Stochastic Heisenberg limit: Optimal estimation of a fluctuating phase

The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cramer-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as 1/omega^p with p>1, the minimum mean-square error in any (single-time) phase estimate scales as N^{-2(p-1)/(p+1)}, where N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit p--> infinity) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2 (Brownian motion), this limit can be attained by phase tracking.Comment: 5+4 pages, to appear in Physical Review Letter

Occurrence and core-envelope structure of 1--4x Earth-size planets around Sun-like stars

Small planets, 1-4x the size of Earth, are extremely common around Sun-like stars, and surprisingly so, as they are missing in our solar system. Recent detections have yielded enough information about this class of exoplanets to begin characterizing their occurrence rates, orbits, masses, densities, and internal structures. The Kepler mission finds the smallest planets to be most common, as 26% of Sun-like stars have small, 1-2 R_e planets with orbital periods under 100 days, and 11% have 1-2 R_e planets that receive 1-4x the incident stellar flux that warms our Earth. These Earth-size planets are sprinkled uniformly with orbital distance (logarithmically) out to 0.4 AU, and probably beyond. Mass measurements for 33 transiting planets of 1-4 R_e show that the smallest of them, R < 1.5 R_e, have the density expected for rocky planets. Their densities increase with increasing radius, likely caused by gravitational compression. Including solar system planets yields a relation: rho = 2.32 + 3.19 R/R_e [g/cc]. Larger planets, in the radius range 1.5-4.0 R_e, have densities that decline with increasing radius, revealing increasing amounts of low-density material in an envelope surrounding a rocky core, befitting the appellation "mini-Neptunes." Planets of ~1.5 R_e have the highest densities, averaging near 10 g/cc. The gas giant planets occur preferentially around stars that are rich in heavy elements, while rocky planets occur around stars having a range of heavy element abundances. One explanation is that the fast formation of rocky cores in protoplanetary disks enriched in heavy elements permits the gravitational accumulation of gas before it vanishes, forming giant planets. But models of the formation of 1-4 R_e planets remain uncertain. Defining habitable zones remains difficult, without benefit of either detections of life elsewhere or an understanding of life's biochemical origins.Comment: 11 pages, 6 figures, accepted for publication Proc. Natl. Acad. Sc

Concepts of quantum non-Markovianity: a hierarchy

Markovian approximation is a widely-employed idea in descriptions of the dynamics of open quantum systems (OQSs). Although it is usually claimed to be a concept inspired by classical Markovianity, the term quantum Markovianity is used inconsistently and often unrigorously in the literature. In this report we compare the descriptions of classical stochastic processes and quantum stochastic processes (as arising in OQSs), and show that there are inherent differences that lead to the non-trivial problem of characterizing quantum non-Markovianity. Rather than proposing a single definition of quantum Markovianity, we study a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, such as quantum white noise, factorization approximation, divisibility, Lindblad master equation, etc.. Others are first proposed in this report, including those we call past-future independence, no (quantum) information backflow, and composability. All of these concepts are defined under a unified framework, which allows us to rigorously build hierarchy relations among them. With various examples, we argue that the current most often used definitions of quantum Markovianity in the literature do not fully capture the memoryless property of OQSs. In fact, quantum non-Markovianity is highly context-dependent. The results in this report, summarized as a hierarchy figure, bring clarity to the nature of quantum non-Markovianity.Comment: Clarifications and references added; discussion of the related classical hierarchy significantly improved. To appear in Physics Report

The quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation

We propose quantum versions of the Bell-Ziv-Zakai lower bounds on the error in multiparameter estimation. As an application we consider measurement of a time-varying optical phase signal with stationary Gaussian prior statistics and a power law spectrum $\sim 1/|\omega|^p$, with $p>1$. With no other assumptions, we show that the mean-square error has a lower bound scaling as $1/{\cal N}^{2(p-1)/(p+1)}$, where ${\cal N}$ is the time-averaged mean photon flux. Moreover, we show that this accuracy is achievable by sampling and interpolation, for any $p>1$. This bound is thus a rigorous generalization of the Heisenberg limit, for measurement of a single unknown optical phase, to a stochastically varying optical phase.Comment: 18 pages, 6 figures, comments welcom

Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates

The ultimate bound to the accuracy of phase estimates is often assumed to be given by the Heisenberg limit. Recent work seemed to indicate that this bound can be violated, yielding measurements with much higher accuracy than was previously expected. The Heisenberg limit can be restored as a rigorous bound to the accuracy provided one considers the accuracy averaged over the possible values of the unknown phase, as we have recently shown [Phys. Rev. A 85, 041802(R) (2012)]. Here we present an expanded proof of this result together with a number of additional results, including the proof of a previously conjectured stronger bound in the asymptotic limit. Other measures of the accuracy are examined, as well as other restrictions on the generator of the phase shifts. We provide expanded numerical results for the minimum error and asymptotic expansions. The significance of the results claiming violation of the Heisenberg limit is assessed, followed by a detailed discussion of the limitations of the Cramer-Rao bound.Comment: 22 pages, 4 figure

Mineral Trioxide Aggregate Material Use in Endodontic Treatment: A Review of the Literature

Objective The purpose of this paper was to review the composition, properties, biocompatibility, and the clinical results involving the use of mineral trioxide aggregate (MTA) materials in endodontic treatment. Methods Electronic search of scientific papers from January 1990 to August 2006 was accomplished using PubMed and Scopus search engines (search terms: MTA, GMTA, WMTA, mineral AND trioxide AND aggregate). Results Selected exclusion criteria resulted in 156 citations from the scientific, peer-reviewed dental literature. MTA materials are derived from a Portland cement parent compound and have been demonstrated to be biocompatible endodontic repair materials, with its biocompatible nature strongly suggested by its ability to form hydroxyappatite when exposed to physiologic solutions. With some exceptions, MTA materials provide better microleakage protection than traditional endodontic repair materials using dye, fluid filtration, and bacterial penetration leakage models. In both animal and human studies, MTA materials have been shown to have excellent potential as pulp-capping and pulpotomy medicaments but studies with long-term follow-up are limited. Preliminary studies suggested a favorable MTA material use as apical and furcation restorative materials as well as medicaments for apexogenesis and apexification treatments; however, long-term clinical studies are needed in these areas. Conclusion MTA materials have been shown to have a biocompatible nature and have excellent potential in endodontic use. MTA materials are a refined Portland cement material and the substitution of Portland cement for MTA products is presently discouraged. Existing human studies involving MTA materials are very promising, however, insufficient randomized, double-blind clinical studies of sufficient duration exist involving MTA for all of its clinical indications. Further clinical studies are needed in these areas

Some interactions among driver, vehicle, and roadway variables in normal driving

Effects of road and vehicle conditions, visual warning signs, direction of turns, night time, and skill on automobile driver performance are studied in several experiments. Considered criteria are variability in speed and acceleration

Improved efficiency of nutrient and water use for high quality field vegetable production using fertigation

Drip-based fertigation may improve the application efficiency of water and nutrients while maintaining or improving marketable yield and quality at harvest and post-harvest. Two plantings of lettuce (Lactuca sativa) were grown in the UK, with six N treatments and two methods of irrigation and N application. The conventional overhead irrigated treatments had all N applied in the base dressing with irrigation scheduled from SMD calculations. The closed loop treatments had nitrogen and irrigation delivered via drip automatically controlled by a sensor and logger system. The work established that water content in the root zone can be monitored in real time using horizontally oriented soil moisture sensors linked to data logging and telemetry, and that these data can be used to automatically trigger drip irrigation for commercially grown field vegetables. When the closed loop irrigation control was combined with fertigation treatments, lettuce crops were grown with savings of up to 60% and 75% of water and nitrogen respectively, compared to standard UK production systems. However, excess supply of N through fertigation rather than solid fertiliser was more detrimental to marketable yield and post harvest quality highlighting that care is needed when selecting N rates for fertigation