Wigner Function Research Articles

Wigner function method for the Gibbons–Hawking and the Unruh effect

Wigner function method for the Gibbons–Hawking and the Unruh effect

Unveiling the nonclassicality within quasi-distribution representations through deep learning

Abstract To unequivocally distinguish genuine quantumness from classicality, a widely adopted approach focuses on the negative values of a quasi-distribution representation as compelling evidence of nonclassicality. Prominent examples include the dynamical process nonclassicality characterized by the canonical Hamiltonian ensemble representation (CHER) and the nonclassicality of quantum states
characterized by the Wigner function. However, to construct a multivariate joint quasi-distribution function with negative values from experimental data is typically highly cumbersome. Here we propose a computational approach utilizing a deep generative model, processing three marginals, to construct the bivariate joint quasi-distribution functions. We first apply our model to tackle the
challenging problem of the CHERs, which lacks universal solutions, rendering the problem ground-truth (GT) deficient. To overcome the GT deficiency of the CHER problem, we design optimal synthetic datasets to train our model. While trained with synthetic data, the physics-informed optimization enables our model to capture the detrimental effect of the thermal fluctuations on nonclassicality, which cannot be obtained from any analytical solutions. This underscores the reliability of our approach. This approach also allows us to predict the Wigner functions subject to thermal noises. Our model predicts the Wigner functions with a prominent accuracy by processing three marginals of probability distributions. Our approach also provides a significant reduction of the experimental efforts of
constructing the Wigner functions of quantum states, giving rise to an efficient alternative way to realize the quantum state tomography.

ToMCCA: a Toy Monte Carlo Coalescence Afterburner

Antinuclei in our Galaxy may stem either from annihilation or decay of dark matter, or from collisions of cosmic rays with the interstellar medium, which constitute the background of indirect dark matter searches. Understanding the formation mechanism of (anti)nuclei is crucial for setting limits on their production in space. Coalescence models, which describe the formation of light nuclei from final-state interaction of nucleons, have been widely employed in high-energy collisions. In this work, we introduce ToMCCA (Toy Monte Carlo Coalescence Afterburner), which allows for detailed studies of the nuclear formation processes without the overload of general-purpose event generators. ToMCCA contains parameterizations of the multiplicity dependence of the transverse momentum distributions of protons and of the baryon-emitting source size, extracted from ALICE measurements in pp collisions at s=5-13\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{s} = 5 - 13$$\\end{document} TeV, as well as of the event multiplicity distributions, taken from the EPOS event generator. ToMCCA provides predictions of the deuteron transverse momentum distributions, with agreement of ∼5%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 5\\%$$\\end{document} with the experimental data. The results of ToMCCA show that the coalescence mechanism in pp collisions depends only on the event multiplicity, not on the collision system or its energy. This allows the model to be utilized for predictions at lower center-of-mass collision energies, which are the most relevant for the production of antinuclei from processes related to dark matter. This model can also be extended to heavier nuclei as long as the target nucleus wavefunction and its Wigner function are known.

Wigner state and process tomography on near-term quantum devices

We present an experimental scanning-based tomography approach for near-term quantum devices. The underlying method has previously been introduced in an ensemble-based NMR setting. Here we provide a tutorial-style explanation along with suitable software tools to guide experimentalists in its adaptation to near-term pure-state quantum devices. The approach is based on a Wigner-type representation of quantum states and operators. These representations provide a rich visualization of quantum operators using shapes assembled from a linear combination of spherical harmonics. These shapes (called droplets in the following) can be experimentally tomographed by measuring the expectation values of rotated axial tensor operators. We present an experimental framework for implementing the scanning-based tomography technique for circuit-based quantum computers and showcase results from IBM quantum experience. We also present a method for estimating the density and process matrices from experimentally tomographed Wigner functions (droplets). This tomography approach can be directly implemented using the Python-based software package DROPStomo.

Photon-modulated displaced thermal states: Negativity and non-Gaussianty

Abstract In this paper, we propose a new non-Gaussian quantum state, i.e., photon-modulated displaced thermal state (DTS), and use the operator ordering method to obtain the normal ordering product of its density operator, and then investigate its normalization, negativity of Wigner function and non-Gaussianity.

