Abstract
Light states composed of multiple entangled photons—such as cluster states—are essential for developing and scalingup quantum computing networks. Photonic cluster states can be obtained from singlephoton sources and entangling gates, but so far this has only been done with probabilistic sources constrained to intrinsically low efficiencies, and an increasing hardware overhead. Here, we report the resourceefficient generation of polarizationencoded, individuallyaddressable photons in linear cluster states occupying a single spatial mode. We employ a single entanglinggate in a fiber loop configuration to sequentially entangle an evergrowing stream of photons originating from the currently most efficient singlephoton source technology—a semiconductor quantum dot. With this apparatus, we demonstrate the generation of linear cluster states up to four photons in a singlemode fiber. The reported architecture can be programmed for linearcluster states of any number of photons, that are required for photonic oneway quantum computing schemes.
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Introduction
Optical quantum technologies include a wide range of applications, from quantum sensing^{1,2}, to quantum communication^{3}, and computing^{4,5,6}. Entanglement is the most common resource for these applications^{7}, exploiting various degreesoffreedom, e.g., polarization, timefrequency, orbital angular momentum, and spatial modes. The generation of high quality large cluster states can be used for photonic oneway quantum computing^{5,6,8}, which is favorable compared to the widely used Turinglike gatebased model in solidstate systems^{4}. Moreover, photonic cluster states have been proposed to implement measurementbased quantum communication networks^{9,10,11}, an architecture that promises long distance quantum communication at higher rates compared to other memorybased counterparts.
Cluster states are a special class of multipartite graph states that show improved robustness to loss compared to GHZstates, or Wstates^{12,13,14}. They can be generated using singlephoton sources and entangling operations. So far, they have been implemented using probabilistic nonlinear sources of photon pairs up to six photons, by the χ^{(2)} parametric downconversion (PDC), and the χ^{(3)} fourwave mixing processes, or, for continuous variable encoding, by multimode squeezing in optical parametric oscillators^{6,15,16,17,18,19}. Photonic GHZ states of up to twelve entangled photons have also been obtained in this manner^{20,21}. The common approach consists of multiple sources and multiple well balanced paths to manipulate the photons. In this approach, it is important to operate in a regime where the probability to generate a pair is low (typically few percents) in order to limit multipair emission. These low source efficiencies make the protocols difficult to scale up to large photon numbers, in addition to an increased resource budget when employing multiplexed schemes^{22,23}. A more scalable way to produce large photonic cluster states has been proposed in 2009^{24}, making use of a single quantum emitter embedding a spin acting as a quantum memory. A proofofconcept experimental demonstration has been provided with a semiconductor quantum dot up to two photons^{25}. However, this was obtained at very low generation rates, and with considerable challenges to allow further scalability, such as the need for longer spin coherence times and efficient polarizationindependent photon extraction.
In this work, we demonstrate an approach for the generation of photonic cluster states that takes advantage of an already proven and available technology for singlephoton generation—semiconductor quantum dots (QDs)^{26}—and a recent proposal for entanglement generation based on temporal delayloops^{27}. Quantum dots generate singlephotons on demand with nearunity indistinguishability^{28,29,30,31,32,33}, and high singlephoton purity. In addition, they can have high infiber brightness (defined as the probability to have a singlephoton coupled into a singlemode fiber per excitation pulse), typically one orderofmagnitude larger than heralded singlephoton sources^{28,29}. This allows for an exponential increase in multiphoton generation rates, which has already been used for Boson sampling^{34,35,36}, and onchip quantum walks^{37}. We employ a fiber delayloop apparatus to sequentially entangle photons successively generated by a bright QD singlephoton source. Our experimental demonstration brings the record for the number of entangled photons from a single emitter from two photons^{32,38,39} to four photons. Linear photonic cluster states of two, three, and four photons are obtained, with a four photon generation rate of ~10 Hz, assuming perfect detection efficiency. Our compact entangling apparatus allows for both entanglement generation and polarization state analysis. Additionally, we define a parameter, the scalingratio, to quantify prospects of scalability, and to allow comparison between different implementations.
