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Tunable biphoton quantum entanglement generated from nonlinear processes is highly desirable for cutting-edge quantum technologies, yet its tunability is substantially constrained by the symmetry of material nonlinear tensors. Here, we overcome this constraint by introducing symmetry-breaking in nonlinear polarization to generate optically tunable biphoton entanglement at picosecond speeds. Asymmetric optical responses have made breakthroughs in classical applications like non-reciprocal light transmission. We now experimentally demonstrate the nonlinear asymmetry response for biphoton entanglement using a semiconductor metasurface incorporating [110] InGaP nano-resonators with structural asymmetry. We realize continuous tuning of polarization entanglement from near-unentangled states to a Bell state. This tunability can also extend to produce tailored hyperentanglement. Furthermore, our nanoscale entanglement source features an ultra-high coincidence-to-accidental ratio of $\approx7\times10^4$, outperforming existing semiconductor flat optics by two orders of magnitude. Introducing asymmetric nonlinear response in quantum metasurfaces opens new directions for tailoring on-demand quantum states and beyond., Comment: 27 pages, 4 figures

The development of active metasurface systems, such as lasing metasurfaces, requires the optimization of multiple modes at the absorption and lasing wavelength bands, including their quality factor, mode profile and angular dispersion. Often, these requirements are contradictory and impossible to obtain with conventional design techniques. Importantly, the properties of the eigenmodes of a metasurface are directly linked to their symmetry, which offers an opportunity to explore mode symmetry as an objective in optimization routines for active metasurface design. Here, we propose and numerically demonstrate a novel multi-objective optimization technique based on symmetry projection operators to quantify the symmetry of the metasurface eigenmodes. We present, as an example, the optimization of a lasing metasurface based on up-converting nano-particles. Our technique allows us to optimize the absorption mode dispersion, as well as the directionality of the lasing emission and therefore offers advantages for novel lasing systems with high directionality and low lasing threshold., Comment: 20 pages, 5 figures

Complex polarization states of photon pairs are indispensable in various quantum technologies. Conventional methods for preparing desired two-photon polarization states are realized through bulky nonlinear crystals, which can restrict the versatility and tunability of the generated quantum states due to the fixed crystal nonlinear susceptibility. Here we present a solution using a nonlinear metasurface incorporating multiplexed silica metagratings on a lithium niobate film of 300 nanometer thickness. We fabricate two orthogonal metagratings on a single substrate with an identical resonant wavelength, thereby enabling the spectral indistinguishability of the emitted photons, and demonstrate in experiments that the two-photon polarization states can be shaped by the metagrating orientation. Leveraging this essential property, we formulate a theoretical approach for generating arbitrary polarization-entangled qutrit states by combining three metagratings on a single metasurface, allowing the encoding of desired quantum states or information. Our findings enable miniaturized optically controlled quantum devices using ultrathin metasurfaces as polarization-entangled photon sources., Comment: 21 pages, 5 figures, journal paper

Metasurfaces consisting of nano-scale structures are underpinning new physical principles for the creation and shaping of quantum states of light. Multi-photon states that are entangled in spatial or angular domains are an essential resource for quantum imaging and sensing applications, however their production traditionally relies on bulky nonlinear crystals. We predict and demonstrate experimentally the generation of spatially entangled photon pairs through spontaneous parametric down-conversion from a metasurface incorporating a nonlinear thin film of lithium niobate. This is achieved through nonlocal resonances with tailored angular dispersion mediated by an integrated silica meta-grating, enabling control of the emission pattern and associated quantum states of photon pairs by designing the grating profile and tuning the pump frequency. We measure the correlations of photon positions and identify their spatial anti-bunching through violation of the classical Cauchy-Schwartz inequality, witnessing the presence of multi-mode entanglement. Simultaneously, the photon-pair rate is strongly enhanced by 450 times as compared to unpatterned films due to high-quality-factor metasurface resonances, and the coincidence to accidental ratio reaches 5000. These results pave the way to miniaturization of various quantum devices by incorporating ultra-thin metasurfaces functioning as room-temperature sources of quantum-entangled photons.

We develop a finite difference approximation of order $\alpha$ for the $\alpha$-fractional derivative. The weights of the approximation scheme have the same rate-matrix type properties as the popular Gr\"unwald scheme. In particular, approximate solutions to fractional diffusion equations preserve positivity. Furthermore, for the approximation of the solution to the skewed fractional heat equation on a bounded domain the new approximation scheme keeps its order $\alpha$ whereas the order of the Gr\"unwald scheme reduces to order $\alpha-1$, contradicting the convergence rate results by Meerschaert and Tadjeran.

