Time dependence for barrier penetration is considered in the phase space. An asymptotic phase-space propagator for nonrelativistic scattering on a one - dimensional barrier is constructed. The propagator has a form universal for various initial state preparations and local potential barriers. It is manifestly causal and includes time-lag effects and quantum spreading. Specific features of quantum dynamics which disappear in the standard semi-classical approximation are revealed. The propagator may be applied to calculation of the final momentum and coordinate distributions, for particles transmitted through or reflected from the potential barrier, as well as for elucidating the tunneling time problem.
We investigate quantum tunneling in smooth symmetric and asymmetric double-well potentials. Exact solutions for the ground and first excited states are used to study the dynamics. We introduce Wigner's quasi-probability distribution function to highlight and visualize the non-classical nature of spatial correlations arising in tunneling.
We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients $R$ or $T$ are then given by the total weight of all classical phase space trajectories corresponding to energies below, or above the top of the barrier given by the Wigner function.
Discrete quantum phase space formalism is used to discuss some basic aspects of the spin tunneling occurring in Fe8 magnetic cluster by means of Wigner functions as well as Husimi distributions. Those functions were obtained for sharp angle states and symmetric combinations of the lowest energy doublet with the application of an external magnetic field. The time evolution of those functions carried out numerically allows one to extract valuable information about the dynamics of the states under consideration, in particular, the coherent oscillations associated with the spin tunneling. It is al…
In this work we revisit the Salecker-Wigner-Peres clock formalism and show that it can be directly applied to the phenomenon of tunneling. Then we apply this formalism to the determination of the tunneling time of a non relativistic wavepacket, sharply concentrated around a tunneling energy, incident on a symmetric double barrier potential. In order to deepen the discussion about the generalized Hartmann effect, we consider the case in which the clock runs only when the particle can be found inside the region \emph{between} the barriers and show that, whenever the probability to find the parti…
The Salecker-Wigner-Peres quantum-clock approach is applied in order to determine the tunneling time of an electron in strong-field tunnel ionization via a time-dependent electric field. Our results show that the ionization of the electron takes a nonvanishing period of time. This tunneling time is of the order of the Keldysh time but strictly larger than the Keldysh time. Comparing the quantum-clock tunneling time to the mean tunneling time as obtained by the virtual-detector approach, one finds that these two complementary methods give very similar results. Due to the asymmetric distribution…
Asymptotic time evolution of a wave packet describing a non-relativistic particle incident on a potential barrier is considered, using the Wigner phase-space distribution. The distortion of the trasmitted wave packet is determined by two time-like parameters, given by the energy derivative of the complex transmission amplitude. The result is consistent with various definitions of the tunneling time (e.g. the Büttiker-Landauer time, the complex time and Wigner's phase time). The speed-up effect and the negative dispersion are discussed, and new experimental implications are considered.
The quantum clock of Salecker-Wigner-Peres is used, by performing a post-selection of the final state, to obtain average transmission and reflection times associated to the scattering of localized wave packets by static potentials in one dimension. The behavior of these average times is studied for a gaussian wave packet, centered around a tunneling wave number, incident on a rectangular barrier and, in particular, on a double delta barrier potential. The regime of opaque barriers is investigated and the results show that the average transmission time does not saturate, showing no evidence of …
A trapped-ion quantum tunneling rotor (QTR) is in a quantum superposition of two different Wigner crystal orientations. In a QTR system, quantum tunneling drives the coherent transition between the two different Wigner crystal orientations. We theoretically study the quantum dynamics of a QTR, particularly when the spin state of one of the ions is flipped. We show that the quantum dynamics of an $\it{N}$-ion QTR can be described by continuous-time cyclic quantum walks. We also investigate the quantum dynamics of the QTR in a magnetic field. Flipping the spin state breaks the rotational symmetr…
An analogy between Wigner resonant tunneling and tunneling across a static potential barrier in a static magnetic field is found. Whereas in the process of Wigner tunneling an electron encounters a classically allowed regions, where a discrete energy level coincides with its energy, in the magnetic field a potential barrier is a constant in the direction of tunneling. Along the tunneling path the certain regions are formed, where, in the classical language, the kinetic energy of the motion perpendicular to tunneling is negative. These regions play a role of potential wells, where a discrete en…
Tunneling of a particle through a potential barrier remains one of the most remarkable quantum phenomena. Owing to advances in laser technology, electric fields comparable to those electrons experience in atoms are readily generated and open opportunities to dynamically investigate the process of electron tunneling through the potential barrier formed by the superposition of both laser and atomic fields. Attosecond-time and angstrom-space resolution of the strong laser-field technique allow to address fundamental questions related to tunneling, which are still open and debated: Which time is s…
I propose to consider photon tunneling as a space-time correlation phenomenon between the emission and absorption of a photon on the two sides of a barrier. Standard technics based on an appropriate counting rate formula may then be applied to derive the tunneling time distribution without any {\em ad hoc} definition of this quantity. General formulae are worked out for a potential model using Wigner-Weisskopf method. For a homogeneous square barrier in the limit of zero tunneling probability a vanishing tunneling time is obtained.
