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Sample quantiles, such as the median, are often better suited than the sample mean for summarising location characteristics of a data set. Similarly, linear combinations of sample quantiles and ratios of such linear combinations, e.g. the interquartile range and quantile-based skewness measures, are often used to quantify characteristics such as spread and skew. While often reported, it is uncommon to accompany quantile estimates with confidence intervals or standard errors. The rquest package provides a simple way to conduct hypothesis tests and derive confidence intervals for quantiles, linear combinations of quantiles, ratios of dependent linear combinations (e.g., Bowley's measure of skewness) and differences and ratios of all of the above for comparisons between independent samples. Many commonly used measures based on quantiles are included, although it is also very simple for users to define their own. Additionally, quantile-based measures of inequality are also considered. The methods are based on recent research showing that reliable distribution-free confidence intervals can be obtained, even for moderate sample sizes. Several examples are provided herein., Comment: 13 pages

When a chemical bond is broken, the molecular structure undergoes a transformation. An ideal experiment should probe the change in the electronic and nuclear structure simultaneously. Here, we present a method for the simultaneous time-resolved imaging of nuclear and electron dynamics by combining Coulomb explosion imaging with strong-field photoelectron momentum imaging. The simplest chemical reaction, H$_2^+$ $\rightarrow$ H$^+ +$ H, is probed experimentally for the delay-dependent kinetic energy release, and numerically for the transient change in the photoelectron spectra during the dissociation process. The three-dimensional Schr\"odinger equation is solved in the fixed-nuclei approximation numerically and the results are compared to those from a simple imaging model. The numerical results reflect the evolution in the electron density in the molecular ion as its bond is first stretched and then brakes apart. Our work shows how simple gas-phase chemical dynamics can be captured in complete molecular movies.

High-harmonic generation in solids allows probing and controlling electron dynamics in crystals on few femtosecond timescales, paving the way to lightwave electronics. In the spatial domain, recent advances in the real-space interpretation of high-harmonic emission in solids allows imaging the field-free, static, potential of the valence electrons with picometer resolution. The combination of such extreme spatial and temporal resolutions to measure and control strong-field dynamics in solids at the atomic scale is poised to unlock a new frontier of lightwave electronics. Here, we report a strong intensity-dependent anisotropy in the high-harmonic generation from ReS$_2$ that we attribute to angle-dependent interference of currents from the different atoms in the unit cell. Furthermore, we demonstrate how the laser parameters control the relative contribution of these atoms to the high-harmonic emission. Our findings provide an unprecedented atomic perspective on strong-field dynamics in crystals and suggest that crystals with a large number of atoms in the unit cell are not necessarily more efficient harmonic emitters than those with fewer atoms.

Photoelectron spectroscopy in intense laser fields has proven to be a powerful tool for providing detailed insights into molecular structure. The ionizing molecular orbital, however, has not been reconstructed from the photoelectron spectra, mainly due to the fact that its phase information can be hardly extracted. In this work, we propose a method to retrieve the phase information of the ionizing molecular orbital with laser-induced electron tunneling spectroscopy. By analyzing the interference pattern in the photoelectron spectrum, the weighted coefficients and the relative phases of the constituent atomic orbitals for a molecular orbital can be extracted. With this information we reconstruct the highest occupied molecular orbital of N$_2$. Our work provides a reliable and general approach for imaging of molecular orbitals with the photoelectron spectroscopy., Comment: 6 pages, 4 figures, including Supplementary Materials

We propose and experimentally demonstrate that the trihydrogen cation (H$_3^+$) can be produced via single photoionization of the molecular hydrogen dimer (H$_4$). Using near-infrared, femtosecond laser pulses and coincidence momentum imaging, we find that the dominant channel after single ionization of the dimer is the ejection of a hydrogen atom within a few hundred femtoseconds, leaving an H3+ cation behind. The formation mechanism is supported and well reproduced by an ab-initio molecular dynamics simulation. This is a new pathway of H$_3^+$ formation from ultracold hydrogen gas that may help explain the unexpected high abundance of H$_3^+$ in the interstellar medium in the universe.

