Continuous-Variable Entanglement Test in Driven Quantum Contacts

The standard entanglement test using the Clauser-Horne-Shimony-Holt inequality is known to fail in mesoscopic junctions at finite temperatures. Since this is due to the bidirectional particle flow, a similar failure is expected to occur in an ac-driven contact. We develop a continuous-variable entanglement test suitable for electrons and holes that are created by the ac drive. At low enough temperatures the generalized Bell inequality is violated in junctions with low conductance or a small number of transport channels and with ac voltages which create few electron-hole pairs per cycle. Our ac-entanglement test depends on the total number of electron-hole pairs and on the distribution of probabilities of pair creations similar to the Fano factor.

Figure 1

Setups for the generalized ac Bell test: (a) normal junction with transmission eigenvalues $\{T_n\}$ and (b) superconductor–normal-metal beam splitter. In both cases, the test observables are the differences between the numbers of spin-up and spin-down particles along directions $\pm \mathit{a}$ and $\pm \mathit{b}$ detected in the leads $A$ (Alice) and $B$ (Bob).

Figure 2

(a) Probabilities of electron-hole pair creations $p_k$ and (b) parameter $F_p=\sum _k{p}_k(1-p_k)/\sum _k{p}_k$ as a function of the amplitude $V_0$ for harmonic drive $V(t)=V_0\cos (\omega t)$. (c) Test of a generalized Bell inequality: Normal junction [solid lines, see Eq. (5) and Fig. 1] and $SN$ beam splitter geometry [dash-dotted lines, see Eq. (6) and Fig. 1]. The generalized Bell inequality is violated for junction conductances and applied voltages that are below the lines shown in (c). Fano factors $F=1$, 0.99, 0.98 ($F_S=2$, 1.96) for the normal (superconducting) junction are shown in the plot. The effective charge is $e^{*}=e$ for the normal and $e^{*}=2e$ for the $SN$ junction.

Figure 3

Violation of a generalized Bell inequality as a function of the harmonic drive amplitude: (a) normal junction with $G/G_Q=0.02$ and (b) $SN$ beam splitter with $G_S/4G_Q=0.02$. Inequality is violated in the shaded regions in which ${\beta }_1$ and ${\beta }_2$ (solid lines) are smaller than $(\sqrt{2}-1)/4$ (dashed line).

Figure 4

Generalized Bell test for a single-channel contact: (a) normal junction and (b) $SN$ beam splitter. Results are shown for different shapes of the drive with the amplitude $V_0$. The biharmonic voltage $V(t)=V_0[0.8\cos (\omega t)+0.1\cos (3\omega t)]$ is an approximation of the triangle-wave drive. The regions to the left of the displayed lines are where the generalized Bell inequality is violated.

Figure 5

Effect of a finite temperature on the violation of the generalized Bell inequality for a harmonic drive. (a) The voltage ranges of violation (to the left of the respective curves) are reduced by a finite temperature. (b) For a fixed conductance a violation is only possible below a certain critical temperature for entanglement generation.