Phase-space Function Research Articles

Detection of Emotions Induced by Colors in Compare of Two Nonlinear Mapping of Heart Rate Variability Signal: Triangle and Parabolic Phase Space (TPSM, PPSM)

Emotions in the word mean understanding, sensing and recognizing something with one’s senses. Emotions are relatively simple responses that the body presents to external or internal stimuli. As the individual’s mental and emotional state changes, the activity of the various parts of the body also changes and increases. This applies to all cultures. So emotion detection from biological signals especially heart rate variability (HRV) signal is considered in this article. The behavior of HRV in two nonlinear phase space have been evaluated and the extracted features of these two mapping have been used for emotion detection. Triangle and Parabolic Phase Space Mapping (TPSM and PPSM) are recently introduced methods which are able to indicate hidden aspects of HRV in response to different emotions. The Lead II of electrocardiogram signal was recorded from 32 female students while they were stimulated by four main psychology colors (blue, green, red, and yellow) to induce emotions which were labeled by SAM test. The results show that the mentioned colors caused emotions pleasure, joy, anger, and sadness, respectively. After rating the results by Kruskal–Wallis test, k-nearest neighbor classifier was used for emotion classification. The obtained accuracy after 10-fold cross validation was 90.91%. The hypotheses of being a relationship between colors and emotions without awareness of subjects have been proved. Furthermore, by studying the usefulness of TPSM and PPSM as new mapping of HRV, significant results have obtained which indicate the advantages of using them. So it is suggested to use and analyze the usage of these kinds of colors in different psychological situations and for biofeedback tests.

Three-dimensional conditional hyperbolic quadrature method of moments

The conditional hyperbolic quadrature method of moments (CHyQMOM) was introduced by Fox et al. [19] to reconstruct 1- and 2-D velocity distribution functions (VDF) from a finite set of integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a hyperbolic closure for the spatial flux term in the corresponding moment equations derived from a kinetic equation for the 3-D VDF. Here, CHyQMOM is extended for 3-D velocity phase space using the modified conditional quadrature method of moments with 16 (or 23) trivariate velocity moments up to fourth order. In order to verify the numerical implementation, it is applied to simulate several canonical particle-laden flows including crossing jets, cluster-induced turbulence (CIT), and vertical channel flow. The numerical results are compared with those from Euler–Lagrange simulations and two other quadrature-based moment methods, namely, anisotropic Gaussian (AG) and 8-node tensor-product (TP) quadrature. The relative advantages and disadvantages of each method are discussed. The crossing-jet problem highlights that CHyQMOM handles particle crossing more accurately than AG. For CIT, the results from all methods are similar, but the computational cost of TP is significantly larger than AG and CHyQMOM, both of which have nearly the same cost.

BAM: bias assignment method to generate mock catalogues

ABSTRACT We present BAM: a novel Bias Assignment Method envisaged to generate mock catalogues. Combining the statistics of dark matter tracers from a high-resolution cosmological N-body simulation and the dark matter density field calculated from down-sampled initial conditions using efficient structure formation solvers, we extract the halo-bias relation on a mesh of a $3\, h^{-1}$ Mpc cell side resolution as a function of properties of the dark matter density field (e.g. local density, cosmic web type), automatically including stochastic, deterministic, local and non-local components. We use this information to sample the halo density field, accounting for ignored dependencies through an iterative process. By construction, our approach reaches ${\sim } 1\, {\rm {per\,cent}}$ accuracy in the majority of the k-range up to the Nyquist frequency without systematic deviations for power spectra (about $k \sim 1\, h$ Mpc−1) using either particle mesh or Lagrangian perturbation theory based solvers. When using phase-space mapping to compensate the low resolution of the approximate gravity solvers, our method reproduces the bispectra of the reference within $10\, {\rm {per\,cent}}$ precision studying configurations tracing the quasi-non-linear regime. BAM has the potential to become a standard technique to produce mock halo and galaxy catalogues for future galaxy surveys and cosmological studies being highly accurate, efficient and parameter free.

