Table Of Content

Scalable performance in solid-state single-photon sources J. C. Loredo,1,∗ N. A. Zakaria,1 N. Somaschi,2 C. Anton,2 L. De Santis,2,3 V. Giesz,2 T. Grange,4 M. A. Broome,1,5 O. Gazzano,2,6 G. Coppola,2 I. Sagnes,2 A. Lemaitre,2 A. Auffeves,4 P. Senellart,2,7 M. P. Almeida,1 and A. G. White1 1Centre for Engineered Quantum Systems, Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia 2CNRS-LPN Laboratoire de Photonique et de Nanostructures, Université Paris-Saclay, Route de Nozay, 91460 Marcoussis, France 3Université Paris-Sud, Université Paris-Saclay, F-91405 Orsay, France 4Université Grenoble-Alpes, CNRS, Institut Néel, “Nanophysique et semiconducteurs” group, F-38000 Grenoble, France 5Present address: Centre of Excellence for Quantum Computation and Communication Technology, 6 School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia 1 6Present address: Joint Quantum Institute, National Institute of Standards and Technology, 0 University of Maryland, Gaithersburg, MD, USA 2 7Département de Physique, Ecole Polytechnique, Université Paris-Saclay, F-91128 Palaiseau, France r The desiderata for an ideal photon source are high brightness, high single-photon purity, and p high indistinguishability. Defining brightness at the first collection lens, these properties have been A simultaneously demonstrated with solid-state sources, however absolute source efficiencies remain 4 closetothe1%level,andindistinguishabilityonlydemonstratedforphotonsemittedconsecutively 1 on the few nanosecond scale. Here we employ deterministic quantum dot-micropillar devices to demonstratesolid-statesingle-photonsourceswithscalableperformance. Inonedevice,anabsolute ] brightnessattheoutputofasingle-modefibreof14%andpuritiesof97.1–99.0%aredemonstrated. h When non-resontantly excited, it emits a long stream of photons that exhibit indistinguishability p upto70%—abovetheclassicallimitof50%—evenafter33consecutivelyemittedphotons,a400ns - t separation between them. Resonant excitation in other devices results in near-optimal indistin- n guishability values: 96% at short timescales, remaining at 88% in timescales as large as 463 ns, a after 39 emitted photons. The performance attained by our devices brings solid-state sources into u a regime suitable for scalable implementations. q [ 2 Photon indistinguishability—responsible for unique indistinguishability, in turn, has been achieved in re- v quantum phenomena with no classical counterpart, no- centyearsunderstrictly-resonantexcitationofthequan- 4 tably photon bunching via interference [1]—has been tumdot[24–26],whereastherecentdevelopmentofelec- 5 demonstrated in various physical systems [2–9], result- tric control on deterministically coupled devices [27]— 6 0 inginabroadrangeofapplicationsinphotonicquantum thuswithscalablefabrication—hasnowenabledstrictly- 0 technologies [10], including quantum teleportation [11, resonantexcitationincombinationwithPurcellenhance- . 12], generation of entangled photon sources [13–15], and ment,resultinginnear-optimalsingle-photonsources[28] 1 0 linear-optics quantum computation [16, 17]. However, with visibilities reaching the 99% mark, simultaneous 6 achieving conclusive indistinguishability, i..e. above 50% state-of-the-artextractionefficiencyof65%andpolarised 1 (theclassicallimit),whilesimultaneouslydisplayinghigh brightness at the first lens around 16%. : v single-photon purity and high absolute brightness is ex- Albeit impressive, the reported efficiencies in these i perimentally challenging. X demonstrations are defined at the first lens, and poor optical collection results in low photon count rates avail- r Semiconductor quantum dots (QDs) inserted in pho- a able in practice. Consequently, absolute brightnesses re- tonic structures [18–22] are a rapidly improving tech- main around the 1% mark, too low for practical scalable nology for generating bright sources of indistinguishable applications [10]. In addition, direct measurements of single-photons. Addressing the excited states of the indistinguishability via two-photon interference, so far, quantum dot using a non-resonant scheme early showed only employed photons consecutively emitted with a few two-photon interference visibilities in the 70% 80% − nanosecond separation, while a key question regarding range [8], yet with limited collection efficiencies. Im- the scalable potential of the developed sources is to de- provements in the efficiency have been made by deter- termine how many consecutive photons exhibit high in- ministically placing the quantum dot in the centre of a distinguishability. A recent work obtained on quantum photonic micro-cavity. Here the acceleration of photon dots in microlenses reported a 40 % drop in the indistin- emission into well defined cavity modes [23], due to Pur- guishability over 10 ns only [29]. cell enhancement, has enabled two-photon interference visibilities in the same range, with simultaneous efficien- In the present work, we demonstrate high absolute ciesatthefirstcollectionlensaround80%[9]. Near-unity brightness and generation of indistinguishable photons 2 consecutively emitted over 463 ns. Our measurements (a) were performed on various quantum dot-micropillar devices,allobtainedusingadeterministic—thusscalable— technology. Using a simple micropillar (Device 1) [9], 1 we demonstrate a high-purity single-photon source with an absolute brightness of 14%. That is, about one in seven laser pulses creates a high-purity single-photon at 0 the output of a single-mode fibre. We also demonstrate 15K robustandconclusivequantuminterferencebetweencon- 13K