477 research outputs found
Breakdown of the few-level approximation in collective systems
The validity of the few-level approximation in dipole-dipole interacting
collective systems is discussed. As example system, we study the archetype case
of two dipole-dipole interacting atoms, each modelled by two complete sets of
angular momentum multiplets. We establish the breakdown of the few-level
approximation by first proving the intuitive result that the dipole-dipole
induced energy shifts between collective two-atom states depend on the length
of the vector connecting the atoms, but not on its orientation, if complete and
degenerate multiplets are considered. A careful analysis of our findings
reveals that the simplification of the atomic level scheme by artificially
omitting Zeeman sublevels in a few-level approximation generally leads to
incorrect predictions. We find that this breakdown can be traced back to the
dipole-dipole coupling of transitions with orthogonal dipole moments. Our
interpretation enables us to identify special geometries in which partial
few-level approximations to two- or three-level systems are valid
Consistency of the Shannon entropy in quantum experiments
The consistency of the Shannon entropy, when applied to outcomes of quantum
experiments, is analysed. It is shown that the Shannon entropy is fully
consistent and its properties are never violated in quantum settings, but
attention must be paid to logical and experimental contexts. This last remark
is shown to apply regardless of the quantum or classical nature of the
experiments.Comment: 12 pages, LaTeX2e/REVTeX4. V5: slightly different than the published
versio
Remarks on Shannon's Statistical Inference and the Second Law in Quantum Statistical Mechanics
We comment on a formulation of quantum statistical mechanics, which
incorporates the statistical inference of Shannon.
Our basic idea is to distinguish the dynamical entropy of von Neumann, , in terms of the density matrix ,
and the statistical amount of uncertainty of Shannon, , with in the representation where the total
energy and particle numbers are diagonal. These quantities satisfy the
inequality . We propose to interprete Shannon's statistical inference
as specifying the {\em initial conditions} of the system in terms of . A
definition of macroscopic observables which are characterized by intrinsic time
scales is given, and a quantum mechanical condition on the system, which
ensures equilibrium, is discussed on the basis of time averaging.
An interesting analogy of the change of entroy with the running coupling in
renormalization group is noted. A salient feature of our approach is that the
distinction between statistical aspects and dynamical aspects of quantum
statistical mechanics is very transparent.Comment: 16 pages. Minor refinement in the statements in the previous version.
This version has been published in Journal of Phys. Soc. Jpn. 71 (2002) 6
Chaos for Liouville probability densities
Using the method of symbolic dynamics, we show that a large class of
classical chaotic maps exhibit exponential hypersensitivity to perturbation,
i.e., a rapid increase with time of the information needed to describe the
perturbed time evolution of the Liouville density, the information attaining
values that are exponentially larger than the entropy increase that results
from averaging over the perturbation. The exponential rate of growth of the
ratio of information to entropy is given by the Kolmogorov-Sinai entropy of the
map. These findings generalize and extend results obtained for the baker's map
[R. Schack and C. M. Caves, Phys. Rev. Lett. 69, 3413 (1992)].Comment: 26 pages in REVTEX, no figures, submitted to Phys. Rev.