Measuring the field of visible light with high spatial resolution has been challenging, as many established methods only detect a focus-averaged signal. Here, we introduce a near-field method for optical field sampling that overcomes that limitation by employing the localization of the enhanced near-field of a nanometric needle tip. A probe field perturbs the photoemission from the tip, which is induced by a pump pulse, generating a field-dependent current modulation that can easily be captured with our electronic detection scheme. The approach provides reliable characterization of near-petahertz fields. We show that not only the spiral wave-front of visible femtosecond light pulses carrying orbital angular momentum (OAM) can be resolved, but also the field evolution with time in the focal plane. Additionally, our method is polarization sensitive, which makes it applicable to vectorial field reconstruction., Comment: 8 pages, 3 figures, supplemental information

We formulate a transparent measure that quantifies chirality in single electron ionization triggered in atoms, which are achiral systems. We do so in the context of Ar driven by a new type of optical fields that consists of two non-collinear laser beams giving rise to chirality that varies in space across the focus of the beams. Our computations account for realistic experimental conditions. To define this measure of chirality, we first find the sign of the electron final momentum scalar triple product $\mathrm{{\bf{p}}_{k}\cdot ({\bf{p}}_{i}\times {\bf p}_{j})}$ and multiply it with the probability for an electron to ionize with certain values for both $\mathrm{p_{k}}$ and $\mathrm{p_{i}p_{j}}$. Then, we integrate this product over all values of $\mathrm{p_{k}}$ and $\mathrm{p_{i}p_{j}}$. We show this to be a robust measure of chirality in electron ionization triggered by globally chiral electric fields., Comment: 6 pages, 4 figures

We present a novel approach to single-shot characterization of the spectral phase of broadband laser pulses. Our method is inexpensive, insensitive to alignment and combines the simplicity and robustness of the dispersion scan technique, that does not require spatio-temporal pulse overlap, with the advantages of single-shot pulse characterization methods such as single-shot frequency-resolved optical gating at a real-time reconstruction rate of several Hz.

For two-electron diatomic molecules, we investigate magnetic field effects in non-sequential double ionization where recollisions prevail. We do so by formulating a three-dimensional semi-classical model that fully accounts for the Coulomb singularities and for magnetic field effects during time propagation. Using this model, we identify a prominent signature of non-dipole effects. Namely, we demonstrate that the recolliding electron back-scatters along the direction of light propagation. Hence, this electron escapes opposite to the direction of change in momentum due to the magnetic field. This is in striking contrast to strongly-driven atoms where the recolliding electron forward-scatters along the direction of light propagation. We attribute these distinct signatures to the different gate that the magnetic field creates jointly with a soft recollision in molecules compared to a hard recollision in atoms. These two different gates give rise, shortly before recollision, to different momenta and positions of the recolliding electron along the direction of light propagation. As a result, we show that the Coulomb forces from the nuclei act to back-scatter the recolliding electron in molecules and forward-scatter it in atoms along the direction of light propagation., Comment: 14 pages, 8 figures

Strong-field methods in solids enable new strategies for ultrafast nonlinear spectroscopy and provide all-optical insights into the electronic properties of condensed matter in reciprocal and real space. Additionally, solid-state media offers unprecedented possibilities to control high-harmonic generation using modified targets or tailored excitation fields. Here we merge these important points and demonstrate circularly-polarized high-harmonic generation with polarization-matched excitation fields for spectroscopy of chiral electronic properties at surfaces. The sensitivity of our approach is demonstrated for structural helicity and termination-mediated ferromagnetic order at the surface of silicon-dioxide and magnesium oxide, respectively. Circularly polarized radiation emanating from a solid sample now allows to add basic symmetry properties as chirality to the arsenal of strong-field spectroscopy in solids. Together with its inherent temporal (femtosecond) resolution and non-resonant broadband spectrum, the polarization control of high harmonics from condensed matter can illuminate ultrafast and strong field dynamics of surfaces, buried layers or thin films., Comment: 24 pages, 5 figures

