Wigner Function Research Articles

Gazeau-Klauder coherent states in position-deformed Heisenberg algebra

In this paper, we present coherent states à la Gazeau-Klauder for a free particle in square well potential within position-deformed Heisenberg algebra . These states satisfy the Klauder’s mathematical requirement to build coherent states. Some statistical properties such as the probability distribution, the intensity correlation function and the Mandel parameter are calculated and analyzed. We find that these states are sub-Poissonian in nature. We also construct for these coherent states, the even cat states and we evaluate its Wigner function which analyses the quasiprobability distribution of these states. We graphically demonstrate that these states exhibit nonclassical behavior.

Visualizing the breakdown of quantum multimodality in coherently driven light-matter interaction

We show that the saturation of a multiphoton transition is accompanied by a gradual collapse of quantum multimodality in the strong-coupling limit of the weakly driven Jaynes-Cummings (JC) model. By means of a perturbative method, we illustrate the prominent role of quantum fluctuations by focusing on the two and three-photon resonance operation in a regime where the steady-state average photon number is below or marginally above unity. We also reveal two coexistent quantum beats in the intensity correlation function of the forwards scattered photons. These beats, originating from the states mediating the cascaded decay, arise as a direct consequence of the distinct JC spectrum. Their interference coordinates with the alternation of positive and negative values of the Wigner function around the phase-space origin in the transient conditioned on a photodetection.

Frequency-Dependent Squeezing from a Detuned Squeezer.

Frequency-dependent squeezing is a promising technique to overcome the standard quantum limit in optomechanical force measurements, e.g., gravitational wave detectors. For the first time, we show that frequency-dependent squeezing can be produced by detuning an optical parametric oscillator from resonance. Its frequency-dependent Wigner function is reconstructed quantum tomographically and exhibits a rotation by 39°, along which the noise is reduced by up to 5.5dB. Our setup is suitable for realizing effective negative-mass oscillators required for coherent quantum noise cancellation.

Multi-particle production in proton-nucleus collisions in the Color Glass Condensate

We compute multi-gluon production in the Color Glass Condensate approach in dilute-dense collisions, pA. We include the contributions that are leading in the overlap area of the collision but keep all orders in the expansion in the number of colors. We use a form of the Lipatov vertices that leads to the Wigner function approach for the projectile previously employed, that we generalise to take into account quantum correlations in the projectile wave function. We compute four gluon correlations and we find that the second order four particle cumulant is negative, so a sensible second Fourier azimuthal coefficient can be defined.

Entropy production in a longitudinally expanding Yang–Mills field with use of the Husimi function: semiclassical approximation

Abstract We investigate the possible thermalization process of the highly occupied and weakly coupled Yang–Mills fields expanding along the beam axis through an evaluation of the entropy, particle number, and pressure anisotropy. The time evolution of the system is calculated by solving the equation of motion for the Wigner function in the semiclassical approximation with initial conditions mimicking the glasma. For the evaluation of the entropy, we adopt Husimi–Wehrl (HW) entropy, which is obtained by using the Husimi function, a positive semidefinite quantum distribution function given by smearing the Wigner function. By numerical calculations at g = 0.1 and 0.2, the entropy production is found to occur together with the particle creation in two distinct stages: In the first stage, the particle number and entropy at low longitudinal momenta grow rapidly. In the second stage, the particle number and entropy of higher longitudinal momentum modes show a slower increase. The pressure anisotropy remains in our simulation and implies that the system is still out of equilibrium.

Generalization of the Fourier Calculus and Wigner Function

In this paper, we consider -periodical functions and, which are defined on the curve given by the equation: |x|^P+|y|^P=1, p>1 on as functions of its length. Considering and as an independent functional system, we construct the theory similar to Fourier analysis with the proper weights. For these weights, we establish an analogous of the Riemannian theorem. The adjoint representations are introduced and dual theory is developed. These Fourier representations can be used for approximation of the oscillation processes.

