Phase-space Function Research Articles

Probing nonclassicality with matrices of phase-space distributions

We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's integral inequality \cite{BA19}. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions beyonds-parametrized ones, and is accessible in experiments. To demonstrate the power of our technique, the quantum characteristics of discrete- and continuous-variable, single- and multimode, as well as pure and mixed states are certified only employing second-order correlations and Husimi functions, which always resemble a classical probability distribution. Moreover, nonlinear generalizations of our approach are studied. Therefore, a versatile and broadly applicable framework is devised to uncover quantum properties in terms of matrices of phase-space distributions.

Strong-chaos-caused negative mobility in a periodic substrate potential.

We numerically investigate negative mobility of an inertial Brownian particle moving in a periodic double-well substrate potential in the presence of a time-periodic force and a constant bias. For the deterministic case, we find from the average velocity that the varying shape parameter and driving forces can cause negative mobility, differential negative mobility, and giant positive mobility. We analyze these findings via the bifurcation diagram and maximal Lyapunov exponent and find that certain chaos can give rise to negative mobility. For the presence of a Gaussian color noise, the results suggest that the noise intensity can enhance or result in negative and positive mobilities, whereas correlation time can enhance, weaken, or even eliminate them. On the basis of the time series, phase-space map, and power spectrum of various attractors, we unveil how these mobilities connect to strong chaotic attractors (SCAs), including both stable attractor and unstable attractors, and propose an underlying mechanism that SCAs can result in the negative mobility, whereas other attractors do not. Our findings may be potentially useful for research on anomalous transports of the particles and on designs of various devices, such as atomic chains, crystals with dislocations, and superconducting nanowires, etc.

The Landau problem and nonclassicality

Exploring the concept of the extended Galilei group [Formula: see text]. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space–time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of [Formula: see text]. This representation gives rise to the spin-0 Schrödinger (Klein–Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli–Schrödinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin-1/2 particles in phase space and show how implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the nonrelativistic Landau levels. Finally, we study the parameter of negativity associated with the nonclassicality of the system.

Analytical Solution for the Gross-Pitaevskii Equation in Phase Space and Wigner Function

In this work, we study symplectic unitary representations for the Galilei group. As a consequence a nonlinear Schrödinger equation is derived in phase space. The formalism is based on the noncommutative structure of the star product, and using the group theory approach as a guide a physically consistent theory is constructed in phase space. The state is described by a quasi-probability amplitude that is in association with the Wigner function. With these results, we solve the Gross-Pitaevskii equation in phase space and obtained the Wigner function for the system considered.

Margolus–Levitin speed limit across quantum to classical regimes based on trace distance**Project supported by the National Natural Science Foundation of China (Grant No. 11775040), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi Province of China (Grant No. 2019L0527), and the Fundamental Research Fund for the Central Universities of China (Grant No. DUT18LK45).

The classical version of Mandelstam–Tamm speed limit based on the Wigner function in phase space was reported by Shanahan et al. [Phys. Rev. Lett. 120 070401 (2018)]. We present the Margolus–Levitin speed limit across the quantum-to-classical transition in phase space based on the trace distance. The Margolus–Levitin speed limit is set by the Schatten L1 norm of the generator of time-dependent evolution for both the quantum and classical domains. As an example, the time-dependent harmonic oscillator is considered to illustrate the result.

Entropy Dynamics of Phonon Quantum States Generated by Optical Excitation of a Two-Level System.

In quantum physics, two prototypical model systems stand out due to their wide range of applications. These are the two-level system (TLS) and the harmonic oscillator. The former is often an ideal model for confined charge or spin systems and the latter for lattice vibrations, i.e., phonons. Here, we couple these two systems, which leads to numerous fascinating physical phenomena. Practically, we consider different optical excitations and decay scenarios of a TLS, focusing on the generated dynamics of a single phonon mode that couples to the TLS. Special emphasis is placed on the entropy of the different parts of the system, predominantly the phonons. While, without any decay, the entire system is always in a pure state, resulting in a vanishing entropy, the complex interplay between the single parts results in non-vanishing respective entanglement entropies and non-trivial dynamics of them. Taking a decay of the TLS into account leads to a non-vanishing entropy of the full system and additional aspects in its dynamics. We demonstrate that all aspects of the entropy’s behavior can be traced back to the purity of the states and are illustrated by phonon Wigner functions in phase space.

Stellar Representation of Non-Gaussian Quantum States.