Results
Experimental scheme
Figure 1a presents the principle of the proposed scheme. A single quantum dot positioned in an optical cavity serves as an efficient singlephoton source. Periodical excitation with optical pulses leads to the emission of a stream of singlephotons. The separated emission times enable individual addressability of each photon. The singlephotons are then sent into an entangling gate, where the time between emissions is tuned to match the length of a delayloop, that serves as a quantum memory. As a result, the combined system constitutes a source of linear cluster states encoded in the polarization degreeoffreedom, and individually addressable in the time domain. With this protocol, linear cluster states of any length can be produced, controlled by the number of consecutive photons sent into the entangling apparatus. Figure 1a depicts the case for four photons. All entangling operations occur at the same entangling gate, hence the low resource requirements of our approach.
Figure 1b depicts the physical implementation of our protocol. The singlephoton sources used here are made of a single InGaAs quantum dot deterministically positioned in an electrically connected pillar cavity^{28,40} with optical resonances around 925 nm. The electrically controlled emittercavity coupling ensures efficient collection of photons through accelerated spontaneous emission into the cavity mode. The QD transition is coherently controlled with resonant excitation pulses from a Ti:Sapphire laser operating at 81 MHz repetition rate. Maximum source efficiency is achieved by setting the excitation at πpulse level, and the single photons are collected in a crossed polarization scheme. In this work, we used several sources with various characteristics: sources based on either neutral or charged excitons, with infiber brightness between 4 and 15%. The detected singlephoton rate outoffiber varies from 0.8 to 3 MHz using standard silicon avalanche photon detectors (APD) with 25% detection efficiency typically. The degree of indistinguishability M for photons generated by a single source at various time delays was measured for all sources, and varies from 0.77 to 0.95, the higher values being obtained with spectral filtering to remove contribution from phonon sidebands (see Supplementary Note 6). The laser pump driving the singlephoton source is modulated by an electrooptic intensity modulator (EOM), that creates the required sequence for entangling the desired number of photons.
The entangling apparatus is implemented in an allfibered compact device, packaged into a standard 19" rack mountable box, see inset in Fig. 1b. It has one singlemode fiber input, a fiber delayloop about 15 m long, and one "fusion" gate^{41} implemented by a polarizing beamsplitter (PBS). The output of this first PBS1 is sent to a singlemode fiber from where the entangled photons emerge. The analysis setup is also included in the same box. It consists of another projecting PBS2, whose two fiber outputs exit the box. Four electricallydriven polarization controllers, labeled EPCi, contain four voltagecontrolled birefringent elements each. The birefringence is induced by mechanically squeezing the fiber. The squeezing axes are oriented at 45° relative to each other. Thus, this configuration allows full control over the photons’ polarization. EPC1 is positioned before the delay loop, and used to align the source photon polarization. The next two polarization controllers EPC2 and EPC3, are inside the loop. EPC2 compensates for the fiber arbitrary polarization rotation between PBS1 and the alignment of the first element of EPC3. As the incoming photon into this first element is perfectly aligned along the ordinary birefringent axis, it only experiences a temporal phase shift rather than a polarization rotation. This is used as a phase scanning mechanism in our setup, described later on. After applying the phase, the other three elements of EPC3 are set to compensate for the arbitrary rotation of the fiber between EPC3 and PBS1. EPC4 is positioned after the delay loop, and is used to align the photon polarization at the output of PBS1 and the input of PBS2. At the output of PBS2, two singlephoton detectors are temporally synchronized to the laser clock frequency, and the time analysis of the state is controlled using a customdesigned FPGA controller. The required temporal synchronization is between the repetition rate of the laser used to excite the source, the generated pulse sequence of the EOM that selects the input photon pulses, and the detection events. Once this is achieved between two photons, any larger number of injected photons will also be synchronized as they all experience the same loop delay time (see Supplementary Note 1).