Study on nonlinear dynamic behavior and stable operation region of single-phase photovoltaic inverter

In order to design a photovoltaic inverter with excellent performance, the second-order photovoltaic inverter circuit model under current loop proportional control is established in this paper, the discrete iterative model of system is also obtained. The nonlinear behavior of system is analysed in detail through bifurcation diagram, folding diagram and the phase diagram of output current (i) and output voltage (uc). The simulation model is constructed in Matlab/Simulink, and the correctness of theoretical analysis is verified through time domain waveform and phase-space map. The influence of bifurcation and chaos on system spectrum is analysed from the frequency domain. According to the Jacobian matrix of system, the stable operation region of system is given. The experiment shows that the correct selection of circuit parameters is important for the stable operation of system.

Non-Gaussianity dynamics of two-mode squeezed number states subject to different types of noise based on cumulant theory**Project supported by the Natural Science Foundation of Hunan Province, China (Grant No. 2017JJ2214) and the Key Project Foundation of the Education Department of Hunan Province, China (Grant No. 14A114.

We provide a measure to characterize the non-Gaussianity of phase-space function of bosonic quantum states based on the cumulant theory. We study the non-Gaussianity dynamics of two-mode squeezed number states by analyzing the phase-averaged kurtosis for two different models of decoherence: amplitude damping model and phase damping model. For the amplitude damping model, the non-Gaussianity is very fragile and completely vanishes at a finite time. For the phase damping model, such states exhibit rich non-Gaussian characters. In particular, we obtain a transition time that such states can transform from sub-Gaussianity into super-Gaussianity during the evolution. Finally, we compare our measure with the existing measures of non-Gaussianity under the independent dephasing environment.

Binary contraction method for the construction of time-dependent dividing surfaces in driven chemical reactions

Transition state theory formally provides a simplifying approach for determining chemical reaction rates and pathways. Given an underlying potential energy surface for a reactive system, one can determine the dividing surface in phase space which separates reactant and product regions, and thereby also these regions. This is often a difficult task, and it is especially demanding for high-dimensional time-dependent systems or when a non-local dividing surface is required. Recently, approaches relying on Lagrangian descriptors have been successful at resolving the dividing surface in some of these challenging cases, but this method can also be computationally expensive due to the necessity of integrating the corresponding phase space function. In this paper, we present an alternative method by which time-dependent, locally recrossing-free dividing surfaces can be constructed without the calculation of any auxiliary phase space function, but only from simple dynamical properties close to the energy barrier.

Gyrokinetics from variational averaging: Existence and error bounds

The gyrokinetic paradigm in the long wavelength regime is reviewed from the perspective of variational averaging (VA). The VA-method represents a third pillar for averaging kinetic equations with highly oscillatory characteristics, besides classical averaging and Hilbert expansions. VA operates on the level of the Lagrangian function and preserves the Hamiltonian structure of the characteristics at all orders. We discuss the methodology of VA in detail by means of charged-particle motion in a strong magnetic field. The application of VA to a broader class of highly oscillatory problems can be envisioned. For the charged particle, we prove the existence of a coordinate map in phase space that leads to a gyrokinetic Lagrangian at any order of the expansion, for general external fields. We compute this map up to third order, independent of the electromagnetic gauge. Moreover, an error bound for the solution of the derived gyrokinetic equation with respect to the solution of the Vlasov equation is provided, allowing us to estimate the quality of the VA-approximation in this particular case.

Mode-Coupling theory for glass transition of active-passive binary mixture

Collective behaviours of active particle systems have gained great research attentions in recent years. Here we present a mode-coupling theory (MCT) framework to study the glass transition of a mixture system of active and passive Brownian particles. The starting point is an effective Smoluchowski equation, which governs the dynamics of the probability distribution function in the position phase space. With the assumption of the existence of a nonequilibrium steady state, we are able to obtain dynamic equations for the intermediate scattering functions (ISFs), wherein an irreducible memory function is introduced which in turn can be written as functions of the ISFs based on standard mode-coupling approximations. The effect of particle activity is included through an effective diffusion coefficient which can be obtained via short time simulations. By calculating the long-time limit of the ISF, the Debye-Waller (DW) factor, one can determine the critical packing fraction ηc of glass transition. We find that for active-passive (AP) mixtures with the same particle sizes, ηc increases as the partial fraction of active particle xA increases, which is in agreement with previous simulation works. For system with different active/passive particle sizes, we find an interesting reentrance behaviour of glass transition, i.e., ηc shows a non-monotonic dependence on xA. In addition, such a reentrance behaviour would disappear if the particle activity is large enough. Our results thus provide a useful theoretical scheme to study glass transition behaviour of active-passive mixture systems in a promising way.

