Abstract
Semiconductor InAs/GaAs quantum dots grown by the Stranski–Krastanov method are among the leading candidates for the deterministic generation of polarizationentangled photon pairs. Despite remarkable progress in the past 20 years, many challenges still remain for this material, such as the extremely low yield, the low degree of entanglement and the large wavelength distribution. Here, we show that with an emerging family of GaAs/AlGaAs quantum dots grown by droplet etching and nanohole infilling, it is possible to obtain a large ensemble of polarizationentangled photon emitters on a wafer without any postgrowth tuning. Under pulsed resonant twophoton excitation, all measured quantum dots emit single pairs of entangled photons with ultrahigh purity, high degree of entanglement and ultranarrow wavelength distribution at rubidium transitions. Therefore, this material system is an attractive candidate for the realization of a solidstate quantum repeater—among many other key enabling quantum photonic elements.
Introduction
Solidstate sources that emit single pairs of entangled photons are a key element in quantum information technology. Polarizationentangled photons from atomic cascades were first used to test Bell’s inequality^{1,2}, but demonstrating scalable applications with single atoms is clearly a technological challenge. In 1988, Shih and Alley^{3} reported that the photon pairs generated from spontaneous parametric down conversion^{4} are polarization entangled and can violate Bell’s inequality, opening the door for various polarizationentanglementbased experiments. However, spontaneous parametric down conversion sources are characterized by Poissonian statistics, that is, a tradeoff has to be made between the source brightness and the multiphoton emission probability. This lack of ondemand singlephoton generation fundamentally limits their applications in complex quantum protocols.
Semiconductor InAs/GaAs quantum dots (QDs) grown by the Stranski–Krastanov method are among the leading candidates for the deterministic generation of polarizationentangled photons^{5,6,7,8,9,10}. As proposed by Benson et al.^{5}, the cascaded emission in single QDs from the biexciton to the ground state via the intermediate exciton states produces polarizationentangled photon pairs , where R and L denote right and lefthanded circular polarization, respectively. In real InAs/GaAs semiconductor QDs, however, the anisotropy in strain, composition and shape reduces the QD symmetry and mixes the two bright exciton states, resulting in two nondegenerate bright exciton states split by the fine structure splitting (FSS)^{11}. The final twophoton state has a timevarying form , where T_{1} is the radiative lifetime of the exciton, S the FSS and H and V horizontal and vertical linear polarization, respectively^{12}. To reduce the phase shift between the HH〉 and VV〉 twophoton components and to obtain a high degree of entanglement, the experimental strategies are to reduce the FSS S and/or the exciton lifetime T_{1}.
This is unfortunately no easy task. In the last decade there have been extensive efforts to generate entangled photons with InAs/GaAs QDs. The probability of finding suitable QDs in an asgrown sample is <10^{−2} (refs 13, 14), thus necessitating the use of postgrowth tuning techniques (such as thermal annealing, the optical Stark effect, magnetic, electric and strain fields) to eliminate the FSS^{15}. On the one hand, the fact that every single QD needs to be independently engineered imposes a great challenge for the practical application of QDbased devices. On the other hand, due to the electron–nuclear spin hyperfine interactions^{16,17}, the degree of entanglement of InAs/GaAs QDbased sources is generally low even at zero FSS. The best result so far yields an entanglement fidelity F=0.82 and concurrence C=0.75 (ref. 18). Alternatively, one can reduce the exciton lifetime T_{1} by using the Purcell enhancement in a cavity, or perform time gating before a significant phase shift T_{1}S/ħ between HH〉 and VV〉 takes place. The practical implementation of the former requires a simultaneous Purcell enhancement of both X and XX emissions^{6}, which is a nontrivial task. The latter discards a large portion of photons and reduces the source brightness significantly. Applying post selection, fidelities up to F=0.86 have been achieved using an InAsP QD containing InP nanowire^{19}.
