Quantum entanglement for acoustic spintronics

We consider the entanglement of spins for two electrons contributing to the acoustoelectric current driven by a surface acoustic wave (SAW) in two adjacent narrow channels by calculating their exchange energy $\left(J\right)$. The channels belong to an acoustic nanocircuit which comprises a network of quasi-one-dimensional pinched-off channels serving as wires along which SAW quantum dots transport electrons. This is motivated by possible practical applications involving quantum information processing and quantum computers. We calculate $J$ as a function of time as the electrons travel side-by-side in the adjacent channels and as a function of the distance between the centers of the channels. The leakage from the state in which the system is prepared, is calculated. The oscillations in the leakage indicate the probability for the electron system to make transitions between the ground and excited states, or for an electron to hop back and forth between channels.

Figure 1

Schematic plot of the SAW beam transporting electrons along a pair of channels. Two adjacent SAW dots in different channels form a pair of coupled quantum dots. The vertical lines in the figure represent maxima of the SAW potential which carries electrons (depicted as full circles) along the channels. The split gates defining the channels are denoted by $G_1$, $G_2$, and $G_3$.

Figure 2

The exchange interaction energy $J$ as a function of time for $\beta =0.5$ and separation $a=0.7{\lambda }_0$ between a pair of narrow channels. $J$ is larger when we take the difference between the energies for the singlet and triplet states, obtained by finding the eigenvalues of the Slater determinant numerically (solid line) compared to the Heitler-London approximation for separated harmonic wells (dashed line). The parameters used in the calculation are given in the text.

Figure 3

The exchange interaction at $t=0$ as a function of the separation between a pair of parallel channels. The solid line is the difference in energies between the singlet and triplet states obtained by numerically solving the Schrödinger equation for a pair of interacting electrons. The dashed line is the result in the Heitler-London model. The parameters used in the calculation are given in the text.

Figure 4

The leakage $1-{\mid c_0\left(t\right)\mid }^2$ as a function of time. Here, $a∕{\lambda }_o=0.7,\beta =0.5$. The oscillations in the leakage are due to the phase factors for the levels involved in our calculations. The inset shows the position of the electron in the potential of Eq. (2) at $t=T∕4$ and $t=3T∕4$.