Entangling protocol
The sequential operation of the entangling apparatus can be described in the following manner. The excitation laser is switched on and off using the EOM to create a series of consecutive laser pulses, corresponding to the temporal sequence of singlephotons to be entangled. Each modulated pump sequence is preceded by two empty cycles in order to ensure that the delay loop is empty at the beginning of the protocol. The generated singlephotons are injected into the sequential entangler, see Fig. 1b, and their state is set to the diagonal polarization relative to PBS1 orientation \(\leftp\right\rangle =\frac{1}{\sqrt{2}}(\lefth\right\rangle +\leftv\right\rangle )\) via EPC1, where \(\lefth\right\rangle\) and \(\leftv\right\rangle\) designate the horizontal and vertical polarization states, respectively. Similarly, the antidiagonal state is defined as \(\leftm\right\rangle =\frac{1}{\sqrt{2}}(\lefth\right\rangle \leftv\right\rangle )\). After the first photon leaves PBS1, it is in a spatial superposition of being transmitted as the \(\lefth\right\rangle\) (reflected as the \(\leftv\right\rangle\)) state. Consequently, the protocol succeeds if the first photon enters the loop, verified by post selection of not detecting a photon at the first timebin. Inside the loop, EPC2 compensates for arbitrary polarization rotations, and EPC3 controls the birefringent phase φ, and rotates the \(\lefth\right\rangle\) state of EPC2 orientation to the \(\leftp\right\rangle\) state of PBS1. Each photon is delayed for τ ≃ 74 ns, corresponding to six laser cycles. When the photon inside the loop arrives to PBS1, it is timed to entangle with a new photon from the single photon source. This is achieved by fine adjustment of the pump laser repetition rate.
Postselecting each photon to exit from a different port of the entangling PBS1, the two diagonal photons are projected onto the maximally entangled state:
where the subscripts 1, 2 refer to the photon detection times τ, 2τ. Photon 1 has left the loop towards the detectors. Photon 2 remains in the loop, where it is rotated by EPC3 to the p/m polarization basis, resulting in the state \(\frac{1}{\sqrt{2}}\left(\left{h}_{1}{p}_{2}\right\rangle +\left{v}_{1}{m}_{2}\right\rangle \right)\), which is a twophoton linear cluster in graph representation^{42}. The conditional detection of photon 1 at time τ and photon 2 at time 2τ or later, verifies the postselection condition.
When a third photon enters the setup, it arrives at the entangling PBS1 at the \(\leftp\right\rangle\) state. The outcome of the entangling PBS1 is:
Thus, the new photon is entangled with the two previous photons into a GHZ state \(\frac{1}{\sqrt{2}}\left(\left{p}_{1}{h}_{2}{p}_{3}\right\rangle +\left{m}_{1}{v}_{2}{m}_{3}\right\rangle \right)\), where the photon remaining in the loop (now photon 3) is rotated by EPC3. This threephoton GHZ state is a linear cluster in graph representation. When a fourth photon enters the setup, repeating the above protocol, the resulting entangled state is not a GHZ state, but the fourphoton linear cluster (LC) state:
The conditional detection of a photon at each of the first n − 1 time slots starting at τ and on, as well as detecting the last nth photon at time nτ or later, verifies the postselection condition for all the projections.
Local unitary operations on each photon may transform \(\left{\psi }_{{\rm{LC}}}^{(4)}\right\rangle\) to other equivalent graph states^{42}. When more photons come in a timely manner, they are entangled into an evergrowing linear cluster state^{27}. Figure 2 depicts the quantum logic circuit implementation corresponding to this protocol.
Quantum state analysis
In our protocol, correlations are detected by photon measurements at consecutive time slots. The polarization analysis procedure is performed by applying X or Z Pauli operators by EPC4 and PBS2 to the first n − 1 photons. The last nth photon inside the loop is projected on PBS1 (the entangling PBS). If projected onto the \(\lefth\right\rangle\) polarization, it exits the loop, and if projected onto \(\leftv\right\rangle\), it stays inside for another cycle. Thus, the last photon polarization is analyzed by its arrival time to either detector.
The projection of n photons results in 2^{n} possible amplitudes. In order to demonstrate the quantum nonlocality of the produced states, the nphoton amplitudes are interfered^{43,44}. This nonlocal interference is achieved by rotating the measurement basis of the first n − 1 photons to the diagonal basis via a Hadamard rotation. In this case, half of the photons’ probability amplitudes interfere constructively while the other half interfere destructively. The difference between these two amplitude groups is indicative of the level of quantum interference. This level can be quantified by \({V}_{n}={\rm{Tr}}\left({X}^{\otimes n}\hat{\rho }\right)\), where \(\hat{\rho }\) is the density matrix of the generated nphoton state. In order to accumulate more information about the nonlocal interference, the phase φ is applied by the operator \({Z}_{\varphi }^{\otimes (n1)}\otimes I\) to the n − 1 photons, where I is the unit operator and \({Z}_{\varphi }=I\cos (\varphi /2)iZ\sin (\varphi /2)\). When this phase is scanned, the different amplitude probabilities, and thus also V_{n} oscillate, revealing more information about the degree of entanglement (see Supplementary Note 2). The maximal value of V_{n}(φ) is used to evaluate the quality of the generated state.
