Abstract
The development of scalable sources of nonclassical light is fundamental to unlocking the technological potential of quantum photonics. Semiconductor quantum dots are emerging as nearoptimal sources of indistinguishable single photons. However, their performance as sources of entangledphoton pairs are still modest compared to parametric down converters. Photons emitted from conventional Stranski–Krastanov InGaAs quantum dots have shown nonoptimal 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 polarizationentangled 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 quantumdot entanglement sources in future quantum technologies.
Introduction
Any prototype of quantum devices that will be used in emerging quantum technologies^{1} has to fulfill a long list of requirements. The ideal source of quantum light, for example, should deliver single and entangled photons deterministically, with high purity, high efficiency, high indistinguishability and high degree of entanglement, and it should be compatible with current photonic integration technologies. Implementing such a source, however, is far from an easy task^{2,3}.
If we restrict the discussion to single photon sources^{4}, that is if we disregard entanglement, semiconductor quantum dots (QDs) have recently demonstrated their capability to fulfill all the requirements on the wish list, thus carrying great promise for the implementation of photonic quantum networks^{1}. This is eventually testified by the increasing interest of the community working with parametric down converters^{5,6,7}, the workhorse sources of nonclassical light that have been used to achieve a wealth of breakthroughs in quantum optics.
Besides single photons, QDs can also generate triggered polarizationentangled photons during the biexciton–exciton (XX–X) radiative cascade^{8}. Having in mind their performances as single photon sources, it would be natural to regard QDs as ideal entanglement sources. The reality, unfortunately, is quite distant from this idea. The main reason is that the development of scalable sources of entangled photons demands approaches that are much more advanced than those employed for single photons. First, the very possibility of using a QD to generate entangled photons poses severe constraints on its structural symmetry^{9}. Second, the efficient extraction of both X and XX photons requires sophisticated photonic microcavities or broadband nanowires^{10,11}. Third, twophoton resonant excitation schemes are needed to achieve truly ondemand generation of photon pairs and to avoid some of the decoherence processes limiting the indistinguishability of the emitted photons^{12,13}. Although the different points have been recently addressed (although never simultaneously on the very same device), nonideal levels of indistinguishability and degree of entanglement were reported so far^{11,12,14,15}. In particular for the latter parameter, all the experiments have shown fidelities to the expected Bell state that rarely exceed 80% (without inefficient postselection techniques)^{10,11,14,15,16}, even under resonant pumping^{12}. Assuming that the X states involved in the cascade are degenerate^{15} and ultraclean QD samples with reduced charge noise^{17} are studied, the degree of entanglement is mainly limited by two effects: recapture processes^{11,16} and random magnetic fields produced by the QD nuclei^{18,19}. The first one arises when the intermediate X levels are reexcited to the XX level before decaying to the ground state. The second effect is instead related to the hyperfine interaction^{20}, which couples the spin of the electron and hole forming the exciton with the spins of the QD nuclei, effectively giving rise to a fluctuating magnetic field (Overhauser field^{21}) that depolarizes randomly the X states. It is well known that these sources of entanglement degradation lower the maximum entanglement fidelity, and there are strategies to compensate for their deleterious effect, at least in principle.
Recapture processes can be suppressed by shortening the X decay time via the Purcell effect^{11} or/and using resonant excitation^{12,13}. Nuclear spinpolarization techniques^{22} can be used to polarize the nuclear ensemble with the drawback of inducing a finite effective magnetic field, which splits the X states^{23}. This splitting can be in turn compensated via additional external perturbations, such as strain, magnetic or electric fields^{9,24}. Even if theoretically feasible, this approach is not straightforward to implement experimentally, as the effect of a magnetic field on the X states depends strongly on its direction and can give rise to offdiagonal terms in the X Hamiltonian that cannot be easily compensated with external fixes.
