Wigner Density Research Articles

Inelastic scattering effects on the inhomogeneous transverse conductivity of liquid metals and plasmas

Abstract On the basis of the quantum Wigner density operator formalism for multicomponent charged particle systems, the wavenumber (k)- and frequency (ω)- dependent transverse conductivity σ^T (k,ω) of simple liquid metals is presented for homogeneous (k = 0) and inhomogeneous (k≠0) electric fields separately. The theory takes into account correlated density fluctuations of electrons and ions, with their mutual inelastic scattering being described by the ionic dynamic structure factors. Since homogeneous and inhomogeneous fields perturb the diagonal and off-diagonal elements of the density matrix, respectively, the corresponding transport equations exhibit a dissimilarity in their collision terms. In both cases, approximate solutions to the transport equations yield a conductivity of generalized Drude form characterized by the k- and ω-dependent complex relaxation time. It is shown that the k→0 limit of the inhomogeneous conductivity, σ^T (k→0,ω), coincides with the homogeneous one in classical ion regime such that βℏω_i≪1, where β is the inverse temperature and ω_i is a characteristic frequency of ionic density fluctuation; this condition may be satisfied for metals well above melting. At lower temperatures, it is suggested that the (homogeneous) DC conductivity σ(0) can exceed the (inhomogeneous) optical conductivity extrapolated to zero frequency, σ^T (k→0,0). This trend is qualitatively consistent with earlier observations for liquid polyvalent metals.

Quarkonium production in high energy pp collisions

We investigate the charmonium and bottomonium production in pp collisions using the Wigner densities formalism. The Wigner density of the quarkonia is approximated by analytical 3-D isotropic harmonic oscillator Wigner densities with the same root-mean-square radius given by the solution of the Schrödinger equation. This approach reproduces quite well the available experimental transverse momentum and rapidity distributions.

Charmonium production in a thermalizing heat bath

Using the Remler formalism for the creation of composed particles, we study charmonium production both in thermalized and thermalizing boxes, which contain charm and anticharm quarks. The thermalizing box studies include the lowering of the box temperature, the spatial diffusion of charm and anticharm quarks, which are initially confined in the central region, as well as the combination of both, what imitates heavy-ion collisions. Comparing numerical and analytical results we demonstrate that the rate of the original Remler formalism has to be supplemented by two rates to obtain, for $t\to \infty$, results, which are consistent with the statistical model predictions: i) a rate, which takes into account the temperature dependence of the Wigner density of the quarkonium during the expansion and, in the case that a heavy quark potential is not implemented in the Monte Carlo approach, ii) a rate which comes from the change of the relative distance between the heavy quark and antiquark. These results provide the basis for future applications of the Remler formalism to heavy-ion collisions.

Dynamical cluster and hypernuclei production in heavy-ion collisions

We study light cluster and hypernuclei production in heavy-ion collisions from SIS to RHIC energies based on the n-body dynamical transport approach PHQMD (Parton-Hadron-Quantum-Molecular-Dynamics). In PHQMD clusters are formed dynamically due to the interactions between baryons described on the basis of Quantum Molecular Dynamics (QMD) which allows to propagate the n-body Wigner density and n-body correlations in phase-space, which is essential for the cluster formation. The clusters are identified by the MST (Minimum Spanning Tree) or the SACA (‘Simulated Annealing Cluster Algorithm’) algorithm which finds the most-bound configuration of nucleons and clusters. Collisions among hadrons as well as Quark-Gluon-Plasma formation and parton dynamics in PHQMD are treated in the same way as in the PHSD (Parton-Hadron-String-Dynamics) transport approach. We study the time evolution of the cluster formation in the expanding medium and the stability of the clusters. We present a comparison of the PHQMD results for d, 3He as well as for the hypernuclei with experimental data.

