Classical Correlation Functions Research Articles

Evaluation of classical correlation functions from 2/3D images on CPU and GPU architectures: Introducing CorrelationFunctions.jl

Correlation functions are becoming one of the major tools for quantification of structural information that is usually represented as 2D or 3D images. In this paper we introduce ▪ open-source package developed in Julia and capable of computing all classical correlation functions based on imaging input data. Images include both binary and multi-phase representations. Our code is capable of evaluating two-point probability S2, phase cross-correlation ρij, cluster C2, lineal-path L2, surface-surface Fss, surface-void Fsv, pore-size P and chord-length p distribution functions on both CPU and GPU architectures. Where possible, we presented two types of computations: full correlation map (correlations of each point with other points on the image, that also allows obtaining ensemble averaged CF) and directional correlation functions (currently in major orthogonal and diagonal directions). Such an implementation allowed for the first time to assemble a completely free solution to evaluate correlation functions under any operating system with well documented application programming interface (API). Our package includes automatic tests against analytical solutions that are described in the paper. We measured execution times for all CPU and GPU implementations and as a rule of thumb full correlation maps on GPU are faster than other methods. However, full maps require more RAM and, thus, are limited to available RAM resources. On the other hand, directional CFs are memory efficient and can be evaluated for huge datasets – this way they are the first candidates for structural data compression of feature extraction. The package itself is available through Julia package ecosystem and on GitHub, the latter source also contains documentation and additional helpful resources such as tutorials. We believe that a single powerful computational tool such as ▪ presented in this paper will significantly facilitate the usage of correlation functions in numerous areas of structural description and research of porous materials, as well as in machine learning applications. We also present some examples as applied to ceramic, soil composite and oil-bearing rock samples based on their 3D X-ray tomography and 2D scanning electron microscope images. Finally, we conclude our paper with discussion of possible ways to further improve presented computational framework. Program summaryProgram Title: CorrelationFunctions.jlCPC Library link to program files:https://doi.org/10.17632/6gb9gfm3dw.1Developer's repository link:https://github.com/fatimp/CorrelationFunctions.jlLicensing provisions: MITProgramming language: JuliaSupplementary material: Numerous Jupiter notebooks with examples are available on the GitHub pageNature of problem: Correlation functions are invaluable universal statistical descriptors of structures used in numerous scientific fields such as astronomy, material science, rock and soil physics, hydrology and biology, to name just a handful of examples. While computational approaches are available in the literature for some functions, they are fragmented and are usually implemented in proprietary interpreted languages for CPU architecture alone.Solution method: We contribute an open source and cross-platform solution with well documented API for computation of all classical correlation functions from both 2D and 3D images on CPU and GPU architectures. The package computes correlation functions using two approaches: computation of correlation maps and computation along predefined directions. These two approaches can be thought of as an execution time - memory trade-off, but the choice may also depend on the application. The computations are based on a) fast Fourier transform with preprocessing steps such as cluster labeling or edge detection, and b) linear scan approach to evaluate correlation functions along predefined directions. Where justified, the algorithms can be executed on both CPU and GPU which results in high execution speed on modern hardware.

ANALISIS FUNGSI KORELASI KLASIK DAN KUANTUM UNTUK MODEL CINCIN: SEBUAH REVIEW

A review of the journal entitled 'Classical And Quantum Correlation Functions For A Ring Model' has been carried out which discusses the solutions of classical and quantum correlation functions for the ring model. Classical and quantum correlation functions are derived for systems of non-interacting particles moving in a circle. demonstrated that the decay behavior of the classical expression for the correlation function can be recovered from a strictly periodic quantum mechanical expression by taking the limit ℏ→0, after the appropriate transformation. The aim of this study is to present clearly and in detail how the position correlation function for a system of particles moving in a circle having a certain average energy and written in a form which shows the nature of the transformation. Next, we examine the use of Poisson addition to represent F(z), and how the correlation function becomes identical to the form given by classical statistical mechanics, which exhibits Gaussian decay. The results of this study indicate that the positional correlation function for a system of particles moving in a circle, and having a certain average energy, can be written in a form which shows the nature of the transformation. Then using the Poisson addition formula to represent F(z), the expression is rewritten to enable the limit ℏ→0 to be taken. It was finally found that the correlation function became identical to the form given by classical statistical mechanics, which exhibits a Gaussian decay.

