Generation of spin entanglement in nonequilibrium quantum dots

We propose schemes for generating spatially separated spin entanglement in systems of two quantum dots with onsite Coulomb repulsion weakly coupled to a joint electron reservoir. An enhanced probability for the formation of spin entanglement is found in nonequilibrium situations with one extra electron on each dot, either in the transient state after rapid changes of the gate voltage or in the steady state with applied bias voltage. In both cases, so-called Werner states with spin-singlet fidelity exceeding $1∕2$ are generated, which indicates entanglement.

Figure 1

The setups: (a) Two quantum dots ($u$ and $d$) are coupled to a joint electron reservoir $\left(L\right)$. (b) In addition to (a), the quantum dots are coupled to two independent reservoirs ($R_u$ and $R_d$) on the right.

Figure 2

Upper panels: time evolution of the probabilities for a singlet and a triplet state. Lower panel: the corresponding Werner fidelity. For the perfectly symmetric setup, $\Delta \epsilon =0$, ${\Gamma }_{L_u}={\Gamma }_{L_d}$, in the absence of spin relaxation, ${\Gamma }_{S\to T}=0$, curve (i), we find $F\equiv 1$. The Werner fidelity is reduced for either (ii) nondegenerate dot energy levels, $\Delta \epsilon ={\Gamma }_L$, (iii) asymmetric coupling ${\Gamma }_{L_d}=0.1{\Gamma }_{L_u}$, and (iv) a finite spin relaxation rate ${\Gamma }_{S\to T}=0.2{\Gamma }_L$. The high-energy cutoff is set to $D=100{\mathrm{k}}_{B}T$.

Figure 3

Time evolution of the total charge during the discharging of an initial singlet state compared to an initial triplet state. The system is assumed to be perfectly symmetric, ${\Gamma }_{L_u}={\Gamma }_{L_d}$, $\Delta \epsilon =0$. Starting from a singlet, the system empties quickly, but it remains singly occupied if we start from a triplet state.

Figure 4

The stationary Werner fidelity $F$ vs bias voltage for $\epsilon =0$ and different ratios of the coupling strengths, ${\Gamma }_R∕{\Gamma }_L=2$, 1, 0.5. The bias is applied symmetrically, $-{\mu }_L=eV∕2={\mu }_{R_u}={\mu }_{R_d}$. The inset shows the corresponding stationary overall probabilities $p_S+p_T$ to find the system doubly occupied.