Based on the above discussion, it is possible to obtain a large ensemble of QDbased polarizationentangled photon emitters by simultaneously incorporating a highly symmetric confinement potential, a short radiative lifetime and a weak electron–nuclear spin hyperfine interaction. The first attempt was reported by Juska et al.^{13}, where arrays of symmetric In_{0.25}Ga_{0.75}As_{1−δ}N_{δ} QDs were grown on the GaAs (111)B surface. They were able to obtain areas with an impressive 15% of entangled photon emitters with fidelities F in the range of 0.5 up to 0.72. Although the FSS is consistently below 4 μeV for these novel QDs, the exciton lifetime is quite long (1.8±0.6 ns). Nevertheless, the violation of Bell’s inequality was recently shown in electrically injected pyramidal QDs^{20}. Kuroda et al.^{21} demonstrated the generation of entangled photons (with fidelity up to F=0.86) using highly symmetric GaAs/AlGaAs QDs grown on the GaAs (111)A surface by droplet epitaxy. Although the exciton lifetime is short (560 ps), the FSS are relatively large (with a mean value of 10±5 μeV) and the hyperfine interaction of the exciton with nuclear spins is significant in this system^{21}.
In this work, we show that a large ensemble of asgrown polarizationentangled photon emitters can be obtained, using an emerging family of GaAs/AlGaAs QDs grown by droplet etching and nanohole infilling. These QDs exhibit very small FSS (with a mean value of 4.8±2.4 μeV) and short radiative lifetime (T_{1}<220 ps). Under pulsed resonant twophoton excitation, a coherent excitation of the biexciton state can be achieved (with pronounced Rabi oscillations up to 7π), and all measured QDs emit single pairs of entangled photons with ultrahigh purity and high degree of entanglement (fidelity F up to 0.91, concurrence C=0.90). The QDs presented in this work offer a deterministic wavelength control and ultranarrow wavelength distribution, specifically tailored to match the optical transitions of rubidium. Thereby, we envision a hybrid quantum repeater that incorporates QDgenerated entangled photon qubits interfaced with a rubidium vapourbased quantum memory.
Results
Sample growth
The QDs presented in this work are fabricated by solidsource molecular beam epitaxy. The in situ droplet etching^{22,23} is used to create selfassembled nanoholes with ultrahigh inplane symmetry^{24,25}, which are subsequently filled and capped to obtain embedded solidstate quantum emitters^{26,27}. Figure 1a shows a sketch of the processes involved in the QD formation. The initial point is a GaAs (001) substrate that has been deoxidized and overgrown with a GaAs buffer layer followed by 200 nm of Al_{x}Ga_{1−x}As. During the growth As_{2} is provided by a cracker cell running at 650 °C. First, Al is deposited under low arsenic pressure (<10^{−8} mbar), forming droplets on the surface at 630 °C. Driven by concentration gradients, the concurring dissolution of As through the droplets and diffusion of Al towards the substrate induces the formation of nanoholes with high symmetry. In a following annealing step at 630 °C the structures crystallize under a reestablished As atmosphere of 10^{−7} mbar. Then, the nanoholes are filled with GaAs and subsequently overgrown by Al_{x}Ga_{1−x}As to obtain the isolated QDs with threedimensional carrier confinement.
Envisioning a hybrid QD–atomic interface as a promising solidstate quantum memory^{28,29}, it is desirable to match the QD emission with atomic transitions, illustrated by the inset in Fig. 1b. For this purpose several samples with varying GaAs infilling amounts have been grown, targeting the Rb D1 and D2 transition lines at a wavelength of 794.9 and 780.2 nm, respectively. Figure 1b shows the exciton wavelength distribution for two different samples with 2 nm (blue) and 2.75 nm (green) GaAs deposited at a growth rate of 0.47 and 0.5 μm h^{−1}, accordingly. The statistics on more than 50 QDs across an area of 1 cm^{2} on each sample show an unprecedented control on the central emission wavelengths, with mean values of 779.8±1.6 nm and 796.3±1.3 nm. The wavelength distributions, or the socalled inhomogeneous broadenings, are among the narrowest for semiconductor QDs and are about 5 times smaller than that of a typical selfassembled InAs/GaAs QD sample. A similarly narrow inhomogeneous broadening can be observed in pyramidal QDs^{20,30}.