Husimi function and phase-space analysis of bilayer quantum Hall systems at ν = 2/λ

We propose localization measures in phase space of the ground state of bilayer quantum Hall systems at fractional filling factors , to characterize the three quantum phases (shortly denoted by spin, canted and ppin) for arbitrary -isospin λ. We use a coherent state (Bargmann) representation of quantum states, as holomorphic functions in the 8-dimensional Grassmannian phase-space (a higher-dimensional generalization of the Haldane’s 2-dimensional sphere ). We quantify the localization (inverse volume) of the ground state wave function in phase-space throughout the phase diagram (i.e. as a function of Zeeman, tunneling, layer distance, etc, control parameters) with the Husimi function second moment, a kind of inverse participation ratio that behaves as an order parameter. Then we visualize the different ground state structure in phase space of the three quantum phases, the canted phase displaying a much higher delocalization (a Schrödinger cat structure) than the spin and ppin phases, where the ground state is highly coherent. We find a good agreement between analytic (variational) and numeric diagonalization results.

Temperature is not an observable in superstatistics

Superstatistics (Beck and Cohen, 2003) is a formalism that attempts to explain the presence of distributions other than the Boltzmann–Gibbs distributions in Nature, typically power-law behavior, for systems out of equilibrium such as fluids under turbulence, plasmas and gravitational systems. Superstatistics postulates that those systems are found in a superposition of canonical ensembles at different temperatures, and sometimes the physical interpretation is one of local thermal equilibrium in the sense of an inhomogeneous temperature distribution in different regions of space or instants of time.Here we show that, in order for superstatistics to be internally consistent, it is impossible to define a phase-space function or microscopic observable B(p,q) corresponding one-to-one to the local value of β=1∕kBT. Thus, unlike energy which is defined by a phase-space function H(p,q) (the Hamiltonian), temperature is not a microscopic observable.An important consequence of our proof is that, in Superstatistics, the identification of temperature with the kinetic energy is limited to the expectation of β and cannot be used to measure the different temperatures in local thermal equilibrium or its fluctuations.

Conditional hyperbolic quadrature method of moments for kinetic equations

The conditional quadrature method of moments (CQMOM) was introduced by Yuan and Fox (2011) [4] to reconstruct a velocity distribution function (VDF) from a finite set of its integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a closure for the spatial flux term in the corresponding kinetic equation. The CQMOM closure for the flux leads to a weakly hyperbolic system of moment equations. In subsequent work by Chalons et al. (2010) [8], the Dirac delta functions were replaced by Gaussian distributions, which make the moment system hyperbolic but at the added cost of dealing with continuous distributions. Here, a hyperbolic version of CQMOM is proposed that uses weighted Dirac delta functions. While the moment set employed for multi-Gaussian and conditional HyQMOM (CHyQMOM) are equivalent, the latter is able to access all of moment space whereas the former cannot (e.g. arbitrary values of the fourth-order velocity moment in 1-D phase space with two nodes). By making use of the properties of CHyQMOM in 2-D phase space, it is possible to control a symmetrical subset of the optimal moments from Fox (2009) [24]. Furthermore, the moment sets for 2-D problems are smaller for CHyQMOM than in the original CQMOM thanks to a judicious choice of the velocity abscissas in phase space.

A Refinement of the Robertson\u2013Schr\xf6dinger Uncertainty Principle and a Hirschman\u2013Shannon Inequality for Wigner Distributions

We propose a refinement of the Robertson–Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state rho . We prove sharp estimates which critically depend on the purity of rho and which are saturated in the Gaussian case.