Figure 3 shows the resulting phasedependent oscillations in V_{n}, for the cases of two, three and four photons. Assuming that the main source for imperfect interference is the twophoton indistinguishability M, the predicted twophoton, and threephoton quantum interference levels are \({V}_{2}(\varphi )=M\cos (\varphi )\), and \({V}_{3}(\varphi )\,=\,{M}^{2}\frac{1\,\,\cos (2\varphi )}{2}\). The corresponding fourphoton value V_{4} has an upper theoretical limit of \(\frac{2}{3\sqrt{3}}\,\simeq\,0.38\), which limits the experimental sensitivity. Therefore, we present \({V}_{4^{\prime} }={\rm{Tr}}\left(X\otimes I\otimes X\otimes X\hat{\rho }\right)\) which can reach 1, as this observable is part of the stabilizer group of \(\left{\psi }_{{\rm{LC}}}^{(4)}\right\rangle\) before the rotation of the last photon (see Supplementary Note 3). The dependence of this fourphoton interference level on φ is \({V}_{4^{\prime} }(\varphi )\,=\,{M}^{2}\frac{1+\cos (2\varphi )}{2}\) (see Supplementary Note 4). Figure 3c shows our experimental results for \({V}_{4^{\prime} }\). The measurements presented in Fig. 3 were obtained with two QD sources. The first one, corresponding to a positively charged QD operated with a 10 pm spectral filtering to reduce the contribution of phonon sideband emission, showed a resulting infiber brightness of about 4%, and M = 0.95 ± 0.01. It allows observing high interference levels for two and three photons. The other one corresponds to a negatively charged QD with an infiber brightness of about 15% but a reduced indistinguishability of M = 0.77 ± 0.01. It allows implementing the scheme up to four photons within a measurement time of 10 h for the complete data set. Although the indistinguishability was below the typical one observed with these QD sources^{45}, this source still allows the generation of fourphoton cluster states with an improved scalability, as discussed later on.
We have repeated the measurements using several of the mentioned sources based on negatively charged, positively charged or neutral dots, the latter two showing higher M. For some measurements, the indistinguishability was further increased using an etalon filter, but at the cost of a reduced count rate. In Fig. 4a, the quantum interference values of these measurements are presented for various values of M, obtained by standard HongOuMandel (HOM) interference^{46,47}, and the g^{(2)}(0) value from secondorder intensity correlations^{48} (see Supplementary Note 6). The solid lines represent the theoretical expected nphoton interference levels, showing good agreement with experiment.
It is possible to define a bounded witness measure \({\mathcal{W}}\,\le\,\tilde{{\mathcal{W}}}\)^{49} that detects entanglement of cluster states when \(\langle \tilde{{\mathcal{W}}}\rangle\,<\,0\). This inequality would also be satisfied for equivalent states up to a local unitary operation, but those equivalent states require large transformations to the cluster state and are highly improbable in our experiment (see Supplementary Note 5). From the experimental results presented in Fig. 3 we obtain the values for the twophoton, threephoton, fourphoton measurements: \(\langle {\tilde{{\mathcal{W}}}}_{2}\rangle =0.32\pm 0.01\), \(\langle {\tilde{{\mathcal{W}}}}_{3}\rangle =0.22\pm 0.01\), \(\langle {\tilde{{\mathcal{W}}}}_{4}\rangle =0.04\pm 0.02\) certifying the generation of cluster states. In addition, the results from the more indistinguishable source are \(\langle {\tilde{{\mathcal{W}}}}_{2}\rangle =0.45\pm 0.02\), \(\langle {\tilde{{\mathcal{W}}}}_{3}\rangle\,=\,0.41\,\pm\,0.02\). All are negative and imply the generated entanglement. Furthermore, as the state fidelity is related to its witness as \({\mathcal{F}}=\frac{1}{2}{\mathcal{W}}\), all of the quoted values imply a better than 50% fidelity.
