Wigner Quasi-distribution Function Research Articles

Quantum dynamics of a driven damped harmonic oscillator in Heisenberg picture: exact results and possible generalizations

In the framework of the Heisenberg picture, an alternative derivation of the reduced density matrix of a driven dissipative quantum harmonic oscillator as the prototype of an open quantum system is investigated. The reduced density matrix for different initial states of the combined system is obtained from a general formula, and different limiting cases are studied. Exact expressions for the corresponding characteristic function in quantum thermodynamics and Wigner quasi-distribution function are found. A possible generalization based on the Magnus expansion of the evolution operator is presented.

A WKB approach to the quantum multiband electron dynamics in the kinetic formalism

We derive a WKB-like asymptotic expansion of the multiband Wigner function. The model, derived in the envelope function theory, is designed to describe the dynamics in semiconductor devices when the interband conduction-valence transition cannot be neglected. We derive a hierarchy of equations that to lowest order consist of two HamiltonJacobi equations corresponding to the classical dynamics of point particles with positive and negative kinetic energy. Our methodology is based on the Van Vleck approach and a WKB-like asymptotic expansion procedure is used to reduce the numerical complexity of the Wigner multiband evolution system. An approximate closed-form solution is obtained by an iterative procedure that exploits the different time scales on which the intraband and interband dynamical variables evolve. The interband tunneling mechanism appearing to the first order of the expansion is expressed in a very simple mathematical form. By exploiting the highly oscillating behaviour of the multiband Wigner functions we derive a asymptotic expression of the interband transition probability. The resulting formulation reveals particularly close to the classical description of the particles motion and this formal analogy is useful to gain new physical insight and to profit of the numerical method developed for classical systems. The approximates evolution equations are used to simulate the evolution of the Wigner quasi-distribution function in a IRTD diode.

Quantum mechanics on phase space: The hydrogen atom and its Wigner functions

Symplectic quantum mechanics (SQM) considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space HΓ, to construct a unitary representation for the Galilei group. From this unitary representation the Schrödinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville–von Neumann equation. In this article the Coulomb potential in three dimensions (3D) is resolved completely by using the phase space Schrödinger equation. The Kustaanheimo–Stiefel(KS) transformation is applied and the Coulomb and harmonic oscillator potentials are connected. In this context we determine the energy levels, the amplitude of probability in phase space and correspondent Wigner quasi-distribution functions of the 3D-hydrogen atom described by Schrödinger equation in phase space.

Quantum mechanics on phase space and the Coulomb potential

Symplectic quantum mechanics (SMQ) makes possible to derive the Wigner function without the use of the Liouville–von Neumann equation. In this formulation of the quantum theory the Galilei Lie algebra is constructed using the Weyl (or star) product with Qˆ=q⋆=q+iħ2∂p,Pˆ=p⋆=p−iħ2∂q, and the Schrödinger equation is rewritten in phase space; in consequence physical applications involving the Coulomb potential present some specific difficulties. Within this context, in order to treat the Schrödinger equation in phase space, a procedure based on the Levi-Civita (or Bohlin) transformation is presented and applied to two-dimensional (2D) hydrogen atom. Amplitudes of probability in phase space and the correspondent Wigner quasi-distribution functions are derived and discussed.

Decoherence of quantum excitation of even/odd coherent states in thermal environment

In this paper, we study the decoherence of quantum excitation (photon-added) even /odd coherent states, $((\hat a{^{\dagger }})^{m} \left | {\alpha _ \pm } \right \rangle )$ , in a thermal environment by investigating the variation of negative part of the Wigner quasidistribution function vs. the rescaled time. For this purpose, at first we obtain the time-dependent Wigner function corresponding to the mentioned states in the framework of standard master equation. Then, the time evolution of the Wigner function associated with photon-added even /odd coherent states, as well as the number of added photons m are analysed. It is shown that, in both states, the negative part of the Wigner function decreases with time. By deriving the threshold value of the rescaled time for single photon-added even /odd coherent states, it is also found that, if the rescaled time exceeds the threshold value, the associated Wigner function becomes positive, i.e., the decoherence occurs completely.

Molecular descriptors and quasi-distribution functions

The concept of molecular quantum similarity is a rather subjective notion which depends on the definition of a molecular descriptor. Despite the vast set of options, today the main descriptor in use is the electron density usually provided by the time-independent Schrödinger equation. In this paper, an alternative definition of quantum similarity in the context of the Wigner formulation of quantum mechanics is suggested, based on a molecular descriptor defined in terms of quasi-distribution functions. We show that this definition has several important advantages over density based similarities. In particular, it offers a higher detailed description of a quantum system, thus refining the concept of similarity. To support our claim, we first present a theoretical example where two systems which appear to be similar, when using the density as a descriptor, become non-similar when their quasi-distributions are involved in the computation of the similarity. Moreover we simulate two quantum systems (utilizing the Wigner Monte Carlo method based on signed particles), i.e. a free Gaussian wave-packet and an H2 molecule, and show that Wigner quasi-distribution functions provide a different perspective on the concept of quantum similarity.

