Detection of nonlocal spin entanglement by light emission from a superconducting $p\ensuremath{-}n$ junction

Detection of nonlocal spin entanglement by light emission from a superconducting $p - n$ junction

Alexander Schroer and Patrik Recher

Phys. Rev. B 92, 054514 – Published 25 August 2015

Abstract

We model a superconducting $p - n$ junction in which the $n$ and the $p$ sides are contacted through two optical quantum dots (QDs), each embedded into a photonic nanocavity. Whenever a Cooper pair is transferred from the $n$ side to the $p$ side, two photons are emitted. When the two electrons of a Cooper pair are transported through different QDs, polarization-entangled photons are created, provided that the Cooper pairs retain their spin singlet character while being spatially separated on the two QDs. We show that a Clauser-Holt-Shimony-Horne (CHSH) Bell-type measurement is able to detect the entanglement of the photons over a broad range of microscopic parameters, even in the presence of parasitic processes and imperfections.

Received 30 December 2014
Revised 22 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.054514

Authors & Affiliations

Alexander Schroer¹ and Patrik Recher^1,2

¹Institut für Mathematische Physik, Technische Universität Braunschweig, D-38106 Braunschweig, Germany
²Laboratory for Emerging Nanometrology Braunschweig, D-38106 Braunschweig, Germany

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Vol. 92, Iss. 5 — 1 August 2015

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Figure 1
Left: Two QDs (1,2) are tunnel coupled to superconducting $p$ -type and $n$ -type leads. Coulomb repulsion on the QDs splits the electrons or holes of incoming CPs, so each QD contains one electron and one hole. Upon recombination polarization-entangled photons are emitted into two nanocavities (gray) with resonance frequency $ω_{0}$ and detected by an optical Bell test. Right: Lowest relevant energy levels of the double-QD system. The SC leads hybridize the empty state and the singlet state, $| 〉 \pm | S 〉$ , and the singly occupied states, $| 1 σ 〉 \pm | 2 σ 〉$ , creating a closed two-photon emission cycle.
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Figure 2
Maximum violation of the CHSH Bell inequality depending on the local and nonlocal pair creation amplitudes $Λ_{ph}$ and $Δ_{ph}$ . In the black regions entanglement cannot be detected, $B_{max} \leq 2$ , but in the white regions $B_{max} \geq 2 \sqrt{2}$ . (a) Perfect detectors, $γ = 1$ , isolate the contribution of a single nonlocal pair and entanglement is detectable for almost all parameters. Right inset: Cavity losses affect $B_{max}$ [shown at different points A, B ( $γ = 1$ ) and C–E ( $γ = 10^{- 4}$ ) in the parameter space], possibly reducing it below the critical value of 2 (lower dashed line). At high losses, however, all surviving coincidences are due to nonlocal pairs and $B_{max}$ approaches the two-particle maximum value $2 \sqrt{2}$ (upper dashed line). Left inset: Probabilities $P$ to find $N$ photons in the cavities at point B/D and at different loss rates $κ$ . At finite cavity losses, odd number states are populated, because photon pairs are broken up. (b) Lossy detectors, $γ = 10^{- 4}$ (see text), are able to detect entanglement if the admixture of local pairs is sufficiently small. The lines indicate the parameters obtained from the microscopic model if ECT is (i) as strong as CAR at infinite (solid) or finite charging energy 2 meV (dashed) on both QDs, or (ii) if ECT is half as strong as CAR. Following these lines in the direction of the arrow corresponds to different cavity detunings, $e V_{sd} / ℏ = ω_{0} + δ ω$ , where $ℏ δ ω$ increases from $- 0.1 g$ to $0.1 g$ .
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Figure 3
(a) Two different device geometries. Left panel: The tunnel contacts between the SC leads and QDs are separated by several Fermi wavelengths. ECT between the QDs is as strong as CP splitting. Right panel: Electrons can tunnel into both QDs from the same point in the SC leads. CP splitting dominates over ECT. (b) Proximity-induced amplitudes of CAR, ${\tilde{Δ}}_{12}$ , and ECT, ${\tilde{t}}_{12}$ , and the renormalization of the on-site energy ${\tilde{t}}_{11}$ at different carrier concentrations $n$ . When optimizing the CP splitting amplitude, ECT cannot be neglected. The QDs are separated by $r = 100$ nm and the SC coherence length is $ξ = 200$ nm (arbitrarily chosen).
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Physical Review B

covering condensed matter and materials physics

Detection of nonlocal spin entanglement by light emission from a superconducting $p - n$ junction

Alexander Schroer and Patrik Recher

Phys. Rev. B 92, 054514 – Published 25 August 2015

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Physical Review B

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Detection of nonlocal spin entanglement by light emission from a superconducting p−n junction

Alexander Schroer and Patrik Recher

Phys. Rev. B 92, 054514 – Published 25 August 2015

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Detection of nonlocal spin entanglement by light emission from a superconducting $p - n$ junction