Together with the high homogeneity, the QDs also exhibit high symmetry due to the negligible intermixing and a virtually strainfree interface between GaAs and AlGaAs. Previous work suggests that a reduction of the amount of deposited Al and an increase of the deposition rate can enhance the nanohole symmetry^{25}. Following this trend, a single pulse of 0.09 nm excess Al at a growth rate of 0.8 μm h^{−1} (corresponding to AlAs growth) was used for our samples. The optimized growth protocols lead to highquality QDs. Figure 1c shows the statistical distribution of the FSS for the GaAs/AlGaAs QD sample studied in this work (blue) and for a typical InAs/GaAs QD sample grown by partial capping and annealing (grey). The total number of measured dots is 45 and 114, respectively. The GaAs QDs feature an average FSS of only 4.8±2.4 μeV that is among the best values reported so far^{13,21,25}. With these superior spectral properties, the investigated samples are promising candidates for the generation of polarizationentangled photons.
Resonant excitation of the biexciton
The major challenge to the realization of entangled photon pair emission from dropletetched GaAs/AlGaAs QDs is the effective excitation of the biexciton (XX) state. In pyramidal GaAs/AlGaAs QDs the biexciton has been observed^{31}, as well as in QDs based on GaAs/AlGaAs quantum well thickness flucutations^{32,33}. So far there are only few reports about the observation of a biexciton in GaAs/AlGaAs QDs grown by local droplet etching^{21,27}, presumably due to the low internal population probability under nonresonant excitation. Due to the optimized growth process we are able to observe strong XX emissions even with aboveband excitations. We select a QD from the sample emitting close to the Rb D2 transition (∼780.2 nm) and excite it by pumping the surrounding higherbandgap AlGaAs with a pulsed laser. The resulting spectrum (Fig. 2a), which is relatively clean in a broad range, shows several different excitonic transitions: The transition with the highest intensity is the exciton emission (X) at λ=778.5 nm. Among several redshifted transitions, the XX emission is the strongest (λ=780.1 nm).
To efficiently excite and to coherently drive the XX transition, we pump the twophoton resonance of the XX state by using a pulsed laser that lies spectrally in between the X and XX transitions. This excitation scheme has already been proven very effective in case of InAs/GaAs QDs^{34}. Making use of tunable notch filters we can effectively suppress the laser background. Hence, a very pure spectrum showing mostly the XX and X emissions can be observed (Fig. 2b). The integrated intensities are the same for both emissions, strongly indicating a close to unity efficiency for the cascaded emission process^{34}.
To obtain the evidence of pure singlephoton emissions from both XX and X, we perform an autocorrelation measurement at πpulse excitation using a standard Hanbury Brown and Twiss setup and the results are shown in Fig. 2c. The autocorrelation function g^{(2)}(τ) plotted over the photon arrival delay τ shows a clear absence of counts at zero delay and proves the ultrahigh purity singlephoton emission. The backgroundcorrected correlation function is measured to be for XX and for X.
Next, we measure the luminescence lifetime T_{1} by recording an intensity correlation between the excitation laser pulse and the arrival time of the photons (see Fig. 2d). The XX shows a simple exponential decay, which is fitted taking into account the convolution with the detector response function. The X decay shows a longer rise time since the state has to be populated first by the decay of the XX state. The extracted lifetimes are T_{1,XX}=112 ps and T_{1,X}=134 ps, and these are among the lowest values recorded for asgrown semiconductor QDs. The ideal lifetimelimited linewidth of the exciton emission is therefore ΔE=4.9 μeV, close to the mean value of the FSS in our sample.
To further evaluate the resonant twophoton excitation scheme, we record the intensity of XX and X photons while changing the excitation power. The result is summarized in Fig. 2e by plotting the intensity over the pulse area θ that is proportional to the square root of the excitation power. Clear Rabi oscillations are observed, which are oscillations of the intensity due to a coherent rotation on the Bloch sphere between the ground state 0〉 and the excited state XX〉. The abscissa is normalized in units of π to the first maximum of the XX intensity, where the pulse area is equal to π. Intensity oscillations up to 7π are observed. The mean intensity is decreasing for higher excitation powers, which may be caused by several different factors like chirp in the excitation pulse or scattering processes in the QD^{34}. Increasing the power also leads to an increase in the oscillation frequency. This is a fingerprint of the twophoton excitation process in clear contrast to onephoton resonant excitation, where the frequency remains constant.