The tunneling dynamics in relativistic strong-field ionization is investigated with the aim to develop an intuitive picture for the relativistic tunneling regime. We demonstrate that the tunneling picture applies also in the relativistic regime by introducing position dependent energy levels. The quantum dynamics in the classically forbidden region features two time scales, the typical time that characterizes the probability density's decay of the ionizing electron under the barrier (Keldysh time) and the time interval which the electron spends inside the barrier (Eisenbud-Wigner-Smith tunneli…
Recent attoclock experiments using the attsecond angular streaking technique enabled the measurement of the tunneling time delay during laser induced strong field ionization. Theoretically the tunneling time delay is commonly modelled by the Wigner time delay concept which is derived from the derivative of the electron wave function phase with respect to energy. Here, we present an alternative method for the calculation of the Wigner time delay by using the fixed energy propagator. The developed formalism is applied to the nonrelativistic as well as to the relativistic regime of the tunnel-ion…
We discuss a family of quasi-distributions (s-ordered Wigner functions of Agarwal and Wolf) and its connection to the so called phase space representation of the Schroedinger equation. It turns out that although Wigner functions satisfy the Schroedinger equation in phase space they have completely different interpretation.
For a circularly polarized single-color field at a central frequency of $2ω$ the final electron momentum distribution upon strong field ionization does not carry any information about the phase of the initial momentum distribution. Adding a weak, co-rotating, circularly polarized field at a central frequency of $ω$ gives rise to a sub-cycle interference pattern (holographic angular streaking of electrons (HASE)). This interference pattern allows for the retrieval of the derivative of the phase of the initial momentum distribution after tunneling $ϕ^{\prime}_{\mathrm{off}}(p_i)$. A trajectory-b…
We propose a methodology to design Wigner representations in phase spaces with nontrivial topology having evolution equations with desired mathematical properties. As an illustration, two representations of molecular rotations are developed to facilitate the analysis of molecular alignment in moderately intense laser fields, reaction dynamics, scattering phenomena and dissipative processes.
We demonstrate that the Wigner function of the Einstein-Podolsky-Rosen state, though positive definite, provides a direct evidence of the nonlocal character of this state. The proof is based on an observation that the Wigner function describes correlations in the joint measurement of the phase space displaced parity operator.
Evaluating the Wigner function of quantum states in the entangled state representation is introduced, based on which we present a new approach for deriving time evolution formula of Wigner function in laser process. Application of this fomula to calculating time evolution of photon number is also presented, as an example, the case when the initial state is photon-added coherent state is discussed.
We have found an effective method of calculating the Wigner function, being a quantum analogue of joint probability distribution of position and momentum, for bound states of nonrelativistic hydrogen atom. The formal similarity between the eigenfunctions of nonrelativistic hydrogen atom in the momentum representation and Klein-Gordon propagators has allowed the calculation of the Wigner function for an arbitrary bound state of the hydrogen atom. These Wigner functions for some low lying states are depicted and discussed.
