Highly indistinguishable and strongly entangled photons from symmetric GaAs quantum dots

The development of scalable sources of non-classical light is fundamental to unlocking the technological potential of quantum photonics. Semiconductor quantum dots are emerging as near-optimal sources of indistinguishable single photons. However, their performance as sources of entangled-photon pairs are still modest compared to parametric down converters. Photons emitted from conventional Stranski–Krastanov InGaAs quantum dots have shown non-optimal levels of entanglement and indistinguishability. For quantum networks, both criteria must be met simultaneously. Here, we show that this is possible with a system that has received limited attention so far: GaAs quantum dots. They can emit triggered polarization-entangled photons with high purity (g(2)(0) = 0.002±0.002), high indistinguishability (0.93±0.07 for 2 ns pulse separation) and high entanglement fidelity (0.94±0.01). Our results show that GaAs might be the material of choice for quantum-dot entanglement sources in future quantum technologies.

Supplementary Figure 1 Evaluation of the visibility of two-photon interference. Two-photon interference measurement of a representative quantum dot. The dashed coloured lines are the fits of peak 0 (red), 1 (violet), 2 (blue), 3 (green) and 4 (orange), respectively. The black line is the sum of all single peak fits. The areas under the peaks 1, 2 and 3 are used for the calculations of the two-photon interference visibility.
In order to extract the visibility of two-photon interference, we fit the experimental data using the following equation: where y 0 is the offset, A i the area of peak i, x 0 the position of the first peak, w the width of the peaks and d the temporal distance between the peaks. From the experimental conditions we expect that the distance d between the peaks is the same.
Furthermore, all the peaks should have the same width, which is mainly determined by the time jitter of the avalanche photodiode (500 ps). In general, one would also expect that peak 0 and 4 as well as 1 and 3 are equal in intensity. However, we have to consider the slightly different intensity between the two excitation pulses as well as the different detection efficiency for both fibre outputs. This is taken into account by leaving the A i as free parameters.
We have then calculated the two-photon interference visibility via To correct for the imperfections of the beam splitter we measured the mode overlap (1 − ε) = 0.96 ± 0.01 (using the fibre beam splitter to perform a Michelson measurement on the shaped excitation laser), the transmission coefficient T = 0.48 ± 0.005 and the reflection coefficient R = 0.52 ± 0.005 (using a power meter) and calculate the corrected visibility with (see Supplementary information of 1 and 2 ): where we put g 2 (0) = 0.

Supplementary Note 3: Decay time and coherence time measurements
We measured the decay time of exciton (X) and XX under two-photon excitation using a detector with a time resolution of around 50 ps. The acquired data are presented in Supplementary Figure 3 where V is the interference fringe visibility, A the amplitude, y 0 the offset and T 2 is the decay constant. The parameters y 0 and A, which under ideal experimental conditions should be 0 and 1 are left as free parameters because of small imperfections of the mode overlap. The calculated coherence time T 2 for X and XX are 305±39 ps and 109±15 ps, respectively. From the lifetime and coherence-times we can obtain an estimate for the visibility of two-photon interference by using (see 4 which is found to be V TPE = 0.6 and V TPE = 0.4 for X and XX, respectively. The data from the two-photon interference experiment (see Supplementary Table 1), yields a higher visibility for the same QD (V TPE = 0.69 and V TPE = 0.76 for X and XX, respectively). This discrepancy can be explained by the presence of decoherence processes on a time scale that is larger than the time delay between the two laser pulses used to generate the interfering photons. A closer inspection of the difference between the determined values indicates that the effect on the XX is more dominant. A possible explanation is that the XX is more sensitive to spectral diffusion mediated by temporally charged defects than the X state, but additional investigations (see for example 5 ) are needed to test this hypothesis.

Supplementary Note 4: Source of entanglement degradation
Although a fidelity of 0.94 is high, it is not yet perfect. We have investigated this in more detail by using a simple model according to 6 . The density matrix of the model system in the [H XX H X , H XX V X ,V XX H X ,V XX V X ] basis is given by: where Using the measured lifetime of the exciton transition for T 1 , the value of the g (2) (0) to estimate k, and considering that the effect of cross-dephasing can be safely neglected 6 , i.e., g H,V = g H,V , the only unknown parameter entering in Supplementary   Equation 7 is the spin-scattering time. Here, we assume that T SS is mainly determined by the Fermi-contact interaction between the confined electron and the nuclear spins 7 , while the heavy-hole dephasing related to a dipole-dipole interaction 7,8 , is assumed to be weaker and is therefore not considered. Taking the values from the literature (T SS = 15 ns for GaAs QDs 9 ), we can therefore estimate the behaviour of the entanglement fidelity as a function of the FSS. This is shown in Supplementary Figure 4 (a) with and without including the effect of the background laser (as estimated by the value of the auto-correlation function for X, that is, k = 1 − g (2) (0) = 0.975 (see Supplementary Figure 5)). The obtained experimental data are in consensus with the theoretical curve and support the supposition that the reduced value of fidelity for QD2 is due to to the background of laser photons. Most importantly, this figure highlights that near-unity values (0.99) of entanglement fidelity can be obtained in QDs with suppressed FSS (s=0). This is in contrast to what is reported for InGaAs QDs where maximum values of around 0.9 were predicted 10 . It is therefore interesting to compare directly the two systems using the very same model. Supplementary Figure 4 (b) shows the results of such a comparison, as obtained by using the literature values for the spin scattering times (T SS = 1.9 ns for InGaAs 11 ) and by assuming identical lifetimes. The calculations indeed confirm that the maximum entanglement fidelity that can be reached in InGaAs QDs is roughly 10% lower than in GaAs QDs, that is, bound to values around 90%. Moreover, the calculations also highlight the importance of having short X lifetimes to reach high values of entanglement at non-zero FSS, although a combination of FSS=0 and short X lifetime is the key to reach the ideal levels of entanglement needed by the envisioned applications. From this perspective, another potential source of technical problems is the rejection of the stray light.
As the laser is spectrally separated from the X and XX line a fibre Bragg grating could be used to filter out the laser emission 12 .
Alternatively, rejection of the straight light can be achieved by decoupling excitation and collection. This solution -which has been already employed in 13 and 14 -can also be used in combination with different concepts for on-chip quantum optics 15 .

5/7 Supplementary Note 5: Evaluation of the entanglement fidelity
For calculating the fidelity from the 6 cross-correlation measurements reported in Fig. 3(b) and (c) of the main text, the raw counts at g (2) (0) within a time window of 1.6 ns (the bunching peak is within this time window) are summed up for all polarization settings. The degree of correlations is calculated via: XX,X − g (2) XX,X g (2) XX,X + g (2) XX,X , where g (2) XX,X is the co-and g (2) XX,X the cross-polarized correlation measurement, respectively, in the base µ (linear, diagonal and circular). In Supplementary Table 2  The fidelity is given by 6 : which yields f = 0.88 ± 0.01 and f = 0.94 ± 0.01 for QD2 and QD3, respectively. The errors are calculated by assuming a Poisson distribution for the correlation counts and propagated by Gaussian error propagation.