Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spin-entangled electrons

Abstract

We propose and analyze a spin-entangler for electrons based on an s-wave superconductor coupled to two quantum dots, each of which is coupled to normal Fermi leads. We show that in the presence of a voltage bias and in the Coulomb blockade regime two correlated electrons provided by the Andreev process can coherently tunnel from the superconductor via different dots into different leads. The spin singlet coming from the Cooper pair remains preserved in this process, and the setup provides a source of mobile and nonlocal spin-entangled electrons. The transport current is calculated and shown to be dominated by a two-particle Breit-Wigner resonance that allows the injection of two spin-entangled electrons into different leads at exactly the same orbital energy, which is a crucial requirement for the detection of spin entanglement via noise measurements. The coherent tunneling of both electrons into the same lead is suppressed by the on-site Coulomb repulsion and/or the superconducting gap, while the tunneling into different leads is suppressed through the initial separation of the tunneling electrons. In the regime of interest the particle-hole excitations of the leads are shown to be negligible. The Aharonov-Bohm oscillations in the current are shown to contain single-and two-electron periods with amplitudes that both vanish with increasing Coulomb repulsion albeit differently fast.We begin with a qualitative description of the entangler and its principal mechanism based on Andreev processes and Coulomb blockade effects. In subsequent sections we introduce the Hamiltonian and calculate the stationary current in detail. We consider an s-wave superconductor that acts as a natural source of spin-entangled electrons, since the electrons form Cooper pairs with singlet spin wave functions. 15 The superconductor, which is held at the chemical potential S , is weakly coupled by tunnel barriers to two separate quantum dots D 1 and D 2 , which themselves are weakly coupled to Fermi liquid leads L 1 and L 2 , respectively, both held at the

Decoherence of Einstein-Podolsky-Rosen pairs in a noisy Andreev entangler

“…We have also introduced the state |Dq = d † 1σ a † 1q−σ |i representing the state with one electron on dot 1 and the other one in lead 2 (L2). At this point we emphasize that virtual states with both electrons in the leads leading to |DD → |pD → |pq → |Dq → |DD are suppressed by a factor γ l /µ < 1 compared to that with only one electron in the leads and therefore can be neglected 7 . Recall that |pq stands for a † 1pσ a † 2q−σ |i , where p is the momentum from lead 1 and q from lead 2; this is the final state |f of the electrons on the leads with energy ǫ f = ǫ q + ǫ p .…”

“…Along the lines of Ref. 7, to calculate the transition rates W DD,f and W i,DD we resort to the T-matrix approach. The on-shell transmission or T-matrix at the energy E is precisely defined as…”

“…Following this line of ideas, it is then interesting to ask to what extent zero-point fluctuations of the electrical environments of the two dots will affect the non-local Cooper pair of charge 2e and hence the stationary current(s) discussed in Ref. 7. Therefore, we now turn our attention to the specific setup of Fig.…”

“…Recently, an interesting setup involving an s-wave superconductor weakly-coupled to two quantum dots, which themselves are coupled to normal leads has been envisioned in Ref. 7. The spin correlations of an s-wave superconductor induce a spin-singlet state between two electrons, each of which can now reside on a separate quantum dot; this results in the formation of a nonlocal Cooper pair described by the quantum state…”

“…Moreover, like in Ref. 7, we require the dot-lead coupling to be much stronger than the superconductor-dot coupling, i.e., |T SD | < |T DL |, so that electrons that enter the dots from the superconductor will leave the quantum dots to the leads much faster that new electrons can be provided to the dots from the superconductor. Additionally, a stationary occupation due to the couplings to the leads is exponentially small if µ > k B T .…”