The so-called stellar formalism allows us to represent the non-Gaussian properties of single-mode quantum states by the distribution of the zeros of their Husimi Q function in phase space. We use this representation in order to derive an infinite hierarchy of single-mode states based on the number of zeros of the Husimi Q function: the stellar hierarchy. We give an operational characterization of the states in this hierarchy with the minimal number of single-photon additions needed to engineer them, and derive equivalence classes under Gaussian unitary operations. We study in detail the topological properties of this hierarchy with respect to the trace norm, and discuss implications for non-Gaussian state engineering, and continuous variable quantum computing.

Screening properties of quark-gluon plasma obtained from distribution and correlation functions of the constituent quasiparticle model

Based on the constituent quasiparticle model of quark-gluon plasma (QGP), the matrix elements of the density operator and the Wigner function in the color phase space are presented in form of color path integrals over Wiener and SU(3) group Haar measures. Monte Carlo calculations of quark and gluon momentum distributions and spatial pair distribution functions have been carried out for the strongly coupled QGP plasma in thermal equilibrium at zero baryon chemical potential. The Debye screening mass and the running coupling constant have been obtained from the spatial pair distribution function and are in agreement with the available lattice QCD data. At densities related to the average interparticle distance more than 0.4 fm the gluon bound states in the form of glueballs have been found. Comparison with the Maxwell Boltzmann distribution shows significant influence of interparticle interaction on high energy asymptotics of the momentum distribution functions, resulting in appearance of quantum tails. The new color pair correlation function has been introduced, and related new color screening mass has been discussed.

Probing the C P nature of the top quark Yukawa at hadron colliders

We analyze the prospects of probing the C P -odd itilde{kappa}overline{t}gamma 5th interaction at the LHC and its projected upgrades, the high-luminosity and high-energy LHC, directly using associated on-shell Higgs boson and top quark or top quark pair production. To this end we first construct a C P -odd observable based on top quark polarization in Wb → th scattering with optimal linear sensitivity to tilde{kappa} . For the corresponding hadronic process pp → thj we present a method of extracting the phase-space dependent weight function that allows to retain close to optimal sensitivity to tilde{kappa} . We project future sensitivity to the signal in pp → t(→ ℓ\U0001d708b)h(→ b overline{b} )j. We also propose novel C P -odd observables for top quark pair production in association with the Higgs, pp → t overline{t} h, with semileptonically decaying tops and h → b overline{b} , that rely solely on measuring the momenta of leptons and b-jets from the decaying tops without having to distinguish the charge of the b-jets. Among the many possibilities we single out an observable that can potentially probe tilde{kappa} ∼ 0.1 at the high-energy LHC with 2\U0001d70e confidence.

Detector-Agnostic Phase-Space Distributions.

The representation of quantum states via phase-space functions constitutes an intuitive technique to characterize light. However, the reconstruction of such distributions is challenging as it demands specific types of detectors and detailed models thereof to account for their particular properties and imperfections. To overcome these obstacles, we derive and implement a measurement scheme that enables a reconstruction of phase-space distributions for arbitrary states whose functionality does not depend on the knowledge of the detectors, thus defining the notion of detector-agnostic phase-space distributions. Our theory presents a generalization of well-known phase-space quasiprobability distributions, such as the Wigner function. We implement our measurement protocol, using state-of-the-art transition-edge sensors without performing a detector characterization. Based on our approach, we reveal the characteristic features of heralded single- and two-photon states in phase space and certify their nonclassicality with high statistical significance.

Plasma in a monopole background does not have a twisted Poisson structure

For a particle in the magnetic field of a cloud of monopoles, the naturally associated 2-form on phase space is not closed, and so the corresponding bracket operation on functions does not satisfy the Jacobi identity. Thus, it is not a Poisson bracket; however, it is twisted Poisson in the sense that the Jacobiator comes from a closed 3-form. The space $\mathcal D$ of densities on phase space is the state space of a plasma. The twisted Poisson bracket on phase-space functions gives rise to a bracket on functions on $\mathcal D$. In the absence of monopoles, this is again a Poisson bracket. It has recently been shown by Heninger and Morrison that this bracket is not Poisson when monopoles are present. In this note, we give an example where it is not even twisted Poisson.

From probabilistic mechanics to quantum theory

We show that quantum theory (QT) is a substructure of classical probabilistic physics. The central quantity of the classical theory is Hamilton’s function, which determines canonical equations, a corresponding flow, and a Liouville equation for a probability density. We extend this theory in two respects: (1) The same structure is defined for arbitrary observables. Thus, we have all of the above entities generated not only by Hamilton’s function but also by every observable. (2) We introduce for each observable a phase space function representing the classical action. This is a redundant quantity in a classical context but indispensable for the transition to QT. The basic equations of the resulting theory take a “quantum-like” form, which allows for a simple derivation of QT by means of a projection to configuration space reported previously [Quantum Stud Math Found 5:219–227, 2018]. We obtain the most important relations of QT, namely the form of operators, Schrödinger’s equation, eigenvalue equations, commutation relations, expectation values, and Born’s rule. Implications for the interpretation of QT are discussed, as well as an alternative projection method allowing for a derivation of spin.