Mixed Quantum-Classical Liouville Equation Treatment of Electronic Spectroscopy of Condensed Systems: Harmonic and Anharmonic Electron-Phonon Coupling.

This Review integrates the use of electronic optical response function theory and the mixed quantum-classical (MQC) Liouville equation (MQCLE), thereby leading to electronic spectroscopy in MQC media. It further sheds light on the applicability, utility, and efficiency of the mixed quantum-classical dynamics (MQCD) formalism, which starts off with the MQCLE, in probing spectroscopy and dynamics of condensed systems, whereby quantum mechanics and classical mechanics are combined systematically. The author has been exploring and implementing MQCD to investigate electron-phonon coupling effects on electronic dephasing in harmonic and anharmonic systems by calculating linear and nonlinear optical transition analytically and numerically dipole moment time correlation functions in an MQC environment, thereby presenting an in depth spectral profile analysis and their shape and symmetry. The distinctive capability of the MQC time correlation functions is that ergodicity and stationarity properties are inherently satisfied as part of the mixed quantum-classical dynamics (MQCD) framework, unlike classical correlation functions. While some research groups have applied MQCLE to calculate vibrational spectra to study hydrogen-bonded complexes in a MQC environment and other groups calculated Optical response function to probe electron transfer dynamics using the basis mapping technique, the approach, purpose, rigor, applications, and path to the end results reported herein are different. Finally, the same framework is employed to study dissipative systems in the MQC limit, whereby the zero-phonon line adopts the correct width and eliminates its asymmetry. While the full quantum mechanical model, like the multimode Brownian oscillator (MBO) model, yields the correct width and inaccurate shape in the low-temperature limit, the MQCD formalism seems to produce an accurate zero-phonon profile. Nonlinear optical signals are also reviewed in MQC media to show the applicability and utility of this approach. The vibronic optical response functions developed here will account for geometry change, frequency change, and anharmonicity upon electronic excitation to accurately probe electronic dephasing, electron-phonon coupling, shape, and symmetry of profiles and present differences and similarities to the MBO model on pure electronic dephasing. Frequency change and anharmonicity are vitally crucial for accurately assessing electron-phonon coupling upon electronic excitation. This is an additional unique result obtained by the author to further demonstrate the applicability and utility of this approach over other approximation schemes in probing electronic dephasing including that of the MBO model.

Experimental violations of Leggett-Garg inequalities on a quantum computer

Leggett-Garg's inequalities predict sharp bounds for some classical correlation functions that address the quantum or classical nature of real-time evolutions. We experimentally observe the violations of these bounds on single- and multiqubit systems, in different settings, exploiting the IBM Quantum platform. In the multiqubit case, we introduce the Leggett-Garg-Bell's inequalities as an alternative to the previous ones. Measuring these correlation functions, we find quantum error mitigation to be essential to spot inequalities violations. Accessing only two qubit readouts, we assess Leggett-Garg-Bell's inequalities to emerge as the most efficient quantum coherence witnesses to be used for investigating quantum hardware, among those introduced. Our analysis highlights the limits of current quantum platforms, showing that the above-mentioned correlation functions deviate from theoretical prediction as the number of qubits and the depth of the circuit grow.

