J C Loredo
email: juan.loredo@u-psud.fr

C Antón

B Reznychenko

P Hilaire

A Harouri

C Millet

H Ollivier

N Somaschi

L De Santis

A Lemaître

I Sagnes

L Lanco

A Auffèves

O Krebs

P Senellart

Generation of non-classical light in a photon-number superposition

The ability to generate light in a pure quantum state is essential for advances in optical quantum technologies. However, obtaining quantum states with control in the photon-number has remained elusive. Optical light fields with zero and one photon can be produced by single atoms, but so far it has been limited to generating incoherent mixtures, or coherent superpositions with a very small one-photon term. Here, we report on the on-demand generation of quantum superpositions of zero, one, and two photons via pulsed coherent control of a single artificial atom. Driving the system up to full atomic inversion leads to the generation of quantum superpositions of vacuum and one photon, with their relative populations controlled by the driving laser intensity. A stronger driving of the system, with 2π-pulses, results in a coherent superposition of vacuum, one, and two photons, with the two-photon term exceeding the one-photon component, a state allowing phase super-resolving interferometry. Our results open new paths for optical quantum technologies with access to the photon-number degree-of-freedom.

Controlling the photon-number in a light pulse has been a primary task enabling progress in optical quantum technologies [START_REF] O'brien | Photonic quantum technologies[END_REF][START_REF] Erhard | Twisted photons: new quantum perspectives in high dimensions[END_REF]. Single-, and N-photon sources [START_REF] Waks | Generation of photon number states[END_REF][START_REF] Pan | Multiphoton entanglement and interferometry[END_REF][START_REF] Wang | Experimental ten-photon entanglement[END_REF] are at the heart of future quantum communication networks [START_REF] Scarani | The security of practical quantum key distribution[END_REF][START_REF] Bunandar | Metropolitan quantum key distribution with silicon photonics[END_REF], sensors [START_REF] Brida | Experimental realization of sub-shot-noise quantum imaging[END_REF][START_REF] Schwartz | Superresolution microscopy with quantum emitters[END_REF], as well as optical quantum computers [START_REF] Knill | A scheme for efficient quantum computation with linear optics[END_REF][START_REF] O'brien | Optical quantum computing[END_REF] and simulators [START_REF] Lanyon | Towards quantum chemistry on a quantum computer[END_REF][START_REF] Loredo | Measuring entanglement in a photonic embedding quantum simulator[END_REF][START_REF] Chen | Efficient measurement of multiparticle entanglement with embedding quantum simulator[END_REF][START_REF] Santagati | Witnessing eigenstates for quantum simulation of hamiltonian spectra[END_REF]. These achievements make use of the interference of indistinguishable single-photons, allowing the realisation of quantum gates [START_REF] O'brien | Demonstration of an all-optical quantum controlled-NOT gate[END_REF][START_REF] Patel | A quantum fredkin gate[END_REF], and protocols such as quantum teleportation [START_REF] Wang | Quantum teleportation of multiple degrees of freedom of a single photon[END_REF] and entanglement swapping [START_REF] Takeda | Entanglement swapping between discrete and continuous variables[END_REF]. The onephoton term has been exploited heretofore, and the vacuum component has been considered detrimental to the overall protocol efficiency, motivating a quest for deterministic sources producing single-photon Fock states with no vacuum component [START_REF] Somaschi | Near-optimal single-photon sources in the solid state[END_REF][START_REF] Ding | On-demand single photons with high extraction efficiency and near-unity indistinguishability from a resonantly driven quantum dot in a micropillar[END_REF][START_REF] Senellart | Highperformance semiconductor quantum-dot single-photon sources[END_REF][START_REF] Kiršanskė | Indistinguishable and efficient single photons from a quantum dot in a planar nanobeam waveguide[END_REF]-a challenging task, to say the least. If vacuum is set instead in a quantum superposition with the single-photon, one could use it to encode quantum information in the photon-numberbecoming a resource for optical quantum information processing. For instance, vacuum within a pure quantum state can be exploited in quantum teleportation [START_REF] Lombardi | Teleportation of a vacuum˘one-photon qubit[END_REF], or quantum random number generators [START_REF] Gabriel | A generator for unique quantum random numbers based on vacuum states[END_REF]. However, obtaining quantum superpositions in the photon-number basis has so far demanded complex quantum state engineering and conditioned state preparation [START_REF] Bimbard | Quantum-optical state engineering up to the two-photon level[END_REF][START_REF] Fuwa | Experimental proof of nonlocal wavefunction collapse for a single particle using homodyne measurements[END_REF].

