Wigner Function Research Articles

Complex scaling method applied to the study of the Swanson Hamiltonian in the broken PT-symmetry phase

In this work, we study the non-PT symmetry phase of the Swanson Hamiltonian in the framework of the Complex Scaling Method. By constructing a bi-orthogonality relation, we apply the formalism of the response function to analyse the time evolution of different initial wave packages. The Wigner Functions, mean value of operators, and the probabilities of survival and persistence for the different wave packages are evaluated as a function of time. We analyse in detail the time evolution in the neighbourhood of Exceptional Points. We derive a continuity equation for the system. We compare the results obtained using the Complex Scaling Method to the ones obtained by working in a Rigged Hilbert Space.

Quadrature Squeezing Enhances Wigner Negativity in a Mechanical Duffing Oscillator

Generating macroscopic nonclassical quantum states is a long-standing challenge in physics. Anharmonic dynamics is an essential ingredient to generate these states, but for large mechanical systems, the effect of the anharmonicity tends to become negligible compared with the effect of decoherence. As a possible solution to this challenge, we propose using a motional squeezed state as a resource to effectively increase the anharmonicity. We analyze the production of negativity in the Wigner distribution of a quantum anharmonic resonator initially in a squeezed state. We find that initial squeezing increases the rate at which negativity is generated. We also analyze the effect of two common sources of decoherence—namely, energy damping and dephasing—and find that the detrimental effects of energy damping are suppressed by strong squeezing. In the limit of large squeezing, which is needed for state-of-the-art systems, we find good approximations for the Wigner function. Our analysis is significant for current experiments attempting to prepare macroscopic mechanical systems in genuine quantum states. We provide an overview of several experimental platforms featuring nonlinear behaviors and low levels of decoherence. In particular, we discuss the feasibility of our proposal with carbon nanotubes and levitated nanoparticles. Published by the American Physical Society 2024

Critical behaviors of non-stabilizerness in quantum spin chains

Non-stabilizerness – commonly known as magic – measures the extent to which a quantum state deviates from stabilizer states and is a fundamental resource for achieving universal quantum computation. In this work, we investigate the behavior of non-stabilizerness around criticality in quantum spin chains. To quantify non-stabilizerness, we employ a monotone called mana, based on the negativity of the discrete Wigner function. This measure captures non-stabilizerness for both pure and mixed states. We introduce Rényi generalizations of mana, which are also measures of non-stabilizerness for pure states, and utilize it to compute mana in large quantum systems. We consider the three-state Potts model and its non-integrable extension and we provide strong evidence that the mutual mana exhibits universal logarithmic scaling with distance in conformal field theory, as is the case for entanglement.

Reconstruction of the Wigner function of electron beams based on coherence measurements

Reconstruction of the Wigner function of electron beams based on coherence measurements

Light and hyper nuclei formation at [formula omitted] 3 GeV Au+Au collisions using Wigner coalescence approach

The production of light nuclei and hyper-nuclei in heavy-ion collisions, particularly at high baryon density, is crucial for understanding the dynamical evolution of the collision system and exploring the internal state of nuclear matter of compacted stellar object. Despite being a topic of ongoing debate, an improved theoretical understanding is necessary. In this work, production of light nuclei (d, t, 3He, 4He) and hyper-nuclei (Λ3H, Λ4H) was investigated using the JAM microscopic transport model combined with an afterburner coalescence process at sNN= 3 GeV Au+Au collisions. The formation of a specific nucleus during the coalescence process is determined by its Wigner function. The comparison of the calculations for pT spectra, average pT, and rapidity distributions to the measurements from the STAR experiment was performed. We investigated the dynamic information carried by light nuclei and determined the averaged spatial distance 〈ΔR〉 and momentum difference 〈ΔP〉 of constituent nucleons (Λ) for each nucleus species.

Mode-selective photon-added two-mode squeezed vacuum state and its entanglement properties

A class of non-Gaussian entangled states is introduced by applying the nonlocal single-photon addition to two-mode squeezed vacuum state and the properties of entanglement are numerically investigated according to linear entropy and Einstein–Podolsky–Rosen (EPR) steering. After the nonlocal single-photon addition operation, Wigner function of the generated state appears some negative region and loses its Gaussian property in phase space. In essence, non-Gaussian entangled states are generated after applying nonlocal single-photon addition. Additionally, by studying the linear entropy and EPR steering, we find that single-photon addition can enhance their entanglement degree and non-Gaussian steering can be witnessed in a large range of squeezing parameter for the second-order quadratures, which may provide a well application in the fields of quantum information processing.

