4,002 research outputs found
A class of exactly solvable models for the Schrodinger equation
We present a class of confining potentials which allow one to reduce the
one-dimensional Schroodinger equation to a named equation of mathematical
physics, namely either Bessel's or Whittaker's differential equation. In all
cases, we provide closed form expressions for both the symmetric and
antisymmetric wavefunction solutions, each along with an associated
transcendental equation for allowed eigenvalues. The class of potentials
considered contains an example of both cusp-like single wells and a
double-well.Comment: 5 pages, 7 figure
Localization of massless Dirac particles via spatial modulations of the Fermi velocity
The electrons found in Dirac materials are notorious for being difficult to
manipulate due to the Klein phenomenon and absence of backscattering. Here we
investigate how spatial modulations of the Fermi velocity in two-dimensional
Dirac materials can give rise to localization effects, with either full
(zero-dimensional) confinement or partial (one-dimensional) confinement
possible depending on the geometry of the velocity modulation. We present
several exactly solvable models illustrating the nature of the bound states
which arise, revealing how the gradient of the Fermi velocity is crucial for
determining fundamental properties of the bound states such as the zero-point
energy. We discuss the implications for guiding electronic waves in few-mode
waveguides formed by Fermi velocity modulation.Comment: 9 pages, 6 figure
One-dimensional Coulomb problem in Dirac materials
We investigate the one-dimensional Coulomb potential with application to a
class of quasirelativistic systems, so-called Dirac-Weyl materials, described
by matrix Hamiltonians. We obtain the exact solution of the shifted and
truncated Coulomb problems, with the wavefunctions expressed in terms of
special functions (namely Whittaker functions), whilst the energy spectrum must
be determined via solutions to transcendental equations. Most notably, there
are critical bandgaps below which certain low-lying quantum states are missing
in a manifestation of atomic collapse.Comment: 7 pages, 5 figure
Bielectron vortices in two-dimensional Dirac semimetals
Searching for new states of matter and unusual quasiparticles in emerging
materials and especially low-dimensional systems is one of the major trends in
contemporary condensed matter physics. Dirac materials, which host
quasiparticles which are described by ultrarelativistic Dirac-like equations,
are of a significant current interest from both a fundamental and applied
physics perspective. Here we show that a pair of two-dimensional massless
Dirac-Weyl fermions can form a bound state independently of the sign of the
inter-particle interaction potential, as long as this potential decays at large
distances faster than Kepler's inverse distance law. This leads to the
emergence of a new type of energetically-favourable quasiparticle: bielectron
vortices, which are double-charged and reside at zero-energy. Their bosonic
nature allows for condensation and may give rise to Majorana physics without
invoking a superconductor. These novel quasiparticles arguably explain a range
of poorly understood experiments in gated graphene structures at low doping.Comment: 9 pages, 2 figure
Massless Dirac fermions in two dimensions: Confinement in nonuniform magnetic fields
We show how it is possible to trap two-dimensional massless Dirac fermions in
spatially inhomogeneous magnetic fields, as long as the formed magnetic quantum
dot (or ring) is of a slowly decaying nature. It is found that a modulation of
the depth of the magnetic quantum dot leads to successive
confinement-deconfinement transitions of vortexlike states with a certain
angular momentum, until a regime is reached where only states with one sign of
angular momentum are supported. We illustrate these characteristics with both
exact solutions and a hitherto unknown quasi-exactly solvable model utilizing
confluent Heun functions.Comment: 7 pages, 3 figure
Optimal traps in graphene
We transform the two-dimensional Dirac-Weyl equation, which governs the
charge carriers in graphene, into a non-linear first-order differential
equation for scattering phase shift, using the so-called variable phase method.
This allows us to utilize the Levinson Theorem to find zero-energy bound states
created electrostatically in realistic structures. These confined states are
formed at critical potential strengths, which leads to us posit the use of
`optimal traps' to combat the chiral tunneling found in graphene, which could
be explored experimentally with an artificial network of point charges held
above the graphene layer. We also discuss scattering on these states and find
the zero angular momentum states create a dominant peak in scattering
cross-section as energy tends towards the Dirac point energy, suggesting a
dominant contribution to resistivity.Comment: 11 pages, 5 figure
Perfect State Transfer: Beyond Nearest-Neighbor Couplings
In this paper we build on the ideas presented in previous works for perfectly
transferring a quantum state between opposite ends of a spin chain using a
fixed Hamiltonian. While all previous studies have concentrated on
nearest-neighbor couplings, we demonstrate how to incorporate additional terms
in the Hamiltonian by solving an Inverse Eigenvalue Problem. We also explore
issues relating to the choice of the eigenvalue spectrum of the Hamiltonian,
such as the tolerance to errors and the rate of information transfer.Comment: 8 pages, 2 figures. Reorganised, more detailed derivations provided
and section on rate of information transfer adde
Doublons, topology and interactions in a one-dimensional lattice
We investigate theoretically the Bose-Hubbard version of the celebrated
Su-Schrieffer-Heeger topological model, which essentially describes a
one-dimensional dimerized array of coupled oscillators with on-site
interactions. We study the physics arising from the whole gamut of possible
dimerizations of the chain, including both the weakly and the strongly
dimerized limiting cases. Focusing on two-excitation subspace, we
systematically uncover and characterize the different types of states which may
emerge due to the competition between the inter-oscillator couplings, the
intrinsic topology of the lattice, and the strength of the on-site
interactions. In particular, we discuss the formation of scattering bands full
of extended states, bound bands full of two-particle pairs (including so-called
`doublons', when the pair occupies the same lattice site), and different
flavors of topological edge states. The features we describe may be realized in
a plethora of systems, including nanoscale architectures such as photonic
cavities and optical lattices, and provide perspectives for topological
many-body physics.Comment: 9 pages, 5 figure
Directionality between driven-dissipative resonators
The notion of nonreciprocity, in essence when going forwards is different
from going backwards, emerges in all branches of physics from cosmology to
electromagnetism. Intriguingly, the breakdown of reciprocity is typically
associated with extraordinary phenomena, which may be readily capitalized on in
the design of (for example) nontrivial electromagnetic devices when Lorentz
reciprocity is broken. However, in order to enable the exploitation of
nonreciprocal-like effects in the next generation of quantum technologies,
basic quantum optical theories are required. Here we present a versatile model
describing a pair of driven-dissipative quantum resonators, where the relative
phase difference between the coherent and incoherent couplings induces an
asymmetry. The interplay between the diverse dissipative landscape - which
encompasses both intrinsic losses and dissipative couplings - and the coherent
interactions leads to some remarkable consequences including highly directional
(or even one-way) energy transport. Our work proffers the tantalizing prospect
of observing dissipation-induced quantum directionality in areas like photonics
or cavity magnonics (spin waves), which may aid the design of unconventional
nanoscopic devices.Comment: 7 pages, 4 figure
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