The text-book model of a quantum emitter is a two level atom-a system shown to generate quantum light in various excitation regimes. Incoherent non-resonant excitation of natural [START_REF] Grangier | Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences[END_REF] and artificial atoms [START_REF] Lounis | Single photons on demand from a single molecule at room temperature[END_REF][START_REF] Michler | A quantum dot single-photon turnstile device[END_REF][START_REF] Michler | Quantum correlation among photons from a single quantum dot at room temperature[END_REF] can produce optical fields with a large single-photon component, but without coherence in the photon-number basis due to the incoherent creation process of the atomic population. In contrast, coherent driving of an atom can in principle be used to transfer the coherence between the atomic ground and excited state to the emitted light field. This has so far been explored in the weak-excitation regime to produce quantum light that exhibits coherence with the driving laser-observed with atoms [START_REF] Jessen | Observation of quantized motion of rb atoms in an optical field[END_REF], as well as semiconductor quantum dots [START_REF] Nguyen | Ultra-coherent single photon source[END_REF][START_REF] Matthiesen | Subnatural linewidth single photons from a quantum dot[END_REF][START_REF] Matthiesen | Phase-locked indistinguishable photons with synthesized waveforms from a solid-state source[END_REF][START_REF] Proux | Measuring the photon coalescence time window in the continuous-wave regime for resonantly driven semiconductor quantum dots[END_REF]. This regime has been shown to produce squeezed light where an atomic dipole-with vanishing population-elastically scatters a coherent superposition of vacuum and a small one-photon term [START_REF] Schulte | Quadrature squeezed photons from a two-level system[END_REF]. Generating a photon-number superposition with large single-photon population requires to create an atomic population-inherently coupled to its environment-that remains insensitive to any decoherence until spontaneous emission takes place. To the best of our knowledge, the generation of photon-number quantum superpositions under strong coherent driving has not been reported so far, neither with natural atoms, nor with artificial ones.

In this work, we report on the on-demand generation of quantum superpositions in the photon-number basis, in light pulses emitted by a single artificial atom. We observe superpositions of zero, one, and two photons emitted from semiconductor quantum dots coupled to optical microcavities [START_REF] Somaschi | Near-optimal single-photon sources in the solid state[END_REF][START_REF] Giesz | Coherent manipulation of a solid-state artificial atom with few photons[END_REF]. We use pulsed coherent driving, beyond full inversion of the atomic population, and perform interferometric measurements with a pathunbalanced Mach-Zehnder interferometer (MZI). As supported by our theoretical calculations, phase-dependent oscillations at the interferometer output demonstrate the production of coherent superpositions of vacuum, one, and two photons. Below π-pulse driving, we obtain superpositions of vacuum and one-photon Fock states, with their relative populations controlled by the driving laser intensity. By driving the quantum dot with 2π-pulses, we obtain a state with the two-photon component larger than the one-photon population-a state allowing phase super-resolving interferometry, and incidentally resembling a small Schrödinger-cat state.