A signed particle formulation of non-relativistic quantum mechanics

A formulation of non-relativistic quantum mechanics in terms of Newtonian particles is presented in the shape of a set of three postulates. In this new theory, quantum systems are described by ensembles of signed particles which behave as field-less classical objects which carry a negative or positive sign and interact with an external potential by means of creation and annihilation events only. This approach is shown to be a generalization of the signed particle Wigner Monte Carlo method which reconstructs the time-dependent Wigner quasi-distribution function of a system and, therefore, the corresponding Schrödinger time-dependent wave-function. Its classical limit is discussed and a physical interpretation, based on experimental evidences coming from quantum tomography, is suggested. Moreover, in order to show the advantages brought by this novel formulation, a straightforward extension to relativistic effects is discussed. To conclude, quantum tunnelling numerical experiments are performed to show the validity of the suggested approach.

The decoherence of the quantum excitation of even/odd coherent states in a photon-loss channel

In this paper, the decoherence of the quantum excitation (the photon-added case) of even/odd coherent states, in a photon-loss channel is studied via the evaluation of the negative part of the Wigner quasi-distribution function in terms of the rescaled time (). We first obtain the explicit form of the time-dependent Wigner functions of the states considered, for arbitrary values of the photon excitations. To achieve this purpose, an equivalent Fokker–Planck-like equation which describes the time evolution of the Wigner function in the framework of the standard master equation of the density matrix is used. Then, by analyzing the time variation of the corresponding Wigner functions, the loss of the nonclassicality behavior and, hence, the decay of the negative part of the Wigner functions as a result of decoherence are clearly shown. In particular, we derive the threshold value of the rescaled time for single-photon-added even and odd coherent states; through this it is observed that the associated Wigner functions become positive whenever the rescaled time exceeds that value, i.e., the decoherence has become complete. By virtue of numerical calculations, it is also illustrated that the negative volume of the corresponding Wigner functions decreases with increasing time. In addition, the states with larger values of the excitation number m generally (but not absolutely always) possess smaller values of the rescaled threshold time.

Time evolution of a Gaussian class of quasi-distribution functions under quadratic Hamiltonian

A Lie algebraic method for propagation of the Wigner quasi-distribution function (QDF) under quadratic Hamiltonian was presented by Zoubi and Ben-Aryeh. We show that the same method can be used in order to propagate a rather general class of QDFs, which we call the "Gaussian class." This class contains as special cases the well-known Wigner, Husimi, Glauber, and Kirkwood-Rihaczek QDFs. We present some examples of the calculation of the time evolution of those functions.

The Wigner function of the relativistic finite-difference oscillator in an external field

The phase-space representation for a relativistic linear oscillator in a homogeneous external field expressed through the finite-difference equation is constructed. Explicit expressions of the relativistic oscillator Wigner quasi-distribution function for the stationary states as well as for the states of thermodynamical equilibrium are obtained and their correct limits are shown.

Quantum-engineered neutron states and phase space manipulations

In neutron interferometry and in spin-echo systems non-classical neutron quantum states can be produced and manipulated. Schrödinger cat-like states exhibit two spatially separated regions which are occupied by the neutron at the same time. When used in scattering experiments this enables a direct observation of the local structure and dynamics of matter. A proper description can be obtained by means of Wigner quasi-distribution functions which are common in quantum optics. More complicated quasi-distribution functions have been verified recently by means of a double loop interferometer. These quantum states have the capacity to observe even higher-order correlation functions characterizing condensed matter. The non-local character of quantum theory has also been demonstrated in the behavior of neutrons within narrow slits where typical confinement effects have been observed. Such Casimir-like and Zeno-like phenomena show how neutrons are an attractive tool for quantum optical experiments opening new possibilities of neutron phase space manipulations. A proper treatment of the neutron field can enhance the available neutron intensities considerably. First cooling to ultra-cold neutron energies by cold solid deuterium moderators increases the phase space density, which can be transferred to higher energies by a proper phase space transformer consisting of fast rotating multi-layer mirrors or crystals. Various proposals in this directions will be discussed.

Quantum-engineered neutron states and phase space manipulations

In neutron interferometry and in spin-echo systems non-classical neutron quantum states can be produced and manipulated. Schrödinger cat-like states exhibit two spatially separated regions which are occupied by the neutron at the same time. When used in scattering experiments this enables a direct observation of the local structure and dynamics of matter. A proper description can be obtained by means of Wigner quasi-distribution functions which are common in quantum optics. More complicated quasi-distribution functions have been verified recently by means of a double loop interferometer. These quantum states have the capacity to observe even higher-order correlation functions characterizing condensed matter. The non-local character of quantum theory has also been demonstrated in the behavior of neutrons within narrow slits where typical confinement effects have been observed. Such Casimir-like and Zeno-like phenomena show how neutrons are an attractive tool for quantum optical experiments opening new possibilities of neutron phase space manipulations. A proper treatment of the neutron field can enhance the available neutron intensities considerably. First cooling to ultra-cold neutron energies by cold solid deuterium moderators increases the phase space density, which can be transferred to higher energies by a proper phase space transformer consisting of fast rotating multi-layer mirrors or crystals. Various proposals in this directions will be discussed.