secutivelyemittedphotonpulsesuptoafirstandthirty- third,separatedby400ns. Interferencevisibilities,under (b) non-resonant excitation, reach maximum values of 70% in short timescales, decreasing to plateaus above 60% at longer temporal separations, and remain above the clas- sicallimitof50%evenathighpump-powers. Usingelec- trically controlled pillar devices [28] (Device 2 and 3) we demonstrate,understrictlyresonant-excitation,indistinguishability reaching near-optimal values: 96% at short timescales, remaining above 88% at 463 ns separation. Device 1 contains self-assembled InGaAs QDs grown by molecular beam epitaxy, positioned in between two layers of GaAs/AlAs distributed Bragg reflectors, con- sisting of 16 (36) pairs acting as a top (bottom) mirror. FIG. 1. Absolute brightness and purity of Device 1. a) De- Note that Device 1 is a pillar from the same batch as tected count rates at T=15 K (red), with the QD in reso- in Ref. [9]. Low-temperature in situ lithography [30] was nance with the cavity mode, and 13 K (blue), with the QD employed to fabricate micropillars centred around a sin- slightly detuned from the cavity. Solid curves represent fits gle QD with 50 nm accuracy. The sample is mounted on toR0(1−exp(−P/P0)),withP0=197µW,andR0=3.8MHz for T=15 K, and R =3.4 MHz for T=13 K. Inset: QD spec- a closed-cycle cryostat and is optically pumped by 5 ps 0 tra with varying temperature. b) Power-dependent g(2)(0) laser pulses at 80 MHz repetition rate with wavelength at T=15 K. Note that even three times above the saturation tunedto905.3nm, correspondingtooneofthequantum pump power the photon purity remians > 97%. Top inset dot excited states in its p-shell. We optimised our col- showstheautocorrelationmeasurementforP=1P ,andbot- 0 lection efficiency by judicious choice of optical elements, tom inset zooms into the zero delay resolving the non-zero achieving an efficiency budget as follows. After emis- g(2)(0) from experimental noise. sion from the micropillar, single-photons travel across the following elements, with measured transmittances ηelem, before reaching detectors: two cryostat windows the saturation measurements in Fig. 1a. The satura- with ηcryo=(96±1)%; a microscope objective (Olympus tion curves are fitted to R0(1−exp(−P/P0)), where LMPLN10XIR) with N.A.=0.3 and ηobj=(91 ± 1)%; a R0 is an asymptotic rate value, and P0 is the satura- dichroic mirror (Alluxa filters) used to separate single- tion power. The inset figure shows Device 1 spectra photons from the laser path, with a measured attenua- with varying temperature T. The energy of the QD tion at 905 nm bounded to >60 dB extinction, while no transition varies like the band gap of the semiconduc- appreciable loss is recorded at wavelengths correspond- tor with temperature [31], whereas the cavity mode en- ing to single-photon emission, we thus consider ηdich=1; ergy follows the temperature variation of the refractive 6 mirrors and 2 lenses, with an overall transmission of index. Adjusting the temperature thus allows tuning ηml=(95 1)%; and a 0.85 nm FWHM band-pass fil- the QD-cavity resonance. For the measurements pre- ± ter (Alluxa filters) with ηbp=(91 1)% used to ensure sented in Fig. 1, the neutral exciton line is brought in ± that any residual scattered laser light is filtered out. resonance at T=15 K. The count-rates in pulsed config- Remaining losses are due to coupling to a single-mode uration reach values as high as 3.6 MHz. In fact, for fibre, where we estimate a fibre-coupling efficiency of this measurement a known loss must be introduced in ηfc=(65 4)%, by comparing collection with a multi- the optical path in order to properly quantify the avail- ± mode fibre assumed to have a unity coupling efficiency. able count-rates, as they are beyond the APD’s (Perkin- Thisresultsinanoveralltransmissionofouropticalsetup Elmer SPCM-AQR-14-FC) linear regime. This allows of ηsetup=(49 3)%. us to accumulate a high amount of statistics with no- ± Wecharacterisethisdeviceintermsofabsolutebright- tably short integration times. For instance, the inset ness and purity, see Fig. 1. We detect large count-rates in Fig. 1b shows a g(2)(∆t) measurement—second-order in a silicon avalanche photodiode (APD), as shown in autocorrelation function with g(2)(0)=0 corresponding 3 to an ideal single-photon state—at P=P , yielding a (a) 0 BPfilter value of g(2)(0)=0.0130 0.0002, where the small er- dichroic ± ror is reached with an integration time of only 29 sec- firstlens onds. We in fact used about half the available counts FBS APD after selecting one linear polarisation emitted by our device. Thus,inoursetup,thesameamountofstatisticsis fine achievedfourtimesfasterwhenthepolariserisremoved. PBS QWP temporal tuning Remarkably, we observe low multi-photon emission at HWP Pol all pump-powers, with a measured maximum value of g(2)(0)=0.0288 0.0002 at P=3P . We thus observe a 0 ± single-photon purity 1 g(2)(0) above 97% even at maximum brightness. Thes−e values were extracted from inte- (b) V5P00ns=(60.31±0.60)%(cid:31) V5P00ns=(0.71±0.01)%⊥ grating raw counts in a 2 ns window—sufficiently larger than the < 0.5 ns lifetime [9]—around the peak at zero delay compared to the average of the 10 adjacent lat- eral peaks, without any background subtraction. Error bars in this work are deduced from assuming poissonian statistics in detected events. Our APD efficiency of 32%—measured using the ap- proachofRef.