Wigner Analysis of Particle Dynamics and Decoherence in Wide Nonharmonic Potentials

We derive an analytical expression of a Wigner function that approximately describes the time evolution of the one-dimensional motion of a particle in a nonharmonic potential. Our method involves two exact frame transformations, accounting for both the classical dynamics of the centroid of the initial state and the rotation and squeezing about that trajectory. Subsequently, we employ two crucial approximations, namely the constant-angle and linearized-decoherence approximations, upon which our results rely. These approximations are effective in the regime of wide potentials and small fluctuations, namely potentials that enable spatial expansions orders of magnitude larger than the one of the initial state but that remain smaller compared to the relevant dynamical length scale (e.g., the distance between turning points). Our analytical result elucidates the interplay between classical and quantum physics and the impact of decoherence during nonlinear dynamics. This analytical result is instrumental to designing, optimizing, and understanding proposals using nonlinear dynamics to generate macroscopic quantum states of massive particles.

Navigating the phase diagram of quantum many-body systems in phase space.

We demonstrate the unique capabilities of the Wigner function, particularly in its positive and negative parts, for exploring the phase diagram of the spin -(1/2-1/2) and spin-(1/2-1) Ising-Heisenberg chains. We highlight the advantages and limitations of the phase-space approach in comparison with the entanglement concurrence in detecting phase boundaries. We establish that the equal angle slice approximation in the phase space is an effective method for capturing the essential features of the phase diagram but falls short in accurately assessing the negativity of the Wigner function for the homogeneous spin-(1/2-1/2) Ising-Heisenberg chain. In contrast, we find for the inhomogeneous spin-(1/2-1) chain that an integral over the entire phase space is necessary to accurately capture the phase diagram of the system. This distinction underscores the sensitivity of phase-space methods to the homogeneity of the quantum system under consideration.

Quantum squeezing in a nonlinear mechanical oscillator

Mechanical degrees of freedom are natural candidates for continuous-variable quantum information processing and bosonic quantum simulations. However, these applications require the engineering of squeezing and nonlinearities in the quantum regime. Here we demonstrate squeezing below the zero-point fluctuations of a gigahertz-frequency mechanical resonator coupled to a superconducting qubit. This is achieved by parametrically driving the qubit, which results in an effective two-phonon drive. In addition, we show that the resonator mode inherits a nonlinearity from the off-resonant coupling with the qubit, which can be tuned by controlling the detuning. We, thus, realize a mechanical squeezed Kerr oscillator, in which we demonstrate the preparation of non-Gaussian quantum states of motion with Wigner function negativities and high quantum Fisher information. This shows that our results can also have applications in quantum metrology and sensing.

Effective algorithms for calculation of quasi-probability distributions of bright “banana” states

Non-Gaussian quantum states, described by negative-valued Wigner functions, are important for both fundamental tests of quantum physics and for emerging quantum information technologies. One of the promising ways of generating a non-Gaussian state from a coherent one is the use of cubic (Kerr) optical nonlinearity, which produces the characteristic banana-like shape of the resulting quantum states. However, the Kerr effect is weak in highly transparent optical materials (dimensionless nonlinearity parameter Γ≲10−6). Therefore, a big number of the photons in the optical mode (n≳106) is necessary to generate an observable non-Gaussianity. In this case, the direct approach to calculation of the Wigner function becomes extremely computationally expensive. In this work, we develop quick algorithms for computing the Husimi and Wigner quasi-probability functions of these non-Gaussian states by means of the Kerr nonlinearity. This algorithm can be used for any realistic values of the photon numbers and the nonlinearity.

Quantum versus classical quenches and the broadening of wave packets

The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.

Phase space geometry of general quantum energy transitions

The mixed density operator for coarsegrained eigenlevels of a static Hamiltonian is represented in phase space by the spectral Wigner function, which has its peak on the corresponding classical energy shell. The action of trajectory segments along the shell determine the phase of the Wigner oscillations in its interior. The classical transitions between any pair of energy shells, driven by a general external time dependent Hamiltonian, also have a smooth probability density. It is shown here that a further contribution to the transition between the corresponding pair of coarsegrained energy levels, which oscillates with either energy, or the driving time, is determined by four trajectory segments (two in the pair of energy shells and two generated by the driving Hamiltonian) that join exactly to form a closed compound orbit (CCO). In its turn, this sequence of segments belongs to the semiclassical expression of a compound unitary operator that combines four quantum evolutions: a pair generated by the static internal Hamiltonian and a pair generated by the driving Hamiltonian. The CCOs are shown to belong to continuous families, which are initially seeded at points where the classical flow generated by both Hamiltonians commute.

Gouy phase and quantum interference with cross-Wigner functions for matter-waves

The Gouy phase is essential for accurately describing various wave phenomena, ranging from classical electromagnetic waves to matter waves and quantum optics. In this work, we employ phase-space methods based on the cross-Wigner transformation to analyze spatial and temporal interference in the evolution of matter waves characterized initially by a correlated Gaussian wave packet. First, we consider the cross-Wigner of the initial wave function with its free evolution, and second for the evolution through a double-slit arrangement. Different from the wave function which acquires a global Gouy phase, we find that the cross-Wigner acquires a Gouy phase difference due to different evolution times. The results suggest that temporal like-Gouy phase difference is important for an accurate description of temporal interference. Furthermore, we propose a technique based on the Wigner function to reconstruct the cross-Wigner from the spatial intensity interference term in a double-slit experiment with matter waves.