Secular instabilities of Keplerian stellar discs

We present idealized models of a razor-thin, axisymmetric, Keplerian stellar disc around a massive black hole, and study non-axisymmetric secular instabilities in the absence of either counter-rotation or loss cones. These discs are prograde mono-energetic waterbags, whose phase space distribution functions are constant for orbits within a range of eccentricities (e) and zero outside this range. The linear normal modes of waterbags are composed of sinusoidal disturbances of the edges of distribution function in phase space. Waterbags which include circular orbits (polarcaps) have one stable linear normal mode for each azimuthal wavenumber m. The m = 1 mode always has positive pattern speed and, for polarcaps consisting of orbits with e < 0.9428, only the m = 1 mode has positive pattern speed. Waterbags excluding circular orbits (bands) have two linear normal modes for each m, which can be stable or unstable. We derive analytical expressions for the instability condition, pattern speeds, growth rates and normal mode structure. Narrow bands are unstable to modes with a wide range in m. Numerical simulations confirm linear theory and follow the non-linear evolution of instabilities. Long-time integration suggests that instabilities of different m grow, interact non-linearly and relax collisionlessly to a coarse-grained equilibrium with a wide range of eccentricities.

Metrologically useful states of spin-1 Bose condensates with macroscopic magnetization

We study theoretically the usefulness of spin-1 Bose condensates with macroscopic magnetization in a homogeneous magnetic field for quantum metrology. We demonstrate Heisenberg scaling of the quantum Fisher information for states in thermal equilibrium. The scaling applies to both antiferromagnetic and ferromagnetic interactions. The effect preserves as long as fluctuations of magnetization are sufficiently small. Scaling of the quantum Fisher information with the total particle number is derived within the mean-field approach in the zero temperature limit and exactly in the high magnetic field limit for any temperature. The precision gain is intuitively explained owing to subtle features of the quasi-distribution function in phase space.

Incomplete Detection of Nonclassical Phase-Space Distributions.

We implement the direct sampling of negative phase-space functions via unbalanced homodyne measurement using click-counting detectors. The negativities significantly certify nonclassical light in the high-loss regime using a small number of detectors which cannot resolve individual photons. We apply our method to heralded single-photon states and experimentally demonstrate the most significant certification of nonclassicality for only two detection bins. By contrast, the frequently applied Wigner function fails to directly indicate such quantum characteristics for the quantum efficiencies present in our setup without applying additional reconstruction algorithms. Therefore, we realize a robust and reliable approach to characterize nonclassical light in phase space under realistic conditions.

Soliton operators in the quantum equivalence of the CP1 and O(3) - σmodels

We discuss some interesting aspects of the well known quantum equivalence between the O(3) - σ and CP1 models in 3D, working inthe canonical and in the path integral formulations. We show first that the canonical quantization in the hamiltonian formulation is freeof ordering ambiguities for both models. We use the canonical map between the fields and momenta of the two models and compute therelevant functional determinant to verify the equivalence between the phase-space partition functions and the quantum equivalence in allthe topological sectors. We also use the explicit form of the map to construct the soliton operator of the O(3) - σ model starting from therepresentation of the operator in the CP1 model, and discuss their properties.

Quantum mechanics on phase space: The hydrogen atom and its Wigner functions

Symplectic quantum mechanics (SQM) considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space HΓ, to construct a unitary representation for the Galilei group. From this unitary representation the Schrödinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville–von Neumann equation. In this article the Coulomb potential in three dimensions (3D) is resolved completely by using the phase space Schrödinger equation. The Kustaanheimo–Stiefel(KS) transformation is applied and the Coulomb and harmonic oscillator potentials are connected. In this context we determine the energy levels, the amplitude of probability in phase space and correspondent Wigner quasi-distribution functions of the 3D-hydrogen atom described by Schrödinger equation in phase space.

