8,808 research outputs found
Pseudo-Hermitian continuous-time quantum walks
In this paper we present a model exhibiting a new type of continuous-time
quantum walk (as a quantum mechanical transport process) on networks, which is
described by a non-Hermitian Hamiltonian possessing a real spectrum. We call it
pseudo-Hermitian continuous-time quantum walk. We introduce a method to obtain
the probability distribution of walk on any vertex and then study a specific
system. We observe that the probability distribution on certain vertices
increases compared to that of the Hermitian case. This formalism makes the
transport process faster and can be useful for search algorithms.Comment: 13 page, 7 figure
Mathematical Aspects of Vacuum Energy on Quantum Graphs
We use quantum graphs as a model to study various mathematical aspects of the
vacuum energy, such as convergence of periodic path expansions, consistency
among different methods (trace formulae versus method of images) and the
possible connection with the underlying classical dynamics.
We derive an expansion for the vacuum energy in terms of periodic paths on
the graph and prove its convergence and smooth dependence on the bond lengths
of the graph. For an important special case of graphs with equal bond lengths,
we derive a simpler explicit formula.
The main results are derived using the trace formula. We also discuss an
alternative approach using the method of images and prove that the results are
consistent. This may have important consequences for other systems, since the
method of images, unlike the trace formula, includes a sum over special
``bounce paths''. We succeed in showing that in our model bounce paths do not
contribute to the vacuum energy. Finally, we discuss the proposed possible link
between the magnitude of the vacuum energy and the type (chaotic vs.
integrable) of the underlying classical dynamics. Within a random matrix model
we calculate the variance of the vacuum energy over several ensembles and find
evidence that the level repulsion leads to suppression of the vacuum energy.Comment: Fixed several typos, explain the use of random matrices in Section
From quantum graphs to quantum random walks
We give a short overview over recent developments on quantum graphs and
outline the connection between general quantum graphs and so-called quantum
random walks.Comment: 14 pages, 6 figure
Quantum Graphs: A simple model for Chaotic Scattering
We connect quantum graphs with infinite leads, and turn them to scattering
systems. We show that they display all the features which characterize quantum
scattering systems with an underlying classical chaotic dynamics: typical
poles, delay time and conductance distributions, Ericson fluctuations, and when
considered statistically, the ensemble of scattering matrices reproduce quite
well the predictions of appropriately defined Random Matrix ensembles. The
underlying classical dynamics can be defined, and it provides important
parameters which are needed for the quantum theory. In particular, we derive
exact expressions for the scattering matrix, and an exact trace formula for the
density of resonances, in terms of classical orbits, analogous to the
semiclassical theory of chaotic scattering. We use this in order to investigate
the origin of the connection between Random Matrix Theory and the underlying
classical chaotic dynamics. Being an exact theory, and due to its relative
simplicity, it offers new insights into this problem which is at the fore-front
of the research in chaotic scattering and related fields.Comment: 28 pages, 13 figures, submitted to J. Phys. A Special Issue -- Random
Matrix Theor
The Generalized Star Product and the Factorization of Scattering Matrices on Graphs
In this article we continue our analysis of Schr\"odinger operators on
arbitrary graphs given as certain Laplace operators. In the present paper we
give the proof of the composition rule for the scattering matrices. This
composition rule gives the scattering matrix of a graph as a generalized star
product of the scattering matrices corresponding to its subgraphs. We perform a
detailed analysis of the generalized star product for arbitrary unitary
matrices. The relation to the theory of transfer matrices is also discussed
- …