Paolo Bordone

The Wigner-function approach to non-equilibrium electron transport

The Wigner-function (WF) approach to quantum electron transport in semiconductors is reviewed in this paper. The main definitions and properties related to the WF are presented, with a discussion of the various forms of the dynamical equations that govern its evolution. Monte Carlo solutions of such equations are also discussed. Interactions of electrons with applied fields, potential profiles, and phonons are analysed in detail. Finally, several physical applications are presented. Each topic has been developed from basic principles for the benefit of interested readers who are not experts in the particular subjects discussed in this paper.

Linear entropy as an entanglement measure in two-fermion systems

We describe an efficient theoretical criterion, suitable for indistinguishable particles to quantify the quantum correlations of any pure two-fermion state, based on the Slater rank concept. It represents the natural generalization of the linear entropy used to treat quantum entanglement in systems of nonidentical particles. Such a criterion is here applied to an electron-electron scattering in a two-dimensional system in order to perform a quantitative evaluation of the entanglement dynamics for various spin configurations and to compare the linear entropy with alternative approaches. Our numerical results show the dependence of the entanglement evolution upon the initial state of the system and its spin components. The differences with previous analyses accomplished by using the von Neumann entropy are discussed. The evaluation of the entanglement dynamics in terms of the linear entropy results to be much less demanding from the computational point of view, not requiring the diagonalization of the density matrix.

Dynamics of quantum correlations in colored-noise environments

We address the dynamics of entanglement and quantum discord for two non interacting qubits initially prepared in a maximally entangled state and then subjected to a classical colored noise, i.e. coupled with an external environment characterized by a noise spectrum of the form

Time evolution of tripartite quantum discord and entanglement under local and nonlocal random telegraph noise

1/f^{\alpha}

EFFECTS OF CLASSICAL ENVIRONMENTAL NOISE ON ENTANGLEMENT AND QUANTUM DISCORD DYNAMICS

. More specifically, we address systems where the Gaussian approximation fails, i.e. the sole knowledge of the spectrum is not enough to determine the dynamics of quantum correlations. We thus investigate the dynamics for two different configurations of the environment: in the first case the noise spectrum is due to the interaction of each qubit with a single bistable fluctuator with an undetermined switching rate, whereas in the second case we consider a collection of classical fluctuators with fixed switching rates. In both cases we found analytical expressions for the time dependence of entanglement and quantum discord, which may be also extended to a collection of flcutuators with random switching rates. The environmental noise is introduced by means of stochastic time-dependent terms in the Hamiltonian and this allows us to describe the effects of both separate and common environments. We show that the non-Gaussian character of the noise may lead to significant effects, e.g. environments with the same power spectrum, but different configurations, give raise to opposite behavior for the quantum correlations. In particular, depending on the characteristics of the environmental noise considered, both entanglement and discord display either a monotonic decay or the phenomena of sudden death and revivals. Our results show that the microscopic structure of environment, besides its noise spectrum, is relevant for the dynamics of quantum correlations, and may be a valid starting point for the engineering of non-Gaussian colored environments.

EFFECT OF MARKOV AND NON-MARKOV CLASSICAL NOISE ON ENTANGLEMENT DYNAMICS

Few studies explored the dynamics of nonclassical correlations besides entanglement in open multipartite quantum systems. Here, we address the time evolution of quantum discord and entanglement in a model of three noninteracting qubits subject to a classical random telegraph noise in common and separated environments. Two initial entangled states of the system are examined, namely the GHZ- and W-type states. The dynamics of quantum correlations results to be strongly affected by the input configuration of the qubits, the type of the system-environment interaction, and the memory properties of the environmental noise. When the qubits are nonlocally coupled to the random telegraph noise, the GHZ-type states partially preserve, at long times, both discord and entanglement, regardless of the correlation time of the environmental noise. The survived entangled states turn out to be also detectable by means of suitable entanglement witnesses. On the other hand, in the same conditions, the decohering effects suppress all the quantum correlation of the W-type states which are thus less robust than the GHZ-type ones. The long-time survival of tripartite discord and entanglement opens interesting perspectives in the use of multipartite entangled states for practical applications in quantum information science.

Entanglement dynamics of electron-electron scattering in low-dimensional semiconductor systems

We address the effect of classical noise on the dynamics of quantum correlations, entanglement and quantum discord (QD), of two non-interacting qubits initially prepared in a Bell state. The effect of noise is modeled by randomizing the single-qubit transition amplitudes. We address both static and dynamic environmental noise corresponding to interaction with separate and common baths in either Markovian and non-Markovian regimes. In the Markov regime, a monotone decay of the quantum correlations is found, whereas for non-Markovian noise sudden death and revival phenomena may occur, depending on the characteristics of the noise. Entanglement and QD show the same qualitative behavior for all kind of noises considered. On the other hand, we find that separate and common environments may play opposite roles in preserving quantum correlations, depending on the noise regime considered.

Wigner-function approach to multiband transport in semiconductors

We analyze the effect of a classical noise into the entanglement dynamics between two particles, initially entangled, subject to continuous time quantum walks in a onedimensional lattice. The noise is modeled by randomizing the transition amplitudes from one site to another. Both Markovian and non-Markovian environments are considered. For the Markov regime an exponential decay of the initial quantum correlation is found, while the loss of coherence of the quantum state increases monotonically with time up to a saturation value depending upon the degrees of freedom of the system. For the non-Markov regime the presence or absence of entanglement revival and entanglement sudden death phenomena is found or deduced depending on the peculiar characteristics of the noise. Our results indicate that the entanglement dynamics in the non-Markovian regime is affected by the persistence of the memory effects of the environment and by its intrinsic features.

Wigner-function formulation for quantum transport in semiconductors: theory and Monte Carlo approach

We perform the quantitative evaluation of the entanglement dynamics in scattering events between two indistinguishable electrons interacting via the Coulomb potential in one- and two-dimensional semiconductor nanostructures. We apply a criterion based on the von Neumann entropy and the Schmidt decomposition of the global state vector suitable for systems of identical particles. From the time-dependent numerical solution of the two-particle wave function of the scattering carriers we compute their entanglement evolution for different spin configurations: two electrons with the same spin, with different spin, and singlet and triplet spin states. The procedure allows us to evaluate the mechanisms that govern entanglement creation and their connection with the characteristic physical parameters and initial conditions of the system. The cases in which the evolution of entanglement is similar to the one obtained for distinguishable particles are discussed.

Analytical expression of genuine tripartite quantum discord for symmetrical X-states

In this work we present a one-dimensional, multi-band model for electron transport in semiconductors that makes use of the Wigner-function formalism and that allows for energy bands of any shape. A simplified two-band model is then derived from the general equations, by using the parabolic band approximation.