[32]—80MHzpumprate,and3.6MHzde- tectedcountratecorrespondstoanabsolutebrightness— the probability-per-laser-pulse of finding a spectrally- (c) V0.5P0=(67.52 0.78)% V0.5P0=(59.97 0.76)% isolated high-purity single-photon at the output of a 12.5ns ± 400ns ± single-mode fibre—of 14%, the highest reported to date. Such absolute brightness represents a clear improvement over what has been previously achieved with quantum dot-based photon sources. For instance, a drastic con- trast between performance at the first lens and actual detected count rates has been common until now, e.g., reporting a brightness as high as 72% while detecting 65 kHz [33], or 143 MHz collected on the first lens but only 72 kHz available on detection [34]. Detected rates of 4.0 MHz at the single-photon level have been reported [35], however without coupling into a single- FIG. 2. Two-photon interference between temporally-distant mode fibre and at the cost of high multi-photon contri- photons. a) A simple unbalanced Mach-Zehnder interferom- bution with g(2)(0)=0.4. In fact, our source greatly ex- eterwithapath-lengthdifferenceof∆τe probestheindistin- guishability of two photons emitted with the same ∆τ tem- ceeds, in terms of absolute brightness, the performance e poral separation. b) Interference histograms of orthogonal- of any other single-photon source from any physical sys- (red) and parallel-polarised (blue) photons with ∆τ =50 ns, e tem, including the well established Spontaneous Para- at saturation of the quantum dot. (Note the suppression at metric DownConversion source—so far considered as the ∆τ ,seetextfordetails). c)Interferenceofparallel-polarised e premierphotonsource—wheretheequivalent(triggered) photons with ∆τe=12.5 ns (blue) and ∆τe=400 ns (orange), absolute brightness is well below 1%. taken at P=0.5P0. A temporal offset of 3.5 ns has been introduced between histograms for clarity. We note that, given our setup collection efficiency of η =49%, Device 1 exhibits—for the neutral exciton setup state—a brightness at the first lens of 29%. Deducing the exciton lifetime from the correlation curves at low result, the probability of the quantum dot to be in the excitation power, we estimate the Purcell factor of the neutral exciton is reduced leading to the measured 29% device to be around F =2, and the fraction of emission p brightness at the first lens. Note that this instability into the cavity mode around 66%. Considering an out- of the charge state was not observed originally in the de- put coupling efficiency of 90%, the measured brightness vicesunderstudy,seeRef.[9],butappearedaftersample inthefirstlenscouldreach60%withaunityprobability accidental freezing. to find the QD in the neutral exciton state. However, as evidenced in the inset of Fig. 1a, the present QD also We now explore the indistinguishability of photons presents an non-negligible probability to emit from the emitted by Device 1 with various temporal distances. positively- or negatively-charged exciton transition that We perform our measurements at T=13 K to reduce are brought in resonance at higher temperatures. As a phonon-induceddephasing[36],whichissufficientlyclose 4 (a) (b) 0.65 0.6 0.55 0.5 FIG. 3. Power- and temporal-dependent two-photon interference. a) Over >100 measured visibilities (red points) showing conclusive quantum interference, i.e. V>0.5, at all measured powers and timescales. Coloured surface is an interpolation to the data. b) Fitted values of V at different ∆τ (bottom axis), for P=0 (red), P=P (green), and P=2P (blue), showing e 0 0 interference between a first and n-th consecutive emitted photon (top axis). Curves are fits to our model in Eq. (2). to the quantum dot cavity resonance at T=15 K. Note WeusethevisibilityV toquantifythedegreeofindis- that contrary to most reports, the phonon sideband here tinguishability of the source. Since the measured visibil- is not filtered out by the 0.85 nm bandpass filter used ity depends both on the photon source and on the appa- to further suppress the laser light. Figure 2a depicts ratususedtocharacteriseitthelattermustbeaccounted our experimental setup. Single-photons are injected for. Ideallytheapparatusisabeamsplitterofreflectivity into an unbalanced Mach-Zehnder interferometer with =0.5; in our experiment =0.471, =0.529, and the R R T a variable fibre-based path-length difference designed to visibility V is thus, match—by using multiple fibres of distinct lengths—an integer multiple of 12.5 ns up to 400 ns. Polarisa- 2+ 2 A /A 0 V = R T − , (1) tion control—polariser (Pol) and a half-wave plate 2 RT (HWP)—and a polarising beamsplitter (PBS) behave as a beamsplitter with tuneable reflectivity, thus balancing where A is taken as the average value of Ak. Note that the photon-flux entering the interference point inside since the g(2)(0) values are intrinsic to the source, and a fibre-beamsplitter (FBS) closing the Mach-Zehnder hence affect any process in which we wish to use it, we configuration. Quarter-wave plates (QWPs) and HWPs do not correct for non-zero g(2)(0) in Eq. (1). The de- are used to tune the polarisation of interfering photons ducedV thereforecorrespondstotherawtwo-photonin- in parallel or orthogonal configuration. Time-correlation terference visibility, and quantifies the degree of photon histograms from the output of this interferometer indistinguishability. reveal the indistinguishability of photons emitted Figure 2b shows histograms for the indistinguisha- with a temporal distance ∆τ . Fully distinguishable bility of orthogonal- and parallel-polarised photons at e photons—e.g., with orthogonal polarisation—meeting ∆τe=50nsandP=P0. InvirtueofEq.