Effect of inelastic scattering on spin entanglement detection through current noise

We study the effect of inelastic scattering on the spin entanglement detection and discrimination scheme proposed by Egues, Burkard, and Loss [Phys. Rev. Lett. 89, 176401 (2002)]. The finite-backscattering beam splitter geometry is supplemented by a phenomenological model for inelastic scattering, the charge-conserving voltage probe model, conveniently generalized to deal with entangled states. We find that the behavior of shot-noise measurements in one of the outgoing leads remains an efficient way to characterize the nature of the nonlocal spin correlations in the incoming currents for an inelastic scattering probability up to 50%. Higher order cumulants are analyzed, and are found to contain no additional useful information on the spin correlations. The technique we have developed is applicable to a wide range of systems with voltage probes and spin correlations.

Figure 1

(Color online) The beam-splitter geometry fed with pairwise nonlocally entangled electron currents or polarized currents. The action of the spin rotation via Rashba spin-orbit coupling in one of the input leads changes noise in the output leads dramatically. Inelastic transport is modeled between the entangler and the spin rotation region by means of one or more fictitious probes. Shot noise measured in terminal $R_1$ as a function of $\theta$ can be used to detect the nature of the incoming electron correlations.

Figure 2

(Color online) In the upper plot (a) we represent the current shot noise in units of $e^{3}V∕h$ in lead $R1$ for the singlet and $m_s=0$ triplet incoming states, as a function of spin rotation angle $\theta$ and decoherence strength $\alpha$. The same for the polarized spin state case is presented in the lower plot (b). Interlead transmission probability between upper and lower arms $T$ is fixed to 0.5, and no backscattering $(T_B=1)$ is assumed.

Figure 3

(Color online) Normalized value of shot noise in lead $R_1$ at zero spin rotation angle as a function of beam-splitter transmission $T_B$ for $T=0.5$. Solid (blue) lines correspond to singlet incoming current whereas dashed (red) lines account for the polarized one. In both cases, different values of inelastic scattering probability have been considered, from $\alpha =0$ (upper curves) to $\alpha =1$ (lower curves) in steps of 0.25. Inset, the same for the oscillation amplitude with $\theta$ of the shot noise.

Figure 4

(Color online) Oscillation range with angle $\theta$ of the skewness in lead $R_1$, in units of $e^{4}V∕h$, as a function of beam-splitter transmission between left and right arms $T_B$. $T$ is fixed to the optimal point $T=0.5$. Different oscillation ranges for different values of inelastic scattering are indicated by different shades of gray, ranging from pale gray for $\alpha =0$ to dark gray for $\alpha =1$ in steps of 0.25. The case of singlet-entangled incoming current is considered in plot (a), whereas in plot (b) the incoming current is in a polarized state. The actual oscillation of skewness with Rashba spin rotation angle for entangled current [solid (blue) lines] and polarized current [dashed (red) lines] is plotted in the inset of (b) (see main text).

Figure 5

(Color online) Single channel wire geometry with fictitious probe and nontransparent contacts.

Figure 6

(Color online) The results for the singlet current skewness in the beam-splitter geometry calculated within the modified Langevin approach begin to deviate slightly from those of the time-resolved (TR) technique, Fig. 4. Note that the deviation is most apparent in the intermediate $\alpha$ limit. Shot noise results are identical. At this level, however, both techniques are qualitatively equivalent.