We report a decreased surface wettability when polymer films on a glass substrate are treated by ultra-fast laser pulses in a back-illumination geometry. We propose that back-illumination through the substrate confines chemical changes beneath the surface of polymer films, leaving the surface blistered but chemically intact. To confirm this hypothesis, we measure the phase contrast of the polymer when observed with a focused ion beam. We observe a void at the polymer-quartz interface that results from the expansion of an ultrafast laser-induced plasma. A modified polymer layer surrounds the void, but otherwise the film seems unmodified. We also use X-ray photoelectron spectroscopy to confirm that there is no chemical change to the surface. When patterned with partially overlapping blisters, our polymer surface shows increased hydrophobicity. The increased hydrophobicity of back-illuminated surfaces can only result from the morphological change. This contrasts with the combined chemical and morphological changes of the polymer surface caused by a front-illumination geometry., Comment: 13 pages, 5 figures

We demonstrate a novel method to measure the temporal evolution of electric fields with optical frequencies. Our technique is based on the detection of transient currents in air plasma. These directional currents result from sub-cycle ionization of air with a short pump pulse, and the steering of the released electrons with the pulse to be sampled. We assess the validity of our approach by comparing it with different state-of-the-art laser-pulse characterization techniques. Notably, our method works in ambient air and facilitates a direct measurement of the field waveform, which can be viewed in real time on an oscilloscope in the exact same way as a radio frequency signal.

Chi-squared tests for lack of fit are traditionally employed to find evidence against a hypothesized model, with the model accepted if the Karl Pearson statistic comparing observed and expected numbers of observations falling within cells is not significantly large. However, if one really wants evidence for goodness of fit, it is better to adopt an equivalence testing approach in which small values of the chi-squared statistic are evidence for the desired model. This method requires one to define what is meant by equivalence to the desired model, and guidelines are proposed. Then a simple extension of the classical normalizing transformation for the non-central chi-squared distribution places these values on a simple to interpret calibration scale for evidence. It is shown that the evidence can distinguish between normal and nearby models, as well between the Poisson and over-dispersed models. Applications to evaluation of random number generators and to uniformity of the digits of pi are included. Sample sizes required to obtain a desired expected evidence for goodness of fit are also provided., Comment: 25 pages, 6 figures, 4 tables this is a substantial revision over the first arkived version. The published version (see below) also includes R code and more examples as supplementary material

We examine a commonly used relative poverty measure called the headcount ratio ($H_p$), defined to be the proportion of incomes falling below the relative poverty line, which is defined to be a fraction $p$ of the median income. We do this by considering this concept for theoretical income populations, and its potential for determining actual changes following transfer of incomes from the wealthy to those whose incomes fall below the relative poverty line. In the process we derive and evaluate the performance of large sample confidence intervals for $H_p$. Finally, we illustrate the estimators on real income data sets.

The ability to measure and control the carrier envelope phase (CEP) of few-cycle laser pulses is of paramount importance for both frequency metrology and attosecond science. Here, we present a phase meter relying on the CEP-dependent photocurrents induced by circularly polarized few-cycle pulses focused between electrodes in ambient air. The new device facilitates compact single-shot, CEP measurements under ambient conditions and promises CEP tagging at repetition rates orders of magnitude higher than most conventional CEP detection schemes as well as straightforward implementation at longer wavelengths.