Generation and quantum state reconstruction of a squeezed vacuum light field resonant on the rubidium D1 line

The squeezed light field is a kind of important continuous variable quantum resource.It has wide applications in precision measurement and quantum information processing.Quantum storage is the foundations of quantum repeater and long distance quantum communication,and alkali metal atoms are an ideal quantum storage medium due to long ground state coherent time. With the rapid development of quantum storage technology in atomic medium,the preparation of the squeezed light which resonates with alkali metal atoms has become one of the research hotspots in the field of quantum information.In this paper,we report the generation of squeezed vacuum at 795 nm (resonant on the rubidium D1 transition line) by using an optical parametric oscillation based on a periodically poled KTiOPO4 crystal. The generated squeezed light field is detected by a balanced homodyne detector,and the squeezing of-3 dB and anti-squeezing of 5.8 dB are observed at a pump power of 45 mW.By using a maximum likelihood estimation,the density matrix of the squeezed light field is reconstructed.The time-domain signals from the balanced homodyne detector are collected to acquire the noise distribution of the squeezed light under different phase angles.The likelihood function is established for the measured quadrature components.An identity matrix is chosen as an initial density matrix,and the density matrix of the squeezed field is obtained through an iterative algorithm.The diagonal elements of the density matrix denote the photon number distribution,which includes not only even photon number states but also odd photon number states.The occurrence of odd photon number states mainly comes from the system losses and the imperfect quantum efficiency of detector.The Wigner function in phase space is calculated through the density matrix,and the maximum value of the Wigner function is 0.309.The standard deviation of the squeezed component is 64.4% of that of the vacuum state,corresponding to the squeezing degree of-3.8 dB.The standard deviation of the anti-squeezing component is 1.64 times that of the vacuum state,corresponding to the anti-squeezing degree of 4.3 dB.We theoretically calculate the photon number distribution and the Wigner function of the vacuum squeezed field,and compare the results obtained by theoretical calculation with those obtained by maximum likelihood reconstruction.The probability of vacuum state|0 obtained by maximum likelihood reconstruction is greater,and the probability of photon number state|n(n=1,2,) is smaller than the corresponding theoretical calculation results.From the theoretical calculation,the maximum value of Wigner function is 0.231,and the short axis and long axis of noise range deduced from the contours of the Wigner function are larger than the results from the maximum likelihood reconstruction.The possible reasons for the discrepancy are as follows. 1) The phase scanning is nonuniform during the measurement of the quadrature components.2) The low-frequency electronic noise is not completely filtered out in the datum acquisition process.3) The datum points of measured quadrature components are not enough.In conclusion,we produce a vacuum squeezed field of 795 nm,and obtain the photon number distribution and the Wigner function in phase space through maximum likelihood estimation and theoretical calculation,respectively.This work will provide an experimental basis for generating the Schrodinger cat state.

Efficient characterization of phase space mapping in axially symmetric optical systems

Phase space mapping, typically between an object and image plane, characterizes an optical system within a geometrical optics framework. We propose a novel conceptual frame to characterize the phase mapping in axially symmetric optical systems for arbitrary object locations, not restricted to a specific object plane. The idea is based on decomposing the phase mapping into a set of bivariate equations corresponding to different values of the radial coordinate on a specific object surface (most likely the entrance pupil). These equations are then approximated through bivariate Chebyshev interpolation at Chebyshev nodes, which guarantees uniform convergence. Additionally, we propose the use of a new concept (effective object phase space), defined as the set of points of the phase space at the first optical element (typically the entrance pupil) that are effectively mapped onto the image surface. The effective object phase space provides, by means of an inclusion test, a way to avoid tracing rays that do not reach the image surface.

Nonclassical thermal-state superpositions: Analytical evolution law and decoherence behavior

Employing the integration technique within normal products of bosonic operators, we present normal product representations of thermal-state superpositions and investigate their nonclassical features, such as quadrature squeezing, sub-Poissonian distribution, and partial negativity of the Wigner function. We also analytically and numerically investigate their evolution law and decoherence characteristics in an amplitude-decay model via the variations of the probability distributions and the negative volumes of Wigner functions in phase space. The results indicate that the evolution formulas of two thermal component states for amplitude decay can be viewed as the same integral form as a displaced thermal state ρ(V,d), but governed by the combined action of photon loss and thermal noise. In addition, the larger values of the displacement d and noise V lead to faster decoherence for thermal-state superpositions.