Electrical current noise of a beamsplitter as a test of spin entanglement

We investigate the spin entanglement in the superconductor-quantum dot system proposed by Recher, Sukhorukov, and Loss, coupling it to an electronic beamsplitter. The superconductor-quantum dot entangler and the beamsplitter are treated within a unified framework and the entanglement is detected via current correlations. The state emitted by the entangler is found to be a linear superposition of nonlocal spin singlets at different energies, a spin-entangled two-particle wave packet. Colliding the two electrons in the beamsplitter, the singlet spin state gives rise to a bunching behavior, detectable via the current correlators. The amount of bunching depends on the relative positions of the single particle levels in the quantum dots and the scattering amplitudes of the beamsplitter. It is found that the bunching-dependent part of the current correlations is of the same magnitude as the part insensitive to bunching, making an experimental detection of the entanglement feasible. The spin entanglement is insensitive to orbital dephasing but suppressed by spin dephasing. A lower bound for the concurrence, conveniently expressed in terms of the Fano factors, is derived. A detailed comparison between the current correlations of the nonlocal spin-singlet state and other states, possibly emitted by the entangler, is performed. This provides conditions for an unambiguous identification of the nonlocal singlet spin entanglement.

Figure 1

Schematic picture of the system. A superconductor (S) is connected, via tunnel barriers to two quantum dots (1 and 2) in the Coulomb blockade regime. The dots are further coupled, via a second pair of tunnel barriers, to normal leads that cross in a forward scattering single-mode beamsplitter. The beamsplitter is characterized by scattering amplitudes $r,t,r^{\prime }$, and $t^{\prime }$. On the other side of the beamsplitter, the normal leads are connected to normal electron reservoirs A and B.

Figure 2

Energy diagram of the entangler-beamsplitter system in Fig. 1. A bias $eV$ is applied between the superconducting reservoir, with chemical potential ${\mu }_S=0$, and the normal reservoirs $A$ and $B$, with the same chemical potential ${\mu }_N=-eV$. There is only one spin-degenerate level of each dot, with energies ${\epsilon }_1$ and ${\epsilon }_2$, respectively, in the energy range $-eV\text{to}eV$. The level width $\gamma$ is determined by the coupling to the normal reservoirs. The bias $eV$ is taken to be much smaller than the superconducting gap, $eV⪡\Delta$, but so large that the broadened levels are well within the bias window, $eV-\mid {\epsilon }_j\mid ⪢\gamma$, $j=1,2$.

Figure 3

Tunneling processes transporting two electrons from the superconductor to the normal leads. Process I, in which the two electrons tunnel through different dots, one through dot 1 and one through dot 2, creates the wanted EPR pair. Process II, in which the two electrons tunnel through the same dot, either both through dot 1 or both through dot 2, is unwanted.

Figure 4

Elementary scattering processes (shown at the beamsplitter) contributing to the cross-correlators $S_{AB}^{\mathrm{I}}$. The two processes (a) and (b) transport a pair of electrons $\mid \uparrow ,E⟩$ and $\mid \downarrow ,-E⟩$ from the superconductor to the reservoirs $A$ and $B$, respectively. The two processes, having the same initial and final state, are indistinguishable and their amplitudes must be added. The correlator $S_{AB}^{\mathrm{I}}$ is proportional to the integral over energy of the (energy-dependent) joint detection probability.

Figure 5

The Fano factor for the cross-correlations $F_{AB}=F_{BA}$ (left panel) and auto-correlations $F_{AA}=F_{BB}$ (right panel) as a function of the normalized energy difference $({\epsilon }_1-{\epsilon }_2)∕\gamma$ for various beamsplitter transparencies.

Figure 6

The Fano factor for the cross-correlations $F_{AB}=F_{BA}$ for phase difference $\varphi =0$ (left panel) $\varphi =\pi ∕2$ (right panel) and as a function of the normalized energy difference $({\epsilon }_1-{\epsilon }_2)∕\gamma$ for various beamsplitter transparencies.

Figure 7

The Fano factor for the auto-correlations $F_{AA}=F_{BB}$ for phase difference $\varphi =0$ (left panel) $\varphi =\pi ∕2$ (right panel) and as a function of the normalized energy difference $({\epsilon }_1-{\epsilon }_2)∕\gamma$ for various beamsplitter transparencies.

Figure 8

Elementary scattering processes (shown at the beamsplitter) contributing to the cross-correlators $S_{AB}^{\mathrm{II}}$. The two processes (a) and (b) transport a pair of electrons $\mid \uparrow ,E⟩$ and $\mid \downarrow ,-E⟩$ from the superconductor to the reservoirs $A$ and $B$, respectively.