We report on the unambiguous observation of the sub-cycle ionization bursts in sequential strong-field double ionization of H$_2$ and their disentanglement in molecular frame photoelectron angular distributions. This observation was made possible by the use of few-cycle laser pulses with a known carrier-envelope phase in combination with multi-particle coincidence momentum imaging. The approach demonstrated here will allow sampling of the intramolecular electron dynamics and the investigation of charge-state specific Coulomb-distortions on emitted electrons in polyatomic molecules., Comment: 5 pages + Supplemental material

Estimation of the four generalized lambda distribution parameters is not straightforward, and available estimators that perform best have large computation times. In this paper, we introduce a simple two-step estimator of the parameters that is comparatively very quick to compute and performs well when compared with other methods. This computational efficiency makes the use of bootstrapping to obtain interval estimators for the parameters possible. Simulations are used to assess the performance of the new estimators and applications to several data sets are included.

We introduce and experimentally demonstrate a method, where the two intrinsic time scales of a molecule, the slow nuclear motion and the fast electronic motion, are simultaneously measured in a photo-electron photo-ion coincidence experiment. In our experiment, elliptically polarized, 750~nm, 4.5~fs laser pulses were focused to an intensity of $9\times10^{14}\mathrm{W/cm}^2$ onto H$_2$. Using coincidence imaging, we directly observe the nuclear wavepacket evolving on the \ssg{} state of H$_2^+$ during its first roundtrip with attosecond temporal and picometer spatial resolution. The demonstrated method should enable insight into the first few femtoseconds of the vibronic dynamics of ionization-induced unimolecular reactions of larger molecules.

The coefficient of variation (CV) is commonly used to measure relative dispersion. However, since it is based on the sample mean and standard deviation, outliers can adversely affect the CV. Additionally, for skewed distributions the mean and standard deviation do not have natural interpretations and, consequently, neither does the CV. Here we investigate the extent to which quantile-based measures of relative dispersion can provide appropriate summary information as an alternative to the CV. In particular, we investigate two measures, the first being the interquartile range (in lieu of the standard deviation), divided by the median (in lieu of the mean), and the second being the median absolute deviation (MAD), divided by the median, as robust estimators of relative dispersion. In addition to comparing the influence functions of the competing estimators and their asymptotic biases and variances, we compare interval estimators using simulation studies to assess coverage., Comment: 30 pages, 2 figures Changed "analogues" to "analogs" in title to match published version. Journal of Applied Statistics (2020)

Double ionization in intense laser fields can comprise electron correlations, which manifest in the non-independent emission of two electrons from an atom or molecule. However, experimental methods that directly access the electron emission times have been scarce. Here, we explore the application of an all-optical streaking technique to strong-field double ionization both theoretically and experimentally. We show that both sequential and non-sequential double ionization processes lead to streaking delays that are distinct from each other and single ionization. Moreover, coincidence detection of ions and electrons provides access to the emission time difference, which is encoded in the two-electron momentum distributions. The experimental data agree very well with simulations of sequential double ionization. We further test and discuss the application of this method to non-sequential double ionization, which is strongly affected by the presence of the streaking field.

The strong coupling between intense laser fields and valence electrons in molecules causes a distortion of the potential energy hypersurfaces which determine the motion of nuclei in a molecule and influences possible reaction pathways. The coupling strength varies with the angle between the light electric field and valence orbital, and thereby adds another dimension to the effective molecular potential energy surface, allowing for the emergence of light-induced conical intersections. Here, we demonstrate in theory and experiment that the full complexity of such light-induced potential energy surfaces can be uncovered. In H$_2^+$, the simplest of molecules, we observe a strongly modulated angular distribution of protons which has escaped prior observation. These modulations directly result from ultrafast dynamics on the light-induced molecular potentials and can be modified by varying the amplitude, duration and phase of the mid-infrared dressing field. This opens new opportunities for manipulating the dissociation of small molecules using strong laser fields.