Optical Signatures of Periodic Charge Distribution in a Mott-like Correlated Insulator State

The elementary optical excitations in two dimensional semiconductors hosting itinerant electrons are attractive and repulsive polarons -- excitons that are dynamically screened by electrons. Exciton-polarons have hitherto been studied in translationally invariant degenerate Fermi systems. Here, we show that electronic charge order breaks the excitonic translational invariance and leads to a direct optical signature in the exciton-polaron spectrum. Specifically, we demonstrate that new optical resonances appear due to spatially modulated interaction between excitons and electrons in an incompressible Mott state. Our observations demonstrate that resonant optical spectroscopy provides an invaluable tool for studying strongly correlated states, such as Wigner crystals and density waves, where exciton-electron interactions are modified by the emergence of new electronic charge or spin order.

Dynamics of light hypernuclei in collisions of $$^{197}$$Au+$$^{197}$$Au at GeV energies

The dynamics of light hypernuclei and nuclear clusters produced in $^{197}$Au+$^{197}$Au collisions has been investigated thoroughly with a microscopic transport model. All possible channels of hyperon production and transportation of hyperons in nuclear medium are implemented into the model. The light complex fragments are recognized with the Wigner density approach at the stage of freeze out in nuclear collisions. The isospin diffusion in the collisions is responsible for the neutron-rich cluster formation. The collective flows of nuclear clusters are consistent with the experimental data from FOPI collaboration. It is found that the influence of the hyperon-nucleon potential on the free hyperons is negligible, but available for the light hypernuclide formation. The directed and elliptic flows of $^{3}_{\Lambda}$H and $^{4}_{\Lambda}$H at incident energies of 2, 2.5, 3, 3.5 and 4 GeV/nucleon are investigated thoroughly and manifest the same structure with the nuclear clusters. The hypernuclear yields are produced in a wide rapidity and momentum regime with increasing the beam energy.

Parton-hadron-quantum-molecular dynamics: A novel microscopic n -body transport approach for heavy-ion collisions, dynamical cluster formation, and hypernuclei production

Cluster and hypernuclei production in heavy-ion collisions is presently under active experimental and theoretical investigation. Since clusters are weekly bound objects, their production is very sensitive to the dynamical evolution of the system and its interactions. The theoretical description of cluster formation is related to the n-body problem. Here we present the novel n-body dynamical transport approach PHQMD (Parton-Hadron-Quantum-Molecular Dynamics) which is designed to provide a microscopic description of nuclear cluster and hypernucleus formation as well as of general particle production in heavy-ion reactions at relativistic energies. In difference to the coalescence or statistical models, often used for the cluster formation, in PHQMD clusters are formed dynamically due to the interactions between baryons described on a basis of Quantum Molecular Dynamics (QMD)which allows to propagate the n-body Wigner density and n-body correlations in phase-space, essential for the cluster formation. The clusters are identified by the MST (Minimum Spanning Tree) or the SACA ('Simulated Annealing Cluster Algorithm') algorithm which finds the most bound configuration of nucleons and clusters. Collisions among hadrons as well as Quark-Gluon-Plasma formation and parton dynamics in PHQMD are treated in the same way as in the established PHSD (Parton-Hadron-String Dynamics)transport approach. In order to verify our approach with respect to the general dynamics we present here the first PHQMD results for general 'bulk' observables such as rapidity distributions and transverse mass spectra for hadrons ($\pi, K, \bar K, p, \bar p, \Lambda, \bar \Lambda$) from SIS to RHIC energies. We find a good description of the 'bulk' dynamics which allows us to proceed with the results on cluster production, including hypernuclei.

Mapping Atomic Motions with Electrons: Toward the Quantum Limit to Imaging Chemistry