(1),andmeasured at a 50:50 beamsplitter result in a 50 % probability =0.471, we obtain VP0 =(0.71 0.01)% in orthogonal of being detected simultaneously at the output of the cRonfiguration(redhisto5g0rnasm),and±VP0 =(60.31 0.60)% 50ns ± beamsplitter. This results in the peak around ∆t=0 of for parallel-polarised photons (blue histogram), where the time-correlation measurement being about half of VP denotes visibility taken at a power P and tempo- ∆τe those at ∆t>0, with the exception of peaks at ∆t=∆τe, ral delay ∆τe. We observe higher visibilities at lower which larger suppression indicates that the interfering powers and shorter delays. For instance, the mea- photons were emitted with a temporal distance ∆τe. In surements in Fig. 2c were taken at P=0.5P0, and re- general it can be shown for a pure single-photon source, veal V0.5P0=(67.52 0.78)% at a temporal delay (blue 12.5ns ± see Supplementary Material, that the areas A∆t cen- histogram) of ∆τe=12.5 ns. Remarkably, we find teredaround∆taregivenbyA =N, A =N(1 2), that indistinguishability is robust in the temporal do- k (cid:0)(cid:0) −∆τe(cid:1) −R (cid:1) A =N(1 2), and A =N 2+ 2 2 V , main. Even after 33 consecutive emitted photons (or- ∆τe −T 0 R T − RT where k= 12.5 ns, 25 ns,..., and excludes peaks ange histogram), at ∆τe=400 ns, it only decreases to at ∆τ , ±N is an ±integration constant, is the V0.5P0=(59.97 0.76)%. That is, less than 8% visibility ± e R 400ns ± beamsplitter reflectivity, and =1 . decreasein 400ns. AllV valueswiththenon-resonant T −R ∼ schemeareobtainedwithoutanybackgroundcorrection. 5 To thoroughly examine the indistinguishability prop- Thedecreaseoftheindistinguishabilitybyfewpercents erties of Device 1, we carried out power- and temporal- for temporally distant photons demonstrates a very lim- dependent measurements, see Fig. 3a. All these mea- itedspectraldiffusioninourmicropillardevices. Thisob- sured V are within the 50% 70% range, thus showing servationisinstrikingcontrasttopreviousmeasurements − conclusive quantum interference at all measured powers onsinglephotonsourcesbasedonalternativeapproaches andtimescales. Thelargeavailablephotonfluxallowsus for efficient photon extraction, such as nanowires [38], or to gather more than 100 visibility values with measure- micro lenses [29]. A significantly lower stability of the ment errors sufficiently small to identify an interesting electrostatic environment of the QD can reasonably be behaviour in this narrow visibility range. At any given attributed to the close proximity of free surfaces in the ∆τ , V is linear in P, see Supplementary Material, and latter. Indeed, as indicated by the observation of three e we simply use V=Vmax+m P to characterise the P- emission lines from the same QD, even the micropillar ∆τe ∆τe dependence of V at fixed ∆τ . Conversely, at fixed P, devices under study do not provide a fully stable charge e V decreases monotonically and asymptotically in ∆τ , statefortheQDs, aneffectthatweobservetobedepen- e flattening to fixed values at longer timescales. dent on the quality of the etched surfaces. This makes We model this behaviour by considering a time- strictly resonant spectroscopy difficult without an addi- dependent wandering of the spectral line as the origin of tional non-resonant excitation, a situation also observed thetemporalmodulation. Thatis,thefrequencyofevery in other micropillar devices [26]. emitted photon ω(t)=ω0+δω(t) varies in time according Therefore, to explore the indistinguishability of tosomewanderingfunctionδω(t)occurringintimescales temporally-distant photons under strictly resonant ex- much larger than the photon lifetime. Our problem is citation, we turn to electrically controlled micropillars then equivalent to finding the mutual interference visi- and present data on two devices, Device 2 and Device 3. bility between independent sources with finite frequency These devices consist of quantum dots deterministically (cid:0) (cid:1) detuning [37], which is given by V(0)/ 1+δω2 in the coupled to micropillars embedded in cylindrical gated r case where V(0) is the degree of indistinguishability for structures with p- and n-contacts respectively defined each source alone (equal value for both), and δωr is the on the top and bottom sides of the device, resulting in ratio of the frequency detuning to the spectral linewidth an effective p-i-n diode structure onto which an electric of the sources (equal linewidth for both). If this mis- field can be applied. (See Ref. [28] for a detailed de- match arises due to spectral wandering within the same scription of the device). We perform our measurements source,thenthetime-averagedrelativedetuningsquared at T=9 K and tune the emission into cavity-resonance is given by 2δωr2(1−exp(−∆τe/τc)), with τc a charac- via an applied bias voltage of −0.3 V. This sample is teristic wandering timescale, see Supplementary Mate- cooled by gas exchange in a closed-cycle cryostat, and rial for more details. We thus derive the visibility of is pumped by shaped 15 ps laser pulses at 82 MHz rep- temporally-distant photons: etition rate. The experimental setup used for photon collection is reported in Ref. [28], and the appartus used V(0) V (∆τe)= 1+2δωr2(cid:0)1−e−∆τe/τc(cid:1). (2) faollrytihdeentteimcaplotroalt-hdaetpeinndFeingt. 