Super-Resolution Imaging by Computationally Fusing Quantum and Classical Optical Information

A high-speed super-resolution computational imaging technique is introduced on the basis of classical and quantum correlation functions obtained from photon counts collected from quantum emitters illuminated by spatiotemporally structured illumination. The structured illumination is delocalized—allowing the selective excitation of separate groups of emitters as the modulation of the illumination light advances. A recorded set of photon counts contains rich quantum and classical information. By processing photon counts, multiple orders of Glauber correlation functions are extracted. Combinations of the normalized Glauber correlation functions convert photon counts into signals of increasing order that contain increasing spatial frequency information. However, the amount of information above the noise floor drops at higher correlation orders, causing a loss of accessible information in the finer spatial frequency content that is contained in the higher-order signals. We demonstrate an efficient and robust computational imaging algorithm to fuse the spatial frequencies from the low-spatial-frequency range that is available in the classical information with the spatial frequency content in the quantum signals. Because of the overlap of low spatial frequency information, the higher signal-to-noise ratio (SNR) information concentrated in the low spatial frequencies stabilizes the lower SNR at higher spatial frequencies in the higher-order quantum signals. Robust performance of this joint fusion of classical and quantum computational single-pixel imaging is demonstrated with marked increases in spatial frequency content, leading to super-resolution imaging, along with much better mean squared errors in the reconstructed images.

Impact of baryons in cosmic shear analyses with tomographic aperture mass statistics

NonGaussian cosmic shear statistics based on weak-lensing aperture mass (Map) maps can outperform the classical shear two-point correlation function (γ-2PCF) in terms of cosmological constraining power. However, reaching the full potential of these new estimators requires accurate modeling of the physics of baryons as the extra nonGaussian information mostly resides at small scales. We present one such modeling based on the Magneticum hydrodynamical simulation for the KiDS-450 and DES-Y1 surveys and a Euclid-like survey. We compute the bias due to baryons on the lensing PDF and the distribution of peaks and voids in Map maps and propagate it to the cosmological forecasts on the structure growth parameter S8, the matter density parameter Ωm, and the dark energy equation of state w0 using the SLICS and cosmo-SLICS sets of dark-matter-only simulations. We report a negative bias of a few percent on S8 and Ωm and also measure a positive bias of the same level on w0 when including a tomographic decomposition. These biases reach ∼5% when combining Map statistics with the γ-2PCF as these estimators show similar dependency on the AGN feedback. We verify that these biases constitute a less than 1σ shift on the probed cosmological parameters for current cosmic shear surveys. However, baryons need to be accounted for at the percentage level for future Stage IV surveys and we propose to include the uncertainty on the AGN feedback amplitude by marginalizing over this parameter using multiple simulations such as those presented in this paper. Finally, we explore the possibility of mitigating the impact of baryons by filtering the Map map but find that this process would require suppressing the small-scale information to a point where the constraints would no longer be competitive.

Nonlinear spectroscopy in the condensed phase: The role of Duschinsky rotations and third order cumulant contributions.

First-principles modeling of nonlinear optical spectra in the condensed phase is highly challenging because both environment and vibronic interactions can play a large role in determining spectral shapes and excited state dynamics. Here, we compute two dimensional electronic spectroscopy (2DES) signals based on a cumulant expansion of the energy gap fluctuation operator, with specific focus on analyzing mode mixing effects introduced by the Duschinsky rotation and the role of the third order term in the cumulant expansion for both model and realistic condensed phase systems. We show that for a harmonic model system, the third order cumulant correction captures effects introduced by a mismatch in curvatures of ground and excited state potential energy surfaces, as well as effects of mode mixing. We also demonstrate that 2DES signals can be accurately reconstructed from purely classical correlation functions using quantum correction factors. We then compute nonlinear optical spectra for the Nile red and methylene blue chromophores in solution, assessing the third order cumulant contribution for realistic systems. We show that the third order cumulant correction is strongly dependent on the treatment of the solvent environment, revealing the interplay between environmental polarization and the electronic-vibrational coupling.