Electron motion on the (sub-)femtosecond time scale constitutes the fastest response in many natural phenomena such as light-induced phase transitions and chemical reactions. Whereas static electron densities in single molecules can be imaged in real-space using scanning tunnelling and atomic force microscopy, probing real-time electron motion inside molecules requires ultrafast laser pulses. Here, we demonstrate an all-optical approach to imaging an ultrafast valence electron wave packet in real-time with a time-resolution of a few femtoseconds. We employ a pump-probe-deflect scheme that allows us to prepare an ultrafast wave packet \textit{via} strong-field ionization and directly image the resulting charge oscillations in the residual ion. This approach extends and overcomes limitations in laser-induced orbital imaging and may enable the real-time imaging of electron dynamics following photoionization such as charge migration and charge transfer processes.

The quantile ratio index introduced by Prendergast and Staudte 2017 is a simple and effective measure of relative inequality for income data that is resistant to outliers. It measures the average relative distance of a randomly chosen income from its symmetric quantile. Another useful property of this index is investigated here: given a partition of the income distribution into a union of sets of symmetric quantiles, one can find the conditional inequality for each set as measured by the quantile ratio index and readily combine them in a weighted average to obtain the index for the entire population. When applied to data for various years, one can track how these contributions to inequality vary over time, as illustrated here for Australian Bureau of Statistics income and wealth data., Comment: 24 pages, 4 figures

Ionization of an atom or molecule by a strong laser field produces sub-optical cycle wave packets whose control has given rise to attosecond science. The final states of the wave packets depend on ionization and deflection by the laser field, which are convoluted in conventional experiments. Here, we demonstrate a technique enabling efficient electron deflection, separate from the field driving strong-field ionization. Using a mid-infrared deflection field permits one to distinguish electron wave packets generated at different field maxima of an intense few-cycle visible laser pulse. We utilize this capability to trace the scattering of low-energy electrons driven by the mid-infrared field. Our approach represents a general technique for studying and controlling strong-field ionization dynamics on the attosecond time scale., Comment: 5 Pages, 4 figures

We use orthogonally polarized two-colour (OTC) laser pulses to separate quantum paths in multiphoton ionization of Ar atoms. Our OTC pulses consist of 400~nm and 800~nm light at a relative intensity ratio of 10:1. We find a hitherto unobserved interference in the photoelectron momentum distribution, which exhibits a strong dependence on the relative phase of the OTC pulse. Analysis of model calculations reveal that the interference is caused by quantum pathways from non-adjacent quarter cycles.

The probability density quantile (pdQ) carries essential information regarding shape and tail behavior of a location-scale family. Convergence of repeated applications of the pdQ mapping to the uniform distribution is investigated and new fixed point theorems are established. The Kullback-Leibler divergences from uniformity of these pdQs are mapped and found to be ingredients in power functions of optimal tests for uniformity against alternative shapes., Comment: 13 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1605.00189

For every discrete or continuous location-scale family having a square-integrable density, there is a unique continuous probability distribution on the unit interval that is determined by the density-quantile composition introduced by Parzen in 1979. These probability density quantiles (pdQs) only differ in shape, and can be usefully compared with the Hellinger distance or Kullback-Leibler divergences. Convergent empirical estimates of these pdQs are provided, which leads to a robust global fitting procedure of shape families to data. Asymmetry can be measured in terms of distance or divergence of pdQs from the symmetric class. Further, a precise classification of shapes by tail behavior can be defined simply in terms of pdQ boundary derivatives., Comment: 21 pages, 8 figures, 8 tables

Using a semi-classical model, we investigate frustrated double ionization (FDI) in $\mathrm{D_3^+}$, a two-electron triatomic molecule, when driven by an intense, linearly polarized, near-infrared (800 nm) laser field. We compute the kinetic energy release of the nuclei and find a good agreement between experiment and our model. We explore the two pathways of FDI and show that, with increasing field strength, over-the-barrier ionization overtakes tunnel ionization as the underlying mechanism of FDI. Moreover, we compute the angular distribution of the ion fragments for FDI and identify a feature that can potentially be observed experimentally and is a signature of only one of the two pathways of FDI., Comment: 5 pages, 4 figures

