35,792 research outputs found
Nearly Cloaking the Full Maxwell Equations
The approximate cloaking is investigated for time-harmonic Maxwell's
equations via the approach of transformation optics. The problem is reduced to
certain boundary effect estimates due to an inhomogeneous electromagnetic
inclusion with an asymptotically small support but an arbitrary content
enclosed by a thin high-conducting layer. Sharp estimates are established in
terms of the asymptotic parameter, which are independent of the material
tensors of the small electromagnetic inclusion. The result implies that the
`blow-up-a-small-region' construction via the transformation optics approach
yields a near-cloak for the electromagnetic waves. A novelty lies in the fact
that the geometry of the cloaking construction of this work can be very
general. Moreover, by incorporating the conducting layer developed in the
present paper right between the cloaked region and the cloaking region,
arbitrary electromagnetic contents can be nearly cloaked. Our mathematical
technique extends the general one developed in [30] for nearly cloaking scalar
optics. In order to investigate the approximate electromagnetic cloaking for
general geometries with arbitrary cloaked contents, new techniques and analysis
tools must be developed for this more challenging vector optics case
Berry phases of quantum trajectories in semiconductors under strong terahertz fields
Quantum evolution of particles under strong fields can be essentially
captured by a small number of quantum trajectories that satisfy the stationary
phase condition in the Dirac-Feynmann path integrals. The quantum trajectories
are the key concept to understand extreme nonlinear optical phenomena, such as
high-order harmonic generation (HHG), above-threshold ionization (ATI), and
high-order terahertz sideband generation (HSG). While HHG and ATI have been
mostly studied in atoms and molecules, the HSG in semiconductors can have
interesting effects due to possible nontrivial "vacuum" states of band
materials. We find that in a semiconductor with non-vanishing Berry curvature
in its energy bands, the cyclic quantum trajectories of an electron-hole pair
under a strong terahertz field can accumulate Berry phases. Taking monolayer
MoS as a model system, we show that the Berry phases appear as the Faraday
rotation angles of the pulse emission from the material under short-pulse
excitation. This finding reveals an interesting transport effect in the extreme
nonlinear optics regime.Comment: 5 page
On Picard Type Theorems and Entire Solutions of Differential Equations
We give a connection between the Picard type theorem of Polya-Saxer-Milliox
and characterization of entire solutions of a differential equation and then
their higher dimensional extensions, which leads further results on both
(ordinary and partial) differential equations and Picard type theorems.Comment: 7 page
Dynamical decoupling for a qubit in telegraph-like noises
Based on the stochastic theory developed by Kubo and Anderson, we present an
exact result of the decoherence function of a qubit in telegraph-like noises
under dynamical decoupling control. We prove that for telegraph-like noises,
the decoherence can be suppressed at most to the third order of the time and
the periodic Carr-Purcell-Merboom-Gill sequences are the most efficient scheme
in protecting the qubit coherence in the short-time limit.Comment: 4 page
Solutions of Kapustin-Witten equations for ADE-type groups
Kapustin-Witten (KW) equations are encountered in the localization of the
topological N=4 SYM theory. Mikhaylov has constructed model solutions of KW
equations for the boundary 't~Hooft operators on a half space. Direct proof of
the solutions boils down to check a boundary condition. There are two
computational difficulties in explicitly constructing the solutions to Lie
algebra of higher rank. The first one is related to the commutation of
generators of Lie algebra. We derived an identity which effectively reduces
this computational difficulty. The second one involves the number of ways from
the highest weights to other weights in the fundamental representation. For
ADE-type gauge groups, we found an amazing formula which can be used to rewrite
the solutions of KW equations. This new formula of solutions bypass above two
computational difficulties.Comment: 26 pages, 3 figure. typos corrected, english improved, references
added, corrected mistakes and rewrote Sec. I
A note on "Extremal graphs with bounded vertex bipartiteness number"
This paper is devoted to present two counterexamples to the theorem from
\cite{MK} Maria R., Katherine T. M., Bernardo S. M., Extremal graphs with
bounded vertex bipartiteness number, Linear Algebra Appl. 493 (2016) 28-36.
