Near-term performance of quantum repeaters with imperfect ensemble-based quantum memories

We study the feasibility of meaningful proof-of-principle demonstrations of several quantum repeater protocols with photon (single-photon and photon-pair) sources and atomic-ensemble-based quantum memories. We take into account nonunit memory efficiencies that decay exponentially with time, which complicates the calculation of repeater rates. We discuss implementations based on quantum dots, parametric downconversion, rare-earth-ion doped crystals, and Rydberg atoms. Our results provide guidance for the near-term implementation of long-distance quantum repeater demonstrations, suggesting that such demonstrations are within reach of current technology.

Figure 1

Sketches of quantum repeater protocols we mainly discuss in this paper. Here we show the two-link version of these repeater protocols. (a) Single-photon source (SPS) with single-photon BSM (1 $+$ 1). (b) Deterministic photon-pair source (dPPS) with two-photon BSM (2 $+$ 2). (c) Nondeterministic photon-pair source (ndPPS) with single-photon BSM ($\tilde{2}+$ 1). (d) Nondeterministic photon-pair source (ndPPS) with two-photon BSM ($\tilde{2}+$ 2).

Figure 2

Rates of various repeater schemes for a total distance of 100 km. The numbers in the contour line represent the corresponding repeater rates in Hz. The plot shows the situation of two links (nesting level is 1). The corresponding repeater protocols and parameter regimes are (a) SPS $+$ single-photon BSM($1+1$) with local beam-splitter transmission probability 0.8 and single-photon emission probability 0.75; (b) dPPS $+$ two-photon BSM ($2+2$) with photon-pair emission probability 0.5; (c) ndPPS $+$ single-photon BSM ($\tilde{2}+1$) with photon-pair emission probability 0.03; and (d) ndPPS $+$ two-photon BSM ($\tilde{2}+2$) with photon-pair emission probability 0.03.

Figure 3

Comparison of repeater implementations with two links: the 1 $+$ 1 scheme with QDs and REIs (A); the 1 $+$ 1 scheme with RAs (B); the 2 $+$ 2 scheme with QDs and REIs (C); the 2 $+$ 2 scheme with RAs (D); the $\tilde{2}+$ 1 scheme with PDC and REIs (E); the $\tilde{2}+$ 2 scheme with PDC and REIs (F).

Figure 4

The relation between $\beta$ and $p_0$ in different $T_0/{\tau }_m$ regimes: $T_0/{\tau }_m=0.001$ (blue dots), 0.01 (yellow triangles), 0.1 (green squares), and 1 (red diamonds). The first three subfigures show results for single-photon BSM (a, b, and c) and two-photon BSM (d). For a single-photon BSM situation, we consider fidelity of entanglement generated state ${\alpha }^{\left(0\right)}$, which depends on the memory efficiency, 0.9 (a), 0.5 (b), and 0.1 (c).

Figure 5

The numerical (blue dot) and theoretical (red solid line) probability distribution for $T_{\mathrm{dif}}$. The regime is considered as a high-memory-lifetime regime ($T_0/{\tau }_m=0.01$), low-memory-lifetime regime ($T_0/{\tau }_m=1$), high-$p_0$ regime ($p_0=0.1$), and low-$p_0$ regime ($p_0=0.01$). The theoretical prediction fits well with the experimental result.

Figure 6

The numerical (blue dot) and theoretical (yellow x) expectation value of $T_{min}$(a and b) and $T_{max}$(c and d). We plot the dependence with $T_0/{\tau }_m$(a and c) and $p_0$(c and d), and find the experimental result and theoretical result fit well.

Figure 7

Average swapping probability in memory cutoff assumption ${\langle p_s\rangle }_{\mathrm{cut}}$(diamond) and exponential decay assumption ${\langle p_s\rangle }_{exp}$(circle). The plots show the dependence of average swapping probability with different ${\tau }_m/T_0$ regime: ${\tau }_m/T_0=1$ (blue), 10 (red), 100 (green), and 1000 (magenta).

Figure 8

Average entanglement distribution time with limited entanglement generation time. The parameters used are ${\eta }_d=0.95$, ${\eta }_m=0.5$, $p_0=0.1$, and ${\tau }_m=10T_0$. The minimum value (maximum rate) is obtained with ${\tau }_0\approx 16T_0$, and $\lambda \approx 1.06$.

Figure 9

The ratio $\lambda$ of Eq. (D9), which represents the improvement achieved by optimizing the memory buffer time.