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Quantum dots as potential sources of strongly entangled photons for quantum networks

The generation and long-haul transmission of highly entangled photon pairs is a cornerstone of emerging photonic quantum technologies, with key applications such as quantum key distribution and distributed quantum computing. However, a natural limit for the maximum transmission distance is inevitably set by attenuation in the medium. A network of quantum repeaters containing multiple sources of entangled photons would allow to overcome this limit. For this purpose, the requirements on the source's brightness and the photon pairs' degree of entanglement and indistinguishability are stringent. Despite the impressive progress made so far, a definitive scalable photon source fulfilling such requirements is still being sought for. Semiconductor quantum dots excel in this context as sub-poissonian sources of polarization entangled photon pairs. In this work we present the state-of-the-art set by GaAs based quantum dots and use them as a benchmark to discuss the challenges to overcome towards the realization of practical quantum networks.


I. INTRODUCTION
After decades of fundamental research, quantum entanglement emerged as a pivotal concept in a variety of fields, such as quantum computation 1 , -communication [2][3][4] andmetrology 5 . A manifold of quantum systems are beeing investigated and photons stand out in many areas due to their robustness against environmental decoherence and their compatibility with existing optical fiber 6 and free-space 7 infrastructure. Non-local correlations were demonstrated in several photonic degrees of freedom, such as time-bin 8,9 , time-energy 10 , orbital angular momentum 11 , polarization 12 , spin-polarization 13,14 or in a combination of them ("hyperentanglement" 15,16 ). In quantum information processing the manipulation and measurement of entangled qubits plays a major role. Applications like quantum key distribution (QKD) with entangled qubits 4,17,18 require high source brightness, high degree of entanglement, transmission through a lownoise quantum channel and finally a straightforward measurement at remote communication partners, all with minimal losses. These prerequisites could be met by polarization entangled photon pairs 19 . Besides the most prominent sources based on spontaneous parametric down-conversion (SPDC) [20][21][22] , also semiconductor quantum dots (QDs) [23][24][25][26][27][28] are capable of generating polarization entangled photon pairs with a fidelity to a maximally entangled state above 0.97 26 . The probabilistic emission characteristics of SPDC sources prohibit a high brightness in combination with a high degree of entanglement 29 . This is not the case for QDs due to their sub-poissionian photon statistics 30 . Furthermore, in a real-world context, applications like QKD require entanglement to be communicated over large distances 6,7 to be practically relevant. Most transmission channels, like optical fibers, underlie damping (about 0.2 dB/km for common telecom-fibers), which severely limits the transmission range. This limitation can be alleviated by exploiting a concept of quantum communication 2 : The interconnection of multiple light sources in quantum networks 2,3 via the realization of a cascaded quantum repeater scheme with entangled photons and quantum memories 3,31,32 . In order to reach this goal, properties of the photon sources beyond the maximum entanglement fidelity become relevant, such as the photon indistinguishability 33 . In this work, we examine the key figures of merit of entangled photon pairs with an emphasis on the distribution of entanglement in a quantum network. We will start from the state-of-the-art focusing on GaAs QDs. Although the emission wavelength of about 785 nm is currently non-ideal for efficient fiber-based applications, GaAs QDs represent an excellent model system for the here discussed ideas due to their performance. Finally, we will outline recent concepts and approaches towards the realization of a viable quantum network.

II. POLARIZATION ENTANGLED PHOTON PAIRS FROM QUANTUM DOTS
A common scheme to generate entangled photon pairs with semiconductor QDs embedded in photonic strucures (see Fig. 1(a)) is by resonantly populating the biexciton (XX) state by a two-photon excitation (TPE) process 34 . The XX radiatively decays via the biexciton-exciton(X)-ground state cascade 35 , as depicted in Fig. 1(b). Ideally, the emitted photon pairs are in the maximum entangled Bell state |φ + = 1/ √ 2 (|HH + |VV ), with |H and |V the horizontal and vertical polarization basis states, respectively. The fidelity f |φ + of the real photon pair's state to |φ + is mostly determined by the fine structure splitting (FSS) S and lifetime T 1,X of the intermediate X state. In the absence of other dephasing mechanisms, the maximum achievable fidelity of an ensemble of photon pairs to a maximum entangled state is given by 36 where g (2) 0 is the multi-photon emission probability. In the case of GaAs QDs obtained by the Al droplet etching technique 24 , a high in-plane symmetry 25 results in average FSS values