Imperfect photon indistinguishability M limits the entanglement length, defined as the longest possible linear cluster state such that positive concurrence is found between the first and last photons of the chain when all the others are measured^{27}. This length is an upper limit for how far can quantum information flow along the linear state during the oneway quantum computation procedure^{5}. We find that for an nphoton entanglement length, the threshold interference value is \({V}_{n}=\frac{1}{3}\), independent of n. All the presented results are above this threshold. Moreover, our results correspond to maximum entanglement lengths between 23 and 5 photons (see Supplementary Note 8). In addition to this criterion, the threephoton genuine entanglement can be evaluated by the V_{3} measure, as the threephoton cluster state matches a GHZ state. The interference threshold value for a threephoton GHZ state is \(\frac{1}{2}\)^{50}, which is well exceeded by all our measurements, with values ranging from V_{3} = 0.62 ± 0.04 to V_{3} = 0.90 ± 0.08, depending on the source used (see Fig. 4a).
Scaling ratio
It is instructive to quantify and compare between the scalability prospects of our scheme using a QD source and those of heralded PDC sources. To this end, we define the scaling ratio r—the reduction factor of detection rates when one photon is added to the protocol. The smaller the scaling ratio, the better the scalability, where the ultimate goal is to reach the ideal value of r = 1, enabling the deterministic entangling of any number of photons. In our demonstration, the entangling operation has a 50% chance to succeed, thus this setup can only reach a value of r = 2. The detection rate for nphoton events is:
where R is the singlephoton repetition rate, and the η’s represent various system efficiencies. Most efficiencies apply to every photon, such as the detector efficiency η_{d}, the system loss without the delay loop η_{s}, the delay loop (memory cycle) loss η_{l}, and the source brightness η_{b}, including both its quantum yield and overall optical collection efficiency into a singlemode fiber including spectral filtering. One other efficiency doesn’t apply to the first photon, the entangling gate efficiency η_{g}. Thus, the scaling ratio is:
Figure 4b presents this scaling ratio as a function of the twophoton interference level for various theoretical and experimental situations. The use of a probabilistic gate with η_{g} = 0.5 limits the scaling ratio to r ≥ 2 (dashed red line) considering that η_{d} = η_{s} = η_{b} = η_{l} = 1. An intrinsic limitation arises when operating the present scheme with heralded PDC sources. For such sources, the twophoton interference reduces when increasing the source efficiency. The solid blue line represents the dependence of the scaling ratio on the twophoton interference level \(r\,=\,2\frac{1\,+\,{V}_{2}}{1\,\,{V}_{2}}\), considering η_{d} = η_{s} = η_{l} = 1 and η_{g} = 0.5 (see Supplementary Note 9). It represents an intrinsic upper limit for PDC sources, a limit that could only be overcome by multiplexing schemes^{23}, yet at the cost of increasingly demanding resources and reduced singlephoton repetition rate R.
The symbols in Fig. 4b present a scaling ratio values for various implementations. The solid blue symbols correspond to an equivalent scheme but implemented with a PDC source and a free space setup^{27}. The halffilled blue symbols correspond to the predicted values where the same experimental scheme would be implemented using the best pulsed PDC source currently available^{20,21}. The red symbols correspond to the present work, with measured efficiency values of η_{d} = 0.25, η_{s} = 0.7, η_{l} = 0.75, 0.04 ≤ η_{b} ≤ 0.15 depending on the source used. The lightblue and lightred data points extrapolate previous implementations, predicted ones and the present experimental results to the case η_{d} = 0.9, since such detection efficiencies are currently available at all considered wavelegnths by replacing the silicon APDs with superconducting nanowire singlephoton detectors^{51}. This allows for a better comparison between different implementations.
The presented comparison shows that our current results, obtained with a lossy and imperfect setup, already approaches the upper limit of lossless PDC sources (see Supplementary Note 9). Further improved scaling ratios are thus expected to be within reach with QD sources in the near future. The QD source brightness can be increased by a factor of up to 2 by changing the excitation schemes following recent propositions^{52,53}. The fibered brightness could also be increased by using a larger numerical aperture collection lens and engineering a better mode matching with the single mode fiber. Moreover, the setup efficiencies η_{l} and η_{s} could also be improved by reducing losses arising mostly from imperfect fiber coupled PBSs, increasing efficiencies to η_{s} = η_{l} = 0.9. In addition, fast polarization elements could double the generation rate by ensuring that the first photon always enters the loop. The cluster state generation rate including all of these improvements will increase by about 2 × 11^{n} relative to our current results. Without changing the pump repetition rate, a 10photon event per 30 s and a 12photon event per 40 min are expected, presenting a scaling ratio of r = 9.
