1,273 research outputs found
Perturbation of an Eigen-Value from a Dense Point Spectrum : An Example
We study a perturbed Floquet Hamiltonian depending on a coupling
constant . The spectrum is assumed to be pure point and
dense. We pick up an eigen-value, namely , and show the
existence of a function defined on such that
for all , 0 is a point of
density for the set , and the Rayleigh-Schr\"odinger perturbation series
represents an asymptotic series for the function . All ideas
are developed and demonstrated when treating an explicit example but some of
them are expected to have an essentially wider range of application.Comment: Latex, 24 pages, 51
Weakly regular Floquet Hamiltonians with pure point spectrum
We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as depending on
the parameter omega. We assume that the spectrum of H is discrete, {h_m (m =
1..infinity)}, with h_m of multiplicity M_m. and that V is an Hermitian
operator, 2pi-periodic in t. Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose
that for some sigma > 0: sum_{m,n such that h_m > h_n} mu_{mn}(h_m -
h_n)^(-sigma) < infinity where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show
that in that case there exist a suitable norm to measure the regularity of V,
denoted epsilon, and positive constants, epsilon_* & delta_*, such that: if
epsilon
|Omega_0| - delta_* epsilon and the Floquet Hamiltonian has a pure point
spectrum for all omega in Omega_infinity.Comment: 35 pages, Latex with AmsAr
Shear-induced breaking of internal gravity waves
Motivated by observations of turbulence in the strongly stratified ocean
thermocline, we use direct numerical simulations to investigate the interaction
of a sinusoidal shear flow and a large-amplitude internal gravity wave. Despite
strong nonlinearities in the flow and a lack of scale separation, we find that
linear ray tracing theory is qualitatively useful in describing the early
development of the flow as the wave is refracted by the shear. Consistent with
the linear theory, the energy of the wave accumulates in regions of negative
mean shear where we observe evidence of convective and shear instabilities.
Streamwise-aligned convective rolls emerge the fastest, but their contribution
to irreversible mixing is dwarfed by shear-driven billow structures that
develop later. Although the wave strongly distorts the buoyancy field on which
these billows develop, the mixing efficiency of the subsequent turbulence is
similar to that arising from Kelvin-Helmholtz instability in a stratified shear
layer. We run simulations at Reynolds numbers of 5000 and 8000, and vary the
initial amplitude of the internal gravity wave. For high values of initial wave
amplitude, the results are qualitatively independent of . Smaller initial
wave amplitudes delay the onset of the instabilities, and allow for significant
laminar diffusion of the internal wave, leading to reduced turbulent activity.
We discuss the complex interaction between the mean flow, internal gravity wave
and turbulence, and its implications for internal wave-driven mixing in the
ocean.Comment: 27 pages, 12 figures, accepted to J. Fluid. Mec
Quantifying mixing and available potential energy in vertically periodic simulations of stratified flows
Turbulent mixing exerts a significant influence on many physical processes in
the ocean. In a stably stratified Boussinesq fluid, this irreversible mixing
describes the conversion of available potential energy (APE) to background
potential energy (BPE). In some settings the APE framework is difficult to
apply and approximate measures are used to estimate irreversible mixing. For
example, numerical simulations of stratified turbulence often use triply
periodic domains to increase computational efficiency. In this setup however,
BPE is not uniquely defined and the method of Winters et al. (1995, J. Fluid
Mech., 289) cannot be directly applied to calculate the APE. We propose a new
technique to calculate APE in periodic domains with a mean stratification. By
defining a control volume bounded by surfaces of constant buoyancy, we can
construct an appropriate background buoyancy profile and
accurately quantify diapycnal mixing in such systems. This technique also
permits the accurate calculation of a finite amplitude local APE density in
periodic domains. The evolution of APE is analysed in various turbulent
stratified flow simulations. We show that the mean dissipation rate of buoyancy
variance provides a good approximation to the mean diapycnal mixing
rate, even in flows with significant variations in local stratification. When
quantifying measures of mixing efficiency in transient flows, we find
significant variation depending on whether laminar diffusion of a mean flow is
included in the kinetic energy dissipation rate. We discuss how best to
interpret these results in the context of quantifying diapycnal diffusivity in
real oceanographic flows.Comment: 28 pages, 10 figures, accepted to J. Fluid Mec
On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum
We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum
of H is semibounded and discrete, and the eigenvalues behave as E_n~n^\alpha,
with 0<\alpha<1. In particular, the gaps between successive eigenvalues decay
as n^{\alpha-1}. V(t) is supposed to be periodic, bounded, continuously
differentiable in the strong sense and such that the matrix entries with
respect to the spectral decomposition of H obey the estimate
|V(t)_{m,n}|0,
p>=1 and \gamma=(1-\alpha)/2. We show that the energy diffusion exponent can be
arbitrarily small provided p is sufficiently large and \epsilon is small
enough. More precisely, for any initial condition \Psi\in Dom(H^{1/2}), the
diffusion of energy is bounded from above as _\Psi(t)=O(t^\sigma) where
\sigma=\alpha/(2\ceil{p-1}\gamma-1/2). As an application we consider the
Hamiltonian H(t)=|p|^\alpha+\epsilon*v(\theta,t) on L^2(S^1,d\theta) which was
discussed earlier in the literature by Howland
Quantum Mutual Information Capacity for High Dimensional Entangled States
High dimensional Hilbert spaces used for quantum communication channels offer
the possibility of large data transmission capabilities. We propose a method of
characterizing the channel capacity of an entangled photonic state in high
dimensional position and momentum bases. We use this method to measure the
channel capacity of a parametric downconversion state, achieving a channel
capacity over 7 bits/photon in either the position or momentum basis, by
measuring in up to 576 dimensions per detector. The channel violated an
entropic separability bound, suggesting the performance cannot be replicated
classically.Comment: 5 pages, 2 figure
Quantum Mutual Information Capacity for High-Dimensional Entangled States
High-dimensional Hilbert spaces used for quantum communication channels offer the possibility of large data transmission capabilities. We propose a method of characterizing the channel capacity of an entangled photonic state in high-dimensional position and momentum bases. We use this method to measure the channel capacity of a parametric down-conversion state by measuring in up to 576 dimensions per detector. We achieve a channel capacity over 7ââbits/photon in either the position or momentum basis. Furthermore, we provide a correspondingly high-dimensional separability bound that suggests that the channel performance cannot be replicated classically
Time Dependent Floquet Theory and Absence of an Adiabatic Limit
Quantum systems subject to time periodic fields of finite amplitude, lambda,
have conventionally been handled either by low order perturbation theory, for
lambda not too large, or by exact diagonalization within a finite basis of N
states. An adiabatic limit, as lambda is switched on arbitrarily slowly, has
been assumed. But the validity of these procedures seems questionable in view
of the fact that, as N goes to infinity, the quasienergy spectrum becomes
dense, and numerical calculations show an increasing number of weakly avoided
crossings (related in perturbation theory to high order resonances). This paper
deals with the highly non-trivial behavior of the solutions in this limit. The
Floquet states, and the associated quasienergies, become highly irregular
functions of the amplitude, lambda. The mathematical radii of convergence of
perturbation theory in lambda approach zero. There is no adiabatic limit of the
wave functions when lambda is turned on arbitrarily slowly. However, the
quasienergy becomes independent of time in this limit. We introduce a
modification of the adiabatic theorem. We explain why, in spite of the
pervasive pathologies of the Floquet states in the limit N goes to infinity,
the conventional approaches are appropriate in almost all physically
interesting situations.Comment: 13 pages, Latex, plus 2 Postscript figure
Fast oxygen dynamics as a potential biomarker for epilepsy
Canadian Institutes of Health Research (GCT: MOP#130495, JGH: MOP#125984).Peer ReviewedChanges in brain activity can entrain cerebrovascular dynamics, though this has not been extensively investigated in pathophysiology. We assessed whether pathological network activation (i.e. seizures) in the Genetic Absence Epilepsy Rat from Strasbourg (GAERS) could alter dynamic fluctuations in local oxygenation. Spontaneous absence seizures in an epileptic rat model robustly resulted in brief dips in cortical oxygenation and increased spectral oxygen power at frequencies greater than 0.08âHz. Filtering oxygen data for these fast dynamics was sufficient to distinguish epileptic vs. non-epileptic rats. Furthermore, this approach distinguished brain regions with seizures from seizure-free brain regions in the epileptic rat strain. We suggest that fast oxygen dynamics may be a useful biomarker for seizure network identification and could be translated to commonly used clinical tools that measure cerebral hemodynamics
Neurophysiology
Contains reports on one research project.Teagle FoundationOffice of Naval ResearchBell Telephone Laboratories, Incorporate
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