Systematically localizable operators for quantum simulations of quantum field theories

Correlations and measures of entanglement in ground state wavefunctions of relativistic quantum field theories are spatially localized over length scales set by the mass of the lightest particle. We utilize this localization to design digital quantum circuits for preparing the ground states of lattice scalar quantum field theories. Controlled rotations that are exponentially localized in their position-space extent are found to provide exponentially convergent wavefunction fidelity. These angles scale with the correlation between sites and the classical two-point correlation function, as opposed to the more localized mutual information or the hyper-localized negativity. We anticipate that further investigations will uncover quantum circuit designs with controlled rotations dictated by the measures of entanglement. This work is expected to impact quantum simulations of systems of importance to nuclear physics, high-energy physics, and basic energy sciences research.

The Triangle Wave Versus the Cosine: How Classical Systems Can Optimally Approximate EPR-B Correlations.

The famous singlet correlations of a composite quantum system consisting of two two-level components in the singlet state exhibit notable features of two kinds. One kind are striking certainty relations: perfect anti-correlation, and perfect correlation, under certain joint settings. The other kind are a number of symmetries, namely invariance under a common rotation of the settings, invariance under exchange of components, and invariance under exchange of both measurement outcomes. One might like to restrict attention to rotations in the plane since those are the ones most commonly investigated experimentally. One can then also further distinguish between the case of discrete rotations (e.g., only settings which are a whole number of degrees are allowed) and continuous rotations. We study the class of classical correlation functions, i.e., generated by classical physical systems, satisfying all these symmetries, in the continuous, planar, case. We call such correlation functions classical EPR-B correlations. It turns out that if the certainty relations and rotational symmetry holds at the level of the correlations, then rotational symmetry can be imposed “for free” on the underlying classical physical model by adding an extra randomisation level. The other binary symmetries are obtained “for free”. This leads to a simple heuristic description of all possible classical EPR-B correlations in terms of a “spinning bi-coloured disk” model. We deliberately use the word “heuristic” because technical mathematical problems remain wide open concerning the transition from finite or discrete to continuous. The main purpose of this paper is to bring this situation to the attention of the mathematical community. We do show that the widespread idea that “quantum correlations are more extreme than classical physics would allow” is at best highly inaccurate, through giving a concrete example of a classical correlation which satisfies all the symmetries and all the certainty relations and which exceeds the quantum correlations over a whole range of settings. It is found by a search procedure in which we randomly generate classical physical models and, for each generated model, evaluate its properties in a further Monte-Carlo simulation of the model itself.

Closed-Form Expressions of Ergodic Capacity and MMSE Achievable Sum Rate for MIMO Jacobi and Rayleigh Fading Channels

Multimode/multicore fibers are expected to provide an attractive solution to overcome the capacity limit of the current optical communication system. In the presence of strong crosstalk between modes and/or cores, the squared singular values of the input/output transfer matrix follow the law of the Jacobi ensemble of random matrices. Assuming that the channel state information is only available at the receiver, we derive a new expression for the ergodic capacity of the MIMO Jacobi fading channel. The proposed expression involves double integrals which can be easily evaluated for a high-dimensional MIMO scenario. Moreover, the method used in deriving this expression does not appeal to the classical one-point correlation function of the random matrix model. Using a limiting transition between Jacobi and associated Laguerre polynomials, we derive a similar formula for the ergodic capacity of the MIMO Rayleigh fading channel. Moreover, we derive a new exact closed-form expressions for the achievable sum rate of MIMO Jacobi and Rayleigh fading channels employing linear minimum mean squared error (MMSE) receivers. The analytical results are compared to the results obtained by Monte Carlo simulations and the related results available in the literature, which shows perfect agreement.