2mae.asurements is conceptu- To obtain a statistically meaningful temporal behaviour, Resonant-excitation allows us to probe two-photon we used the fitted values of V at different ∆τ , for pow- interference in a regime excelling in indistinguisha- e ers P=0, P=P , and P=2P . These values are plot- bility performance. Indeed, for Device 2 we obtain 0 0 ted in Fig. 3b and are in good agreement with our Vπ =(95.0 1.0)% at a short temporal separation, de- 12.2ns ± model in Eq. (2). In the limit of low powers, we obtain creasingonlytoVπ =(90.6 1.7)%atlongtimescales, 158.5ns ± V(0)=(72.8 2.4)%,τ =(45.5 19.1)ns,andδω =(29.4 seeFigs.4a,and4b. Weobserveahighsingle-photonpu- c r ± ± ± 3.1)%;whereasathighpowers,atP=2P ,theseparame- rity quantified by g(2)(0)=0.015 0.007 at π-pulse, see 0 ± tersareV(0)=(59.0 2.0)%,andδω =(19.3 4.5)%. The Fig. 4c, where the non-vanishing g(2)(0) primarily con- r ± ± maximum degree of indistinguishability V(0) decreases sists of background noise and thus a value 1 g(2)(0) of − only by 13.8% with increasing power, evidencing a slight 98.5% represents a lower bound on the intrinsic single- increase of pure dephasing of the exciton transition. On photonpurity. Indistinguishabilitymeasurementsatvar- the contrary, the relative amplitude of the spectral wan- ious temporal distances, see Fig. 4d, reveal plateaus at dering decreases by 34%, evidencing that spectral diffu- high values: Up to a first and fourteenth photon, sepa- sionissignificantlyreducedathigherpowers, asrecently ratedby 150ns,exhibitanindistinguishabilitygreater ∼ observed in nanowire based devices [38]. Note that the than90%. ThecurveisafittoEq.(2),withamaximum largerelativeerrorinτ isduetoasmallrelativedecayin indistinguishability value of V(0)=96.6%, τ =54.4 ns, c c V, an uncertainty that increases with increasing power. and δω =17.8%. The reproducibility of our results, r Thus—although it is reasonable to assume that τ itself thanks to a deterministic fabrication, is evidenced by c is power-dependent—we extracted τ only at P=0 and similar indistinguishability values obtained on Device 3: c useditasafixedparameterforthefitsathigherpowers. Vπ =(96.1 0.8)% at a short temporal delay, and 12.2ns ± 6 (a) (b) Vπ =(95.0 1.0)% Device 2 Vπ =(90.6 1.7)% Device 2 12.2ns ± 158.5ns ± (c) (d) Device 2 Device 2 Device 3 FIG. 4. Temporal-dependent indistinguishability under strictly resonant excitation. Two-photon interference histograms with Device 2 of parallel-polarised photons at a) ∆τ =12.2 ns, and b) ∆τ =158.5 ns, under a π-pulse preparation. c) Second-order e e autocorrelation measurement at π-pulse. d) Indistinguishability between a first and n-th consecutive emitted photon from Device 2 (blue) and Device 3 (red). Indistinguishability remains robust in the temporal domain, decreasing only by 4.4% in ∼159ns(downto90.6%)forDevice 2, andby8.3%in∼463ns(downto87.8%)forDevice 2 . Thecurveisafitofthedata from Device 2 to Eq. (2). Vπ =(87.8 1.6)% for a first and thirty-ninth photon indistinguishability decreases only a few percent—about 463ns ± separated by 463 ns. These values of indistinguishability 8% at low powers and less than 4% at higher powers— are corrected for the measured background noise aris- before flattening to fixed values at longer timescales. ing from detector dark counts: The experimental setup This contrasts favourably to previous works, where pho- usedfortheseresonant-excitationmeasurementspresents ton indistinguishability has been observed to decrease a low collection efficiency, thus an integration of raw de- by 40% in only 10 ns [29]. Moreover, under strictly- tected counts that includes the background noise, which resonant excitation, photon indistinguishability between at zero delay is as large as non-vanishing counts due to a first and thirty-ninth photon remained at 88%. Inter- photon dinstinguishability, would under-estimate the in- estingly, the observation of only small reductions in the trinsic degrees of indistinguishability in our devices, see temporal domain indicate that non-unity indistinguisha- the Supplementary Material for details on this method. bility under non-resonant excitation is mainly caused by No correction for non-vanishing g(2)(0) was included. homogenous broadening of the spectral linewidth (gov- Notethatahighabsolutebrightnesswiththisrecently erning coherence times at short temporal delays), and developed technology is yet to be achieved. However, alimitedinhomogeneousbroadening(governingeffective since the mode profile of connected pillars is the same as coherence times at longer temporal delays). The rela- isolated ones [27] and a photon extraction efficiency at tive amplitude of the spectral diffusion at saturation is thefirstlensof65%hasbeenreportedonthissample[28], similar for both resonant and non-resonant excitation. the same experimental methods as before should allow However, Device 1 operates in a limited Purcell regime even higher absolute efficiencies than the 14% reported whereas Devices 2 and 3 operate with a Purcell factor here. around 7 10, leading to an increased radiative exciton − linewidth. From this, we conclude that, although the Weprovidedherestrongevidencethatoursourcesemit application of an electrical bias in n-i-p diode structures long streams of indistinguishable photons. Under non- allowsagoodcontroloftheQDchargestates,itdoesnot resonant excitation, even a first and a thirty-third con- lead to a significant decrease in the spectral wandering secutive photon, separated by 400 ns, display conclusive phenomena. The excellent indistinguishability observed quantum interference. For a fixed pump power, photon 7 in Devices 2 and 3 arises mainly from reduced pure de- Funding Information phasingoftheexcitonstate,increasedPurcellfactorand reduced time jitter in a resonant excitation scheme. Centre for Engineered Quantum Systems (CE110001013); Centre for Quantum Computation Our reported indistinguishability values correspond to and Communication Technology (CE110001027); Asian the longest temporal delays here studied, at a partic- OfficeofAerospaceResearchandDevelopment(FA2386- ular pump repetition rate of 80 MHz: It only rep- 13-1-4070); ARC Discovery Early Career Research resents a lower bound on the