Generalized hydrodynamics and approach to generalized Gibbs equilibrium for a classical harmonic chain

Abstract We study the evolution of a classical harmonic chain with nearest-neighbor interactions starting from domain wall initial conditions. The initial state is taken to be either a product of two Gibbs ensembles (GEs) with unequal temperatures on the two halves of the chain or a product of two generalized Gibbs ensembles (GGEs) with different parameters in the two halves. For this system, we construct the Wigner function and demonstrate that its evolution defines the generalized hydrodynamics (GHD) describing the evolution of the conserved quantities. We solve the GHD for both finite and infinite chains and compute the evolution of conserved densities and currents. For a finite chain with fixed boundaries, we show that these quantities relax as ∼ 1 / t to their respective steady-state values given by the final expected GE or GGE state, depending on the initial conditions. Exact expressions for the Lagrange multipliers of the final expected GGE state are obtained in terms of the steady-state densities. In the case of an infinite chain, we find that the conserved densities and currents at any finite time exhibit ballistic scaling while, at infinite time, any finite segment of the system can be described by a current-carrying non-equilibrium steady state (NESS). We compute the scaling functions analytically and show that the relaxation to the NESS occurs as ∼ 1 / t for the densities and as ∼ 1 / t 2 for the currents. We compare the analytical results from hydrodynamics with those from exact microscopic numerics and find excellent agreement.

Entanglement of Coupled Harmonic Oscillators without and with Tunneling Effect and Correction Factor

Seeing its simplicity of processing, the system of two coupled harmonic oscillators have witnessed a quick growth of research conducted to quantum information. This approach is explicitly investigated in this paper, in particularly entanglement concept is proved to be intimately related to the tunneling effect. We examine analytically entanglement dynamics by introducing the Lewis and Riesenfeld invariant operator in the Heisenberg picture approach to compute the density matrix on the based of an exact treatment. We use Wigner function of the mixed state as an essential tool to move to linear entanglement entropies and we compute the corresponding correction factor. We follow numerically the evolution of entanglement dynamics without and under tunneling through the quantum potential barrier by introducing two particular models between simple and damped coupled harmonic oscillators. Entanglement dynamics is considered to be average without tunneling, it grows upon encountering the potential barrier and remains moderately constant inside barrier. This specificity is extended for both models but with higher values for damped coupled harmonic oscillators consequently the damping effect rapidly increases entanglement. Correction factor is also considered for both models, it show that; low temperature and high potential barriers make the system more disruptive. An increase of the coupling parameter of the system increase correction factor. Damping make the correction factor more important so damping disturbs more the system. Interference effect increase correction factor and it shows an interference between the values of the barrier penetration integral.

Classical echoes of quantum boundary conditions

Abstract We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit.

Phase-space representation of coherent states generated through SUSY QM for tilted anisotropic Dirac materials

In this paper, we examine the electron interaction within tilted anisotropic Dirac materials when subjected to external electric and magnetic fields possessing translational symmetry. Specifically, we focus on a distinct non-zero electric field magnitude, enabling the decoupling of the differential equation system inherent in the eigenvalue problem. Subsequently, employing supersymmetric quantum mechanics facilitates the determination of eigenstates and eigenvalues corresponding to the Hamiltonian operator. To delve into a semi-classical analysis of the system, we identify a set of coherent states. Finally, we assess the characteristics of these states using fidelity and the phase-space representation through the Wigner function.

Modeling local decoherence of a spin ensemble using a generalized Holstein–Primakoff mapping to a bosonic mode

We show how the decoherence that occurs in an entangling atomic spin–light interface can be simply modeled as the dynamics of a bosonic mode. Although one seeks to control the collective spin of the atomic system in the permutationally invariant (symmetric) subspace, diffuse scattering and optical pumping are local, making an exact description of the many-body state intractable. To overcome this issue we develop a generalized Holstein–Primakoff approximation for collective states which is valid when decoherence is uniform across a large atomic ensemble. In different applications the dynamics is conveniently treated as a Wigner function evolving according to a thermalizing diffusion equation, or by a Fokker–Planck equation for a bosonic mode decaying in a zero-temperature reservoir. We use our formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction in preparing nonclassical spin states for application in quantum metrology.

Experimental preparation of multiphoton-added coherent states of light

Conditional addition of photons represents a crucial tool for optical quantum state engineering and it forms a fundamental building block of advanced quantum photonic devices. Here we report on experimental implementation of the conditional addition of several photons. We demonstrate the addition of one, two, and three photons to input coherent states with various amplitudes. The resulting highly nonclassical photon-added states are completely characterized with time-domain homodyne tomography, and the nonclassicality of the prepared states is witnessed by the negativity of their Wigner functions. We experimentally demonstrate that the conditional addition of photons realizes approximate noiseless quantum amplification of coherent states with sufficiently large amplitude. We also investigate certification of the stellar rank of the generated multiphoton-added coherent states, which quantifies the non-Gaussian resources required for their preparation. Our results pave the way towards the experimental realization of complex optical quantum operations based on combination of multiple photon additions and subtractions.

