Local and nonlocal entanglement for quasiparticle pairs induced by Andreev reflection

Abstract

We investigate local and nonlocal entanglement of particle pairs induced by direct and crossed Andreev reflections at the interfaces between a superconductor and two normal conductors. It is shown theoretically that both local and nonlocal entanglement can be quantified by concurrence and detected from the violation of a Bell inequality of spin current correlators, which are determined only by normal reflection and Andreev reflection eigenvalues. There exists a one-to-one correspondence between the concurrence and the maximal Bell-CHSH parameter in the tunneling limit.

Current and noise correlations in a double-dot Cooper-pair beam splitter

“…[5][6][7][8][9][10] Bell inequality measurements based on noise crossed correlations have been proposed to detect this entanglement. 5,[11][12][13][14] Over the last decade, CAR has been studied extensively for a variety of geometries. The early proposals pointed out that positive crossed correlations can be generated either by selecting the energies of the outgoing electrons or alternatively by filtering their spins.…”

Andreev Reflection and Creation of Spin-Entangled Charge Carriers in Double Barrier Junctions

“…σ y and T stands for the Pauli matrix and the matrix transpose, respectively. The evolution of the input state |Ψ in into the output state |Ψ out can be connected through unitary transformation, which can be expressed in terms of the scattering matrixŜ [13]. The scattering matrix matches the wave functions at both M/S and S/M interfaces in a double barrier junction [15].…”

“…It is convenient to write the input state |Ψ in of the holes (electrons), injected to the superconducting layer, in terms of the stream of hole (electron) pairs with the opposite spins. For example, for the hole pairs we can write [13,14]:…”

“…Thus, the outgoing state, to the first order of the scattering matrix elements, can be written as a superposition of the vacuum state |0 and the local and non-local spin singlet entangled states [13]:…”

“…In our further discussion, we take into account only the Andreev reflections and neglect the EC. A relation between the input (â) and output (b) matrices of the electron (e) and hole (h) annihilation operators can be written in the following form [13]:…”