number of photons we Award (DE120101899); ERC Starting Grant (277885 can generate—limited by radiative lifetimes in the or- QD-CQED);FrenchAgenceNationalepourlaRecherche der of a few hundred picoseconds—that can be further (ANR DELIGHT, ANR USSEPP); French RENATECH used in quantum information processing protocols with network Labex NanoSaclay; European Unions Seventh solid-state sources [39]. Previous works investigating Framework Programme FP7 (618078 WASPS) noise spectra in resonance fluorescence have shown ev- idence of long streams of near transform-limited photons [40] in timescales potentially reaching seconds [41]. In fact, Device 2 has recently been shown to emit pho- Acknowledgments tons with near transform-limited linewidth in a millisec- ond timescale [42], in which case we would expect that J. C. L. and A. G. W. thank the team from the ourdevicesareproducingatleasthundredsofthousands Austrian Institute of Technology for kindly providing of highly indistinguishable single-photons. the time-tagging modules. M. P. A. thanks Halina Rubinsztein-Dunlop for the generous loan of equipment. Ourfindingsareespeciallyrelevantinimplementations The CNRS-LPN authors are very thankful to Anna with time-bin encoded degrees of freedom, such as some Nowak for her help with the technology. A. G. W. ac- recentlyproposedschemesoflinear-opticsquantumcom- knowledges support from a UQ Vice Chancellors Re- puting with time-bin encoding [43, 44], where the in- search and Teaching Fellowship distinguishability of temporally-distant photons will di- rectly determine quantum fidelities of the implemented protocols. Scaling solid-state multi-photon sources by combining multiple independent emitters remains challenging,asatomicgrowthaccuracyorcomplexindividual ∗ Corresponding author: [email protected] [1] C.K.Hong,Z.Y.Ou, andL.Mandel,Phys.Rev.Lett. electriccontrolovermultipledevicesisneeded. Thesere- 59, 2044 (1987). quirementscanbecircumventedbymakinguseofasingle [2] T. Legero, T. Wilk, M. Hennrich, G. Rempe, and photon source emitting a long temporal stream of highly A. Kuhn, Phys. Rev. Lett. 93, 070503 (2004). indistinguishable photons that can be demultiplexed by [3] L.-M. Duan and C. Monroe, Rev. Mod. Phys. 82, 1209 fast active optics. (2010). [4] H. Bernien, L. Childress, L. Robledo, M. Markham, A high absolute brightness will be critical for success- D. Twitchen, and R. Hanson, Phys. Rev. Lett. 108, 043604 (2012). fully implementing multi-photon experiments with these [5] A. Kiraz, M. Ehrl, T. Hellerer, O. E. Mu¨stecaplıo˘glu, sources, where their downconversion counterparts cur- C. Braüchle, and A. Zumbusch, Phys. Rev. Lett. 94, rentlyrequireexperimentalrunsofhundredsofhours[45, 223602 (2005). 46]. The key role of high emission yields in these devices [6] C.Lang,C.Eichler,L.Steffen,J.M.Fink,M.J.Woolley, has been made explicit in the recent demonstration of a A. Blais, and A. Wallraff, Nat Phys 9, 345 (2013). solid-state based multi-photon experiment [47], realised [7] T.Pittman,B.Jacobs, andJ.Franson,OpticsCommu- with Device 1, where integration times outperformed nications 246, 545 (2005). [8] C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and those in equivalent downconversion implementations by Y. Yamamoto, Nature 419, 594 (2002). two-orders of magnitude. Achieving high absolute effi- [9] O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, ciencies, and thus allowing the scaling of multi-photon A.Nowak,E.Galopin,I.Sagnes,L.Lanco,A.Lemaˆıtre, experiments to larger photon numbers, becomes feasible and P. Senellart, Nat Commun 4, 1425 (2013). due to Purcell-enhancement of deterministically-coupled [10] J.L.O’Brien,A.Furusawa, andJ.Vuckovic,Nat.Pho- quantum dot-micropillar devices [9, 27, 28, 37, 48]. This ton. 3, 687 (2009). necessaryconditionisunlikelytobefoundbychancewith [11] D.Bouwmeester,J.-W.Pan,K.Mattle,M.Eibl,H.We- infurter, and A. Zeilinger, Nature 390, 575 (1997). non-deterministic approaches, with reported [48] device [12] X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, yields of 0.01% [26]. Thus, the deterministic fabrica- ∼ L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, Nature 518, tion, high absolute brightness, and long timescale indis- 516 (2015). tinguishability of our devices will enable large-scale ap- [13] P.G.Kwiat,E.Waks,A.G.White,I.Appelbaum, and plications that have been heretofore impossible. P. H. Eberhard, Phys. Rev. A 60, R773 (1999). 8 [14] T. Kim, M. Fiorentino, and F. N. C. Wong, Phys. Rev. S. Reitzenstein, Applied Physics Letters 107, 041105 A 73, 012316 (2006). (2015), http://dx.doi.org/10.1063/1.4927429. [15] A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, [35] S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, A.Lemaitre,I.Sagnes,J.Bloch,P.Voisin, andP.Senel- P. M. Petroff, and D. Bouwmeester, Nat Photon 1, 704 lart, Nature 466, 217 (2010). (2007). [16] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, [36] S.Unsleber,D.P.S.McCutcheon,M.Dambach,M.Ler- 46 (2001). mer, N. Gregersen, S. Ho¨fling, J. Mørk, C. Schneider, [17] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. and M. Kamp, Phys. Rev. B 91, 075413 (2015). Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 [37] V. Giesz, S. L. Portalupi, T. Grange, C. Antoń, (2007). L. De Santis, J. Demory, N. Somaschi, I. Sagnes, [18] P.Michler,A.Kiraz,C.Becher,W.V.Schoenfeld,P.M. A. Lemaˆıtre, L. Lanco, A. Auffèves, and P. Senellart, Petroff,L.Zhang,E.Hu, andA.Imamoglu,Science290, Phys. Rev. B 92, 161302 (2015). 