Recognition of Lung Volume Condition based on Phase Space Mapping Using Electrical Impedance Tomography.

The purpose of this paper is to identify differences between abnormal and normal lung signals gathered by an EIT device, which is a new, non-invasive system that seeks the electrical conductivity and permittivity inside a body. Lung performances in patients are investigated using Phase Space Mapping technique on Electrical EIT signals. The database used in this paper contains 82 registered records of 52 individuals with proper lung volume. The results of this paper show that as the delay parameter (τ) increases, the SD1 parameter of phase space mapping indicates a significant difference between normal and abnormal lung volumes. The value of the SD1 parameter with τ = 6 in the case that the lung volume is in a normal condition is 342.57 ± 32.75 while it is 156.71 ± 26.01 in non-optimal mode. This method can be used to identify the patients’ lung volumes with chronic respiratory illnesses and is an accurate assessment of the diverse methods to treat respiratory system illnesses in addition to saving various therapeutic costs and dangerous consequences that are likely to occur by using improper treatment methods. It can also reduce the required treatment durations.

Basis set convergence of Wilson basis functions for electronic structure

There are many ways to numerically represent chemical systems in order to compute their electronic structure. Basis functions may be localized in real-space (atomic orbitals), in momentum-space (plane waves), or in both components of phase-space. Such phase-space localized basis functions in the form of wavelets have been used for many years in the electronic structure. In this paper, we turn to a phase-space localized basis set first introduced by Wilson [Generalized Wannier Functions (Cornell University, 1987), preprint]. We provide the first full study of this basis and its numerical implementation. To calculate electronic energies of a variety of small molecules and states, we utilize the sum-of-products form, Gaussian quadratures, and introduce methods for selecting sample points from a grid of phase-space localized Wilson basis. Both full configuration interaction and Hartree-Fock implementations are discussed and implemented numerically. As with many grid based methods, describing both tightly bound and diffuse orbitals is challenging so we have considered augmenting the Wilson basis set as projected Slater-type orbitals. We have also compared the Wilson basis set against the recently introduced wavelet transformed Gaussians (gausslets). Throughout, we give comments on the implementation and use small atoms and molecules to illustrate the convergence properties of the Wilson basis.

Quantification of thermally-driven flows in microsystems using Boltzmann equation in deterministic and stochastic contexts

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow—an observation attributed to the thermostress convection effects at the microscale. The dynamics of the overall thermostress convection process is governed by the Boltzmann equation—an integrodifferential equation describing the evolution of the molecular distribution function in six-dimensional phase space—which models dilute gas behavior at the molecular level to accurately describe a wide range of flow phenomena. Approaches for solving the full Boltzmann equation with general intermolecular interactions rely on two perspectives: one stochastic in nature often delegated to the direct simulation Monte Carlo (DSMC) method and the others deterministic by virtue. Among the deterministic approaches, the discontinuous Galerkin fast spectral (DGFS) method has been recently introduced for solving the full Boltzmann equation with general collision kernels, including the variable hard/soft sphere models—necessary for simulating flows involving diffusive transport. In this work, the deterministic DGFS method, Bhatnagar-Gross-Krook (BGK), Ellipsoidal statistical BGK (ESBGK), and Shakhov kinetic models, and the widely used stochastic DSMC method, are utilized to assess the thermostress convection process in micro in-plane Knudsen radiometric actuator—a microscale compact low-power pressure sensor utilizing the Knudsen forces. The BGK model underpredicts the heat-flux, shear-stress, and flow speed; the S-model overpredicts; whereas, ESBGK comes close to the DSMC results. On the other hand, both the statistical/DSMC and deterministic/DGFS methods, segregated in perspectives, yet, yield inextricable results, bespeaking the ingenuity of Graeme Bird who laid down the foundation of practical rarefied gas dynamics for microsystems.

A new perspective for nonadiabatic dynamics with phase space mapping models.