Recent advances in ultrafast electron and X-ray diffraction have pushed imaging of structural dynamics into the femtosecond time domain, that is, the fundamental time scale of atomic motion. New physics can be reached beyond the scope of traditional diffraction or reciprocal space imaging. By exploiting the high time resolution, it has been possible to directly observe the collapse of nearly innumerable possible nuclear motions to a few key reaction modes that direct chemistry. It is this reduction in dimensionality in the transition state region that makes chemistry a transferable concept, with the same class of reactions being applicable to synthetic strategies to nearly arbitrary levels of complexity. The ability to image the underlying key reaction modes has been achieved with resolution to relative changes in atomic positions to better than 0.01 Å, that is, comparable to thermal motions. We have effectively reached the fundamental space-time limit with respect to the reaction energetics and imaging the acting forces. In the process of ensemble measured structural changes, we have missed the quantum aspects of chemistry. This perspective reviews the current state of the art in imaging chemistry in action and poses the challenge to access quantum information on the dynamics. There is the possibility with the present ultrabright electron and X-ray sources, at least in principle, to do tomographic reconstruction of quantum states in the form of a Wigner function and density matrix for the vibrational, rotational, and electronic degrees of freedom. Accessing this quantum information constitutes the ultimate demand on the spatial and temporal resolution of reciprocal space imaging of chemistry. Given the much shorter wavelength and corresponding intrinsically higher spatial resolution of current electron sources over X-rays, this Perspective will focus on electrons to provide an overview of the challenge on both the theory and the experimental fronts to extract the quantum aspects of molecular dynamics.

Sampling the thermal Wigner density via a generalized Langevin dynamics.

The Wigner thermal density is a function of considerable interest in the area of approximate (linearized or semiclassical) quantum dynamics where it is employed to generate initial conditions for the propagation of appropriate sets of classical trajectories. In this paper, we propose an original approach to compute the Wigner density based on a generalized Langevin equation. The stochastic dynamics is nontrivial in that it contains a coordinate-dependent friction coefficient and a generalized force that couples momenta and coordinates. These quantities are, in general, not known analytically and have to be estimated via auxiliary calculations. The performance of the new sampling scheme is tested on standard model systems with highly nonclassical features such as relevant zero point energy effects, correlation between momenta and coordinates, and negative parts of the Wigner density. In its current brute force implementation, the algorithm, whose convergence can be systematically checked, is accurate and has only limited overhead compared to schemes with similar characteristics. We briefly discuss potential ways to further improve its numerical efficiency.

Evolution of quantum states via Weyl expansion in dissipative channel**Project supported by the National Natural Science Foundation of China (Grant No. 11664017), the Outstanding Young Talent Program of Jiangxi Province, China (Grant No. 20171BCB23034), the Degree and Postgraduate Education Teaching Reform Project of Jiangxi Province, China (Grant No. JXYJG-2013-027), and the Science Fund of the Education Department of Jiangxi Province,

Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.

Coherent State-Based Path Integral Methodology for Computing the Wigner Phase Space Distribution.

The accurate evaluation of the Wigner phase space density for multidimensional system remains a challenging task. Path integral Monte Carlo methods offer a numerically exact approach for obtaining the Boltzmann density in coordinate space, but the Fourier-type integral required to construct the Wigner distribution generally leads to poor convergence. This paper describes a path integral method for constructing the Wigner density which substantially mitigates the Monte Carlo sign problem and thus is applicable to systems with many degrees of freedom. The starting point is the path integral representation of the coherent state density, which does not involve a Fourier integral and thus converges rapidly. We then use the relation between the coherent state and Wigner densities to construct the Wigner function, taking advantage of destructive phase cancellation to truncate the infinite series and thus confine the integrand, avoiding highly oscillatory regions. We also describe the use of information-guided noise reduction (IGNoR) to improve the Monte Carlo statistics in the most challenging regimes. The method is applied to strongly anharmonic one-dimensional models, a system-bath Hamiltonian, as well as the formamide molecule within an ab initio quartic potential, and the results are compared to those obtained by various approximate methods. These calculations suggest that the coherent state-based path integral method described in this paper offers an efficient, numerically exact approach for constructing the Wigner phase space density in systems of many degrees of freedom, and thus will be useful for quantizing the initial condition in classical trajectory-based simulations of dynamical properties.

Quasiclassical Correlation Functions from the Wigner Density Using the Stability Matrix.