Moreover, the corrected theorem and proof are presented
Nonlinear optical response induced by non-Abelian Berry curvature in time-reversal-invariant insulators
We propose a general framework of nonlinear optics induced by non-Abelian
Berry curvature in time-reversal-invariant (TRI) insulators. We find that the
third-order response of a TRI insulator under optical and terahertz light
fields is directly related to the integration of the non-Abelian Berry
curvature over the Brillouin zone. We apply the result to insulators with
rotational symmetry near the band edge. Under resonant excitations, the optical
susceptibility is proportional to the flux of the Berry curvature through the
iso-energy surface, which is equal to the Chern number of the surface times
. For the III-V compound semiconductors, microscopic calculations based
on the six-band model give a third-order susceptibility with the Chern number
of the iso-energy surface equal to three
Imaginary geometric phases of quantum trajectories
A quantum object can accumulate a geometric phase when it is driven along a
trajectory in a parameterized state space with non-trivial gauge structures.
Inherent to quantum evolutions, a system can not only accumulate a quantum
phase but may also experience dephasing, or quantum diffusion. Here we show
that the diffusion of quantum trajectories can also be of geometric nature as
characterized by the imaginary part of the geometric phase. Such an imaginary
geometric phase results from the interference of geometric phase dependent
fluctuations around the quantum trajectory. As a specific example, we study the
quantum trajectories of the optically excited electron-hole pairs, driven by an
elliptically polarized terahertz field, in a material with non-zero Berry
curvature near the energy band extremes. While the real part of the geometric
phase leads to the Faraday rotation of the linearly polarized light that
excites the electron-hole pair, the imaginary part manifests itself as the
polarization ellipticity of the terahertz sidebands. This discovery of
geometric quantum diffusion extends the concept of geometric phases.Comment: 5 pages with 3 figure
Cosmology emerging as the gauge structure of a nonlinear quantum system
Berry phases and gauge structures in parameter spaces of quantum systems are
the foundation of a broad range of quantum effects such as quantum Hall effects
and topological insulators. The gauge structures of interacting many-body
systems, which often present exotic features, are particularly interesting.
While quantum systems are intrinsically linear due to the superposition
principle, nonlinear quantum mechanics can arise as an effective theory for
interacting systems (such as condensates of interacting bosons). Here we show
that gauge structures similar to curved spacetime can arise in nonlinear
quantum systems where the superposition principle breaks down. In the canonical
formalism of the nonlinear quantum mechanics, the geometric phases of quantum
evolutions can be formulated as the classical geometric phases of a harmonic
oscillator that represents the Bogoliubov excitations. We find that the
classical geometric phase can be described by a de Sitter universe. The
fundamental frequency of the harmonic oscillator plays the role of the cosmic
scale factor and the classical geometric phase is an integral of a differential
angle 2-form, which is half of the curvature 2-form of the associated de Sitter
universe. While the gauge structure of a linear quantum system presents
monopole singularity at energy level degeneracy points, nonlinear quantum
systems, corresponding to their quantum critical surfaces in the parameter
spaces, exhibits a conic singularity in their gauge structure, which mimics the
casual singularity at the big bang of the de Sitter universe. This finding
opens up a new approach to studying the gauge and topological structures of
interacting quantum systems and sets up a new stage for quantum simulation of
fundamental physics
Inner Product of Irreducible Finite-dimensional Representations of Classical Groups
In this paper, we study the inner product of states corresponding to weights
of irreducible finite-dimensional representations of classical groups. We
propose a recursion algorithm for calculating the inner product effectively. As
applications, we discuss the unitary of the representations space. We construct
the norms of a special kind of states. We also determine the inner product of
states in the minuscule representations. These results can be used to study the
construction of the solutions to Kapustin-Witten equations basing on the
fundamental solutions of Toda systems.Comment: 19 pages, 5 figure
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