The discrete Chebyshev–Meckler–Mermin–Schwarz polynomials and spin algebra

The polynomials discovered by Chebyshev and subsequently related to spin transition probabilities by Meckler [Meckler, Phys. Rev. 111, 1447 (1958)] and Mermin and Schwarz [Found. Phys. 12, 101 (1982)] are studied, and their application to phase space representations of spin states and operators is examined. In particular, a formula relating the end-point value of the polynomials to scale factors relating different phase space representations of spherical harmonic operators is found. This formula is applied to illustrative calculations of Wigner functions for a single spin and the singlet state of a pair of spins.

Uncertainty principle via variational calculus on modulation spaces

We approach uncertainty principles of Cowling-Price-Heis-enberg-type as a variational principle on modulation spaces. In our discussion we are naturally led to compact localization operators with symbols in modulation spaces. The optimal constant in these uncertainty principles is the smallest eigenvalue of the inverse of a compact localization operator. The Euler-Lagrange equations for the associated functional provide equations for the eigenfunctions of the smallest eigenvalue of these compact localization operators. As a by-product of our proofs we derive a generalization to mixed-norm spaces of an inequality for Wigner and Ambiguity functions due do Lieb.

Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 XXZ model

We investigate the non-equilibrium dynamics of a one-dimensional spin-1/2 XXZ model at zero-temperature in the regime |\Delta|< 1|Δ|<1, initially prepared in a product state with two domain walls i.e, |\downarrow\dots\downarrow\uparrow\dots\uparrow\downarrow\dots\downarrow \rangle|↓…↓↑…↑↓…↓⟩. At early times, the two domain walls evolve independently and only after a calculable time a non-trivial interplay between the two emerges and results in the occurrence of a split Fermi sea. For \Delta=0Δ=0, we derive exact asymptotic results for the magnetization and the spin current by using a semi-classical Wigner function approach, and we exactly determine the spreading of entanglement entropy exploiting the recently developed tools of quantum fluctuating hydrodynamics. In the interacting case, we analytically solve the Generalized Hydrodynamics equation providing exact expressions for the conserved quantities. We display some numerical results for the entanglement entropy also in the interacting case and we propose a conjecture for its asymptotic value.

Nanomechanical cat states generated by a dc voltage-driven Cooper pair box qubit

We study a nanoelectromechanical system consisting of a Cooper pair box qubit performing nanomechanical vibrations between two bulk superconductors. We demonstrate that a bias voltage applied to the superconductors may generate states represented by entanglement between qubit states and quantum ’cat states’, i.e. a superposition of the coherent states of the nanomechanical oscillator. We characterize the formation and development of such states in terms of the corresponding Wigner function and entropy of entanglement. Also, we propose an experimentally feasible detection scheme for the effect, in which the average current that attains the specific features created by the entanglement is measured.

Quantum transport for electron quasiparticles interacting with a disordered binary alloy: A Wigner approach

Quantum transport is developed in the Wigner function representation for a Bloch electron quasiparticle interacting with a disordered binary alloy in the presence of a homogeneous electric field of arbitrary time dependence and amplitude. The electron quasiparticle is described by a single-band effective Hamiltonian, and the homogeneous electric field is treated in the vector potential gauge. The methodology for the quantum transport analysis proceeds by first transforming the Liouville equation to one in which the interaction Hamiltonian (the binary alloy Hamiltonian) appears quadratically. The basis states employed in evaluating the requisite matrix elements are the instantaneous eigenstates of the electron quasiparticle Hamiltonian in the presence of the electric field. The Wigner quantum transport equations are derived, and the binary alloy collision term is suitably ensemble averaged over the disordered binary alloy matrix elements. In addition, the general drift and diffusion terms are exactly obtained, resulting in the complete Wigner-Boltzmann equation for the binary alloy system. In approximating the collision term for the two separate cases of parabolic energy dispersion and the long-wavelength limit, it is found that the reduced Wigner-Boltzmann equation includes the manifestation of the intracollisional field effect and other quantum generalities. As a contrast to the actual random alloy treatment, attention is given to the canonical problem of quantum transport for a virtual crystal (VC). As an alternative to the random binary alloy scattering problem, the VC Hamiltonian adopted for this treatment is derived by ensemble averaging the random binary alloy Hamiltonian. The resulting Wigner transport equation for the VC case is descriptive of Bloch dynamics in a graded semiconductor alloy.