Evaluating the degree of entanglement
After realizing an efficient coherent control over the XX decay in GaAs QDs we now evaluate the degree of entanglement in the polarization of the emitted photons. A QD with a FSS of S=2.3 μeV is chosen in the experiment, since it represents a large portion (∼22%) of QDs in the sample (see Fig. 1c). The QD is excited with πpulses for an efficient preparation of the biexciton state. To measure the degree of polarization correlation we send the stream of XX and X photons onto a 50:50 beam splitter. Each subsequent signal arm contains a quarterwave plate, a halfwave plate and a polarizer to select the polarization in an arbitrary basis. After spectral selection of XX and X photons in the first and second signal arm, respectively, they are sent to singlephoton detectors. Coincidence counting hardware is used to obtain the secondorder correlation function between XX and X photons for the selected polarization direction.
Figure 3a shows six crosscorrelation measurements obtained for three bases of copolarized and crosspolarized photons: the rectilinear (HV), diagonal (DA) and circular (RL) basis. As expected for an ideal entangled twophoton state , a strong bunching (antibunching) at τ=0 is observed for copolarization (crosspolarization) in the rectilinear and orthogonal bases, whereas this behaviour is reversed for the circular basis set. The correlation contrast for a chosen basis set is given by^{17}
with denoting the secondorder correlation function at zero delay in collinear, and in orthogonal bases. For the three illustrated basis sets the following contrasts are obtained:
The fidelity F of the measured quantum state to the ideal state can then be obtained by^{17}
which exceeds the classical limit F=0.5 by more than 12 standard deviations.
A more comprehensive picture of the measured entangled twophoton state is given by the density matrix representation. We performed crosscorrelation measurements for 16 different base combinations to account for the 16 unknown variables in the density matrix ρ. The measured values for g^{(2)}(0) are then used to construct a density matrix following the procedure presented in ref. 35. Since the thereby obtained density matrix violates important basic properties like positive semidefiniteness, the maximum likelihood estimation is employed to find the appropriate density matrix that is the closest to the measured results. The resulting matrix is shown in Fig. 3, split into the real part (Fig. 3b) and imaginary part (Fig. 3c). The strongest features are observed in the outerdiagonal realpart matrix elements, which are close to 0.5, while all other elements are close to zero. This is in agreement with the expected entangled state whose density matrix should have only nonzero values of 0.5 in the outerdiagonal elements. The small (but nonzero) real values in the offdiagonal elements indicate a weak spin scattering process in the QD. The finite imaginary offdiagonal values represent a small phase difference between HH〉 and VV〉, presumably caused by the joint effect of a finite FSS and an accumulated phase due to the optical setup. From this density matrix, we obtain a fidelity F (after background corrections^{34}) to the state of
which is very close to the value of 0.88±0.03 obtained from the 6 crosscorrelation measurements in Fig. 3a.
Another measure for nonclassical properties of a quantum state is the concurrence C^{35}. Using the acquired density matrix, a value of C=0.90 (raw data without correction: C=0.81) is obtained. This is not only surpassing the best value measured for InAs/GaAs QDs with zero FSS^{18}, but is also the highest value obtained for any QD entangled photon source so far. The high values for fidelity and concurrence are especially remarkable considering the finite fine structure splitting of S=2.3 μeV, which already significantly degrades the entanglement in case of InAs/GaAs QDs^{17,36}.
Since the phase shift T_{1}S/ħ between HH〉 and VV〉 states is significantly reduced due to the very short lifetime T_{1} in this system, we expect that the generation of entangled photons should be also possible for QDs with even higher values of S. Therefore, we select six dots representing the whole range of FSS measured in the sample. By measuring six crosscorrelations in three basis sets for each dot, their entanglement fidelities F are obtained. Figure 4 shows the values of F plotted as a function of the FSS (black circles), overlaid on the FSS distribution in the sample (grey histogram). The data from Zhang et al.^{36} including a Lorentzian fit are shown as a reference for typical InAs/GaAs QDs (orange line). Remarkably, all of the measured dots show a clear signature of entangled photon emission with F>0.5. Even the QD with S=9.8 μeV, which represents the QDs with the largest FSS in our sample, shows a fidelity F=0.59±0.05. We want to highlight that the measured dots were not preselected according to certain conditions apart from their FSS. All measurements lead to the conclusion that nearly 100% of the QDs in this sample show entangled photon emission.
