k Anyons on a Ladder
We study the effects of braiding interactions on single anyon dynamics using
a quantum walk model on a quasi-1-dimensional ladder filled with stationary
anyons. The model includes loss of information of the coin and nonlocal fusion
degrees of freedom on every second time step, such that the entanglement
between the position states and the exponentially growing auxiliary degrees of
freedom is lost. The computational complexity of numerical calculations reduces
drastically from the fully coherent anyonic quantum walk model, allowing for
relatively long simulations for anyons which are spin-1/2 irreps of SU(2)_k
Chern-Simons theory. We find that for Abelian anyons, the walk retains the
ballistic spreading velocity just like particles with trivial braiding
statistics. For non-Abelian anyons, the numerical results indicate that the
spreading velocity is linearly dependent on the number of time steps. By
approximating the Kraus generators of the time evolution map by circulant
matrices, it is shown that the spatial probability distribution for the k=2
walk, corresponding to Ising model anyons, is equal to the classical unbiased
random walk distribution.Comment: 12 pages, 4 figure
Non-positivity of the Wigner function and bounds on associated integrals
The Wigner function shares several properties with classical distribution
functions on phase space, but is not positive-definite. The integral of the
Wigner function over a given region of phase space can therefore lie outside
the interval [0,1]. The problem of finding best-possible upper and lower bounds
for a given region is the problem of finding the greatest and least eigenvalues
of an associated Hermitian operator. Exactly solvable examples are described,
and possible extensions are indicated.Comment: 5 pages, Latex2e fil
Superconducting Coherent States for an Extended Hubbard Model
An extended Hubbard model with phonons is considered. q-coherent states
relative to the superconducting quantum symmetry of the model are constructed
and their properties studied. It is shown that they can have energy expectation
lower than eigenstates constructed via conventional processes and that they
exhibit ODLRO.Comment: 7 pages, 3 figure
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