In this work, we look at this problem from a different perspective. Instead of using Stranski–Krastanov InGaAs QDs—the standard choice in QD photonics—we focus on highly symmetric GaAs QDs grown via the droplet etching method^{25,26,27,28}. In this type of artificial atoms, the effect of the Overhauser field is much weaker than in InGaAs QDs^{18}. This is not only due to the absence of indium in the QD nuclei, which has a large nuclear spin of 9/2 compared to gallium with 3/2. GaAs/AlGaAs QDs are indeed free of intrinsic strain fields and composition gradients that have a relevant role on the hyperfine interactions^{20}. Moreover, the light and heavyhole mixing, an effect which also depolarizes X states^{29,30}, is minimized to values <5% due to the highly symmetric shape of the investigated QDs^{26}. The use of these novel QDs is not the only ingredient to push the degree of entanglement to its limits. Differently from previous attempts on droplet epitaxy GaAs QDs^{16}, we take advantage of twophoton resonant excitation schemes to coherently populate the XX state^{12}. On the one hand, this technique limits the effect of reexcitation. On the other hand, it allows us to demonstrate high visibilities of twophoton interference for consecutive photons emitted by a GaAs QD.
Results
Twophoton resonant excitation
We begin with the characterization of our (GaAs)/AlGaAs QDs, which were obtained by infilling nanoholes created by in situ droplet etching of AlGaAs layers (see Fig. 1a–1c and methods)^{25,26}. Atomic force microscopy of a representative nanohole (see Fig. 1c) reveals a highly symmetric shape. This shape, which has a strong impact on the QD structural symmetry, is of fundamental importance for the generation of entangled photons. Figure 1d shows a typical microphotoluminescence spectrum of a single GaAs QD under nonresonant excitation. It consists of an isolated X line at high energy and a bunch of neutral and charged multiexcitonic transitions at lower energy (roughly 4 meV below the X). While the detailed discussion of the origin of these states is beyond the scope of this work, we would like to point out that the XX transition is almost always overwhelmed by them and cannot be singled out easily under nonresonant pumping, even at the highest excitation power. This observation is qualitatively ascribed to the large number of confined states in the employed QDs, which lead to many possible charge configurations competing with the XX at high excitation power. In order to make the XX line visible and to gain coherent control over this state, we use a twophoton resonant excitation scheme^{13,31,32}. More specifically, we shape the excitation laser into 9 ps pulses (see Methods section) and we tune its energy to half of the energy between the XX and ground state (0) (see inset of Fig. 1e).
The effect of the twophoton resonant excitation in the emission spectrum is shown in Fig. 1e. The multiexcitonic states disappear and the XX line becomes visible with an intensity equal to that of the X. Two additional transitions appear (C_{1} and C_{2}), which are most probably related to a charged exciton and biexciton. These states, whose detailed investigation is left to future studies, are present in all the QDs we measured. To verify whether the QD is driven resonantly, we measured the power dependence of the intensity of X and XX (see Fig. 2a). Both transitions show welldefined Rabi oscillations^{31}, thus readily confirming that our excitation scheme allows for coherent control of the XX state. We note that, while the XX state and the crystal ground state are coherently coupled by the laser field, X emission stems from spontaneous decay of the XX state. Therefore the X population reflects that of the XX state and shows Rabi oscillations with the same periodicity. The damping of the Rabi oscillations with increasing laser pulse area most probably arises from phononinduced dephasing^{12}. For all subsequent measurements, the power has been set to the πpulse of the XX.
Highly indistinguishable photons
We now characterize the properties of the photons emitted by our QDs in terms of their purity and indistinguishability. In Fig. 2b,c, we report autocorrelation measurements performed on the X and XX lines, which allow us to estimate the value of the secondorder correlation function at zero time delay, being g^{(2)}(0)=0.007±0.004 and g^{(2)}(0)=0.002±0.002 for X and XX, respectively. This result clearly indicates an extremely high purity of our source.
Next, we study the photon indistinguishability of consecutive photons emitted by our QDs by performing a Hong–Ou–Mandeltype twophoton interference experiment^{33}. The QDs are excited every 12.5 ns by two πpulses separated by 2 ns and emit ideally two XX–X photon pairs per cycle. The photons are then guided to an unbalanced Mach–Zehnder interferometer equipped with 2 ns delay and are let to interfere at the beam splitter (BS) in co and crosspolarized configuration.