Using the scaling ratio, a comparison between our approach to the generation of cluster states from a source of entangled photon pairs^{6,15} is not possible. Such schemes use setups that can only generate states of a fixed number of photons, and have to be physically extended in order to accommodate for more. Nevertheless, there are other suggested methods that can be examined using the scaling ratio parameter. Using ancillary photons and linear optical elements have good prospects for scalability, as it can achieve deterministic photon entanglement. However, the deterministic generation of a fourphoton cluster state with the most efficient scheme, requires 26 entangled photon pairs^{41}, thus becoming advantageous only for scaling ratio values r < 1.06 (see Supplementary Note 10). A better scaling ratio than r = 2 is also possible when using a spin–photon interface to remove the probabilistic operation of the entangling gate^{24}. Such approach remains very challenging experimentally: a first implementation reached a scaling ratio of 700 for a twoqubit fidelity of 0.73 ± 0.06^{25}. As long as our method includes the 50% postselection bottleneck, its scaling ratio is limited to r ≥ 2. Nevertheless, it does not require entangled photon sources nor ancillary photons nor spin–photon interaction.
Discussion
In conclusion, we have reported the generation of multiphoton linear cluster states using a single quantum emitter coupled to a compact entanglingloop configuration. The measurement of nonlocal quantum interference demonstrates genuine multiparticle entanglement up to four photons. As our protocol relies only on a single quantum emitter and a single entangling gate, the scheme can provide the best possible scalability ratios using linear optics. The present experimental demonstration, although using both imperfect quantum dot sources and entangling apparatus, already demonstrates a scaling ratio on a par with the best possible level predicted for heralded PDC singlephoton sources. Straightforward technical improvements, both on the source operation side and on the setup design, will allow reaching larger photon numbers in the near future. Additional delay loops could be used to generate cluster states of higher dimensionality. Finally, removing the last bottleneck of the 50% probabilistic entangling operation is also foreseeable, considering recent progress in engineering of photon–photon interactions using natural or artificial atoms^{54,55,56,57}. Thus, the present multiphoton entanglement scheme promises a path for scaling up quantum computation and communication protocols^{58,59,60}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nat. Photon. 5, 222–229 (2011).
Pittman, T. B., Shih, Y. H., Strekalov, D. V. & Sergienko, A. V. Optical imaging by means of twophoton quantum entanglement. Phys. Rev. A 52, R3429–R3432 (1995).
Bennett, C. H. & Brassard, G. Quantum cryptography: public key distribution and coin tossing. Proc. IEEE Int. Conf. Comput. Syst. Signal Process. 175–179 (1984).
Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001).
Raussendorf, R. & Briegel, H. J. A oneway quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001).
Walther, P. et al. Experimental oneway quantum computing. Nature 434, 169–176 (2005).
Einstein, A., Podolsky, B. & Rosen, N. Can quantummechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935).
Rudolph, T. Why I am optimistic about the siliconphotonic route to quantum computing. APL Photonics 2, 030901 (2017).
Zwerger, M., Dür, W. & Briegel, H. J. Measurementbased quantum repeaters. Phys. Rev. A 85, 062326 (2012).
Zwerger, M., Briegel, H. J. & Dür, W. Measurementbased quantum communication. Appl. Phys. B 122, 50 (2016).
Azuma, K., Tamaki, K. & Lo, H.K. Allphotonic quantum repeaters. Nat. Commun. 6, 6787 (2015).
Dür, W. & Briegel, H. J. Stability of macroscopic entanglement under decoherence. Phys. Rev. Lett. 92, 180403 (2004).
Hein, M., Dür, W. & Briegel, H. J. Entanglement properties of multipartite entangled states under the influence of decoherence. Phys. Rev. A 71, 032350 (2005).
Greenberger, D. M., Horne, M. A., Shimony, A. & Zeilinger, A. Bell’s theorem without inequalities. Am. J. Phys. 58, 1131–1143 (1990).