Quantum Kinetics in Classical Electromagnetic Fields by Means of Wigner Quasi-Distribution Function

Quantum analogs of Hamilton-Jacobi and Boltzmann kinetic equations are formulated for arbitrary classical electromagnetic fields in the gaugeinvariant form. Equations are applied to the problem of the electric current flow in a quasi-two-dimensional electron system. Strong deviations from the quasiclassical description are possible. A positive magnetoresistivity in a magnetic field parallel to the current is predicted.

The truncated Wigner method for Bose-condensed gases: limits of validity andapplications11Presented at the Theory of Quantum Gases and Quantum Coherence:The Salerno BEC Workshop, 3–5 June 2001. (A compilation of articles from thisworkshop was published in J. Phys. B: At. Mol. Opt. Phys. 34 (2001) issue23.)

We study the truncated Wigner method applied to a weakly interactingspinless Bose-condensed gas which is perturbed away from thermalequilibrium by a time-dependent external potential. The principleof the method is to generate an ensemble of classical fields ψ(r)which samples the Wigner quasi-distribution function of the initial thermalequilibrium density operator of the gas, and then to evolve each classicalfield with the Gross–Pitaevskii equation. In the first part of the paper weimprove the sampling technique over our previous work (Sinatra et al2000J. Mod. Opt. 47 2629–44) and we test its accuracy against the exactlysolvable model of the ideal Bose gas. In the second part of the paper weinvestigate the conditions of validity of the truncated Wigner method. Forshort evolution times it is known that the time-dependent Bogoliubovapproximation is valid for almost pure condensates. The requirement that thetruncated Wigner method reproduces the Bogoliubov prediction leads to theconstraint that the number of field modes in the Wigner simulation must besmaller than the number of particles in the gas. For longer evolution timesthe nonlinear dynamics of the noncondensed modes of the field plays animportant role. To demonstrate this we analyse the case of a three-dimensionalspatially homogeneous Bose-condensed gas and we test the ability of thetruncated Wigner method to correctly reproduce the Beliaev–Landau dampingof an excitation of the condensate. We have identified the mechanismwhich limits the validity of the truncated Wigner method: the initialensemble of classical fields, driven by the time-dependent Gross–Pitaevskiiequation, thermalizes to a classical field distribution at a temperature Tclasswhich is larger than the initial temperatureT of the quantum gas.When Tclasssignificantly exceeds Ta spurious damping is observed in the Wigner simulation. This leads to the secondvalidity condition for the truncated Wigner method, Tclass − T ≪ T,which requires that the maximum energy ϵmaxof the Bogoliubov modes in the simulation does not exceed a few kB T.

A Wigner Quasi-distribution Function for Charged Particles in Classical Electromagnetic Fields

A gauge-invariant Wigner quasi-distribution function for charged particles in classical electromagnetic fields is derived in a rigorous way. Its relation to the axial gauge is discussed, as well as the relation between the kinetic and canonical momenta in the Wigner representation. Gauge-invariant quantum analogs of Hamilton–Jacobi and Boltzmann kinetic equations are formulated for arbitrary classical electromagnetic fields in terms of the “slashed” derivatives and momenta, introduced for this purpose. The kinetic meaning of these slashed quantities is discussed. We introduce gauge-invariant conditional moments and use them to derive a kinetic momentum continuity equation. This equation provides us with a hydrodynamic representation for quantum transport processes and a definition of the “collision force.” The hydrodynamic equation is applied for the rotation part of the electron motion. The theory is illustrated by its application in three examples: Wigner quasi-distribution function and equations for an electron in a magnetic field and harmonic potential; Wigner quasi-distribution function for a charged particle in periodic systems using the kq representation; two Wigner quasi-distribution functions for heavy-mass polaron in an electric field.

Charged-particle-beam propagator in wave-electron optics: phase-space and tomographic pictures.

Within the framework of the thermal-wave model, the quantumlike description of electron optics in terms of the propagator is given. First we briefly review the standard description in configuration space by analogy to quantum mechanics and in connection with recent investigations of charged-particle-beam transport that have used the concept of propagator. Then new insights are given by extension of the analysis of the particle-beam propagator to the phase-space context for which our system is described by the Wigner quasi-distribution function, as well as to the tomography context for which our system is described by the marginal distribution. Furthermore, the integrals of motion of a charged-particle beam and their relation to the propagator concept are discussed. Finally, the perturbation theory for a charged-particle-beam propagator is developed in the above-described two contexts and is applied to some simple optical devices.

Decoherence in neutron interferometry

The coherence properties of a neutron are analyzed by making use of the Wigner quasi-distribution function. We discuss, in particular, highly non-classical, Schrödinger-cat-like neutron states that can be obtained in an interferometer or in a magnetic field. The dephasing and decoherence effects are quantitatively defined by introducing a “decoherence parameter”, that enables one to emphasize some peculiar aspects of irreversibility and decoherence in neutron scattering.