Coherent driving and photon statistics

We investigate semiconductor devices consisting of a single quantum dot (QD) positioned with nanometerscale accuracy at the centre of a connected-pillar cavity [START_REF] Somaschi | Near-optimal single-photon sources in the solid state[END_REF][START_REF] Dousse | Controlled light-matter coupling for a single quantum dot embedded in a pillar microcavity using far-field optical lithography[END_REF][START_REF] Nowak | Deterministic and electrically tunable bright single-photon source[END_REF]. The QD layer is inserted in a p-i-n diode structure, and electrical contacts are defined to control the QD resonance through the confined Stark effect. We note that the experimental results reported here have been observed on various QD-cavity devices. We focus hereafter on two devices: a neutral (QD1) and a charged (QD2) exciton coupled to the cavity mode, see Methods. QD1 (QD2) is excited resonantly with linearly-polarised 40 ps (15 ps) laser pulses at 925 nm, and its emission is collected using a crossed-polarisation scheme that separates it from the laser, see Fig. 1a. Figure 1b shows the detected countrates for QD1 as a function of the excitation pulse area A, evidencing well defined Rabi oscillations. The signal is damped by spontaneous emission due to the relatively long 40 ps excitation pulses [START_REF] Giesz | Coherent manipulation of a solid-state artificial atom with few photons[END_REF] as compared to the measured emission decay time of 166±16 ps. Second-order autocorrelation functions g (2) (∆t) measured along Rabi cycles evidence a dis- A single semiconductor QD is kept in a cryostat at 9 K, and is excited under pulsed resonant excitation. The QD emitted state |Ψ is separated from the laser in a cross-polarisation scheme, by using a polarising beamsplitter (PBS), a quarter-(Q), and half-wave plate (H). b Rabi oscillation of the QD coherent driving. The emission is collected by a single-mode fibre and directly detected with an APD. c Second-order autocorrelation function g (2) (∆t) measured at π-pulse with QD1, and d at 2π-pulse driving with QD2.

tinct and complementary behaviour between π-, and 2πpulse driving. A pronounced antibunched photon statistics at π-pulse is observed for both QD1 and QD2, with g

π (0)=0.037±0.002 for QD1, see Fig. 1c. Such observations show light wavepackets consisting mostly of either vacuum or one photon. However, bunched statistics is observed at 2π-pulse for QD2, with g

(2) 2π (0)=2.98±0.11, see Fig. 1d. This evidences, as recently observed [START_REF] Fischer | Signatures of two-photon pulses from a quantum two-level system[END_REF], wavepackets containing two-photon populations. In the following, we investigate the nature of light in the photonnumber degree-of-freedom: whether it contains photon Fock-states emitted in a mixture or in a pure quantum state.

Quantum superposition of zero and one photon

The Hong-Ou-Mandel (HOM) effect [START_REF] Hong | Measurement of subpicosecond time intervals between two photons by interference[END_REF] describes two single-photons simultaneously impinging on a beamsplitter. If the photons are polarisation, spatially, and frequency indistinguishable, they bunch at the output of the beamsplitter-a behaviour exclusively of quantum mechanical origin. This requires that the interfering photons are in the same pure quantum state in these degreesof-freedom.

As discussed now, interference can also be used to unravel coherences in the Fock-state basis. Consider a beamsplitter with inputs a, b, and outputs c, d, onto which pure states of photon-number superpositions impinge. These are in the form

|Ψ a = √ p 0 |0 a + √ p 1 e iα |1 a ,
and

|Ψ b = √ p 0 |0 b + √ p 1 e i(α+φ) |1 b , with p 0 +p 1 =1, p 0,1
the vacuum and one-photon populations, and φ a relative phase between the states. When p 1 =1, their quantum interference leads to the well known twophoton output state (|2 c 0 d -|0 c 2 d ) / √ 2-the HOM effect. However, as soon as p 1 <1, the output state shows other photon terms that lead to a mean photon-number N c,d =p 1 (1 ± p 0 cos φ) at the beamsplitter outputs, see Supplementary Information. That is, if states are pure in the photon-number basis, their interference leads to oscillations measured at the output of the interferometer device, with a visibility amplitude equal to the vacuum population p 0 .