Accounting for zero-point energy in the initial conditions of classical trajectory calculations of time correlation functions requires sampling from a quantized phase space distribution, which is often chosen as the Weyl-Wigner transform of a thermalized operator. The numerical construction of the latter and its use as a sampling function can be challenging. We show that the operator dependence of the phase space distribution can be transferred to the dynamics, allowing sampling from the simpler Wigner phase space density. The method involves augmenting the classical equations of motion with additional differential equations for elements of the stability matrix. We also propose a local harmonic approximation for the dynamical derivatives, which significantly reduces the computational cost required to obtain correlation functions of nonlinear operators. We illustrate the method with application to linear and nonlinear correlation functions of model Hamiltonians. While the local harmonic approximation is not always successful in predicting nonlinear correlation functions of one degree of freedom, it quantitatively captures the full quasiclassical results for systems in contact with dissipative environments.

Production of primordial J/ψ in relativistic p+p and heavy-ion collisions

$J/\ensuremath{\psi}$ observables of $p+p$ collisions at $\sqrt{s}=200\phantom{\rule{0.16em}{0ex}}\mathrm{GeV}$ and 2.76 TeV are calculated using the pythia event generator for the production of $c$ and $\overline{c}$ and the Wigner density formalism. The charm quark momentum from pythia is tuned in such a way that the transverse momentum and rapidity distribution correspond to the fixed-order next-to-leading logarithm calculations and hence to the experimental data for open charm mesons. Using the Wigner density of the charmonium wave function we calculate for each charm quark pair the probability to form a charmonium. As a result, the experimental data on the total $J/\ensuremath{\psi}$ yield, its rapidity and transverse momentum distribution as well as the fractions of feed-down from excited states of charmonium are reproduced. Applying the same approach to relativistic heavy-ion collisions, we find, if quark gluon plasma (QGP) effects are ignored, a considerable enhancement of the primordial $J/\ensuremath{\psi}$'s.

State-specific tunneling lifetimes from classical trajectories: H-atom dissociation in electronically excited pyrrole.

A trajectory method of calculating tunneling probabilities from phase integrals along straight line tunneling paths, originally suggested by Makri and Miller [J. Chem. Phys. 91, 4026 (1989)] and recently implemented by Truhlar and co-workers [Chem. Sci. 5, 2091 (2014)], is tested for one- and two-dimensional ab initio based potentials describing hydrogen dissociation in the (1)B1 excited electronic state of pyrrole. The primary observables are the tunneling rates in a progression of bending vibrational states lying below the dissociation barrier and their isotope dependences. Several initial ensembles of classical trajectories have been considered, corresponding to the quasiclassical and the quantum mechanical samplings of the initial conditions. It is found that the sampling based on the fixed energy Wigner density gives the best agreement with the quantum mechanical dissociation rates.

Wigner phase space distribution via classical adiabatic switching.

Evaluation of the Wigner phase space density for systems of many degrees of freedom presents an extremely demanding task because of the oscillatory nature of the Fourier-type integral. We propose a simple and efficient, approximate procedure for generating the Wigner distribution that avoids the computational difficulties associated with the Wigner transform. Starting from a suitable zeroth-order Hamiltonian, for which the Wigner density is available (either analytically or numerically), the phase space distribution is propagated in time via classical trajectories, while the perturbation is gradually switched on. According to the classical adiabatic theorem, each trajectory maintains a constant action if the perturbation is switched on infinitely slowly. We show that the adiabatic switching procedure produces the exact Wigner density for harmonic oscillator eigenstates and also for eigenstates of anharmonic Hamiltonians within the Wentzel-Kramers-Brillouin (WKB) approximation. We generalize the approach to finite temperature by introducing a density rescaling factor that depends on the energy of each trajectory. Time-dependent properties are obtained simply by continuing the integration of each trajectory under the full target Hamiltonian. Further, by construction, the generated approximate Wigner distribution is invariant under classical propagation, and thus, thermodynamic properties are strictly preserved. Numerical tests on one-dimensional and dissipative systems indicate that the method produces results in very good agreement with those obtained by full quantum mechanical methods over a wide temperature range. The method is simple and efficient, as it requires no input besides the force fields required for classical trajectory integration, and is ideal for use in quasiclassical trajectory calculations.