An example of the measurements for the XX are shown in Fig. 2d. For crosspolarized photons, the three central peaks around 0 time delay show the same amplitude, as expected for distinguishable photons. Contrarily, the strong suppression of the central peak for copolarized photons clearly indicates the occurrence of twophoton interference and pinpoints to a large indistinghuishability. A similar behaviour is observed for X photons, as shown in Fig. 2e. By Lorentzian fitting (see Supplementary Fig. 1 and Supplementary Note 1) and by taking into account the imperfections of the BS^{12,34,35,36}, we estimate visibilities of twophoton interference of 0.86±0.09 and 0.93±0.07 for X and XX photons, respectively (for details on the calculations, see Supplementary Note 1 and Supplementary Table 1). These values are among the highest reported so far for QDs driven under twophoton excitation and are very close to the values achieved for nearoptimal singlephoton sources^{35,37}. While the slightly lower value of visibility of the X can be explained by the time jitter introduced by the XX radiative decay, we emphasize that this is not a general feature of our QDs.
We performed the same measurements in eight additional QDs, and for some of them, we report the relevant results in Fig. 2e. Large visibility (exceeding 70%) of twophoton interference has been found in 80% of the QDs we measured, for both X and XX photons. The observed fluctuations can be explained by the different chargenoise configuration in each specific QD, as also reported in the literature^{35}. It is a wellknown problem that spectral diffusion induced by charge fluctuations can lead to a significant drop in the interference visibility at an excitation pulse separation >2 ns^{38}. For this reason, we compared the visibility at 2 and 12.5 ns pulse separation (see Supplementary Fig. 2) and observed a drop of about 30% at a longer separation time (for details, see Supplementary Note 2). This effect would reduce the maximum possible twophoton interference visibility of two remote QDs^{36}, which is a major issue for the realization of a quantum repeater. A possible solution to this problem is to apply an electric field across the QD so as to stabilize the charge environment^{35,39}.
Strongly entangled photons
Finally, we complete our study by demonstrating that our QDs emit photon pairs with a high degree of entanglement. Here we take advantage of two features of our QDs: First, their high structural symmetry (see Fig. 1a1c), which results in an extremely small average finestructure splitting (FSS) between the two X states^{25} and second the short radiative decay time of X (T_{1} around 250 ps, see Supplementary Fig. 3 and Supplementary Note 3). It is indeed well known that only if the FSS is smaller or comparable to the radiative linewidth of the transitions, a large degree of entanglement can be measured in a time integrated experiment^{40}. The used QDs have an average FSS of 3.9±1.8 μeV, as reported earlier^{25}. The FSS of QD1–3 are, respectively, 6.5±0.5 μeV, 1.3±0.5 μeV and 1.2±0.5 μeV. Therefore, we do not expect a high degree of entanglement for QD1 (see Supplementary Fig. 4) and only QD2 and QD3 are retained for the following study.
We start out with QD2 and perform a full quantumstate tomography^{41} to reconstruct the twophoton density matrix (see Fig. 3a) with the aid of a maximal likelihood method^{41}. In order to quantify the degree of entanglement, we extract the concurrence ζ from the density matrix and find ζ=0.84±0.05. Closer inspection of the density matrix reveals that the eigenstate associated with the largest eigenvalue (0.92±0.03) is not exactly the expected Bell state ^{40}, but rather , with α=0.12. The phase is most probably induced by the residual FSS and/or by the measurement setup and in particular by the presence of background photons of the resonant laser that could not be suppressed completely during quantum state tomography (see Supplementary Fig. 5).
In order to demonstrate that the high entanglement degree is not a feature of one specific QD, we extend the study to QD3. In this case, we took special care in rejecting the resonant laser and we extracted the fidelity to the expected Bell state via
where C are the correlation visibilities in the linear, diagonal and circular bases (see Supplementary Note 5). In fact, for QD2 (see Fig. 3b) we confirmed that the value of the fidelity obtained with this formula (0.88±0.01) is very close to the one extracted via full reconstruction of the density matrix (0.92±0.03) and, since α is small, to the largest eigenvalue (0.92±0.03). We have therefore used the six measurements reported in Fig. 3c to extract the fidelity from equation (1), which is found to be 0.94±0.01 (for details, see Supplementary Table 2). This value—which is higher than the one measured for QD2 due to improved rejection of the laser (see Supplementary Fig. 5 for comparison of the g^{(2)}(0) values)—is very close to the maximum achievable (see Supplementary Fig. 4). It is worth mentioning that we obtain this result without any temporal/spectral filtering of the emitted photons and without the aid of the Purcell effect^{37}, which can in principle be used to shorten the X radiative decay time even further, and to improve the level of entanglement and indistinghuishability to the maximum. The high degree of entanglement allows for violation of Bell’s inequality. Using the following expressions:
we obtain S_{RC}=2.52±0.02, S_{DC}=2.59±0.01 and S_{RD}=2.64±0.01. The latter parameter shows violation of Bell’s inequality by >60 standard deviations, a result that we achieve without any temporal or spectral filtering.