Deformation quantization for systems with second-class constraints in deformed fermionic phase space

In order to quantize systems involving second-class constraints, one should use Dirac bracket instead of Poisson bracket. Furthermore, one can specify a star product in which the term linear in [Formula: see text] is proportional to the Dirac bracket. In this way an oscillator system in a deformed fermionic phase space is analyzed and the corresponding energy level and Wigner functions are evaluated according to scheme of deformation quantization. We also study the entanglement entropy induced by the deformation of the fermionic phase space.

Quantum statistics of light emitted from a pillar microcavity

A quantum behavior of the light emitted by exciton polaritons excited in a pillar semiconductor microcavity with embedded quantum well is investigated. Considering the bare excitons and photon modes as coupled quantum oscillators allows for an accurate accounting of the nonlinear and dissipative effects. In particular, using the method of the quantum states representation in a quantum phase space via quasiprobability functions (namely, a P-function and a Wigner function), we study the impact of the laser and the exciton-photon detuning on the second order correlation function of the emitted photons. We determine the conditions under which the phenomena of bunching, giant bunching, and antibunching of the emitted light emerge. In particular, we predict the effect of a giant bunching for the case of a large exciton to photon population ratio. Within the domain of parameters supporting a bistability regime we demonstrate the effect of bunching of photons.

Quantum kinetic theory for spin transport of quarks with background chromo-electromagnetic fields

We derive the quantum kinetic equations for massive and massless quarks coupled with the background chromo-electromagnetic fields from the Wigner-function approach with the n expansion and effective power-counting scheme. For each case, one obtains coupled color-singlet and color-octet kinetic equations, which also involve the scalar and axial-vector components for the charge and spin transport. These kinetic equations delineate entangled evolution of the corresponding distribution functions decomposed in color space. At weak coupling, we derive the close form of the color-singlet kinetic equations for spin transport, which incorporates the diffusion term and the source term that triggers dynamical spin polarization led by correlation functions of color fields. Also, the non-dynamical source term is found in the axial Wigner function. The induced spin polarization and axial charge currents by these source terms are discussed under physical assumptions for color-field correlators in near-equilibrium quark gluon plasmas. In the constant-field limit, we further obtain non-vanishing axial Ward identities, from which we extract the pseudo-scalar condensate for massive quarks at finite temperature.

Realistic non-Gaussian-operation scheme in parity-detection-based Mach-Zehnder quantum interferometry

We theoretically analyze phase sensitivity using parity detection based Mach Zehnder interferometer (MZI) with the input states generated by performing non-Gaussian operations, viz., photon subtraction, photon addition, and photon catalysis on a two-mode squeezed vacuum (TMSV) state. Since these non-Gaussian operations are probabilistic, it is of utmost importance to take the success probability into account. To this end, we consider the realistic model of photon subtraction, addition, and catalysis and derive a single expression of the Wigner function for photon subtracted, added, and catalyzed TMSV state. The Wigner function is used to evaluate the lower bound on the phase sensitivity via quantum Cramer-Rao bound and parity detection based phase sensitivity in MZI. We identify the ranges of squeezing and transmissivity parameters where the non-Gaussian states provide better phase sensitivity than the TMSV state. On qualitatively taking the success probability into account, it turns out that the photon addition is the most advantageous non-Gaussian operation. We hope that the generalized Wigner function derived in this work will be useful in various quantum information protocols and state characterization.

Quantum properties of nonclassical states generated by an optomechanical system with catalytic quantum scissors

A scheme is proposed to investigate the non-classical states generated by a quantum scissors device (QSD) operating on the the cavity mode of an optomechanical system. When the catalytic QSD acts on the cavity mode of the optomechanical system, the resulting state contains only the vacuum, single-photon and two-photon states depending upon the coupling parameter of the optomechanical system as well as the transmission coefficients of beam splitters (BSs). Especially, the output state is just a class of multicomponent cat state truncations at time t = 2π by choosing the appropriate value of coupling parameter. We discuss the success probability of such a state and the fidelity between the output state and input state via QSD. Then the linear entropy is used to investigate the entanglement between the two subsystems, finding that QSD operation can enhance their entanglement degree. Furthermore, we also derive the analytical expression of the Wigner function (WF) for the cavity mode via QSD and numerically analyze the WF distribution in phase space at time t = 2π. These results show that the high non-classicality of output state can always be achieved by modulating the coupling parameter of the optomechanical system as well as the transmittance of BSs.