Thermally enhanced Wigner oscillations in two-electron 1D quantum dots

Motivated by a recent experiment (Pecker et al 2013 Nat. Phys. 9 576), we study the stability, with respect to thermal effects, of Friedel and Wigner density fluctuations for two electrons trapped in a one-dimensional quantum dot. Diagonalizing the system exactly, the finite-temperature average electron density is computed. While the weak and strong interaction regimes display a Friedel oscillation or a Wigner molecule state at zero temperature, which as expected smear and melt as the temperature increases, a peculiar thermal enhancement of Wigner correlations in the intermediate interaction regime is found. We demonstrate that this effect is due to the presence of two different characteristic temperature scales: TF, dictating the smearing of Friedel oscillations, and TW, smoothing Wigner oscillations. In the early Wigner molecule regime, for intermediate interactions, TF < TW leading to the enhancement of the visibility of Wigner oscillations. These results complement those obtained within the Luttinger liquid picture, valid for larger numbers of particles.

Computing thermal Wigner densities with the phase integration method.

We discuss how the Phase Integration Method (PIM), recently developed to compute symmetrized time correlation functions [M. Monteferrante, S. Bonella, and G. Ciccotti, Mol. Phys. 109, 3015 (2011)], can be adapted to sampling/generating the thermal Wigner density, a key ingredient, for example, in many approximate schemes for simulating quantum time dependent properties. PIM combines a path integral representation of the density with a cumulant expansion to represent the Wigner function in a form calculable via existing Monte Carlo algorithms for sampling noisy probability densities. The method is able to capture highly non-classical effects such as correlation among the momenta and coordinates parts of the density, or correlations among the momenta themselves. By using alternatives to cumulants, it can also indicate the presence of negative parts of the Wigner density. Both properties are demonstrated by comparing PIM results to those of reference quantum calculations on a set of model problems.

Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime

Optomechanical systems can exhibit self-sustained limit cycles where the quantum state of the mechanical resonator possesses nonclassical characteristics such as a strongly negative Wigner density, as was shown recently in a numerical study by Qian et al. [Physical Review Letters, 109, 253601 (2012)]. Here we derive a Fokker-Planck equation describing mechanical limit cycles in the quantum regime which correctly reproduces the numerically observed nonclassical features. The derivation starts from the standard optomechanical master equation, and is based on techniques borrowed from the laser theory due to Haake's and Lewenstein. We compare our analytical model with numerical solutions of the master equation based on Monte-Carlo simulations, and find very good agreement over a wide and so far unexplored regime of system parameters. As one main conclusion, we predict negative Wigner functions to be observable even for surprisingly classical parameters, i.e. outside the single-photon strong coupling regime, for strong cavity drive, and rather large limit cycle amplitudes. The general approach taken here provides a natural starting point for further studies of quantum effects in optomechanics.

Quantum Signatures of the Optomechanical Instability

In the past few years, coupling strengths between light and mechanical motion in optomechanical setups have improved by orders of magnitude. Here we show that, in the standard setup under continuous laser illumination, the steady state of the mechanical oscillator can develop a nonclassical, strongly negative Wigner density if the optomechanical coupling is comparable to or larger than the optical decay rate and the mechanical frequency. Because of its robustness, such a Wigner density can be mapped using optical homodyne tomography. This feature is observed near the onset of the instability towards self-induced oscillations. We show that there are also distinct signatures in the photon-photon correlation function g(2)(t) in that regime, including oscillations decaying on a time scale not only much longer than the optical cavity decay time but even longer than the mechanical decay time.

The Quantum State of Inflationary Perturbations

This article reviews the properties of the quantum state of inflationary perturbations. After a brief description of the inflationary background, the wavefunction of the Mukhanov-Sasaki variable is calculated and shown to be that of a strongly squeezed state. The corresponding Wigner function and density matrix, which are convenient tools to characterize the properties of a quantum state, are also evaluated. Finally, the issue of definite outcomes for inflation is discussed.