The measurements of Fig. 3c readily suggest the origin of this high level of entanglement. In fact, an almost perfect antibunching is observed for crosslinear, crossdiagonal and cocircular base. This result clearly indicates a strong suppression of X spinflip processes^{42} and of recapture, a result which is a consequence of the resonant excitation scheme used. Moreover, the fact that the correlation visibility is lower in the circular than in the linear/diagonal base suggests that the X dephasing induced by the hyperfine interaction is quite limited in these QDs^{19}. In fact, a theoretical model^{42}, which takes into account the effect of the Overhauser field on dephasing of the intermediate X states (see Supplementary Note 4) shows that this is exactly the reason why our GaAs QDs show such a high level of entanglement. The same model can also be used to highlight that the small deviation from unity is mainly ascribable to the nonzero values of the FSS. More specifically, we estimate that in GaAs QDs with suppressed FSS—reachable, for example via threeaxial strain engineering^{24}—the fidelity can reach values as high as 0.99, roughly 10% higher than what can be theoretically achieved with InGaAs QDs with FSS=0 with the same X decay time (see Supplementary Note 4).
Discussion
Our results show that the GaAs/AlGaAs material system, which has led to breakthrough discoveries in condensed matter physics and optoelectronics (such as the fractional quantum Hall effect^{43}) has also a great potential as entanglement resources for long distance quantum communication and in particular for quantum repeaters^{1,3}. In contrast to parametric downconverted sources of nonclassical light, low temperatures are usually required to use them as highly indistinguishable entangled photon sources^{38}, but this is not really a hurdle considering recent advances in the fabrication of compact, inexpensive and lowvibration cryocoolers^{44}. Beside the nearoptimal level of entanglement and indistinguishability of the emitted photons, our QDs emit in the spectral range of absorption resonances of rubidium atoms, a system which has proven its potential as longlived and efficient quantum memory^{45}. Moreover, the availability of downconversion techniques^{46} in combination with methods to convert polarization entanglement into timebin entanglement^{47,48} make our QDs suited for quantum communication at telecom wavelengths. A key ingredient to achieve high singlephoton purity, high twophoton interference and high level of entanglement is the twophoton resonant excitation. This excitation method requires stabilization of the laser energy and power, as small fluctuations in these parameters dramatically affect the biexciton preparation fidelity, thus making the scheme less attractive for the envisioned applications. However, we would like to point out that two alternative excitation regimes exist: phononassisted population^{49,50} and rapid adiabatic passage^{51,52}. While the first method uses an incoherent population of the XX via acoustic phonons, the second one allows the XX state to be coherently populated via chirped laser pulses. Both schemes are more robust against small variations of the energy of the laser as well as fluctuations of the laser pump power. In addition, to really use these QDs in reallife applications, such as quantum repeater networks, the next step is to integrate them in photonic structures for boosting the flux of the XX and X photons and to gain control over the energy of the emitted photons using external perturbations^{24,53,54,55}. We envisage that the implementation of these tasks in the GaAs QD system will soon lead to the fabrication of an ideal source of entangled photons.
Methods
Sample growth
The sample is grown by solid state molecular beam epitaxy. A GaAs (001) substrate is overgrown with a 270 nm buffer layer, on which three pairs of alternating Al_{0.2}Ga_{0.8}As/AlAs (with a thickness of 56 and 65 nm) are deposited. On top of this layer structure, which forms a low reflectivity back mirror of a planar cavity, a 120 nm thick layer of Ga_{0.6}Al_{0.4}As is grown. This layer is hosting the nanoholes, which are fabricated at 600 °C using aluminum droplet etching^{25}. The QDs are then obtained by depositing 2.5 nm of GaAs that fills the nanoholes during a 2 min annealing process. This layer is subsequently capped with 120 nm of Al_{0.4}Ga_{0.6}As acting as top barrier. Finally, two pairs of Al_{0.2}Ga_{0.8}As/AlAs layers were grown to complete the cavity. Due to the limited number of pairs used for the distributed Bragg reflector mirrors, we do not expect any appreciable Purcell effect but only a slight enhancement of the light extraction efficiency.