Another outstanding feature of this material system are the significantly higher fidelities compared with that of the typical InAs/GaAs QDs, mostly originating from the weak electron–nuclear spin hyperfine interactions in this type of QDs^{16,17,37}. To better understand the obtained values, we plot two theoretical curves showing the fidelity over the FSS for radiative lifetimes of T_{1}=120 ps (red curve) and T_{1}=220 ps (blue curve), the typical range for the measured exciton lifetimes. We modelled the fidelity following the work by Hudson et al.^{17} that includes the influence of the FSS and lifetime τ as well as crossdephasing and spin scattering:
with
Here, k denotes the probability that the measured photon pairs originate from the dot. We estimate it to be k=0.97 due to the measured autocorrelation measurements presented in Fig. 2b. The factor denotes the fraction of the QD emission that is unaffected by both crossdephasing and spinscattering processes, while only considers spinscattering processes. Since the presented data show no trend that would lead to F<0.5 for large FSS, we expect the influence of spin scattering processes on the entanglement to be negligible in this material system. Considering spin scattering due to the Overhauser field of the nuclear spins present in the dot, a spinscattering time of T_{SS}=15 ns can be assumed^{38}. This is, however, two orders of magnitudes longer than the measured radiative lifetimes and therefore barely contributes to the degradation of the fidelity. On the other hand, in typical InAs/GaAs QDs this effect can be significantly stronger in case of high concentrations of spin9/2 indium^{38}, leading to much lower fidelity values for InAs/GaAs QDs in Fig. 4 (ref. 36). Since the fidelities at small FSS are very high for the GaAs/AlGaAs QDs, we neglect crossdephasing processes in the model. It is clear that the trend in all our fidelity measurements can be well represented by the employed model.
Discussion
In this work, we propose a new type of solidstate polarizationentangled photon source based on an emerging family of GaAs/AlGaAs QDs. These QDs can be grown with unprecedented wavelength control, ultrasmall FSS and short radiative lifetime. The efficient and coherent excitation of the biexciton state in the GaAs/AlGaAs QDs is achieved by employing a resonant twophoton excitation scheme. The combination of a highly symmetric confinement potential, a short radiative lifetime and a weak electron–nuclear spin interaction in this material system enables entanglement fidelities up to F=0.91 and a concurrence of C=0.90. These are among the highest values ever reported for QDbased entangled photon sources. Most remarkably, the whole set of measurements draws an unambiguous conclusion that we have obtained a large ensemble of entangled photon emitters on a single semiconductor wafer, with almost 100% of QDs in the sample having fidelities F>0.5. Moreover, a great fraction of QDs are expected to exhibit high fidelities F>0.8 without any postgrowth tuning.
We envision that a number of key enabling quantum photonic elements can be practically implemented by using this novel material system. A particularly important example is a quantum repeater as the backbone for longrange quantum communication. One requirement for a quantum repeater is the storage of entangled photon pairs within a quantum memory at a millisecond timescale^{39,40}. Promising candidates for exceedingly long coherent storage are ensembles of lasercooled atoms^{41,42,43,44}, single trapped atoms^{45}, impuritydoped crystals^{46,47} and optomechanical systems^{48}. A potential material system for the storage of photons in the telecom wavelength range is erbiumdoped Y_{2}SiO_{5} (ref. 49). However, the storage capabilities of the latter do not yet exceed the nanosecond timescale. Thus, the QDs presented in this work are specifically tailored to match the optical transitions of rubidium that is among the most mature storage candidates. Thereby, we envision a hybrid quantum repeater that incorporates QDgenerated entangled photon qubits that can be mapped reversibly in and out of a rubidium vapourbased quantum memory.
