166 research outputs found
A study of Wigner functions for discrete-time quantum walks
We perform a systematic study of the discrete time Quantum Walk on one
dimension using Wigner functions, which are generalized to include the
chirality (or coin) degree of freedom. In particular, we analyze the evolution
of the negative volume in phase space, as a function of time, for different
initial states. This negativity can be used to quantify the degree of departure
of the system from a classical state. We also relate this quantity to the
entanglement between the coin and walker subspaces.Comment: 16 pages, 8 figure
Ill-defined Topological Phases in Dispersive Photonic Crystals
In recent years there has been a great interest in topological materials and
in their fascinating properties. Topological band theory was initially
developed for condensed matter systems, but it can be readily applied to
arbitrary wave platforms with little modifications. Thus, the topological
classification of optical systems is usually regarded as being mathematically
equivalent to that of condensed matter systems. Surprisingly, here we find that
both the particle-hole symmetry and the dispersive nature of nonreciprocal
photonic materials may lead to situations where the usual topological methods
break-down and the Chern topology becomes ill-defined. It is shown that due to
the divergence of the density of photonic states in plasmonic systems the gap
Chern numbers can be non-integer notwithstanding that the relevant parametric
space is compact. In order that the topology of a dispersive photonic crystal
is well defined, it is essential to take into account the nonlocal effects in
the bulk-materials. We propose two different regularization methods to fix the
encountered problems. Our results highlight that the regularized topologies may
depend critically on the response of the bulk materials for large k
Thermodynamic Limits of Electronic Systems
We review thermodynamic limits and scaling limits of electronic structure
models for condensed matter. We discuss several mathematical ways to implement
these limits in three models of increasing chemical complexity and mathematical
difficulty: (1) Thomas-Fermi like models; (2) Hartree-Fock like models; and (3)
Kohn-Sham density functional theory models
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Computational methods for understanding the role of electric fields in quantum confined materials
The invention of pseudopotential-density functional theory to solve for the electronic structure of materials is one of the major successes of modern computational physics. A code based on this formalism was used to solve for the electronic structure of systems with limited dimensionality. The code solves for the electronic structure problem on a real-space grid without the use of an explicit basis. This scheme is particularly well suited for studying molecules, clusters, and nanostructures. The code was applied to assess how an applied electric field changes the properties of two different systems: the change of vibrational modes with the field in molecules or clusters and tuning the electronic gap with the field in 2D materials.
Three approaches were employed to study the effect of electric fields on the vibrations of small molecules. The approaches used perturbation theory, a finite field method, and an ab initio molecular dynamics approach. This work provides a better understanding of experimental techniques to probe the local electric field in complex materials as in photovoltaics and biomolecules.
The second part of this thesis leverages mixed boundary conditions to study the effects of finite electric fields on two-dimensional materials such as phosphorene. These results demonstrate the ability to tune the band gap and drive semiconductor to metallic transitions in novel two-dimensional materials. This property may enable the creation of nanoscale transistors and sensors to power the next generation of electronic devices.Physic
Fast numerical methods for waves in periodic media
Periodic media problems widely exist in many modern
application areas like
semiconductor nanostructures (e.g.\ quantum dots and nanocrystals),
semi-conductor superlattices,
photonic crystals (PC) structures,
meta materials or Bragg gratings of surface
plasmon polariton (SPP) waveguides, etc.
Often these application problems are modeled by partial differential
equations with periodic coefficients and/or periodic geometries.
In order to numerically solve these periodic structure problems efficiently one usually confines the spatial domain to a bounded computational domain
(i.e.\ in a neighborhood of the region of physical interest).
Hereby, the usual strategy is to introduce so-called
\emph{artificial boundaries} and impose suitable boundary conditions.
For wave-like equations, the ideal boundary conditions should not only lead to well-posed problems,
but also mimic the perfect absorption of waves traveling out of the computational domain
through the artificial boundaries.
In the first part of this chapter we present a novel analytical impedance expression
for general second order ODE problems with periodic coefficients.