By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of Wigner function (W). A non-integral representation is presented in terms of a quadratic form V*FV, where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending in a simple recursive way on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of Wigner function and the spacial uncertainty of the coherent state basis we us…
Phase-space representations based on coherent states (P, Q, Wigner) have been successful in the creation of stochastic differential equations (SDEs) for the efficient stochastic simulation of high dimensional quantum systems. However many problems using these techniques remain intractable over long integrations times. We present a number-phase Wigner representation that can be unraveled into SDEs. We demonstrate convergence to the correct solution for an anharmonic oscillator with small dampening for significantly longer than other phase space representations. This process requires an effectiv…
We propose an approach which allows to construct and use a potential function written in terms of an angle variable to describe interacting spin systems. We show how this can be implemented in the Lipkin-Meshkov-Glick, here considered a paradigmatic spin model. It is shown how some features of the energy gap can be interpreted in terms of a spin tunneling. A discrete Wigner function is constructed for a symmetric combination of two states of the model and its time evolution is obtained. The physical information extracted from that function reinforces our description of phase oscillations in a …
By introducing a new entangled state representation, we show that the Laguerre-Gaussian (LG) mode is just the wave function of the common eigenvector of the orbital angular momentum and the total photon number operators of 2-d oscillator, which can be generated by 50:50 beam splitter with the phase difference phi=Pi/2ϕ between the reflected and transmitted fields. Based on this and using the Weyl ordering invariance under similar transforms, the Wigner representation of LG is directly obtained, which can be considered as the generalized Wigner transform of Hermite Gaussian modes.
We show how to represent the state and the evolution of a quantum computer (or any system with an $N$--dimensional Hilbert space) in phase space. For this purpose we use a discrete version of the Wigner function which, for arbitrary $N$, is defined in a phase space grid of $2N\times 2N$ points. We compute such Wigner function for states which are relevant for quantum computation. Finally, we discuss properties of quantum algorithms in phase space and present the phase space representation of Grover's quantum search algorithm.
Based on the correspondence between Collins diffraction formula (optical Fresnel transform) and the transformation matrix element of a three-parameters two-mode squeezing operator in the entangled state representation (Opt. Lett. 31 (2006) 2622) we further explore the relationship between output field intensity determined by the Collins formula and the input field's probability distribution along an infinitely thin phase space strip both in spacial domain and frequency domain. The entangled Wigner function is introduced for recapitulating the result.
We present a phase space description of the process of quantum teleportation for a system with an $N$ dimensional space of states. For this purpose we define a discrete Wigner function which is a minor variation of previously existing ones. This function is useful to represent composite quantum system in phase space and to analyze situations where entanglement between subsystems is relevant (dimensionality of the space of states of each subsystem is arbitrary). We also describe how a direct tomographic measurement of this Wigner function can be performed.
Phase-space representations as given by Wigner functions are a powerful tool for representing the quantum state and characterizing its time evolution in the case of infinite-dimensional quantum systems and have been widely used in quantum optics and beyond. Continuous phase spaces have also been studied for finite-dimensional quantum systems such as spin systems. However, much less is known for finite-dimensional, coupled systems, and we present a complete theory of Wigner functions for this case. In particular, we provide a self-contained Wigner formalism for describing and predicting the tim…
We construct an explicit Wigner function for N-mode squeezed state. Based on a previous observation that the Wigner function describes correlations in the joint measurement of the phase-space displaced parity operator, we investigate the non-locality of multipartite entangled state by the violation of Zukowski-Brukner N-qubit Bell inequality. We find that quantum predictions for such squeezed state violate these inequalities by an amount that grows with the number N.
We calculate the tunneling process of a Dirac particle across two square barriers separated a distance $d$, as well as the scattering by a double cusp barrier where the centers of the cusps are separated a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that, the presence of transmission resonances modifies the Lorentizian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range …
We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current~$\bm J$ as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. Our central result is an explicit, elementary proof which shows that only systems up to quadratic in their pote…
We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically generalised to accommodate new features which arise when the configuration space changes from $n$-dimensional Euclidean space ${\cal R}^n$ to a Lie group $G$. The notion of canonical momentum is carefully analysed, and the meanings of marginal probability distributions and their recovery from the Wigner distribution are clarified. For the case of compact $G$ an explic…
By introducing the thermo entangled state representation, we convert the calculation of Wigner function (WF) of density operator to an overlap between "two pure" states in a two-mode enlarged Fock space. Furthermore, we derive a new WF evolution formula of any initial state in self-Kerr Medium with photon loss and find that the photon number distribution for any initial state is independent of the coupling factor with Kerr Medium, where the number state is not affected by the Kerr nonlinearity and evolves into a density operator of binomial distribution.