Reference priors for high energy physics
Bayesian inferences in high energy physics often use uniform prior
distributions for parameters about which little or no information is available
before data are collected. The resulting posterior distributions are therefore
sensitive to the choice of parametrization for the problem and may even be
improper if this choice is not carefully considered. Here we describe an
extensively tested methodology, known as reference analysis, which allows one
to construct parametrization-invariant priors that embody the notion of minimal
informativeness in a mathematically well-defined sense. We apply this
methodology to general cross section measurements and show that it yields
sensible results. A recent measurement of the single top quark cross section
illustrates the relevant techniques in a realistic situation
Black-hole radiation, the fundamental area unit, and the spectrum of particle species
Bekenstein and Mukhanov have put forward the idea that, in a quantum theory
of gravity a black hole should have a discrete mass spectrum with a concomitant
{\it discrete} line emission. We note that a direct consequence of this
intriguing prediction is that, compared with blackbody radiation, black-hole
radiance is {\it less} entropic. We calculate the ratio of entropy emission
rate from a quantum black hole to the rate of black-hole entropy decrease, a
quantity which, according to the generalized second law (GSL) of
thermodynamics, should be larger than unity. Implications of our results for
the GSL, for the value of the fundamental area unit in quantum gravity, and for
the spectrum of massless particles in nature are discussed.Comment: 4 page
Macroscopic Interference Effects in Resonant Cavities
We investigate the possibility of interference effects induced by macroscopic
quantum-mechanical superpositions of almost othogonal coherent states - a
Schroedinger cats state - in a resonant microcavity. Despite the fact that a
single atom, used as a probe of the cat state, on the average only change the
mean number of photons by one unit, we show that this single atom can change
the system drastically. Interference between the initial and almost orthogonal
macroscopic quantum states of the radiation field can now take place.
Dissipation under current experimental conditions is taken into account and it
is found that this does not necessarily change the intereference effects
dramatically.Comment: 20 pages, 3 figure
Use and Abuse of the Fisher Information Matrix in the Assessment of Gravitational-Wave Parameter-Estimation Prospects
The Fisher-matrix formalism is used routinely in the literature on
gravitational-wave detection to characterize the parameter-estimation
performance of gravitational-wave measurements, given parametrized models of
the waveforms, and assuming detector noise of known colored Gaussian
distribution. Unfortunately, the Fisher matrix can be a poor predictor of the
amount of information obtained from typical observations, especially for
waveforms with several parameters and relatively low expected signal-to-noise
ratios (SNR), or for waveforms depending weakly on one or more parameters, when
their priors are not taken into proper consideration. In this paper I discuss
these pitfalls; show how they occur, even for relatively strong signals, with a
commonly used template family for binary-inspiral waveforms; and describe
practical recipes to recognize them and cope with them.
Specifically, I answer the following questions: (i) What is the significance
of (quasi-)singular Fisher matrices, and how must we deal with them? (ii) When
is it necessary to take into account prior probability distributions for the
source parameters? (iii) When is the signal-to-noise ratio high enough to
believe the Fisher-matrix result? In addition, I provide general expressions
for the higher-order, beyond--Fisher-matrix terms in the 1/SNR expansions for
the expected parameter accuracies.Comment: 24 pages, 3 figures, previously known as "A User Manual for the
Fisher Information Matrix"; final, corrected PRD versio
Frequency Tracking and Parameter Estimation for Robust Quantum State-Estimation
In this paper we consider the problem of tracking the state of a quantum
system via a continuous measurement. If the system Hamiltonian is known
precisely, this merely requires integrating the appropriate stochastic master
equation. However, even a small error in the assumed Hamiltonian can render
this approach useless. The natural answer to this problem is to include the
parameters of the Hamiltonian as part of the estimation problem, and the full
Bayesian solution to this task provides a state-estimate that is robust against
uncertainties. However, this approach requires considerable computational
overhead. Here we consider a single qubit in which the Hamiltonian contains a
single unknown parameter. We show that classical frequency estimation
techniques greatly reduce the computational overhead associated with Bayesian
estimation and provide accurate estimates for the qubit frequencyComment: 6 figures, 13 page
Analyzing symmetry breaking within a chaotic quantum system via Bayesian inference
Bayesian inference is applied to the level fluctuations of two coupled
microwave billiards in order to extract the coupling strength. The coupled
resonators provide a model of a chaotic quantum system containing two coupled
symmetry classes of levels. The number variance is used to quantify the level
fluctuations as a function of the coupling and to construct the conditional
probability distribution of the data. The prior distribution of the coupling
parameter is obtained from an invariance argument on the entropy of the
posterior distribution.Comment: Example from chaotic dynamics. 8 pages, 7 figures. Submitted to PR
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