Quantum dots as potential sources of strongly entangled photons: Perspectives and challenges for applications in quantum networks

After decades of fundamental research, quantum entanglement emerged as a pivotal concept in a variety of fields, such as quantum computation,¹ ‐communication,^2–4 and -metrology.⁵ A manifold of quantum systems are being investigated and photons stand out in many areas due to their robustness against environmental decoherence and their compatibility with existing optical fiber⁶ and free-space⁷ 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 (“hyper-entanglement”^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 low-noise 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 downconversion (SPDC),^20–22 semiconductor quantum dots (QDs)^23–28 are also 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 so far 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, 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. All of the general concepts introduced in the following, however, are also valid for different material systems, such as InGaAs QDs,^23,34–37 whose emission wavelength can be extended to the telecom C-band, where the attenuation in silica fibers has a minimum. In the final section, we will outline recent approaches toward the realization of a viable quantum network.

The source brightness of QDs is mostly bound to the extraction efficiency, 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 $η E ( 2 )$ from less than $10 − 4$ to about 0.01 is to embed the QDs in a lambda cavity defined between two distributed Bragg reflectors and adding a solid immersion lens on top,⁴⁸ see Fig. 1(a). A pair extraction efficiency of 0.373(2) has been recently reported for GaAs QDs embedded in antenna structures consisting of a semiconductor membrane with a back metal mirror and a top solid immersion lens made of GaP.⁴⁹ Recently, circular Bragg resonators (CBRs) have demonstrated values of $η E ( 2 ) = 0.65 ( 4 )$⁵⁰ and Purcell enhancement up to a factor 11.3.⁵¹ Although a non-ideal entanglement fidelity due to the high FSS was reported in Ref. 50, 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 these considerations, we realize that fiber-based networks with QDs can greatly benefit from the robustness against PMD, especially compared to SPDC sources. However, for emission wavelengths significantly below the telecom bands, as in the case of GaAs QDs and most used InGaAs QDs coherently grown on GaAs substrates, the range remains severely limited. For this purpose, different material systems for QDs emitting in the telecom bands are being developed. An entanglement distribution experiment with an InGaAs QD emitting in the telecom O-band over a distance of about 18 km has already been demonstrated,⁶ while the quality of QDs emitting in the telecom C-band is rapidly approaching that of dots emitting at shorter wavelengths.^34,37,57 As an alternative, frequency conversion techniques can be utilized to adapt the emission wavelengths, although at the cost of efficiency.³⁵ 

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.^50,51 Figure 2 illustrates concatenated entanglement swapping processes with a depth of $L = { 1 , 2 , 3 }$⁠, i.e., a chain of quantum relays forming the backbone of quantum repeaters. The number of QDs required is $2 L$⁠, while the range covered scales with $2 L l 0$⁠, with l₀ being the total length of both fibers departing from one QD. This example serves as a demonstration on how the entanglement fidelity evolves over multiple layers of swapping operations with photons generated by QDs. Figure 2(a) depicts the final entanglement fidelity as a function of the Purcell factor P and the energy jitter δE in multiples of the natural X linewidth of $δ E 0 =$ 2.4 μeV at 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,^69,70 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 QDs⁶³ with a type of BSM which can detect two Bell states^40,71 (⁠ $| ψ + ⟩$ 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 δE (see the supplementary material 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,⁶⁴ which enhances the XX decay rate by a factor of 7 compared to the X, so that $P X = P$ and $P X X = 7 P$⁠. 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 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.⁶⁹ 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 δE for both the X and the XX (see the supplementary material for details). We assume a frequency selective cavity with fixed Purcell enhancement of $P X = 2$ and $P XX = 10$⁠. This asymmetry in enhancement could be achieved by carefully designing the lateral size of the previously introduced CBRs.⁵⁰ 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 δE the interference visibility again drops for $δ E f$ values below the inhomogeneous broadening δE. In addition, in the presence of a finite FSS, the BSM success rate drops when the filtered linewidth is on the order of the FSS or below.⁶³ From the simulations, we can observe that with a low inhomogeneous broadening (⁠ $< 0.1 δ E 0$⁠) and a moderate frequency filtering of about $1.5 δ E 0$ 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,⁷² 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.^73,74 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,⁷⁵ featuring performances that equalize, if not outperform, those of cold atomic ensembles^76,77 or trapped emitters in terms of efficiency⁷⁸ and coherence times.⁷⁹ These memories have shown a full quantum storage protocol with telecom-heralded quantum states of light,⁸⁰ and the first photonic quantum state transfer between nodes of different nature.⁸¹ Furthermore, atomic frequency comb quantum memories were the first to be successfully interfaced with single photons emitted from a quantum dot.⁸² 