Lu, C.Y. et al. Experimental entanglement of six photons in graph states. Nat. Phys. 3, 91–95 (2006).
Bell, B. A. et al. Experimental demonstration of a graph state quantum errorcorrection code. Nat. Commun. 5, 3658 (2014).
Reimer, C. et al. Highdimensional oneway quantum processing implemented on dlevel cluster states. Nat. Phys. 15, 148–153 (2019).
Larsen, M. V., Guo, X., Breum, C. R., NeergaardNielsen, J. S. & Andersen, U. L. Deterministic generation of a twodimensional cluster state. Science 366, 369–372 (2019).
Asavanant, W. et al. Generation of timedomainmultiplexed twodimensional cluster state. Science 366, 373–376 (2019).
Wang, H. et al. Experimental tenphoton entanglement. Phys. Rev. Lett. 117, 210502 (2016).
Zhong, H.S. et al. 12photon entanglement and scalable scattershot boson sampling with optimal entangledphoton pairs from parametric downconversion. Phys. Rev. Lett. 121, 250505 (2018).
Mendoza, G. J. et al. Active temporal and spatial multiplexing of photons. Optica 3, 127–132 (2016).
Kaneda, F. & Kwiat, P. G. Highefficiency singlephoton generation via largescale active time multiplexing. Sci. Adv. 5, eaaw8586 (2019).
Lindner, N. H. & Rudolph, T. Proposal for pulsed ondemand sources of photonic cluster state strings. Phys. Rev. Lett. 103, 113602 (2009).
Schwartz, I. et al. Deterministic generation of a cluster state of entangled photons. Science 354, 434–437 (2016).
Senellart, P., Solomon, G. & White, A. Highperformance semiconductor quantumdot singlephoton sources. Nat. Nanotechol. 12, 1026–1039 (2017).
Pilnyak, Y. et al. Simple source for large linear cluster photonic states. Phys. Rev. A 95, 022304 (2017).
Somaschi, N. et al. Nearoptimal singlephoton sources in the solid state. Nat. Photon. 10, 340–345 (2016).
Ding, X. et al. Ondemand single photons with high extraction efficiency and nearunity indistinguishability from a resonantly driven quantum dot in a micropillar. Phys. Rev. Lett. 116, 020401 (2016).
Loredo, J. C. et al. Scalable performance in solidstate singlephoton sources. Optica 3, 433–440 (2016).
Wang, H. et al. Neartransformlimited single photons from an efficient solidstate quantum emitter. Phys. Rev. Lett. 116, 213601 (2016).
He, Y.M. et al. Ondemand semiconductor singlephoton source with nearunity indistinguishability. Nat. Nanotechnol. 8, 213–217 (2013).
Snijders, H. et al. Fibercoupled cavityQED source of identical single photons. Phys. Rev. Appl. 9, 031002 (2018).
Loredo, J. C. et al. Boson sampling with singlephoton fock states from a bright solidstate source. Phys. Rev. Lett. 118, 130503 (2017).
Wang, H. et al. Highefficiency multiphoton boson sampling. Nat. Photon. 11, 361–365 (2017).
Wang, H. et al. Boson sampling with 20 input photons and a 60mode interferometer in a 10^{14} dimensional Hilbert space. Phys. Rev. Lett. 123, 250503 (2019).
Antón, C. et al. Interfacing scalable photonic platforms: solidstate based multiphoton interference in a reconfigurable glass chip. Optica 6, 1471–1477 (2019).
Gazzano, O. et al. Entangling quantumlogic gate operated with an ultrabright semiconductor singlephoton source. Phys. Rev. Lett. 110, 250501 (2013).
Fattal, D. et al. Entanglement formation and violation of bell’s inequality with a semiconductor single photon source. Phys. Rev. Lett. 92, 037903 (2004).
Nowak, A. K. et al. Deterministic and electrically tunable bright singlephoton source. Nat. Commun. 5, 3240 (2014).
Browne, D. E. & Rudolph, T. Resourceefficient linear optical quantum computation. Phys. Rev. Lett. 95, 010501 (2005).
Hein, M., Eisert, J. & Briegel, H. J. Multiparty entanglement in graph states. Phys. Rev. A 69, 062311 (2004).