Macroscopic quantum tunneling is described using the master equation for the reduced Wigner function of an open quantum system at zero temperature. Our model consists of a particle trapped in a cubic potential interacting with an environment characterized by dissipative and normal and anomalous diffusion coefficients. A representation based on the energy eigenfunctions of the isolated system, i.e. the system uncoupled to the environment, is used to write the reduced Wigner function, and the master equation becomes simpler in that representation. The energy eigenfunctions computed in a WKB appr…
The quantum dissipative dynamics of a tunneling process through double barrier structures is investigated on the basis of a rigorous treatment for the first time. We employ a Caldeira-Leggett Hamiltonian with an effective potential calculated self-consistently, accounting for the electron distribution. With this Hamiltonian, we use the reduced hierarchy equations of motion in the Wigner space representation to study the effects of non-Markovian thermal fluctuations and dissipation at finite temperature in a rigorous manner. Hysteresis, double plateau-like behavior, and self-excited current osc…
The Wigner representation of parametric down conversion in the Heisenberg picture is applied to the study of the Rome teleportation experiment. We investigate the physical meaning of the zeropoint inputs at the different areas of the experimental setup. In particular, we establish a quantitative relationship between the zeropoint sets of modes that are needed for the preparation of the quantum state to be teleported, the idle channels inside the one-photon polarization-momentum Bell-state analyser, and the possibility of performing teleportation of a polarization state whith certainty.
A finite Hilbert space can be associated to a periodic phase space, that is, a torus. A finite subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure …
We propose a general expression for the probability distribution of real-valued tunneling times of a localized particle, as measured by the Salecker-Wigner-Peres quantum clock. This general expression is used to obtain the distribution of times for the scattering of a particle through a static rectangular barrier and for the tunneling decay of an initially bound state after the sudden deformation of the potential, the latter case being relevant to understand tunneling times in recent attosecond experiments involving strong field ionization.
The relation of the Wigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner--Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of…
The electron dynamics in the classically forbidden region during relativistic tunnel-ionization is investigated. The classical forbidden region in the relativistic regime is identified by defining a gauge invariant total energy operator. Introducing position dependent energy levels inside the tunneling barrier, we demonstrate that the relativistic tunnel-ionization can be well described by a one-dimensional intuitive picture. This picture predicts that, in contrast to the well-known nonrelativistic regime, the ionized electron wave packet arises with a momentum shift along the laser's propagat…
Simulation of conditional master equations is important to describe systems under continuous measurement and for the design of control strategies in quantum systems. For large bosonic systems, such as BEC and atom lasers, full quantum field simulations must rely on scalable stochastic methods whose convergence time is restricted by the use of representations based on coherent states. Here we show that typical measurements on atom-optical systems have a common form that allows for an efficient simulation using the number-phase Wigner (NPW) phase-space representation. We demonstrate that a stoch…
We present an operational definition of the Wigner function. Our method relies on the Fresnel transform of measured Rabi oscillations and applies to motional states of trapped atoms as well as to field states in cavities. We illustrate this technique using data from recent experiments in ion traps [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)] and in cavity QED [B. Varcoe et al., Nature 403, 743 (2000)]. The values of the Wigner functions of the underlying states at the origin of phase space are W(0)=+1.75 for the vibrational ground state and W(0)=-1.4 for the one-photon number state…
We study numerically the coordinate wave functions and the Wigner functions of the coherent phase states (CPS), paying the main attention to their differences from the standard (Klauder--Glauber--Sudarshan) coherent states, especially in the case of high mean values of the number operator. In this case, the CPS can possess a strong coordinate (or momentum) squeezing, which is, roughly, twice weaker than for the vacuum squeezed states. The Robertson--Schrödinger invariant uncertainty product in the CPS logarithmically increases with the mean value of the number operator (whereas it is constant …