We reveal a novel regime of photon-pair generation driven by the interplay of multiple bound states in the continuum resonances in nonlinear metasurfaces. This non-degenerate photon-pair generation is derived from the hyperbolic topology of the transverse phase-matching and can enable orders-of-magnitude enhancement of the photon rate and spectral brightness, as compared to the degenerate regime. We show that the entanglement of the photon-pairs can be tuned by varying the pump polarization, which can underpin future advances and applications of ultra-compact quantum light sources., Comment: 6 pages, 4 figures

In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials this paper explains how to use the elementary procedure of Euler summation to sum these divergent series and thereby to make sense of the formal statement of the completeness of the formal sum that represents the reconstruction of the Hamiltonian., Comment: 5 pages, version to appear in American Journal of Physics

Metasurfaces based on resonant nanophotonic structures have enabled novel types of flat-optics devices often outperforming the capabilities of bulk components, yet these advances remain largely unexplored for quantum applications. We show that non-classical multi-photon interferences can be achieved at the subwavelength scale in all-dielectric metasurfaces. We simultaneously image multiple projections of quantum states with a single metasurface, enabling a robust reconstruction of amplitude, phase, coherence, and entanglement of multi-photon polarization-encoded states. One- and two-photon states are reconstructed through nonlocal photon correlation measurements with polarization-insensitive click-detectors positioned after the metasurface, and the scalability to higher photon numbers is established theoretically. Our work illustrates the feasibility of ultra-thin quantum metadevices for the manipulation and measurement of multi-photon quantum states with applications in free-space quantum imaging and communications., Comment: 19 pages, 14 figures

In the case of a linear state space model, we implement an MCMC sampler with two phases. In the learning phase, a self-tuning sampler is used to learn the parameter mean and covariance structure. In the estimation phase, the parameter mean and covariance structure informs the proposed mechanism and is also used in a delayed-acceptance algorithm. Information on the resulting state of the system is given by a Gaussian mixture. In on-line mode, the algorithm is adaptive and uses a sliding window approach to accelerate sampling speed and to maintain appropriate acceptance rates. We apply the algorithm to joined state and parameter estimation in the case of irregularly sampled GPS time series data., Comment: adaptive MCMC method application. need further work

Smoothing splines can be thought of as the posterior mean of a Gaussian process regression in a certain limit. By constructing a reproducing kernel Hilbert space with an appropriate inner product, the Bayesian form of the V-spline is derived when the penalty term is a fixed constant instead of a function. An extension to the usual generalized cross-validation formula is utilized to find the optimal V-spline parameters., Comment: a draft. not peer-reviewed yet

Trajectory reconstruction is the process of inferring the path of a moving object between successive observations. In this paper, we propose a smoothing spline -- which we name the V-spline -- that incorporates position and velocity information and a penalty term that controls acceleration. We introduce a particular adaptive V-spline designed to control the impact of irregularly sampled observations and noisy velocity measurements. A cross-validation scheme for estimating the V-spline parameters is given and we detail the performance of the V-spline on four particularly challenging test datasets. Finally, an application of the V-spline to vehicle trajectory reconstruction in two dimensions is given, in which the penalty term is allowed to further depend on known operational characteristics of the vehicle., Comment: 25 pages, submitted

We demonstrate the active tuning of all-dielectric metasurfaces exhibiting high-quality factor (high-Q) resonances. The active control is provided by embedding the asymmetric silicon meta-atoms with liquid crystals, which allows the relative index of refraction to be controlled through heating. It is found that high quality factor resonances ($Q=270\pm30$) can be tuned over more than three resonance widths. Our results demonstrate the feasibility of using all-dielectric metasurfaces to construct tunable narrow-band filters., Comment: 4 pages, 6 figures

We present an efficient and flexible method for computing likelihoods of phenotypic traits on a phylogeny. The method does not resort to Monte-Carlo computation but instead blends Felsenstein's discrete character pruning algorithm with methods for numerical quadrature. It is not limited to Gaussian models and adapts readily to model uncertainty in the observed trait values. We demonstrate the framework by developing efficient algorithms for likelihood calculation and ancestral state reconstruction under Wright's threshold model, applying our methods to a dataset of trait data for extrafloral nectaries (EFNs) across a phylogeny of 839 Labales species., Comment: 23 pages, 3 figures

Glyph-based visualization is an effective tool for depicting multivariate information. Since sorting is one of the most common analytical tasks performed on individual attributes of a multi-dimensional data set, this motivates the hypothesis that introducing glyph sorting would significantly enhance the usability of glyph-based visualization. In this paper, we present a glyph-based conceptual framework as part of a visualization process for interactive sorting of multivariate data. We examine several technical aspects of glyph sorting and provide design principles for developing effective, visually sortable glyphs. Glyphs that are visually sortable provide two key benefits: 1) performing comparative analysis of multiple attributes between glyphs and 2) to support multi-dimensional visual search. We describe a system that incorporates focus and context glyphs to control sorting in a visually intuitive manner and for viewing sorted results in an Interactive, Multi-dimensional Glyph (IMG) plot that enables users to perform high-dimensional sorting, analyse and examine data trends in detail. To demonstrate the usability of glyph sorting, we present a case study in rugby event analysis for comparing and analysing trends within matches. This work is undertaken in conjunction with a national rugby team. From using glyph sorting, analysts have reported the discovery of new insight beyond traditional match analysis., Comment: 10 pages, 6 figures