Kwiat, P. G., Vareka, W. A., Hong, C. K., Nathel, H. & Chiao, R. Y. Correlated twophoton interference in a dualbeam Michelson interferometer. Phys. Rev. A 41, 2910–2913 (1990).
Ou, Z. Y., Zou, X. Y., Wang, L. J. & Mandel, L. Observation of nonlocal interference in separated photon channels. Phys. Rev. Lett. 65, 321–324 (1990).
Ollivier, H. et al. Reproducibility of high performance quantum dot singlephoton sources. ACS Photon. 7, 1050–1059 (2020).
Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044–2046 (1987).
Santori, C., Fattal, D., Vučković, J., Solomon, G. S. & Yamamoto, Y. Indistinguishable photons from a singlephoton device. Nature 419, 594–597 (2002).
HanburyBrown, R. & Twiss, R. Q. Correlation between photons in two coherent beams of light. Nature 177, 27–29 (1956).
Tóth, G. & Gühne, O. Entanglement detection in the stabilizer formalism. Phys. Rev. A 72, 022340 (2005).
Mermin, N. D. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65, 1838–1840 (1990).
Marsili, F. et al. Detecting single infrared photons with 93% system efficiency. Nat. Photon. 7, 210–214 (2013).
Wang, H. et al. Towards optimal singlephoton sources from polarized microcavities. Nat. Photon. 13, 770–775 (2019).
Huber, T. et al. Filterfree singlephoton quantum dot resonance fluorescence in an integrated cavitywaveguide device. Optica 5, 380–385 (2020).
Shomroni, I. et al. Alloptical routing of single photons by a oneatom switch controlled by a single photon. Science 345, 903–906 (2014).
Tiecke, T. G. et al. Nanophotonic quantum phase switch with a single atom. Nature 508, 241–244 (2014).
Reiserer, A., Kalb, N., Rempe, G. & Ritter, S. A. A quantum gate between a flying optical photon and a single trapped atom. Nature 508, 237–240 (2014).
De Santis, L. et al. A solidstate singlephoton filter. Nat. Nanotechnol. 12, 663–667 (2017).
Humphreys, P. C. et al. Linear optical quantum computing in a single spatial mode. Phys. Rev. Lett. 111, 150501 (2013).
Motes, K. R., Gilchrist, A., Dowling, J. P. & Rohde, P. P. Scalable boson sampling with timebin encoding using a loopbased architecture. Phys. Rev. Lett. 113, 120501 (2014).
Rohde, P. P. Simple scheme for universal linearoptics quantum computing with constant experimental complexity using fiber loops. Phys. Rev. A 91, 012306 (2015).
Acknowledgements
The authors thank the Israeli Science Foundation for supporting this work under grants 793/13 and 2085/18. This work was also supported by the FETOPEN QLUSTER, the ERC PoC PhoW, the QuantERA ERANET Cofund in Quantum Technologies, project HIPHOP, the French RENATECH network, a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d’Avenir" programme (Labex NanoSaclay, reference: ANR10LABX0035). J.C.L. and C.A. acknowledge support from Marie SkłodowskaCurie Individual Fellowships SMUPHOS and SQUAPH, respectively. H. O. Acknowledges support from the Paris ÎledeFrance Région in the framework of DIM SIRTEQ. M.E. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG) (project no. 401390650).
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The fiber system was designed by D.I., Y.P., L.C., and H.S.E. The experiments were conducted by D.I., Y.P., J.C.L., and C.A. with help from N.S., P.H., H.O., M.E., L.V., and C.M. Data analysis was carried out by Y.P, D.I., C.A., and J.C.L. The theoretical framework was done by Y.P., D.I., L.C., L.V., and H.S.E. The cavity devices were fabricated by A.H. and N.S. from samples grown by A.L., and the etching was done by I.S. The manuscript was written by Y.P., D.I., J.C.L., C.A., P.S., and H.S.E. with input from all authors. The project was supervised by L.L., P.S., and H.S.E.
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N.S. is cofounder, and P.S. is scientific advisor and cofounder, of the singlephotonsource company Quandela. The remaining authors do not declare any competing interests.
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Istrati, D., Pilnyak, Y., Loredo, J.C. et al. Sequential generation of linear cluster states from a single photon emitter. Nat Commun 11, 5501 (2020). https://doi.org/10.1038/s41467020193414
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DOI: https://doi.org/10.1038/s41467020193414
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