Spin-Entangled Currents Created by a Triple Quantum Dot

We propose a simple setup of three coupled quantum dots in the Coulomb blockade regime as a source for spatially separated currents of spin-entangled electrons. The entanglement originates from the singlet ground state of a quantum dot with an even number of electrons. To preserve the entanglement of the electron pair during its extraction to the drain leads, the electrons are transported through secondary dots. This prevents one-electron transport by energy mismatch, while joint transport is resonantly enhanced by conservation of the total two-electron energy.

Figure 1

(a) Setup of the triple quantum dot entangler. Three leads $l_i$ $(i=C,L,R)$ at chemical potential ${\mu }_i$ are coupled to three quantum dots $D_i$ in the Coulomb blockade regime. Each dot contains an even number of electrons and can accept only zero, one ($D_L$ and $D_R$), and two ($D_C$) excess electrons. A spin singlet is formed in $D_C$ when two electrons tunnel incoherently, each with a rate $\alpha$, from the source lead $l_C$ into $D_C$. Each of the electrons can subsequently tunnel coherently to $D_L$ and $D_R$ with tunneling amplitudes $T_L$ and $T_R$. Finally, the electrons tunnel out (with rates ${\gamma }_L$ and ${\gamma }_R$) to the drain leads $l_L$ and $l_R$, creating two currents of entangled electrons. (b) Energy level diagram, for each electron. Unentangled currents, which arise from one-electron transport, are suppressed by the energy differences ${ϵ}_{L,R}-{ϵ}_C\pm U$. The joint transport of both electrons is favored by a resonance in the total two-electron energy: ${ϵ}_L+{ϵ}_R\simeq 2{ϵ}_C$.

Figure 2

The entangler states $\{0,C,CC,LC,CR,LR,L,R\}$ and their transitions. Double arrows indicate coherent oscillations of one electron between two dots with overlap matrix element $T_L$ and $T_R$ (in the shaded area). Single arrows indicate incoherent tunneling from ($\alpha$) or to (${\gamma }_L$ and ${\gamma }_R$) the leads. The dashed lines indicate the three types of transitions that must be avoided to ensure the joint transport of the singlet pair $CC$ to the leads (see text; for clarity we do not show the transitions obtained by replacing $L$ by $R$).

Figure 3

Currents and qualities of the entangler, with $\alpha =0.1,\gamma =1,T_0=10,U=1000$ in $\mu \mathrm{e}\mathrm{V}$. (a) Total current in the left ($I_L$) and right ($I_R$) leads. The symmetric current $I_L=I_R$ at the central peak is a signature of the two-electron resonance, where the current is dominated by the entangled current $I_E$. The peaks at $\delta {ϵ}_L=\pm U$ are resonances for the transport of one electron. The inset shows $I_E$ and the total current $I=I_L+I_R$, around $\delta {ϵ}_L=0$. (b) Qualities $Q$ and $\tilde{Q}$ around the resonance at $\delta {ϵ}_L=0$, where the entangled current dominates. The width of the resonance defined by $Q,\tilde{Q}>Q_{\mathrm{I}}^{\min }=10$ is $|\delta {ϵ}_L|<6\mu \mathrm{e}\mathrm{V}$, as given by Eq. (7) (gray region). (c) $Q$ and $\tilde{Q}$ vs the tunneling amplitude $T_0$, at resonance ($\delta {ϵ}_L=0$). In gray, the region where the quality of the entangler is $Q,\tilde{Q}>Q_{\mathrm{I}\mathrm{I}}^{\min }=100$; see Eq. (8). (d) At resonance, the entangled current $I_E\simeq I$ saturates to $\simeq e\alpha =20\mathrm{p}\mathrm{A}$ when $T_0\gg 5\mu \mathrm{e}\mathrm{V}$.