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Simulation of entanglement dynamics for a scattering between a free and a bound carrier in a quantum wire

Paolo Bordone, Andrea Bertoni, Jacoboni

Published Jan 1, 2004• 4 citations
Publisher

Abstract

We present a numerical analysis of the creation of entanglement between an electron freely propagating along a quantum wire and a charged particle bound to a specific site by a harmonic potential. The latter can be considered as a simplified model of an ionized impurity with an internal structure or of a localized phonon mode. Its dynamics is coupled to that of the free electron through a screened Coulomb potential. The time-dependent Schrodinger equation is solved numerically for the two-particle wave function thus describing the coherent evolution of the bipartite system. The two particle are considered distinguishable. As initial condition the product of a minimum uncertainty wave packet and the ground state of the harmonic oscillator are considered.

DOI
10.1109/iwce.2004.1407404

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We present a numerical analysis of the creation of entanglement between an electron freely propagating along a quantum wire and a charged particle bound to a specific site by a harmonic potential. The latter can be considered as a simplified model of an ionized impurity with an internal structure or of a localized phonon mode. Its dynamics is coupled to that of the free electron through a screened Coulomb potential. The time-dependent Schrodinger equation is solved numerically for the two-particle wave function thus describing the coherent evolution of the bipartite system. The two particle are considered distinguishable. As initial condition the product of a minimum uncertainty wave packet and the ground state of the harmonic oscillator are considered.
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.
On the basis of the full analytical solution of the overall unitary dynamics, the time evolution of entanglement is studied in a simple bipartite model system evolving unitarily from a pure initial state. The system consists of two particles in one spacial dimension bound by harmonic forces and having its free center of mass initially localized in space in a minimum uncertainty wave packet. The existence of such initial states in which the bound particles are not entangled is pointed out. The entanglement of the two particles is shown to be independent of the wavepacket mean momentum, and to increase monotonically in a time scale distinct from that of the spreading of the center of mass wavepacket.
We study by means of time-dependent numerical simulations the behavior of the entanglement stemming from the Coulomb scattering between two charged particles subject to a pulse of sinusoidal potential. We show that the splitting of the spatial wavefunction brought about by the interaction with the potential pulse plays a key role in the appearance of quantum correlation, thus leading under specific conditions to a peculiar behavior. The dependence of the final entanglement upon the physical parameters describing the pulse is discussed. Our results can be applied to a number of physical systems, such as electron-electron scattering in semiconductors or cold-ions dynamics in external fields.
As the key requirement of a quantum computer is the creation and exploitation of entanglement, a detailed study of entangled states has been carried out with reference to a solid-state system based on coupled quantum wires. A brief review of the basic gates is given first, based on preliminary investigations, followed by the analysis of electrons running along coupled quantum wires. The particle dynamics has numerically been simulated by means of a time-dependent Schrodinger solver applied to a two-particle system. Results are reported showing entangled states created by means of suitable quantum-gate networks. Starting from these fundamental results, the complexity of some interesting circuits is addressed, showing both the application of the universal set of quantum gates to complex algorithms and the computational speedup achieved within the proposed implementation.
A solid-state implementation of a universal set of gates for quantum computation is proposed and analysed using a time-dependent 2D Schrödinger solver. The qubit is defined as the state of an electron propagating along a couple of quantum wires. The wires are suitably coupled through a potential barrier with variable height and/or width. It is shown how a proper design of the system allows the implementation of any one-qubit transformation. The two-qubit gate is realized through a Coulomb coupler able to entangle the quantum states of two electrons running in two wires of two different qubits. The simulated devices are GaAs—AlGaAs heterostructures that should be on the borderline of present semiconductor technology. An estimate of decoherence effects due to phonon scattering is also presented.
In this paper we numerically solve the time dependent Schrödinger equation for scenarios using wave packets. These examples include the free wave packet, which we use to show the difference between group and phase velocities, the packet in a harmonic oscillator potential with non-trivial initial conditions in one and two dimensions, which is compared with their classical analogs to show how Ehrenfest theorem holds. We also include simulations of the diffraction through the single and double slit potentials, the refraction with a step potential and the dispersion by a central potential. The aim of this paper is to illustrate with simulations, nowadays easy to implement, scenarios that can help explaining the basics of the wave-particle duality.