below 5 µeV, while the weak lateral carrier confinement 37 causes radiative lifetimes of about T 1,XX = 120 ps and T 1,X = 270 ps. The wavelength of the emitted light hereby lies around 780 nm (see Fig. 1(c)), with the XX and X photons separated by about 2 nm (4 meV), allowing them to be split by color. In contrast to SPDC-based sources 38 the pair-generation probability ε and the g (2) 0 of QDs are decoupled due to the sub-poissonian emission characteristics 35 . This led to demonstrated values of g (2) 0 = 8(2) × 10 −5 , with ε = 0.9 (measured via cross-correlation measurements between XX and X 39 ) under pulsed TPE 30 , as illustrated by the corresponding auto-correlation measurement in Fig. 1(d). Figure 1(e) displays a resulting two-photon density matrix ρ in the polarization space of the XX and X photons for an as-grown GaAs QD with S ≈ 0.4 µeV, acquired by fullstate tomography 40 under pulsed TPE at an excitation rate of R = 80 MHz. The fidelity deduced from this matrix as f |φ + = φ + |ρ|φ + is 0.97(1) By utilizing a specifically designed piezo-electric actuator 41 , capable of restoring the inplane symmetry and erasing the FSS of the QDs by strain, fidelity values up to 0.978(5) were demonstrated 26 . These results suggest that a modest Purcell enhancement of a factor 3 could alleviate remaining dephasing effects and push the fidelity up to 0.99, which would match with the best results from SPDC sources 38 . The minimum time delay 1/R between the pulses depends on the lifetimes T 1,XX and T 1,X , allowing for excitation rates up to R ≈ 1 GHz (without Purcell enhancement). This makes GaAs QDs a viable source for applications like QKD with entangled photons 4,17,18,42 , where high source brightness and near-unity entanglement fidelity are required in order to reach practical secure key rates in the high MHz or even GHz range. In order to reach this goal, however, a high photon-pair extraction efficiency is required, which is naturally limited in semiconductor structures due to total internal reflection at the air/semiconductor interface. A simple approach to increase the pair extraction efficiency η  (4) 44 and Purcell enhancement up to a factor 11.3 39 . Although a non-ideal entanglement fidelity due to the high FSS was reported in Ref. 44, these structures are compatible with the aforementioned strain tuning techniques, which could cancel the FSS to create a bright source of highly entangled photon pairs, applicable for QKD with key rates potentially in the GHz range.
A widely discussed and researched topic is the distribution of entanglement over basically arbitrary distances, for which sources operating at high pair emission rates are especially relevant. One approach is free-space transmission via satellites, where recently a distance of 1120 km was covered 7 . From the practical point of view it would be desirable to exploit the existing and well-established telecom optical fiber networks 6 . The obvious effect of fibers on the transmitted light is a uniform damping, which is about 0.2 dB/km for typical fibers in the telecom C-Band (1550 nm wavelength). When transmitting polarization entangled photons through optical fibers, however, also polarization mode dispersion 46 (PMD) has to be taken into account. PMD causes the principal states of polarization (PSPs) of the photons' wave packets to drift apart in time, leading to a degradation of the entanglement. The latter is proportional to the ratio of the total induced PMD τ between the two entangled photons and the length of the photon wave packets, given by 2 T 1 . The maximum achievable fidelity to a perfectly entangled state, derived from the 2-qubit density matrix in polarization space 46 , is then given by: For simplicity, we assume that T 1 = min(T 1,X , T 1,XX ), which represents a worst case scenario. If the two entangled photons experience exactly the opposite relative drift due to a maximum mismatch of the input modes with the PSPs, the PMD from the individual fibers add up to τ = τ 1 + τ 2 . A typical value for the PMD of a single mode fiber is D = 0.1 ps/ √ km (e.g. the type "SMF-28e+" from "Corning"), thus with the given lifetimes and a length of l = 200 km for each the X and the XX photon's fibers, the maximum possible fidelity for T 1 = 120 ps is still f PMD (2D √ l) > 0.99. For T 1 = 10 ps, the f PMD degrades to about 0.98. For T 1 = 1 ps, which is approximately the case for sources based on the SPDC process, the maximum fidelity drops to about 0.79, unless resorting to lossy measures like frequency filtering 47 or conversion to time-bin entanglement 9,48 . If the input polarization modes are aligned with the PSPs, the total PMD reads τ = τ 1 −τ 2 . Therefore, given that τ 1 = τ 2 , a configuration exists which exactly cancels out the effect of PMD and preserves the entanglement. This could be achieved by realigning the input modes to the PSPs during operation by polarization controllers 49 .
An additional effect to consider in the context of a fiberbased network is chromatic dispersion, which leads to a temporal broadening of the wavefunctions 50,51 . This lowers the success probability of two-photon interference, which we will discuss in the next section.

