816 research outputs found
Non-monotonous short-time decay of the Loschmidt echo in quasi-one-dimensional systems
We study the short-time stability of quantum dynamics in quasi-one-dimensional systems with respect to small localized perturbations of the potential. To this end, we address, analytically and numerically, the decay of the Loschmidt echo (LE) during times short compared to the Ehrenfest time. We find that the LE is generally a non-monotonous function of time and exhibits strongly pronounced minima and maxima at the instants of time when the corresponding classical particle traverses the perturbation region. We also show that, under general conditions, the envelope decay of the LE is well approximated by a Gaussian, and we derive explicit analytical formulas for the corresponding decay time. Finally, we demonstrate that the observed non-monotonicity of the LE decay is only pertinent to one-dimensional (and, more generally, quasi-one-dimensional systems), and that the short-time decay of the LE can be monotonous in higher number of dimensions
Diffraction in time: An exactly solvable model
In recent years, matter-wave interferometry has attracted growing attention due to its unique suitability for high-precision measurements and the study of fundamental aspects of quantum theory. Diffraction and interference of matter waves can be observed not only at a spatial aperture (such as a screen edge, slit, or grating), but also at a time-domain aperture (such as an absorbing barrier, or “shutter,” that is being periodically switched on and off). The wave phenomenon of the latter type is commonly referred to as “diffraction in time.” Here, we introduce a versatile, exactly solvable model of diffraction in time. It describes time evolution of an arbitrary initial quantum state in the presence of a time-dependent absorbing barrier, governed by an arbitrary aperture function. Our results enable a quantitative description of diffraction and interference patterns in a large variety of setups, and may be used to devise new diffraction and interference experiments with atoms and molecules
Nonparametric estimation of multivariate convex-transformed densities
We study estimation of multivariate densities of the form
for and for a fixed monotone function and an unknown
convex function . The canonical example is for ; in this case, the resulting class of densities [\mathcal
{P}(e^{-y})={p=\exp(-g):g is convex}] is well known as the class of log-concave
densities. Other functions allow for classes of densities with heavier
tails than the log-concave class. We first investigate when the maximum
likelihood estimator exists for the class for
various choices of monotone transformations , including decreasing and
increasing functions . The resulting models for increasing transformations
extend the classes of log-convex densities studied previously in the
econometrics literature, corresponding to . We then establish
consistency of the maximum likelihood estimator for fairly general functions
, including the log-concave class and many others. In
a final section, we provide asymptotic minimax lower bounds for the estimation
of and its vector of derivatives at a fixed point under natural
smoothness hypotheses on and . The proofs rely heavily on results from
convex analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOS840 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Lyapunov spreading of semi-classical wave packets for the Lorentz Gas: theory and applications
We consider the quantum mechanical propagator for a particle moving in a -dimensional Lorentz gas, with fixed, hard sphere scatterers. To evaluate this propagator in the semi-classical region, and for times less than the Ehrenfest time, we express its effect on an initial Gaussian wave packet in terms of quantities analogous to those used to describe the exponential separation of trajectories in the classical version of this system. This result relates the spread of the wave packet to the rate of separation of classical trajectories, characterized by positive Lyapunov exponents. We consider applications of these results, first to illustrate the behavior of the wave-packet auto-correlation functions for wave packets on periodic orbits. The auto-correlation function can be related to the fidelity, or Loschmidt echo, for the special case that the perturbation is a small change in the mass of the particle. An exact expression for the fidelity, appropriate for this perturbation, leads to an analytical result valid over very long time intervals, inversely proportional to the size of the mass perturbation. For such perturbations, we then calculate the long-time echo for semi-classical wave packets on periodic orbits. This paper also corrects an earlier calculation for a quantum echo, included in a previous version of this paper. We explain the reasons for this correction
The Quantum Normal Form Approach to Reactive Scattering: The Cumulative Reaction Probability for Collinear Exchange Reactions
The quantum normal form approach to quantum transition state theory is used
to compute the cumulative reaction probability for collinear exchange
reactions. It is shown that for heavy atom systems like the nitrogen exchange
reaction the quantum normal form approach gives excellent results and has major
computational benefits over full reactive scattering approaches. For light atom
systems like the hydrogen exchange reaction however the quantum normal approach
is shown to give only poor results. This failure is attributed to the
importance of tunnelling trajectories in light atom reactions that are not
captured by the quantum normal form as indicated by the only very slow
convergence of the quantum normal form for such systems.Comment: 8 pages, 4 figure
Cycloaddition Chemistry of a Silylene‐Nickel Complex toward Organic π‐Systems: From Reversibility to C−H Activation
The versatile cycloaddition chemistry of the Si−Ni multiple bond in the acyclic (amido)(chloro)silylene→Ni0 complex 1, [(TMSL)ClSi→Ni(NHC)2] (TMSL=N(SiMe3)Dipp; Dipp=2,6‐iPr2C6H4; NHC=C[(iPr)NC(Me)]2), toward unsaturated organic substrates is reported, which is both reminiscent of and expanding on the reactivity patterns of classical Fischer and Schrock carbene–metal complexes. Thus, 1:1 reaction of 1 with aldehydes, imines, alkynes, and even alkenes proceed to yield [2+2] cycloaddition products, leading to a range of four‐membered metallasilacycles. This cycloaddition is in fact reversible for ethylene, whereas addition of an excess of this olefin leads to quantitative sp2‐CH bond activation, via a 1‐nickela‐4‐silacyclohexane intermediate. These results have been supported by DFT calculations giving insights into key mechanistic aspects.DFG, 390540038, EXC 2008: UniSysCatTU Berlin, Open-Access-Mittel - 202
Imputation Estimators Partially Correct for Model Misspecification
Inference problems with incomplete observations often aim at estimating
population properties of unobserved quantities. One simple way to accomplish
this estimation is to impute the unobserved quantities of interest at the
individual level and then take an empirical average of the imputed values. We
show that this simple imputation estimator can provide partial protection
against model misspecification. We illustrate imputation estimators' robustness
to model specification on three examples: mixture model-based clustering,
estimation of genotype frequencies in population genetics, and estimation of
Markovian evolutionary distances. In the final example, using a representative
model misspecification, we demonstrate that in non-degenerate cases, the
imputation estimator dominates the plug-in estimate asymptotically. We conclude
by outlining a Bayesian implementation of the imputation-based estimation.Comment: major rewrite, beta-binomial example removed, model based clustering
is added to the mixture model example, Bayesian approach is now illustrated
with the genetics exampl
Origin of the exponential decay of the Loschmidt echo in integrable systems
We address the time decay of the Loschmidt echo, measuring the sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using a semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, similar to the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral properties of the unperturbed and perturbed Hamilton operators and the initial state. In particular, we show that the decay rate, unlike in the case of the chaotic dynamics, is directly proportional to the strength of the Hamiltonian perturbation. Finally, we compare our analytical predictions against the results of a numerical computation of the Loschmidt echo for a quantum particle moving inside a one-dimensional box with Dirichlet-Robin boundary conditions, and find the two in good agreement
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