Complexity growth and the Krylov-Wigner function

For any state in a D-dimensional Hilbert space with a choice of basis, one can define a discrete version of the Wigner function — a quasi-probability distribution which represents the state on a discrete phase space. The Wigner function can, in general, take on negative values, and the amount of negativity in the Wigner function has an operational meaning as a resource for quantum computation. In this note, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. We introduce the Krylov-Wigner function, i.e., the Wigner function defined with respect to the Krylov basis (with appropriate phases), and show that this choice of basis minimizes the early time growth of Wigner negativity in the large D limit. We take this as evidence that the Krylov basis (with appropriate phases) is ideally suited for a dual, semi-classical description of chaotic quantum dynamics at large D. We also numerically study the time evolution of the Krylov-Wigner function and its negativity in random matrix theory for an initial pure state. We observe that the negativity broadly shows three phases: it rises gradually for a time of OD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ O\\left(\\sqrt{D}\\right) $$\\end{document}, then hits a sharp ramp and finally saturates close to its upper bound of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{D} $$\\end{document}.

Complex-valued Wigner entropy of a quantum state

It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call the complex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.

Twisted splitting and propagation factor of superimposed twisted Hermite-Gaussian Schell-model beams in turbulent atmosphere

We believe this to be a new superposition twisted Hermite-Gaussian Schell-model (STHGSM) beam hat is proposed. Analytic formulas for the intensity distribution and propagation factor of the STHGSM beam in non-Kolmogorov turbulence are derived by utilizing the generalized Huygens-Fresnel principle (HFP) and the Wigner function. The evolution characteristics of STHGSM beams propagating are numerically calculated and analyzed. Our findings indicate that the light intensity of the STHGSM beam gradually undergoes splitting and rotation around the axis during propagation through non-Kolmogorov turbulence, eventually evolving into a diagonal lobe shape at a certain distance of transmission. The anti-turbulence capability of the beam strengthens with higher beam order or twist factor values.

Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems

Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space D=dn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=d^n$$\\end{document} with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space D=d0...dn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=d_0...d_{n-1}$$\\end{document} with d0,...,dn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0,...,d_{n-1}$$\\end{document} odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from O(D2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(D^2)$$\\end{document} to O(DlogD)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(D\\log D)$$\\end{document}. The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.

Photon quantum kinetic equations and collective modes in an axion background

We develop a quantum kinetic theory for photons in the presence of an axion background and in the collisioness limit. In deriving the classical regime of our quantum kinetic equations, we observe that they capture well-known features of axion electrodynamics. By projecting the Wigner function onto a polarization basis, relating the Wigner matrix function with the Stokes parameters, we establish the dispersion relations and transport equations for each polarization space component. Additionally, we investigate how the axion background affects the dispersion relations of photon collective modes within an electron-positron plasma at equilibrium temperature T. While the plasmon remains unaffected, we find that the axion background breaks the degeneracy of transverse collective modes at order egaγT(∂a), where e represents the electron charge, gaγ denotes the photon-axion coupling, and ∂a represents the scale associated with variations in the axion field. Published by the American Physical Society 2024

Noise‐Tailored Constructions for Spin Wigner Function Kernels

AbstractThe effective use of noisy intermediate‐scale quantum devices requires error mitigation to improve the accuracy of sampled measurement distributions. The more accurately the effects of noise on these distributions can be modeled, the more closely error mitigation will be able to approach theoretical bounds. The characterization of noisy quantum channels and the inference of their effects on general observables are challenging problems, but in many cases a change in representation can greatly simplify the analysis. Here, spin Wigner functions for multiqudit systems are investigated. This study generalizes previous kernel constructions, capturing the effects of several probabilistic unitary noise models in few parameters.

Searching for enhancement in coalescence of in-jet (anti-)deuterons in proton-proton collisions

Recent measurements from ALICE report that “in-jet” nucleons carry a higher probability of forming a deuteron via coalescence than the nucleons from the underlying event (UE). This study makes use of an event shape classifier to separate the “in-jet” deuterons and the deuterons in the UE produced in high multiplicity proton-proton collisions at s=13 TeV. Event shape variables such as transverse spherocity allow the categorization of hard and soft components of an event, which can be divided into two respective classes; “jetty” and “isotropic”. The “jetty” deuterons minus the contribution of the deuterons from the “isotropic” event are taken as “in-jet” deuterons, and the coalescence mechanism is tested. The coalescence is performed with a Wigner function formalism, augmented as an afterburner to . The possible enhancement of the coalescence probability of “in-jet” deuterons is investigated by calculating the coalescence parameter (B2) in different spherocity classes in high-multiplicity pp collisions. Published by the American Physical Society 2024