2282 (2000). [38] M. E. Reimer, G. Bulgarini, R. W. Heeres, B. J. Witek, [19] J. Vuckovic, D. Fattal, C. Santori, G. S. Solomon, and M. Versteegh, D. Dalacu, L. J., P. J. Poole, and Y. Yamamoto, Applied Physics Letters 82, 3596 (2003). V. Zwiller, arXiv:1407.2833 (2014). [20] M. Pelton, C. Santori, J. Vuc˘ković, B. Zhang, G. S. [39] O. Gazzano, M. P. Almeida, A. K. Nowak, S. L. Por- Solomon, J. Plant, and Y. Yamamoto, Phys. Rev. Lett. talupi,A.Lemaˆıtre,I.Sagnes,A.G.White, andP.Senel- 89, 233602 (2002). lart, Phys. Rev. Lett. 110, 250501 (2013). [21] T. Lund-Hansen, S. Stobbe, B. Julsgaard, [40] A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, H. Thyrrestrup, T. Su¨nner, M. Kamp, A. Forchel, D. Reuter, A. D. Wieck, M. Poggio, and R. J. War- and P. Lodahl, Phys. Rev. Lett. 101, 113903 (2008). burton, Nat Phys 9, 570 (2013). [22] K.H.Madsen,S.Ates,J.Liu,A.Javadi,S.M.Albrecht, [41] A. V. Kuhlmann, J. H. Prechtel, J. Houel, A. Ludwig, I. Yeo, S. Stobbe, and P. Lodahl, Phys. Rev. B 90, D.Reuter,A.D.Wieck, andR.J.Warburton,NatCom- 155303 (2014). mun 6 (2015). [23] J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, [42] V. Giesz, N. Somaschi, G. Hornecker, T. Grange, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, B. Reznychenko, L. De Santis, J. Demory, C. Gomez, 1110 (1998). I. Sagnes, A. Lemaitre, O. Krebs, N. D. Lanzil- [24] Y.-M. He, Y. He, Y.-J. Wei, D. Wu, M. Atature, lotti, L. Lanco, A. Auffeves, and P. Senellart, C.Schneider,S.Hofling,M.Kamp,C.-Y.Lu, andJ.-W. arXiv:1512.04725 (2015). Pan, Nat Nano 8, 213 (2013). [43] P.C.Humphreys,B.J.Metcalf,J.B.Spring,M.Moore, [25] Y.-J. Wei, Y.-M. He, M.-C. Chen, Y.-N. Hu, Y. He, X.-M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. D. Wu, C. Schneider, M. Kamp, S. Höfling, C.-Y. Lu, Walmsley, Phys. Rev. Lett. 111, 150501 (2013). and J.-W. Pan, Nano Letters 14, 6515 (2014), pMID: [44] P. P. Rohde, Phys. Rev. A 91, 012306 (2015). 25357153. [45] M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, [26] X. Ding, Y. He, Z.-C. Duan, N. Gregersen, M.-C. Chen, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, S.Unsleber,S.Maier,C.Schneider,M.Kamp,S.Ho¨fling, E.F.Galvaõ,A.Crespi,R.Ramponi,R.Osellame, and C.-Y.Lu, andJ.-W.Pan,Phys.Rev.Lett.116,020401 F. Sciarrino, Science Advances 1, 1400255 (2015). (2016). [46] C. Zhang, Y.-F. Huang, Z. Wang, B.-H. Liu, C.-F. Li, [27] A. K. Nowak, S. L. Portalupi, V. Giesz, O. Gazzano, and G.-C. Guo, Phys. Rev. Lett. 115, 260402 (2015). C. Dal Savio, P. F. Braun, K. Karrai, C. Arnold, [47] J. C. Loredo, M. A. Broome, P. Hilaire, O. Gazzano, L. Lanco, I. Sagnes, A. Lemaˆıtre, and P. Senellart, Nat I.Sagnes,A.Lemaitre,M.P.Almeida,P.Senellart, and Commun 5, 3240 (2014). A. G. White, arXiv:1603.00054 (2016). [28] N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, [48] S. Unsleber, Y.-M. He, S. Maier, S. Gerhardt, C.-Y. Lu, M.P.Almeida,G.Hornecker,S.L.Portalupi,T.Grange, J.-W. Pan, M. Kamp, C. Schneider, and S. Hofling, C. Antoń, J. Demory, C. Go´mez, I. Sagnes, N. D. arXiv:1512.07453v1 (2015). Lanzillotti-Kimura, A. Lema´ıtre, A. Auffeves, A. G. White,L.Lanco, andP.Senellart,NatPhotonadvance online publication, (2016). [29] A. Thoma, P. Schnauber, M. Gschrey, M. Seifried, J. Wolters, J.-H. Schulze, A. Strittmatter, S. Rodt, A. Carmele, A. Knorr, T. Heindel, and S. Reitzenstein, Phys. Rev. Lett. 116, 033601 (2016). [30] A.Dousse,L.Lanco,J.Suffczyn´ski,E.Semenova,A.Mi- ard, A. Lemaˆıtre, I. Sagnes, C. Roblin, J. Bloch, and P. Senellart, Phys. Rev. Lett. 101, 267404 (2008). [31] E. Grilli, M. Guzzi, R. Zamboni, and L. Pavesi, Phys. Rev. B 45, 1638 (1992). [32] R. H. Hadfield, Nat Photon 3, 696 (2009). [33] J.Claudon,J.Bleuse,N.S.Malik,M.Bazin,P.Jaffren- nou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. Gerard, Nat Photon 4, 174 (2010). [34] A. Schlehahn, M. Gaafar, M. Vaupel, M. Gschrey, P. Schnauber, J.-H. Schulze, S. Rodt, A. Strittmatter, W. Stolz, A. Rahimi-Iman, T. Heindel, M. Koch, and 9 (cid:0) (cid:1)(cid:0) (cid:1) SUPPLEMENTARY MATERIAL known relation V= 1 A /AV=0 2+ 2 /(2 ), − 0 0 R T RT with A relating the coincidence rate at zero delay of 0 Areas in time-correlation histograms photons with non-zero V indistinguishability. Here we deduce the area distribution of the time- Visibility power-dependence correlationmeasurementsdescribedinthemaintext. For simplicity, we first consider two (fully-distinguishable) Following the main text, the interference visibil- single-photons distributed in time-bins t ,t , entering an unbalanced Mach-Zehnder interfero{me1ter2}composed ity V of two photons separated in time by ∆τe exhibits a linear-dependence in the pump power P. of a first 50:50 beamsplitter and a second beamsplitter For a given ∆τ , we measure V at various val- with reflectance (transmittance =1 ). Our task e is to find all posRsible output distriTbutio−nsRleading to a ues of P, up to three saturation powers P=3P0, and fit the data to V=Vmax+m P. Figure 6 coincidence detection between events separated in time ∆τe ∆τe shows the power-dependence of V for ∆τ =12.5 ns, by ∆t. There are two timescales relevant in such coin- e ∆τ =50 ns, and ∆τ =400 ns. The fitted parameters cidence measurements: the difference in occupied time- e e areVmax =(70.3 0.3)%, m = (6.1 0.2)%atshort binsδt=t t ,andthetemporaldelayinsidetheunbal- 12.5ns ± 12.5ns − ± anced int|e2r−fer1o|meter ∆. By inspecting this reduced sce- timescales; V5m0nasx=(65.0±0.3)%, m50ns=−(4.4±0.2)% at moderate timescales; and Vmax=(60.8 0.3)%, nario,wecanfindthatthereare8eventsleadingtoaco- 400ns ± m = (3.6 0.2)% at the longest timescales explored incidencedetection, asdepictedinFig.5. Thisresultsin 400ns − ± in this work. local patterns of peak areas A given by: A = 2, ∆t −δt−∆ R A =2 , and A = 2, the local pattern around −δt −δt+∆ RT