A scoring rule is a loss function measuring the quality of a quoted probability distribution $Q$ for a random variable $X$, in the light of the realized outcome $x$ of $X$; it is proper if the expected score, under any distribution $P$ for $X$, is minimized by quoting $Q=P$. Using the fact that any differentiable proper scoring rule on a finite sample space ${\mathcal{X}}$ is the gradient of a concave homogeneous function, we consider when such a rule can be local in the sense of depending only on the probabilities quoted for points in a nominated neighborhood of $x$. Under mild conditions, we characterize such a proper local scoring rule in terms of a collection of homogeneous functions on the cliques of an undirected graph on the space ${\mathcal{X}}$. A useful property of such rules is that the quoted distribution $Q$ need only be known up to a scale factor. Examples of the use of such scoring rules include Besag's pseudo-likelihood and Hyv\"{a}rinen's method of ratio matching., Comment: Published in at http://dx.doi.org/10.1214/12-AOS972 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We investigate proper scoring rules for continuous distributions on the real line. It is known that the log score is the only such rule that depends on the quoted density only through its value at the outcome that materializes. Here we allow further dependence on a finite number $m$ of derivatives of the density at the outcome, and describe a large class of such $m$-local proper scoring rules: these exist for all even $m$ but no odd $m$. We further show that for $m\geq2$ all such $m$-local rules can be computed without knowledge of the normalizing constant of the distribution., Comment: Published in at http://dx.doi.org/10.1214/12-AOS971 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper we examine inefficiencies and information disparity in the Japanese stock market. By carefully analysing information publicly available on the internet, an `outsider' to conventional statistical arbitrage strategies--which are based on market microstructure, company releases, or analyst reports--can nevertheless pursue a profitable trading strategy. A large volume of blog data is used to demonstrate the existence of an inefficiency in the market. An information-based model that replicates the trading strategy is developed to estimate the degree of information disparity., Comment: 2 figures

We investigated the possibility that nonlinear gravitational effects influence the preheating era after inflation, using numerical solutions of the inhomogeneous Einstein field equations. We compared our results to perturbative calculations and to solutions of the nonlinear field equations in a rigid (unperturbed) spacetime, in order to isolate gravitational phenomena. We confirm the broad picture of preheating obtained from the nonlinear field equations in a rigid background, but find gravitational effects have a measurable impact on the dynamics. The longest modes in the simulation grow much more rapidly in the relativistic calculation than with a rigid background. We used the Weyl tensor to quantify the departure from homogeneity in the universe. We saw no evidence for the sort of gravitational collapse that leads to the formation of primordial black holes., Comment: Talk given at Marcel Grossmann Meeting IX. 3 pages, 1 figure

Starting from the canonical ensemble over the space of pure quantum states, we obtain an integral representation for the partition function. This is used to calculate the magnetisation of a system of N spin-1/2 particles. The results suggest the existence of a new type of first order phase transition that occurs at zero temperature in the absence of spin-spin interactions. The transition arises as a consequence of quantum entanglement. The effects of internal interactions are analysed and the behaviour of the magnetic susceptibility for a small number of interacting spins is determined., Comment: 4 pages, 2 figures

We investigate the possibility that nonlinear gravitational effects influence the preheating era after inflation. Our work is based on numerical solutions of the inhomogeneous Einstein field equations, and is free of perturbative approximations. The one restriction we impose is to limit the inhomogeneity to a single spatial direction. We compare our results to perturbative calculations and to solutions of the nonlinear field equations in a rigid (unperturbed) spacetime, in order to isolate gravitational phenomena. We consider two types of initial conditions: where only one mode of the field perturbation has a non-zero initial amplitude, and where all the modes begin with a non-zero amplitude. Here we focus on preheating following inflation driven by a scalar field with a quartic potential. We confirm the broad picture of preheating obtained from the nonlinear field equations in a rigid background, but gravitational effects have a measurable impact on the dynamics for both sets of initial data. The rigid spacetime results predict that the amplitude of a single initially excited mode drops rapidly after resonance ends, whereas in the relativistic case the amplitude is roughly constant. With all modes initially excited, the longest modes in the simulation grow much more rapidly in the relativistic calculation than with a rigid background. However, we see no evidence for the sort of gravitational collapse associated with the formation of primordial black holes. The numerical codes described here are easily extended to more complicated resonant models, which we will examine in the future., Comment: 16 pages, uses epsf to import postscript figures (minor changes - typos fixed, extra reference added and data analysis slightly expanded)