This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary
conditions is then used for computing the bound states of the Schr\"odinger operator with
periodic potentials at infinity.
Other potential applications are associated with the exact artificial boundary conditions
for some time-dependent problems with periodic structures.
As an example, a two-dimensional hyperbolic equation modeling the TM polarization of
the electromagnetic field with a periodic dielectric permittivity is considered.
In the second part of this chapter we present a new numerical technique for solving periodic structure problems. This novel approach possesses several advantages.
First, it allows for a fast evaluation of the Sommerfeld-to-Sommerfeld operator for periodic
array problems. Secondly,
this computational method can also be used for bi-periodic structure problems with local defects.
In the sequel we consider several problems, such as the exterior elliptic problems with
strong coercivity, the time-dependent Schr\"odinger equation and the Helmholtz equation
with damping.
Finally, in the third part we consider
periodic arrays that are structures consisting of geometrically identical
subdomains, usually called periodic cells.
We use the Helmholtz equation as a model equation and consider
the definition and evaluation of the exact boundary mappings for general
semi-infinite arrays that are periodic in one direction for any real wavenumber.
The well-posedness of the Helmholtz equation is established via the
\emph{limiting absorption principle} (LABP).
An algorithm based on the doubling procedure of the second part of this chapter
and an extrapolation method is proposed to construct the
exact Sommerfeld-to-Sommerfeld boundary mapping.
This new algorithm benefits from its robustness and the
simplicity of implementation.
But it also suffers from the high computational cost and the
resonance wave numbers.
To overcome these shortcomings, we propose another algorithm based
on a conjecture about the asymptotic behaviour of
limiting absorption principle solutions.
The price we have to pay is the resolution of some generalized eigenvalue problem,
but still the overall computational cost is significantly reduced.
Numerical evidences show that this algorithm presents theoretically
the same results as the first algorithm.
Moreover, some quantitative comparisons between these two algorithms are given
Fast numerical methods for waves in periodic media
Periodic media problems widely exist in many modern application areas
like semiconductor nanostructures (e.g. quantum dots and nanocrystals),
semi-conductor superlattices, photonic crystals (PC) structures, meta
materials or Bragg gratings of surface plasmon polariton (SPP) waveguides,
etc. Often these application problems are modeled by partial differential
equations with periodic coefficients and/or periodic geometries. In order to
numerically solve these periodic structure problems efficiently one usually
confines the spatial domain to a bounded computational domain (i.e. in a
neighborhood of the region of physical interest). Hereby, the usual strategy
is to introduce so-called artificial boundaries and impose suitable boundary
conditions. For wave-like equations, the ideal boundary conditions should not
only lead to w ell-posed problems, but also mimic the perfect absorption of
waves traveling out of the computational domain through the artificial
boundaries ..
Extended study for unitary fermions on a lattice using the cumulant expansion technique
A recently developed lattice method for large numbers of strongly interacting
nonrelativistic fermions exhibits a heavy tail in the distributions of
correlators for large Euclidean time {\tau} and large number of fermions N,
which only allows the measurement of ground state energies for a limited number
of fermions using standard techniques. In such cases, it is suggested that
measuring the log of the correlator is more efficient, and a cumulant expansion
of this quantity can be exactly related to the correlation function. The
cumulant expansion technique allows us to determine the ground state energies
of up to 66 unpolarized unitary fermions on lattices as large as
7214^3, and up to 70 unpolarized unitary fermions trapped in a harmonic
potential on lattices as large as 7264^3. We have also improved our
lattice action with a Galilean invariant form for the four-fermion interaction,
which results in predictive volume scaling of the lowest energy of three
fermions in a periodic box and in good agreement of our results for N \leq 6
trapped unitary fermions with those from other benchmark calculations.Comment: 7 pages, 6 figures, Presented at 29th International Symposium on
Lattice Field Theory (Lattice2011), Squaw Valley, Lake Tahoe, CA, USA, 10-16
July 201
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