Based on the recently developed unified theoretical framework [J. Liu, J. Chem. Phys. 145(20), 204105 (2016)], we propose a new perspective for studying nonadiabatic dynamics with classical mapping models (CMMs) of the coupled multistate Hamiltonian onto the Cartesian phase space. CMMs treat the underlying electronic state degrees of freedom classically with a simple physical population constraint while employing the linearized semiclassical initial value representation to describe the nuclear degrees of freedom. We have tested various benchmark condensed phase models where numerically exact results are available, which range from finite temperature to more challenging zero temperature, from adiabatic to nonadiabatic domains, and from weak to strong system-bath coupling regions. CMMs demonstrate overall reasonably accurate dynamics behaviors in comparison to exact results even in the asymptotic long time limit for various spin-boson models and site-exciton models. Further investigation of the strategy used in CMMs may lead to practically useful approaches to study nonadiabatic processes in realistic molecular systems in the condensed phase.

The general scalar–tensor Hamiltonian with deformed covariance

We derive the scalar–tensor Hamiltonian constraint to all orders of momenta when the canonical constraint algebra is deformed by a phase space function as predicted by some studies into loop quantum cosmology. We find that the momenta and spatial derivatives from gravity and matter fields must combine in a very specific form to maintain a deformed general covariance. This suggests that the deformation should be equally caused by matter field derivatives as it is by gravitational curvature. We demonstrate the cosmological consequences of a variety of deformation functions. Some cause a big bounce at high energy densities and others cause a sudden singularity.

Rapid trajectory design in complex environments enabled by reinforcement learning and graph search strategies

Designing trajectories in dynamically complex environments is challenging and easily becomes intractable. Recasting the problem may reduce the design time and offer global solutions by leveraging phase space mapping patterns available as accessible regions, and the application of search techniques from combinatorics. A computationally efficient search process produces potential trajectory concepts to meet unique design requirements over a broad range of mission types, including low-thrust scenarios. A successful framework is summarized in terms of four components: (i) Accessible regions — establishing reachable regions within the design space for a given thruster/engine capability: (ii) Database exploitation — discretization of well known dynamical structures to form a searchable 2D or 3D volume or map: (iii) Automated pathfinding — exploiting machine learning techniques to determine the transport sequence to deliver an efficient path: (iv) Convergence/optimization — once the transport sequence is determined as a globally efficient concept, it is optimized locally by traditional numerical strategies.

Quantum field theory in phase space

The tilde conjugation rule in thermofield dynamics, equivalent to the modular conjugation in a [Formula: see text]-algebra, is used to develop unitary representations of the Poincaré group, where the Hilbert space has the phase space content, a symplectic Hilbert space. The state is described by a quasi-amplitude of probability, which is a sort of wave function in phase space, associated with the Wigner function. The quantum field theory in phase space is then constructed, including the quantization rules for the Klein–Gordon and the Dirac fields, the derivation of the electrodynamics in phase space and elements of a relativistic quantum kinetic theory. Towards a physical interpretation of the theory, propagators are associated with the corresponding Wigner functions. The Feynman rules follow accordingly with vertices similar to those of usual non-Abelian quantum field theories.

Theoretical Investigation on the Microscopic Mechanism of Lattice Thermal Conductivity of ZnXP2 (X = Si, Ge, and Sn).

Thermal conductivity is an important physical parameter for the application of nonlinear optical single crystal materials. The underlying science of thermal transport behavior is not well established both experimentally and theoretically. In the present work, we have studied the microscopic picture of lattice thermal conductivity of ZnXP2 (X = Si, Ge, Sn), chalcopyrite ABC2 type infrared optical crystals, by using a harmonic and anharmonic lattice dynamic method and phonon Boltzmann transport equation based on first-principle calculations. With the mass of atom X increased, the phonon frequencies and phonon group velocities of ZnXP2 (X = Si, Ge, Sn) are shown not surprisingly to be decreased. Nevertheless, the phonon lifetime of ZnXP2 is unexpectedly increased, which is the governing mechanism for the increased thermal conductivity as 12.5 W/(m·k), 31.6 W/(m·k), and 35.4 W/(m·k), for ZnSiP2, ZnGeP2, and ZnSnP2, respectively, at 300 K. The contributions of optical phonons (with the frequency below 150 cm-1) to the total thermal conductivity are remarkable, reaching 18%, 31%, and 34% for three compounds, due to the significantly increased phonon lifetime in the frequency range 50-150 cm-1. To explore the physical insights of phonon lifetime and phonon anharmonicity, three-phonon scattering phase space and electronic localization function analysis of the X-P bond are provided. The results show that the covalent nature of X-P bonds is enhanced with the increased mass of atom X = Si, Ge, Sn, which induces the reduction of three-phonon scattering phase space in the frequency range 50-150 cm-1, leading to the enhancement of the phonon lifetime and thermal conductivity of ZnXP2.