We provide a scheme for efficient simulation of a broad class of quantum optics experiments. Our efficient simulation extends the continuous variable Gottesman-Knill theorem to a large class of non-Gaussian mixed states, thereby identifying that these non-Gaussian states are not an enabling resource for exponential quantum speed-up. Our results also provide an operationally motivated interpretation of negativity as non-classicality. We apply our scheme to the case of noisy single-photon-added-thermal-states to show that this class admits states with positive Wigner function but negative P -fun…
We present the quantum theory of coherent Ising machines based on networks of degenerate optical parametric oscillators (DOPOs). In a simple model consisting of two coupled DOPOs, both positive-$P$ representation and truncated Wigner representation predict quantum correlation and inseparability between the two DOPOs in spite of the open-dissipative nature of the system. Here, we apply the truncated Wigner representation method to coherent Ising machines with thermal, vacuum, and squeezed reservoir fields. We find that the probability of finding the ground state of a one-dimensional Ising model…
An explicit expression is obtained for the phase-time corresponding to tunneling of a (non-relativistic) particle through two rectangular barriers, both in the case of resonant and in the case of non-resonant tunneling. It is shown that the behavior of the transmission coefficient and of the tunneling phase-time near a resonance is given by expressions with "Breit-Wigner type" denominators. By contrast, it is shown that, when the tunneling probability is low (but not negligible), the non-resonant tunneling time depends on the barrier width and on the distance between the barriers only in a ver…
We introduce an efficient method to reconstruct the Wigner function of many-mode continuous variable systems. It is based on convex optimization with semidefinite programs, and also includes a version of the maximum entropy principle, in order to yield unbiased states. A key ingredient of the proposed approach is the representation of the state in a truncated Fock basis. As a bonus, the discrete finite representation allows to easily quantify the entanglement.
We introduce a new class of unitary transformations based on the su(1,1) Lie algebra that generalizes, for certain particular representations of its generators, well-known squeezing transformations in quantum optics. To illustrate our results, we focus on the two-mode bosonic representation and show how the parametric amplifier model can be modified in order to generate such a generalized squeezing operator. Furthermore, we obtain a general expression for the bipartite Wigner function which allows us to identify two distinct sources of entanglement, here labelled by dynamical and kinematical e…
We study the tomography of propagators for spin systems in the context of finite-dimensional Wigner representations, which completely characterize and visualize operators using shapes assembled from linear combinations of spherical harmonics. The Wigner representation of a propagator can be experimentally recovered by measuring expectation values of rotated axial spherical tensor operators in an augmented system with an additional ancilla qubit. The methodology is experimentally demonstrated for standard one-qubit quantum gates using nuclear magnetic resonance spectroscopy. In particular, this…
We apply the Wigner function formalism to the study of two-photon polarization-momentum hyperentanglement generated in parametric down conversion. It is shown that the consideration of a higher number of degrees of freedom is directly related to the extraction of additional uncorrelated sets of zeropoint modes at the source. We present a general expression for the description of the quantum correlations corresponding to the sixteen Bell base states, in terms of four beams whose amplitudes are correlated through the stochastic properties of the zeropoint field. A detailed analysis of the two ex…
In this work we study the behaviour of Wigner phase delay time for tunneling in the reflection mode. Our system consists of a circular loop connected to a single wire of semi-infinite length in the presence of Aharonov-Bohm flux. We calculate the analytical expression for the saturated delay time. This saturated delay time is independent of Aharonov- Bohm flux and the width of the opaque barrier thereby generalizing the Hartman effect. This effect implies superluminal group velocities as a consequence. We also briefly discuss the concept called "space collapse or space destroyer".
The properties of an alternative definition of quantum entropy, based on Wigner functions, are discussed. Such definition emerges naturally from the Wigner representation of quantum mechanics, and can easily quantify the amount of entanglement of a quantum state. It is shown that smoothing of the Wigner function induces an increase in entropy. This fact is used to derive some simple rules to construct positive definite probability distributions which are also admissible Wigner functions
We develop the Wigner phase space representation of a kicked particle for an arbitrary but periodic kicking potential. We use this formalism to illustrate quantum resonances and anti--resonances.