We want to mention at this point that recently an alternative repeater scheme⁸³ was proposed, which eliminates the necessity of quantum memories, but instead shifts the challenge toward the realization of large-scale photonic cluster states.

We conclude that bright and nearly on-demand sources of highly entangled photon pairs are on the verge of becoming reality. The ground work has been laid through the development of semiconductor quantum dots (QDs) emitting highly entangled photons,²⁶ of advanced optical cavity structures^50,51 and technology capable of manipulating the symmetry and emission energy of QDs.⁴⁴ On-chip integration of QDs⁸⁴ and the implementation of electric excitation schemes⁸⁵ 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,^6,34,37,57 and existing sources with emission at shorter wavelengths could be adapted by frequency conversion.³⁵ Recently, a basic GHz-clocked quantum relay with QDs emitting directly in the Telecom-C band was demonstrated.⁸⁶ One of the greatest, yet rewarding challenges is the interfacing of dissimilar sources of entangled photons for multi-photon applications³⁶ and long-haul entanglement distribution^6,7 in quantum networks.^2,3 The physical limits to the indistinguishability^8,65 set by the currently employed cascade for entangled photon pair generation^26,39 and fluctuations stemming from the solid state environment of QDs^25,66 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-⁴⁴ and electric⁸⁷ degrees of freedom independently. Considering the quantum relay chains depicted in Fig. 2, three strain degrees of freedom can cancel the fine structure splitting (FSS) 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 toward the demonstration of a functional quantum network will be the interconnection of two dissimilar quantum dots via entanglement swapping.^62,63 Several groups are currently developing devices which merge the concepts of circular Bragg reflectors (CBRs)^50,51 with the tuning of the in-plane stress tensors⁴⁴ of the QDs. CBRs are ideally suited for this purpose, since they are fabricated on a dielectric-metal structure which can be virtually placed on any other substrate, and in particular, on piezoelectric actuators. By carefully designing the dimensions of the CBRs, these devices could provide the necessary magnitude and asymmetry of the Purcell factors for the XX and the X emission for a high photon indistinguishability, while eliminating remaining FSS and matching the energies of the involved QD's photons via the three strain degrees of freedom. These concepts are compatible with alternative material systems emitting at telecom wavelength,^6,34,37,57 which will allow to reach longer distances with optical fibers and possibly to utilize the established telecom fiber network. The next steps could be to interface the photons performing the Bell state measurement with quantum memories^{72,73,75,78–82} 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.

See the supplementary material for theoretical considerations about the evolution of entangled states in a chain of quantum relays.

Christian Schimpf is a recipient of a DOC Fellowship of the Austrian Academy of Sciences at the Institute of Semiconductor Physics at Johannes Kepler University, Linz, Austria. This project has received funding from the Austrian Science Fund (FWF): FG 5, P 29603, P 30459, I 4380, I 4320, and I 3762, the Linz Institute of Technology (LIT) and the LIT Secure and Correct Systems Lab funded by the state of Upper Austria and the European Union's Horizon 2020 research and innovation program under Grant Agreement Nos. 820423 (S2QUIP), 899814 (Qurope), 871130 (ASCENT+), and 679183 (SQPRel).

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