Exponential frequency correlation function and its application in Doppler shift estimation

A Doppler shift estimation problem of a noisy received signal consisting of multiple moving targets that have similar radial velocities is addressed here. Traditional Doppler shift estimation algorithms based on Fourier transform (FT) frequently exhibit low resolution. Here, an exponential frequency correlation function (EFCF) is proposed by introducing a sensitivity exponent p that controls its output signal-to-noise ratio (SNR) and velocity resolution into the expression of a classical frequency correlation function (FCF). On the basis of the proposed EFCF, a novel Doppler shift estimation algorithm can adaptively select the exponent of the EFCF to achieve the optimum Doppler shift estimation effect in an echo signal based on the elimination algorithm. Moreover, the proposed method is performed on the basis of the fast FT and product operation. Thus, it can be easily implemented. Numerical simulations demonstrate that the performance of the proposed method is significantly improved compared with the existing algorithms.

How to calculate starch lamellar features with improved accuracy by small angle X-ray scattering

Starch lamellar structures unraveled by small angle X-ray scattering (SAXS) have been critically correlated with its functionalities. However, the lamellar features calculated by the classical linear correlation function based on SAXS data are not well convincingly analyzed, because the thickness of amorphous lamellae calculated by previous studies is smaller than the smallest detectable size extrapolated from the selected data. The main reason causes this unconvincing result is the unsuitable data used. Appropriate data are selected in this study for better portraying the linear correlation function profile and then suitable principles are proposed to convincingly analyze starch lamellar features. For verifying the validity of selected data and calculated results, two criterions are established, i.e., (i) the thickness of amorphous lamellae evaluated from the correlation profiles should be larger than the smallest detectable size extrapolated from the selected data according to the Woolf-Bragg's equation and (ii) the value of the thickness of semi-crystalline lamellae calculated from the linear correlation function profile should close to that obtained from the original SAXS curve. Accordingly, the data of 0.10–2.75 nm−1 are optionally selected for calculating starch lamellar parameters. The results of this study provide a promising pathway to unravel starch lamellar features with improved accuracy.

Aging in Metropolis dynamics of the REM: a proof

We consider Metropolis dynamics of the Random Energy Model. We prove that the classical two-time correlation function that allows one to establish aging converges almost surely to the arcsine law distribution function, as predicted in the physics literature, in the optimal domain of the time-scale and temperature parameters where this result can be expected to hold. To do this we link the two-time correlation function to a certain continuous-time clock process which, after proper rescaling, is proven to converge to a stable subordinator almost surely in the random environment and in the fine $$J_1$$ -topology of Skorohod. This fine topology then enables us to deduce from the arcsine law for stable subordinators the asymptotic behavior of the two-time correlation function that characterizes aging.

Quantum correlations for a simple kicked system with mixed phase space.

We investigate both the classical and quantum dynamics for a simple kicked system (the standard map) that classically has mixed phase space. For initial conditions in a portion of the chaotic region that is close enough to the regular region, the phenomenon of sticking leads to a power-law decay with time of the classical correlation function of a simple observable. Quantum mechanically, we find the same behavior, but with a smaller exponent. We consider various possible explanations of this phenomenon, and settle on a modification of the Meiss-Ott Markov tree model that takes into account quantum limitations on the flux through a turnstile between regions corresponding to states on the tree. Further work is needed to better understand the quantum behavior.

Experimental cancellation of aberrations in intensity correlation in classical optics

We study the classical correlation function of spatially incoherent beams with a phase aberration in the beam path. On the basis of our experimental measurements and in the optical coherence theory, we show that the effects of phase disturbances, independently of their kind and without need of coordinate inversion, can be canceled out if the same phase is aligned in the signal and reference beam path. These results can be useful for imaging and microscopy through random media.

Direct determination of discrete harmonic bath parameters from molecular dynamics simulations.

We present a direct procedure for determining the parameters of a discrete harmonic bath modeling the influence of a complex condensed phase environment on the system of interest. The procedure employs an efficient discretization of the spectral density into modes that correspond to equal fractions of the reorganization energy. The new procedure uses directly the classical correlation function (available from molecular dynamics calculations) as input, avoiding numerical computation of the spectral density by means of a discrete Fourier transform. Convergence is obtained using a shorter time length of the correlation function, leading to significant computational savings. © 2016 Wiley Periodicals, Inc.