Ratios of quantiles are often computed for income distributions as rough measures of inequality, and inference for such ratios have recently become available. The special case when the quantiles are symmetrically chosen; that is, when the p/2 quantile is divided by the (1-p/2), is of special interest because the graph of such ratios, plotted as a function of p over the unit interval, yields an informative inequality curve. The area above the curve and less than the horizontal line at one is an easily interpretable coefficient of inequality. The advantages of these concepts over the traditional Lorenz curve and Gini coefficient are numerous: they are defined for all positive income distributions, they can be robustly estimated and distribution-free confidence intervals for the inequality coefficient are easily found. Moreover the inequality curves satisfy a median-based transference principle and are convex for many commonly assumed income distributions., Comment: 20 pages, 9 figures

The classical Lorenz curve is often used to depict inequality in a population of incomes, and the associated Gini coefficient is relied upon to make comparisons between different countries and other groups. The sample estimates of these moment-based concepts are sensitive to outliers and so we investigate the extent to which quantile-based definitions can capture income inequality and lead to more robust procedures. Distribution-free estimates of the corresponding coefficients of inequality are obtained, as well as sample sizes required to estimate them to a given accuracy. Convexity, transference and robustness of the measures are examined and illustrated., Comment: 28 pages, 9 figures

Ratios of sample percentiles or of quantiles based on a single sample are often published for skewed income data to illustrate aspects of income inequality, but distribution-free confidence intervals for such ratios are to our knowledge not in the literature. Here we derive and compare two large-sample methods for obtaining such intervals. They both require good distribution-free estimates of the quantile density at the quantiles of interest, and such estimates have recently become available. Simulation studies for various sample sizes are carried out for Pareto, lognormal and exponential distributions, as well as fitted generalized lambda distributions, to determine the coverage probabilities and widths of the intervals. Robustness of the estimators to contamination or a positive proportion of zero incomes is examined via influence functions. The motivating example is Australian household income data where ratios of quantiles measure inequality, but of course these results apply equally to data from other countries., Comment: 34 pages, 5 figures

A standard approach to confidence intervals for quantiles requires good estimates of the quantile density. The optimal bandwidth for kernel estimation of the quantile density depends on an underlying location-scale family only through the quantile optimality ratio (QOR), which is the starting point for our results. While the QOR is not distribution-free, it turns out that what is optimal for one family often works quite well for families having similar shape. This allows one to rely on a single representative QOR if one has a rough idea of the distributional shape. Another option that we explore assumes the data can be modeled by the highly flexible generalized lambda distribution (GLD), already studied by others, and we show that using the QOR for the estimated GLD can lead to more than competitive intervals. Confidence intervals for the difference between quantiles from independent populations are also considered, with an application to heart rate data., Comment: 27 pages, 8 figures

When conducting a meta-analysis of standardized mean differences (SMDs), it is common to assume equal variances in the two arms of each study. This leads to Cohen's $d$ estimates for which interpretation is simple. However, this simplicity should not be used as a justification for the assumption of equal variances in situations where evidence may suggest that it is incorrect. Until now, researchers have either used an $F$-test for each individual study as a justification for the equality of variances or perhaps even conveniently ignored such tools altogether. In this paper we propose using a meta-analysis of F-test statistics to estimate the ratio of variances prior to the combination of SMD's. This procedure allows some studies to be included that might otherwise be omitted by individual fixed level tests for unequal variances, sometimes occur even when the assumption of equal variances holds. The estimated ratio of variances, as well as associated confidence intervals, can be used as guidance as to whether the assumption of equal variances is violated. The estimators considered include variance stabilization transformations (VST) of the $F$-test statistics as well as MLE estimators. The VST approaches enable the use of QQ-plots to visually inspect for violations of equal variances while the MLE estimator easily allows for the introduction of a random effect. When there is evidence of unequal variances, this work provides a means to formally justify the use of less common methods such as log ratio of means when studies are measured on a different scale.