Entanglement, a fundamental concept in quantum mechanics, is crucial for quantum technologies, particularly quantum networking. Photons, with their non-interacting nature and long coherence, are ideal carriers for quantum communication. However, their lack of mutual interaction makes entanglement distribution challenging. Second-order interference between indistinguishable photons partially addresses this, making indistinguishability a key requirement for entanglement distribution and photonic quantum repeaters. The exponential loss due to photon loss and the probabilistic nature of interference remain significant barriers to building a quantum network. Quantum repeaters, using entanglement swapping and purification, offer a solution by enabling entanglement distribution over long distances. Measurement-based quantum repeaters, utilizing multiphoton entangled states (cluster or graph states), provide redundancy against photon loss and probabilistic Bell measurements. The success of this architecture hinges on generating deterministic and indistinguishable photonic graph states, which presents a major scientific and technological challenge. This research addresses this challenge by developing a device capable of deterministically producing such states.

Single and entangled photons are typically generated via spontaneous emission from atoms or atom-like quantum emitters. Semiconductor quantum dots (QDs) offer advantages due to their integrability into electro-optical devices and high efficiency and emission rates, making them leading candidates for generating quantum light. Lindner and Rudolph proposed generating one-dimensional photonic cluster states using semiconductor QDs, employing a single QD-confined electron spin precessing in a magnetic field and driven by laser pulses. Each excitation deterministically emits a photon entangled with the QD spin state. This process repeats to generate a large cluster. The entanglement robustness depends on the ratio of radiative and spin precession rates, and spin coherence. Photon indistinguishability is determined by the optical transition. Schwartz et al. modified this proposal using a dark exciton, but the instability of the final state led to spectral broadening and distinguishable photons. This work builds upon previous research by addressing the limitations of using the electron spin for cluster state generation.

This study uses a heavy-hole (HH) spin as an entangler. The HH remains in a stable ground state after photon emission, leading to highly indistinguishable photons. Fine-tuning the external magnetic field controls spin precession and photon emission rates, optimizing entanglement robustness. A periodic sequence of controlled-NOT (CNOT) and Hadamard gates on the HH generates the cluster state. The device uses a sequence of resonantly tuned, linearly polarized laser π-pulses to achieve deterministic cluster state generation. Each pulse adds an entangled photon. Process tomography of a single cycle characterizes the entire cluster state. The experiments involve applying three cycles, detecting three photons, and using six single-photon detectors to project their polarization. Time-resolved single-photon detection rates, degree-of-circular-polarization (D) measurements of sequentially detected photons, and four cluster state witnesses (ω1, ω2, ω3, ω4) are measured and analyzed. The indistinguishability (ID) between sequential photons is measured using a Hong-Ou-Mandel interferometer setup. Localizable entanglement (LE) quantifies the entanglement robustness, with its characteristic decay length (ζLE) determining the entanglement length. A state-evolution model is used for comparison and validation of results. The sample consists of an InGaAs self-assembled semiconductor QD embedded in a planar microcavity. The experimental setup includes a pulsed laser, optical elements for polarization control and projection, spectral filters, and single-photon detectors. Extended Data Fig. 1 provides a schematic of the experimental system.

The study demonstrates a continuously generated string of indistinguishable photons entangled in a cluster state at gigahertz rates. At zero magnetic field, the photon indistinguishability reaches 95%. An optimized entanglement length of approximately ten photons is achieved. The four measured cluster state witnesses show good agreement with the calculated values from the state-evolution model. The localizable entanglement characteristic length (ζLE) reaches about 10 at an optimal magnetic field of 0.09 T. The photon generation is deterministic (every laser pulse produces a photon), although detection is not fully deterministic due to system efficiency limitations (currently better than 1%). The overall single-photon detection rate is about 5 MHz, limited by light harvesting efficiency, coupling efficiency, detector efficiency, and optical element transmission. The multi-photon detection rate is lower due to detector dead time. The study shows that the indistinguishability remains high for times relevant to cluster fusion. This high degree of indistinguishability, combined with the deterministic nature and high generation rate, is crucial for scaling up the cluster state.

The results demonstrate a significant advancement in generating high-quality photonic cluster states, addressing a key challenge in building scalable photonic quantum technologies. The high indistinguishability of the photons is crucial for enabling cluster fusion and creating higher-dimensional graph states. The deterministic nature of the photon generation at gigahertz rates significantly improves the efficiency of quantum information processing protocols. The experimental findings are consistent with the theoretical model, providing confidence in the understanding of the underlying physics. The achievement of a ten-photon entanglement length represents a substantial improvement over previous studies and opens new avenues for applications in quantum computing and communication.

This research demonstrates a significant advance in generating high-quality photonic cluster states with deterministic gigahertz rates and high photon indistinguishability. The achievement of a ten-photon entanglement length and near-perfect indistinguishability is crucial for building larger, more complex graph states for quantum computation and communication. Future work should focus on improving the overall system efficiency to enhance the multiphoton detection rate and further extend the entanglement length, perhaps through the use of a three-dimensional microcavity to increase brightness and improve the other figures of merit.

The current system's efficiency limitations, primarily in light harvesting, coupling, and detection, restrict the multiphoton detection rate. Improving the system efficiency is crucial to fully exploit the potential of the device for practical applications. The measurement of indistinguishability is limited to consecutive photons; further investigation is needed to verify the indistinguishability between more distant photons in the cluster. The study focuses on linear polarization; exploring other polarization schemes could offer further insights and improvements.