Modeling Electronic-Nuclear Interactions for Excitation Energy Transfer Processes in Light-Harvesting Complexes.

An accurate approach for computing intermolecular and intrachromophore contributions to spectral densities to describe the electronic-nuclear interactions relevant for modeling excitation energy transfer processes in light harvesting systems is presented. The approach is based on molecular dynamics (MD) calculations of classical correlation functions of long-range contributions to excitation energy fluctuations and a separate harmonic analysis and single-point gradient quantum calculations for electron-intrachromophore vibrational couplings. A simple model is also presented that enables detailed analysis of the shortcomings of standard MD-based excitation energy fluctuation correlation function approaches. The method introduced here avoids these problems, and its reliability is demonstrated in accurate predictions for bacteriochlorophyll molecules in the Fenna-Matthews-Olson pigment-protein complex, where excellent agreement with experimental spectral densities is found. This efficient approach can provide instantaneous spectral densities for treating the influence of fluctuations in environmental dissipation on fast electronic relaxation.

Quantum discord length is enhanced while entanglement length is not by introducing disorder in a spin chain.

Classical correlation functions of ground states typically decay exponentially and polynomially, respectively, for gapped and gapless short-range quantum spin systems. In such systems, entanglement decays exponentially even at the quantum critical points. However, quantum discord, an information-theoretic quantum correlation measure, survives long lattice distances. We investigate the effects of quenched disorder on quantum correlation lengths of quenched averaged entanglement and quantum discord, in the anisotropic XY and XYZ spin glass and random field chains. We find that there is virtually neither reduction nor enhancement in entanglement length while quantum discord length increases significantly with the introduction of the quenched disorder.

Quantum dynamics of hydrogen atoms on graphene. I. System-bath modeling.

An accurate system-bath model to investigate the quantum dynamics of hydrogen atoms chemisorbed on graphene is presented. The system comprises a hydrogen atom and the carbon atom from graphene that forms the covalent bond, and it is described by a previously developed 4D potential energy surface based on density functional theory ab initio data. The bath describes the rest of the carbon lattice and is obtained from an empirical force field through inversion of a classical equilibrium correlation function describing the hydrogen motion. By construction, model building easily accommodates improvements coming from the use of higher level electronic structure theory for the system. Further, it is well suited to a determination of the system-environment coupling by means of ab initio molecular dynamics. This paper details the system-bath modeling and shows its application to the quantum dynamics of vibrational relaxation of a chemisorbed hydrogen atom, which is here investigated at T = 0 K with the help of the multi-configuration time-dependent Hartree method. Paper II deals with the sticking dynamics.

Classical Description of the Vibrational Spectroscopy, Structure, and Electrostatics of the Halide Solvation Shell with the POLIR Potential.

POLIR (POLarizable for IR) (J. Chem. Phys. 2008, 129, 034504) is a polarizable, flexible, and transferable water potential which describes the IR spectrum in the O-H stretch region via the classical dipole correlation function with simple quantum corrections. POLIR also reproduces experimental spectral shifts in solutions of Ca(2+), Mg(2+), and Cu(2+) (J. Am. Chem. Soc. 2011, 133, 9441-9450). Here we present an extended investigation of POLIR water in the solvation shell of the halides F(-), Cl(-), Br(-), and I(-) using various interaction potentials, polarizabilities, and short-range electrostatic damping parameters. Our results indicate that the correlation of the first solvation shell dipoles produces IR spectra that are in agreement with experiment; that is, vibrational spectral shifts may be obtained with classical mechanics and simple corrections. Calculated ion induced dipoles agree with quantum simulations. Further analysis shows that ion-dependent shifts in the spectra may be attributed to the hydrogen bond O···O-H angle distribution within the first solvation shell, decomposed into ion-water and water-water contributions.