232 research outputs found
Nonstandard Drinfeld-Sokolov reduction
Subject to some conditions, the input data for the Drinfeld-Sokolov
construction of KdV type hierarchies is a quadruplet (\A,\Lambda, d_1, d_0),
where the are -gradations of a loop algebra \A and \Lambda\in \A
is a semisimple element of nonzero -grade. A new sufficient condition on
the quadruplet under which the construction works is proposed and examples are
presented. The proposal relies on splitting the -grade zero part of \A
into a vector space direct sum of two subalgebras. This permits one to
interpret certain Gelfand-Dickey type systems associated with a nonstandard
splitting of the algebra of pseudo-differential operators in the
Drinfeld-Sokolov framework.Comment: 19 pages, LaTeX fil
Quantum projection filter for a highly nonlinear model in cavity QED
Both in classical and quantum stochastic control theory a major role is
played by the filtering equation, which recursively updates the information
state of the system under observation. Unfortunately, the theory is plagued by
infinite-dimensionality of the information state which severely limits its
practical applicability, except in a few select cases (e.g. the linear Gaussian
case.) One solution proposed in classical filtering theory is that of the
projection filter. In this scheme, the filter is constrained to evolve in a
finite-dimensional family of densities through orthogonal projection on the
tangent space with respect to the Fisher metric. Here we apply this approach to
the simple but highly nonlinear quantum model of optical phase bistability of a
stongly coupled two-level atom in an optical cavity. We observe near-optimal
performance of the quantum projection filter, demonstrating the utility of such
an approach.Comment: 19 pages, 6 figures. A version with high quality images can be found
at http://minty.caltech.edu/papers.ph
On Four-Point Functions of Half-BPS Operators in General Dimensions
We study four-point correlation functions of half-BPS operators of arbitrary
weight for all dimensions d=3,4,5,6 where superconformal theories exist. Using
harmonic superspace techniques, we derive the superconformal Ward identities
for these correlators and present them in a universal form. We then solve these
identities, employing Jack polynomial expansions. We show that the general
solution is parameterized by a set of arbitrary two-variable functions, with
the exception of the case d=4, where in addition functions of a single variable
appear. We also discuss the operator product expansion using recent results on
conformal partial wave amplitudes in arbitrary dimension.Comment: The discussion of the case d=6 expanded; references added/correcte
A natural Finsler--Laplace operator
We give a new definition of a Laplace operator for Finsler metric as an
average with regard to an angle measure of the second directional derivatives.
This definition uses a dynamical approach due to Foulon that does not require
the use of connections nor local coordinates. We show using 1-parameter
families of Katok--Ziller metrics that this Finsler--Laplace operator admits
explicit representations and computations of spectral data.Comment: 25 pages, v2: minor modifications, changed the introductio
Diffraction-limited ultrabroadband terahertz spectroscopy
Diffraction is the ultimate limit at which details of objects can be resolved in conventional optical spectroscopy and imaging systems. In the THz spectral range, spectroscopy systems increasingly rely on ultra-broadband radiation (extending over more 5 octaves) making a great challenge to reach resolution limited by diffraction. Here, we propose an original easy-to-implement wavefront manipulation concept to achieve ultrabroadband THz spectroscopy system with diffraction-limited resolution. Applying this concept to a large-area photoconductive emitter, we demonstrate diffraction-limited ultra-broadband spectroscopy system up to 14.5âTHz with a dynamic range of 103. The strong focusing of ultrabroadband THz radiation provided by our approach is essential for investigating single micrometer-scale objects such as graphene flakes or living cells, and besides for achieving intense ultra-broadband THz electric fields
Time separation as a hidden variable to the Copenhagen school of quantum mechanics
The Bohr radius is a space-like separation between the proton and electron in
the hydrogen atom. According to the Copenhagen school of quantum mechanics, the
proton is sitting in the absolute Lorentz frame. If this hydrogen atom is
observed from a different Lorentz frame, there is a time-like separation
linearly mixed with the Bohr radius. Indeed, the time-separation is one of the
essential variables in high-energy hadronic physics where the hadron is a bound
state of the quarks, while thoroughly hidden in the present form of quantum
mechanics. It will be concluded that this variable is hidden in Feynman's rest
of the universe. It is noted first that Feynman's Lorentz-invariant
differential equation for the bound-state quarks has a set of solutions which