T δt; andA = 2, A =2 , andA = 2, thelo- δt−∆ δt δt+∆ Visibilities of temporally-distant photons − R RT T calpatternaroundδt. Fromthis,wefindsimplerulesfor the time-correlation measurement of an array of single- The interference visibility of two photons from two photons distributed in arbitrary time-bins t passing { i} sources a and b reads [37]: through a ∆-unbalanced Mach-Zehnder: rule 1: Find all possible temporal delays δt relating (cid:18) γ γ (cid:19) (γ +γ +γ∗+γ∗) V = a b a b a b , (3) etiaocnh.pair of photons within the given time-bin distribu- γa+γb [(γa+γb+γa∗+γb∗)/2]2+δω2 rule 2: Around each δt, assign the relative fre- where the γ are the radiative rates, γ∗ the pure de- ± i i quency of events 2,2 , 2 at temporal delays phasing rates, and δω the frequency detuning between {R RT T } ∆t= δt ∆, δt, δt+∆ . the two sources. If the interfering photons are emitted {± − ± ± } We note that these two simple rules describe differ- by the same quantum dot, we assume that γa=γb=γ ent interesting histograms relevant in the literature. For and γa∗=γb∗=γ∗ are constant, but only the frequency instance, by simply identifying the involved parameters, ω=ω0+δω(t) varies over time (i.e. spectral wandering) one can find histograms of g(2)(∆t) measurements of ar- around a central value ω0. This model makes sense here bitrary n Fock states by considering n single-photons as the timescale over which ω varies is much larger than occupyin|g(cid:105)the same time-bin, resulting in distributions the radiative lifetime. Then Eq. (3) reduces to: agreeing with g(2)(0)=1 1/n, or the well known 5-peak (cid:28) (cid:29) − V(0) structuresintwo-photoninterferenceexperimentsinvolv- V = , (4) 1+δω2 ingpairsofphotonsseparatedby∆τe <12.5nsrepeated r every 12.5 ns. where we have used V(0)=γ/(γ+γ∗) the ”intrinsic” de- Now, the experiment described in the main text gree of indistinguishability, and δω =δω/(γ + γ∗) the r is the particular case of an infinitely long stream ratio between the frequency detuning and the spectral of single-photons separated by a fixed δt=12.5 ns, linewidth γ+γ∗. and passing through an unbalanced interferometer One can define a time correlation function for the fre- with ∆=∆τ . Under this consideration, and follow- e quency fluctuations as ing rule 1 and rule 2, we derive the distribution of areas A∆t, given by: Ak=N(cid:0)(cid:0), A−∆τe(cid:1)=N(1−R2(cid:1)), F(∆τe)=<δω(t)δω(t+∆τe)>=<δω2 >f(∆τe), (5) A =N(1 2), and A =N 2+ 2 2 V , ∆τe −T 0 R T − RT with k= 12.5 ns, 25 ns,..., excluding peaks at ∆τ , then, the frequency difference as a function of the delay e ± ± ± and N an integration constant. The visibility term V in ∆τ can be expressed as e A appearsfromnoticing(invirtueofrule 1 andrule 2) 0 that the area at ∆t=0 for fully-distinguishable photons <δω2(∆τe)>=<(δω(t+∆τe) δω(t))2 > − isAV=0=N( 2+ 2),andthenonesimplyusesthewell- =2<δω2 >(1 f(∆τ )). (6) 0 R T − e 10 δt (δt+∆) δt δt ∆ ∆ − ∆ − ∆t= δt ∆ ∆t=δt ∆ − − − 50:50 R:T frequency: 2 50:50 R:T frequency: 2 R R δt δt δt δt ∆ − ∆ 50:50 : 50:50 : RT RT ∆t= δt ∆t=δt − δt δt frequency:2 δt δt frequency:2 ∆ − RT ∆ RT 50:50 R:T 50:50 R:T δt (δt ∆) δt δt+∆ ∆ − − ∆ ∆t= δt+∆ ∆t=δt+∆ 50:50 R:T frequen−cy: 2 50:50 R:T frequency: 2 T T FIG. 5. Two consecutive single-photons separated by δt passing through a ∆-unbalanced Mach-Zehnder interferometer. 8 outcomedistributions,occurringwithagivenrelativefrequency,leadtoacoincidencesignalbetweeneventsseparatedintime by ∆t. The relative delay ∆t is positive if a detector in the upper output fires first, and it is negative in the opposite case. so that: 12.5ns (cid:28) (cid:29) 0.70 V(0) 50ns V(∆τ )= 400ns e 1+δω2(∆τ ) r e V(0) 0.60 = V 1+ δω2(∆τ ) (cid:104) r e (cid:105) V(0) = (cid:0) (cid:1) (8) 0.50 classicallimit 1+2δωr2 1−e−∆τe/τc 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Extraction of visibility under resonant excitation P(cid:144)P 0 Here we describe the methods to extract the raw FIG. 6. Power-dependence of V for ∆τ =12.5 ns (orange), e ∆τ =50 ns (purple), and ∆τ =400 ns (brown). Curves are and corrected two-photon interference visibilities under e e fitstoV=Vmax+m P. V isabove50%(theclassicallimit) strictly-resonant excitation and π-pulse preparation, see ∆τe ∆τe at all powers and timescales here explored. Fig. 7. Figure 7a shows the interference histogram of two photons separated by ∆τ =12.2 ns, from which a e (cid:0) (cid:1) visibilityisextractedviaV= 2+ 2 A /A /(2 ), 0 R T − RT where A is the area of the peak around ∆t=0, and A is 0 A common assumption is to assume an exponential cor- taken as the average area of 14 adjacent peaks (exclud- relation function ing the peak at ∆τe). These areas are taken as the inte- grated counts within a temporal window of 2 ns (consid- erably longer than the subnanosecond lifetimes) around ∆t=k 12.2ns,withk=0,2,3,...,15,seeFig.7b. There- f(∆τ )=e−∆τe/τc, (7) × e sultingintegratedareasareshowninFig.7c,fromwhich we extract a raw Vπ =(89.0 1.5)%. As described in 12.2ns ± themaintext,theremainingnon-vanishingareaat∆t=0 with τ a characteristic wandering timescale. Which isindeedquitesmallanditisontheorderofexperimen- c is expected for a Markovian dynamics of the environ- tal noise. We take into account this noise by integrating nement. An additional input which is required is the coincidencecountswithina2nswindowbutnowlocated distribution for δω. Generally one assumes a Gaussian inbetweenpeaks,thatisat∆t=(m+1/2) 12.2ns,with × distribution,butforsimplicity,andwithoutlossofgener- m=1,2,...,14, see Fig.7d. After subtracting the average ality,wetakeatwo-valuedistributionδω = √<δω2 >, of these background counts to the areas in Fig.7c, we ±