To fully reach that goal, however, high source brightness and photon indistinguishability also have to be ensured, for example by implementing QDs into microcavities. Additionally, entangled photons from different sources have to be spectrally matched to meet the requirements for quantum interference. Different strategies have been experimentally implemented to address this issue^{50,51}. Furthermore, efficient frequency conversion of the photons after storage to the telecom wavelength range would be desirable for practical longrange quantum communication.
During the preparation of the manuscript we noted a similar work by Huber et al.^{52} that was executed at the same time. Both works are rather complementary to each other: while their focus lies on high entanglement fidelities and good indistinguishability values of a few selected dots, we show an exceptionally high yield to obtain strongly entangled photon sources for rubidium hybrid systems.
Methods
Optical excitation
The photoluminescence experiments were conducted at T=4 K by placing the sample in either a helium bath or a helium flow cryostat. As excitation laser for the above band and twophoton excitation, a pulsed Ti:Sa laser with 76 MHz repetition rate was used, which generated pulses with a duration of 3 ps. To spectrally narrow the laser pulse it was sent to a homebuilt pulseshaping setup before it was coupled into a singlemode fibre. The excitation laser was then sent to the sample in the cryostat using a beam sampler and focused by a lens or an objective that was also used for the collection of the QD emission. We used halfball solid immersion lenses to increase the photon collection from the sample. The fluorescence signal was coupled into a polarizationmaintaining singlemode fibre. To suppress the resonant laser background, two consecutive tunable notch filters were employed before signal detection with a spectrometer.
Correlation measurements
To measure entanglement, the collected light from the QD was split by a 50:50 beam splitter into two arms, each containing a quarterwave plate, a halfwave plate and a polarizer. The two beams were then coupled into polarizationmaintaining singlemode fibres. After eliminating the residual laser using notch filters, each light path was fed into a monochromator to select the XX or X transition, respectively. The streams of photons were then detected by avalanche photodiodes, whose signals were processed by a timecorrelated single photon counter. The integration times in the presented measurements ranged from 20 to 30 min for each correlation measurement, with count rates in the range of 2,500 c.p.s. up to 5,000 c.p.s. and a time binning of 64 ps.
Characterization of optical properties
We measured the FSS of the sample by rotating the halfwave plate in the entanglement measurement setup by α while rotating the quarterwave plate by 2α. By obtaining highresolution spectra for multiple values of α it was possible to fit the emission lines and determine an oscillation amplitude of the peaks spectral centre position with subμeV accuracy that corresponds to the FSS. The FSS in the InAs/GaAs QD reference sample was determined using the same method, but simply by rotating a halfwave plate in front of a linear polarizer. The lifetimes T_{1} were obtained using an avalanche photodiode with a short response function that was measured by using 3 ps laser pulses to be FWHM ≈100 ps. The measured response function was used to obtain the convoluted theoretical fits.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
Additional information
How to cite this article: Keil, R. et al. Solidstate ensemble of highly entangled photon sources at rubidium atomic transitions. Nat. Commun. 8, 15501 doi: 10.1038/ncomms15501 (2017).
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Acknowledgements
The work was financially supported by the ERC Starting Grant No. 715770 (QDNOMS), the BMBF Q.ComH (16KIS0106) and the European Union Seventh Framework Programme 209 (FP7/2007–2013) under Grant Agreement No. 601126 210 (HANAS). We thank E. Zallo, N. Akopian, K.D. Jöns and A. Rastelli for help and fruitful discussions, and Y. Li, X. Zhang, B. Eichler, R. Engelhard, S. Nestler, S. Baunack and S. Harazim for technical support.
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F.D. conceived the experiment and supervised the project together with O.G.S. who directed the research. The samples were grown by R.K. and the optical measurements were performed by M.Z. and R.K. with help from B.H. and Y.C. Data Analysis was performed by R.K. and M.Z. with help from J.Z. The manuscript was written by R.K., M.Z. and F.D. with input from all the authors.
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Keil, R., Zopf, M., Chen, Y. et al. Solidstate ensemble of highly entangled photon sources at rubidium atomic transitions. Nat Commun 8, 15501 (2017). https://doi.org/10.1038/ncomms15501
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