Photonic spin Hall effect from quantum kinetic theory in curved spacetime

Based on quantum field theory, we formulate the Wigner function and quantum kinetic theory for polarized photons in curved spacetimes which admit a covariantly constant timelike vector. From this framework, the photonic chiral/zilch vortical effects are reproduced in a rigidly rotating coordinate. In a spatially inhomogeneous coordinate, we derive the spin Hall effect for the photon helicity current and energy current in equilibrium. Our derivation reveals that such photonic Hall effect are related to the photonic vortical effect via the Lorentz invariance and their transport coefficients match each other.

Experimental Demonstration of Remotely Creating Wigner Negativity via Quantum Steering.

Non-Gaussian states with Wigner negativity are of particular interest in quantum technology due to their potential applications in quantum computing and quantum metrology. However, how to create such states at a remote location remains a challenge, which is important for efficiently distributing quantum resource between distant nodes in a network. Here, we experimentally prepare an optical non-Gaussian state with negative Wigner function at a remote node via local non-Gaussian operation and shared Gaussian entangled state existing quantum steering. By performing photon subtraction on one mode, Wigner negativity is created in the remote target mode. We show that the Wigner negativity is sensitive to loss on the target mode, but robust to loss on the mode performing photon subtraction. This experiment confirms the connection between the remotely created Wigner negativity and quantum steering. As an application, we present that the generated non-Gaussian state exhibits metrological power in quantum phase estimation.

Spectra of neutron wave functions in Earth’s gravitational field

Abstract The time evolution of a quantum wave packet in the linear gravity potential is known as Quantum Bouncing Ball. The qBounce collaboration recently observed such a system by dropping wave packets of ultracold neutrons by a height of roughly 30 microns. In this article, space and momentum spectra as well as Wigner functions of the neutron wave functions in the gravitational field of the Earth are analyzed. We investigate the quantum states in the “preparation region”, into which they transition after exiting a narrow double-mirror system and where we would expect to observe free fall and bounces in classical physics. For this, we start from the stationary solutions and eigenvalues of the Schrödinger equation in terms of Airy functions and their zeros. Subsequently, we examine space and momentum distributions as well as Wigner functions in phase space for pure and mixed quantum states. The possible influence of Yukawa-like forces for small distances of several micrometers from the mirror is included through first order perturbation calculations. Those allow us to study the resulting modifications of space and momentum distributions, and phase space functions.

X(3872) production in relativistic heavy-ion collisions

Heavy-ion collisions provide a unique opportunity to study the nature of X(3872) compared with electron-positron and proton-proton (antiproton) collisions. We investigate the centrality and momentum dependence of X(3872) in heavy-ion collisions via the Langevin equation and instant coalescence model (LICM). When X(3872) is treated as a compact tetraquark state, the tetraquarks are produced via the coalescence of heavy and light quarks near the quantum chromodynamic (QCD) phase transition due to the restoration of the heavy quark potential at $T\rightarrow T_c$. In the molecular scenario, loosely bound X(3872) is produced via the coalescence of $D^0$-$\bar D^{*0}$ mesons in a hadronic medium after kinetic freeze-out. We employ the LICM to explain both $D^0$ and $J/\psi$ production as a benchmark. Then we give predictions regarding X(3872) production and the nuclear modification factor $R_{AA}^{X(3872)}$. We find that the total yield of tetraquark is several times larger than the molecular production in Pb-Pb collisions. Although the geometric size of the molecule is huge, the coalescence probability is small due to strict constraints on the relative momentum between $D^0$ and $\bar D^{*0}$ in the molecular Wigner function, which significantly suppresses the molecular yield.