III. QUANTUM DOTS IN A QUANTUM REPEATER BASED NETWORK
Although transport of highly polarization entangled photons through fibers is possible 52 , the exponential damping will inevitably lead to an insufficient qubit rate. A possible solution to this problem is the realization of a quantum repeater scheme. The common approach is based on the DLCZ protocol (and its variants), that relies on spin-photon entangle-  53 . However, the probabilistic nature of the entangling scheme limits the entanglement creation 54 . Although an improved, deterministic version of the spin-photon based scheme was developed 55 , the achieved rates are still modest. An alternative scheme relies on the use of entangled photon pair sources, like QDs, interfaced with quantum memories capable of receiving and storing entangled states to increase the qubit rate 32 . This scheme relies on a cascade of entanglement swapping processes 56,57 among entangled photon pairs emitted by independent emitters. The teleportation of the entanglement is enabled by two-photon interference to perform a so-called Bell state measurement (BSM). The success of a BSM strongly depends on the photon indistinguishability, which, in turn, depends on the photon sources and can be experimentally accessed by probing the interference visibility in a Hong-Ou-Mandel (HOM) experiment 50 . For maximum visibility the spatio-temporal wave packets of the two photons involved in the BSM have to be identical and pure, i.e. no other physical system should contain information about the photon's origin. The latter point plays a crucial role in the case of QDs exploiting the decay cascade for entangled photon generation. The XX and X photons are correlated in their decay times 8,58,59 , which limits the maximum possible indistinguishability for both the XX and X photons to Figure 1(f) shows the result of a HOM experiment with two XX photons generated by a GaAs QD, upon excitation with two pulses separated by a 2 ns delay. The resulting visibility of 0.69 is typical for both the XX and X photons under TPE 43 and is close to the maximum according to Eq. (3). As a com-parison: For single X photons generated by resonant excitation a visibility of over 0.9 is achieved for the same QDs 60,61 . When interfacing two dissimilar QDs, the inherently stochastic nature of the epitaxial growth has to be considered, which primarily leads to varying emission energies. Further, imperfections in the solid-state environment of the QDs lead to inhomogeneous broadening due to charge noise 62 . This results in a jitter of the central emission energy by δ E (full width at half maximum) around a mean value at time-scales typically in the microsecond to millisecond timescale 63 . The jitter leads to a degradation of the indistinguishability described by 64 : with σ = δ E/2 √ 2ln2 the standard deviation of the Gaussian distribution of the energy jitter and Re[F(z)] the real part of the Faddeeva function of z = ih/(2 π √ 2 σ T 1 ). Figure 1(g) shows the result of a HOM experiment with two X photons from GaAs QDs with δ E ≈ 4 µeV and a resulting visibility of V = 0.51(5) 45 . The center wavelengths were previously matched by tuning the energy of one QD via a monolithic piezo-electric actuator.
We will now discuss possible solutions to overcome the two major indistinguishability degrading mechanisms in QDs discussed so far: The partial temporal entanglement in the XX-X decay cascade (Eq.(3)) and frequency jitter (Eq.(4)). Both effects are influenced by the radiative lifetimes T 1,XX and T 1,X , which can be modified by exploiting the Purcell effect in a cavity 39,44 . range covered scales with 2 L l 0 , with l 0 the total length of both fibers departing from one QD. This example serves as a demonstration how the entanglement fidelity evolves over multiple layers of swapping operations with photons generated by QDs. Fig. 2(a) depicts the final entanglement fidelity as a function of the Purcell factor P and the energy jitter δ E. In all simulations the natural linewidth of the exciton is set to 2.4 µeV for P = 1, corresponding to T 1,X = 270 ps. Values of P > 15 are unpractical, as the total relaxation time of the QD then approaches the typical excitation pulse width of about 10 ps. This primarily leads to an increasing re-excitation probability 65,66 , which is detrimental to the indistinguishability and the entanglement. In addition, PMD effects in optical fibers start to become relevant for such short wave packets. For the calculation of the fidelity we utilize the density matrix formalism for describing one entanglement swapping process with QDs 57 with a type of BSM which can detect two Bell states 36,67 (ψ + and ψ − ). In order to model a chain of entanglement swapping processes the formalism is applied recursively, assuming uncorrelated BSM success probabilities in successive steps. We simultaneously account for varying lifetimes caused by P and a decreased BSM success rate due to δ E (see supplementary for details). From the simulations we observe that already for two swapping processes the homogeneous Purcell enhancement alone cannot recover the entanglement fidelity sufficiently, as it merely alleviates the impact from inhomogeneous broadening on the indistinguishability, but the visibility degrading effect from the XX-X cascade is still at full force. Figure 2(b) depicts the case for an energy selective cavity 58 , which enhances the XX decay rate by a factor of 7 compared to the X, so that P X = P and P XX = 7P. This approach could strongly increase the BSM success rate and therefore the resulting entanglement fidelity. However, the finite temporal width of the excitation pulse, whose minimum value is set by the the limited spectral separation between X and XX and the necessity of suppressing laser