Frame representation of quantum systems with finite-dimensional Hilbert space

Abstract There exist many attempts to define a Wigner function for quantum systems with finite-dimensional Hilbert space, each of them coming with its advantages and limitations. The existing finite versions have simple definitions, but they are based only on the existence of a formal analogy with the continuous-variable Wigner function and do not allow an intuitive state analysis. The continuous versions have more complicated definitions, but they are closer to the original Wigner function and allow a visualization of the quantum states. The version based on the concept of tight frame we present is finite, but it has certain properties and applications similar to those of continuous versions. It allows us to present a new graphical representation of qubit states, and to define new parameters concerning them. An important advantage of frame representation follows from the use of redundant information. The values taken by the frame version of Wigner function are not independent. They have to satisfy a large number of mathematical relations, useful in error detection and correction.

Simulating Quantum Computation: How Many “Bits” for “It”?

A recently introduced classical simulation method for universal quantum computation with magic states operates by repeated sampling from probability functions [M. Zurel PRL 260404 (2020)]. This method is closely related to sampling algorithms based on Wigner functions, with the important distinction that Wigner functions can take negative values obstructing the sampling. Indeed, negativity in Wigner functions has been identified as a precondition for a quantum speed-up. However, in the present method of classical simulation, negativity of quasiprobability functions never arises. This model remains probabilistic for all quantum computations. In this paper, we analyze the amount of classical data that the simulation procedure must track. We find that this amount is small. Specifically, for any number n of magic states, the number of bits that describe the quantum system at any given time is 2n2+O(n). Published by the American Physical Society 2024

Intrinsic squeezing of mechanical motion of a harmonically trapped atom with spin-orbit coupling

In this paper we investigated one-dimensional harmonically trapped atom with Raman-induced spin-orbit coupling by mapping to quantum-like model. Using displacement Fock states in quantum optics and variational methods we found the odd-parity superposition state of left- and right-displaced oscillator state is a good approximate solution to the ground state, which captures the essential physics of the system. The ground state wave function owes nodes which is remarkably different from the conventional no-node theory in quantum mechanics. The results show the atomic motion of center of mass exhibits intrinsic squeezing properties with relation to the Raman and spin-orbit coupling strength characterized by the variance in position and momentum, information entropy, as well as the Wigner function of the ground state. The dynamics of center of mass are also studied which demonstrates intuitive Zitterbewegung oscillations in momentum space and real space. Our results could be used for observing Zitterbewegung oscillations in neutral atomic gases and engineering nonclassic states.

Extended Weyl-Wigner phase-space framework for nonlinear systems: Typical and modified prey-predator-like dynamics.

The extension of the phase-space Weyl-Wigner quantum mechanics to the subset of Hamiltonians in the form of H(q,p)=K(p)+V(q) [with K(p) replacing single p^{2} contributions] is revisited. Deviations from classical and stationary profiles are identified in terms of Wigner functions and Wigner currents for Gaussian and gamma/Laplacian distribution ensembles. The procedure is successful in accounting for the exact pattern of quantum fluctuations when compared with the classical phase-space pattern. General results are then specialized to some specific Hamiltonians revealing nonlinear dynamics, and suggest a novel algorithm to treat quantum modifications mapped by Wigner currents. Our analysis shows that the framework encompasses, for instance, the quantized prey-predator-like scenarios subjected to statistical constraints.

Precise measurement of spatial coherence and axial brightness based on the Wigner function reconstruction in transmission electron microscopes with field emission guns and a thermionic emission gun.

The spatial coherence and the axial brightness of a cold field emission gun, a Schottky field emission gun and a $\mathrm{LaB}_6$ thermionic gun are precisely measured. By analyzing the Airy pattern from a selected area aperture, various parameters including the spatial coherence length are determined. Using the determined coherence length, the axial brightness of the field emission guns is estimated using the equation which we previously derived based on the discussion of the Wigner function of an electron beam. We also make some extensions in the method to be applicable to the measurements of the thermionic gun, which has anisotropic intensity distribution in most case unlike the field emission guns. Not conventional average brightness but the axial brightness measured for the three kinds of emitters are compared accurately and precisely without influenced by the measurement conditions.