Measurement setup
All the measurements are performed at a sample temperature of 5 K in a heliumflow cryostat. For nonresonant excitation, a 488 nm continuous wave laser is used. The resonant twophoton excitation is performed with a titanium sapphire (TiSa) femtosecond laser featuring a bandwidth of 100 fs and a repetition rate of 80 MHz. The TiSa is shaped by a 4fpulseshaper setup into 9 ps pulses. The excitation lasers are focussed via an objective with a numerical aperture of 0.42 onto the sample, on which a hyperspherical solid immersion lens, made of zirconia, is placed. We estimate that the extraction efficiency of our device (with solid immersion lens on top of the sample) is around 1%. The same objective is used for the collection of the photoluminescence signal.
To reduce the back scattered laser light in resonant excitation, tunable notch filters with a bandwidth of 0.4 nm are placed in the beam path. Further stray light reduction is achieved by coupling the signal into polarization maintaining singlemode fibres placed after a nonpolarizing BS. Depending on the QD, the addition of a continuous wave laser at extremely low (not measurable) power to the resonant laser increases the X and XX count rate. This nonresonant laser—which does not give rise to background photons—most probably fills traps in the QD surrounding.
After the fibres, light is spectrally filtered by one or two independent spectrometers and detected by a chargecoupled device camera or—during photon correlation spectroscopy—by avalanche photodiodes connected to the correlation electronics. The temporal resolution of these detectors is about 500 ps.
The measurements of the FSS are carried out with a rotating λ/2 wave plate in front of a fixed polarizer at the entrance of a double spectrometer. Using Lorentzian fitting of both the X and XX lines, the FSS can be measured with subμeV resolution. For quantum state tomography, polarizers and properly oriented λ/2 and λ/4 wave plates are inserted in the X and XX path, before the two fibre inputs.
For the twophoton interference experiment, the femtosecond laser pulses are sent into an unbalanced Mach–Zehnder interferometer with a 2 ns delay to create two pulses every 12.5 ns. Both laser pulses are shaped with the 4fpulseshaper described above. A polarization maintaining fibre BS is used for twophoton interference of QD photons. The measurements for colinear and crosslinear configuration are obtained by placing polarizers before the input of the fibre BS. The latter is characterized with the same pulseshaped laser used to excite the QDs and shows a mode overlap (1−ɛ)=0.96±0.01, while the reflection and transmission coefficient are found to be 0.52±0.005 and 0.48±0.005, respectively.
Data availability
The data that support the findings of this study are available from the corresponding authors upon request.
Additional information
How to cite this article: Huber, D. et al. Highly indistinguishable and strongly entangled photons from symmetric GaAs quantum dots. Nat. Commun. 8, 15506 doi: 10.1038/ncomms15506 (2017).
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Acknowledgements
This work was financially supported by European Union’s Horizon 2020 research and innovation programme (SPQRel grant agreement no. 679183), European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 601126 (HANAS) and Austrian Science Fund (FWF): P 29603. We thank Dr Javier MartínSánchez, Dr Klaus Jöns, Florian Sipek, Mathias Gartner and Maximilian Prilmüller for fruitful discussions and for help at the very early state of the project.
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D.H. and M.R. performed the measurements with the help from R.T., D.H. performed the data analysis with the help from M.R., J.S.W. and R.T. Y.H. grew the sample with support of A.R. and O.G.S. H.H. and M.R. characterized the sample. D.H. and R.T. wrote the manuscript with the help from all the authors. R.T. and A.R. conceived the project. R.T. designed the experiments and supervised the project.
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Huber, D., Reindl, M., Huo, Y. et al. Highly indistinguishable and strongly entangled photons from symmetric GaAs quantum dots. Nat Commun 8, 15506 (2017). https://doi.org/10.1038/ncomms15506
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DOI: https://doi.org/10.1038/ncomms15506
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