stray light, sets a lower limit to the lifetimes -and therefore an upper limit to the Purcell enhancement -in order to limit re-excitation 65 . A compromise could be achieved by mild frequency filtering of the X photon, as illustrated in Fig. 2(c). Filtering partially erases the temporal information held by the X photon, leading to the same outcome as prolonging the X lifetime and hence decreasing the XX/X lifetime ratio as with the selective Purcell enhancement. In the simulations we assume a filter with Lorentzian transmission characteristics and a FWHM of δ E f and an energy jitter with a FWHM of δ E for both the X and the XX (see supplementary for details). We assume a frequency selective cavity with fixed Purcell enhancement of P X = 2 and P XX = 10. As a result of the filtering, the effective lifetime of the X signal increases while simultaneously reducing the impact of the energy jitter. Note that for the here investigated values of δ E the interference visibility again drops for δ E f values below the inhomogeneous broadening δ E. In addition, in the presence of a finite FSS, the BSM success rate drops when the filtered linewidth is in the order of the FSS or below 57 . From the simulations we can observe that with a low inhomogeneous broadening (< 0.2 µeV) and a moderate frequency filtering of about 4 µeV one could achieve an entanglement fidelity of approximately 0.93 at L = 2 and 0.85 at L = 3. A complete repeater scheme requires also quantum memories 68 which can store and retrieve a photonic qubit with high fidelity. To address the noise and bandwidth limitation of quantum memories two groups invented a cascaded absorption memory scheme, which is intrinsically noise-free. 69,70 Furthermore the possibility to use an off-resonant Raman transition in this cascaded scheme allows for large storage bandwidth, limited mainly by the available control laser power. Currently the main drawback of these schemes is the limited storage time, which is determined by the radiative lifetime of the upper state of the cascade (below 100 ns). Another promising approach is to use rare-earth doped crystals as quantum memories 71 , featuring performances that equalize, if not outperform, those of cold atomic ensembles 72,73 or trapped emitters in terms of efficiency 74 and coherence times 75 . These memories have shown a full quantum storage protocol with telecom-heralded quantum states of light 76 , and the first photonic quantum state transfer between nodes of different nature 77 . Furthermore, atomic frequency comb quantum memories were the first to be successfully interfaced with single photons emitted from a quantum dot 78 .

IV. FUTURE OUTLOOK
We conclude that bright and nearly on-demand sources of highly entangled photon pairs are on the verge of becoming reality. The groundwork has been laid through the development of semiconductor quantum dots (QDs) emitting highly entangled photons 26 , of advanced optical cavity structures 39,44 and technology capable of manipulating the symmetry and emission energy of QDs 41,79 . On-chip integration of QDs 80 and the implementation of electric excitation schemes 81 can further increase the practicability in emerging quantum technology.
The optimal wavelength (about 1550 nm) for transporting entangled photons through fibers is currently determined by the established telecom fiber infrastructure. Material systems to obtain QDs emitting at this wavelength are under development 6,82,83 and existing sources with emission at shorter wavelengths could be adapted by frequency downconversion 51 . Recently, a basic GHz-clocked quantum relay with QDs emitting directly in the Telecom-C band was demonstrated 84 . One of the greatest, yet rewarding challenges is the interfacing of dissimilar sources of entangled photons for multi-photon applications 85 and long-haul entanglement distribution 6,7 . The physical limits to the indistinguishability 8,59 set by the currently employed cascade for entangled photon pair generation 26,35 and fluctuations stemming from the solid state environment of QDs 25,62 pose intricate challenges for the years to come. As demonstrated in this work, the application of selective Purcell enhancement together with mild frequency filtering could alleviate the limit of indistinguishability of the entangled photon pairs. The different emission energies and radiative lifetimes of the biexciton (XX) or exciton (X) in QDs could be matched by utilizing strain- 41 and electric 86 degrees of freedom independently. Considering the quantum relay chains depicted in Fig. 2, three strain degrees of freedom can cancel the fine structure splitting and adapt the central energy of the XX or X to the next neighbor's. The electric degree of freedom can simultaneously be used to fine-tune the respective radiative lifetime and therefore the shape of the photonic wave-packet. By repeating this strategy through the whole relay chain for each QD, one could optimize the resulting entanglement fidelity of the final photon pair.
With these tools at hand, the next leap towards the demonstration of a functional quantum network will be interconnection of two dissimilar quantum dots via entanglement swapping 56,57 . The following steps could be to interface the photons performing the Bell state measurement with quantum memories 68,69,71,[74][75][76][77][78] and use the resulting entangled photon pairs for quantum key distribution 4,17,18 with efficiencies beating the direct transmission through fibers. From there on, the goal is to expand the system to a chain of multiple QDs and implement a quantum repeater scheme 3,31,32 in order to enhance the resulting entanglement fidelity and efficiency compared to a repeater-less distribution scheme.

DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.