Exploring the possibility of a complex-valued non-Gaussianity measure for quantum states of light

We consider a quantity that is the differential relative entropy between a generic Wigner function and a Gaussian one. We prove that said quantity is minimized with respect to its Gaussian argument, if both Wigner functions in the argument of the Wigner differential entropy have the same first and second moments, i.e., if the Gaussian argument is the Gaussian associate of the other, generic Wigner function. Therefore, we introduce the differential relative entropy between any Wigner function and its Gaussian associate and we examine its potential as a non-Gaussianity measure. The proposed, phase-space based non-Gaussianity measure is complex-valued, with its imaginary part possessing the physical meaning of the Wigner function’s negative volume. At the same time, the real part of this measure provides an extra layer of information, rendering the complex-valued quantity a measure of non-Gaussianity, instead of a quantity pertaining only to the negativity of the Wigner function. We prove that the measure (both the real and imaginary parts) is faithful, invariant under Gaussian unitary operations, and find a sufficient condition on its monotonic behavior under Gaussian channels. We provide numerical results supporting the aforesaid condition. In addition, we examine the measure’s usefulness to non-Gaussian quantum state engineering with partial measurements.

Missing-Mass Measurement of the 12C(K-, K+) Reaction at 1.8 GeV/c with the Superconducting Kaon Spectrometer

Abstract We performed a measurement of the inclusive missing-mass spectrum of the $^{12}$C$(K^-, K^+)$ reaction at an incident beam momentum of 1.8 GeV/c. This measurement was carried out by using the Superconducting Kaon Spectrometer (SKS) and the K1.8 beamline spectrometer at the Hadron Experimental Facility in J-PARC. From the missing-mass of the $^{12}$C$(K^-, K^+)$ reaction, the binding energy of a $\Xi ^-$ hyperon in a core $^{11}$B nucleus, $B_{\Xi ^-}$, can be calculated. Our experimental setup yielded a good energy resolution of 8.2 MeV (full width at half maximum), which allowed us to observe significant enhancements in the proximity of the $^{12}_{\Xi }$Be production threshold region. In order to extract information from the missing-mass spectrum, we employed several fitting parameters assumptions. A good agreement with the spectrum shape was obtained by adding two Gaussian functions, with the constant experimental resolution for the $\Xi$-hypernuclear states, to the background distribution. The peak positions were obtained to be $B_{\Xi ^-} = 8.9 \pm 1.4$ (stat.) $^{+3.8}_{-3.1}$ (syst.) MeV and $B_{\Xi ^-} = -2.4 \pm 1.3$ (stat.) $^{+2.8}_{-1.2}$ (syst.) MeV. Another model assumption, one Breit–Wigner function with $B_{\Xi ^-} = -2.7 \pm 2.2$ (stat.) $^{+0.5}_{-0.7}$ (syst.) MeV and $\Gamma = 4.1 \pm 2.1$ (stat.) $^{+1.2}_{-0.7}$ (syst.) MeV, also yielded a similar $\chi ^2$ value.

Lorentz-violating extension of Wigner function formalism and chiral kinetic theory

The quantum kinetic equation for the gauge-invariant Wigner function, constructed from spinor fields that obey the Dirac equation modified by CPT and Lorentz symmetry-violating terms, is presented. The equations for the components of the Wigner function in the Clifford algebra basis are accomplished. Focusing on the massless case, an extended semiclassical chiral kinetic theory in the presence of external electromagnetic fields is developed. We calculate the chiral currents and establish the anomalous magnetic and separation effects in a Lorentz-violating background. The chiral anomaly within the context of extended quantum electrodynamics is elucidated. Finally, we derive the semiclassical Lorentz-violating extended chiral transport equation. Published by the American Physical Society 2024

Wigner Function Method for Describing Electromagnetic Field in Plasma-Like Media with Spatial Dispersion and Resonant Dissipation

The paper gives a systematic presentation of the Wigner function method (Weyl formalism) for modeling the propagation and absorption of electromagnetic waves in anisotropic and gyrotropic dissipative media with spatial dispersion. A general kinetic equation for the Wigner function (tensor) is formulated, and its asymptotic expansion up to the second order for smoothly inhomogeneous and weakly dissipative media is constructed. As a result, a modification of the method of the kinetic equation for rays is proposed, based on the stochastic description of rays, making it possible to increase the accuracy of numerical modeling of wave problems with strong transverse inhomogeneity of the absorption coefficient without increasing the amount of calculations. The technique developed can be used to describe the propagation, absorption, and scattering of electron-cyclotron waves in high-temperature plasma of magnetic traps for controlled fusion in cases where standard modeling methods do not provide the necessary accuracy.