describe all essential features of hadronic physics. These solutions explicitly
depend on the time separation between the quarks. This set also forms the
mathematical basis for two-mode squeezed states in quantum optics, where both
photons are observable, but one of them can be treated a variable hidden in the
rest of the universe. The physics of this two-mode state can then be translated
into the time-separation variable in the quark model. As in the case of the
un-observed photon, the hidden time-separation variable manifests itself as an
increase in entropy and uncertainty.Comment: LaTex 10 pages with 5 figure. Invited paper presented at the
Conference on Advances in Quantum Theory (Vaxjo, Sweden, June 2010), to be
published in one of the AIP Conference Proceedings serie
Cherenkov radiation emitted by ultrafast laser pulses and the generation of coherent polaritons
We report on the generation of coherent phonon polaritons in ZnTe, GaP and
LiTaO using ultrafast optical pulses. These polaritons are coupled modes
consisting of mostly far-infrared radiation and a small phonon component, which
are excited through nonlinear optical processes involving the Raman and the
second-order susceptibilities (difference frequency generation). We probe their
associated hybrid vibrational-electric field, in the THz range, by
electro-optic sampling methods. The measured field patterns agree very well
with calculations for the field due to a distribution of dipoles that follows
the shape and moves with the group velocity of the optical pulses. For a
tightly focused pulse, the pattern is identical to that of classical Cherenkov
radiation by a moving dipole. Results for other shapes and, in particular, for
the planar and transient-grating geometries, are accounted for by a convolution
of the Cherenkov field due to a point dipole with the function describing the
slowly-varying intensity of the pulse. Hence, polariton fields resulting from
pulses of arbitrary shape can be described quantitatively in terms of
expressions for the Cherenkov radiation emitted by an extended source. Using
the Cherenkov approach, we recover the phase-matching conditions that lead to
the selection of specific polariton wavevectors in the planar and transient
grating geometry as well as the Cherenkov angle itself. The formalism can be
easily extended to media exhibiting dispersion in the THz range. Calculations
and experimental data for point-like and planar sources reveal significant
differences between the so-called superluminal and subluminal cases where the
group velocity of the optical pulses is, respectively, above and below the
highest phase velocity in the infrared.Comment: 13 pages, 11 figure
Subcycle Quantum Electrodynamics
Besides their stunning physical properties which are unmatched in a classical
world, squeezed states of electromagnetic radiation bear advanced application
potentials in quantum information systems and precision metrology, including
gravitational wave detectors with unprecedented sensitivity. Since the first
experiments on such nonclassical light, quantum analysis has been based on
homodyning techniques and photon correlation measurements. These methods
require a well-defined carrier frequency and photons contained in a quantum
state need to be absorbed or amplified. They currently function in the visible
to near-infrared and microwave spectral ranges. Quantum nondemolition
experiments may be performed at the expense of excess fluctuations in another
quadrature. Here we generate mid-infrared time-locked patterns of squeezed
vacuum noise. After propagation through free space, the quantum fluctuations of
the electric field are studied in the time domain by electro-optic sampling
with few-femtosecond laser pulses. We directly compare the local noise
amplitude to the level of bare vacuum fluctuations. This nonlinear approach
operates off resonance without absorption or amplification of the field that is
investigated. Subcycle intervals with noise level significantly below the pure
quantum vacuum are found. Enhanced fluctuations in adjacent time segments
manifest generation of highly correlated quantum radiation as a consequence of
the uncertainty principle. Together with efforts in the far infrared, this work
opens a window to the elementary quantum dynamics of light and matter in an
energy range at the boundary between vacuum and thermal background conditions.Comment: 19 pages, 4 figure
Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two
We prove an important partial case of the pseudo-Riemannian version of the
projective Lichnerowicz conjecture stating that a complete manifold admitting
an essential group of projective transformations is the round sphere (up to a
finite cover).Comment: 32 pages, one .eps figure. The version v1 has a misprint in Theorem
1: I forgot to write the assumption that the degree of mobility is greater
than two. The versions v3, v4 have only cosmetic changes wrt v
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