The previous example describes the idealised case of pure states-instances non-existing in the physical world. To account for impurity in the photon-number basis, we consider that each light wavepacket impinging on the beamsplitter is described by a density matrix ρ S =λρ pure +(1-λ)ρ mixed , with ρ pure =|Ψ i Ψ i | a pure state (i=a, b), ρ mixed =diag{p 0 , p 1 } a diagonal matrix, and 0≤λ≤1 a parameter tuning the photon-number purity. Moreover, limited purity in the frequency domain is taken into account by the non-unity mean wave-packet overlap M between interfering photons. It can be shown, see Supplementary Information, that such interfering input states result in where n c,d =N c,d / (N c +N d ) oscillate with a visibility v=λ 2 p 0 √ M . We observe, from Eq. 1, that if the interfering states are distinguishable (M =0), or if the state is emitted in a statistical mixture of photon-numbers (λ=0), then v vanishes. Thus, observing v =0 implies that neither case is true: the state contains quantum coherences in the photon-number basis.

n c,d = 1 2 (1 ± v cos φ) , (1)
Coherent driving of a two-level system creates a quantum superposition of ground and excited state, with a relative phase governed by the classical phase of the driving laser. If this coherence is transferred to the emitted light state through spontaneous emission, we obtain a photonic state with coherences between the vacuum and one-photon components. We test this hypothesis by performing the above described interferometric measurements. To do so, we utilise an unbalanced MZI with a path-length difference matching the temporal separation of consecutive emitted wavepackets from the quantum dot to temporally overlap them on a beamsplitter, see Figure 2a. The free space part of the MZI leads to small path variations in the order of the photon wavelength, acting as the previously described phase φ.

Figure 2b

shows our measurements of n c,d for pulse areas A=0.61π, and A=0.14π. The single detector counts undergo clear oscillations with time, as the optical phase φ freely evolves in time within the interferometerevidencing quantum coherence in the photon-number basis. As predicted, the amplitude of the oscillations increases with the vacuum population, controlled here by choosing the driving pulse area. Figure 2c shows the extracted oscillation visibilities, obtained from maxima and minima of n c,d with respect to φ, for different values of single-photon countrates (bottom axis) as the pulse area varies within 0<A≤π (top axis). We observe the expected increase in visibility when increasing the vacuum part. The visibility v also depends on the mean wave-packet overlap M , which is extracted from coincidences counts at the MZI output. We measured M π =0.903±0.008 at π-pulse excitation, a value limited by a small residual phonon sideband [START_REF] Grange | Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics[END_REF] since no spectral filtering was used. We can then tune M via the relative photon polarisation, see Fig. 2d, and observe that the oscillation visibility vanishes for distinguishable photons, as expected. We observe that v is linear in the single-photon countrates, see Fig. 2c-accordingly, proportional to vacuum-from which we deduce an average λ=0.965±0.018 for all pulse areas up to π-pulse. The state purity in the photon-number basis P=Tr ρ 2 is extracted knowing p 0 , p 1 , and λ. We obtain an average value of P=0.968±0.008 in the full [0-π] pulse area range, see Fig. 2e, evidencing the high degree of purity. These states are produced on-demand: for each excitation pulse, the device emits a photon-number superposition, with p 0 +p 1 =1.

To support the model described above, we consider the situation where a two-level system, with ground |g and excited state |e , is coupled to a single spatial mode of the optical field, i.e., a one-dimensional atom [START_REF] Auffèves-Garnier | Giant optical nonlinearity induced by a single two-level system interacting with a cavity in the purcell regime[END_REF], a model that has been shown to account well for the system under study [START_REF] Giesz | Coherent manipulation of a solid-state artificial atom with few photons[END_REF]. We calculate the light-field generated by the QD by solving the Lindblad equation that accounts for the evolution of a two-level system-treating the incoming laser field, the interaction unitary Hamiltonian, as well as the non-unitary dynamics of spontaneous emission and pure dephasing, see Supplementary Information. We obtain a system output state that can be written as the density matrix ρ S . This matrix is time integrated over the whole light pulse, an approach that is valid for excitation pulses well below the spontaneous emission time.

We theoretically obtain the population p 1 (respectively, p 0 ), and coherences λ √ p 0 p 1 of ρ S from parameters within the one-dimensional atom model. Pure dephasing contributes to reducing the mean wave packet overlap of the emitted photons, as well as the populations. The solid blue line in Fig. 2e shows the calculated populations p 1 , and the solid green line the corresponding coherences for the case of maximally pure states (λ=1).

To the best of our knowledge, our observations report for the first time the on-demand direct generation of highly-pure optical quantum states in the photonnumber basis. Such photon-number quantum superpositions were demonstrated for microwave photons, using quantum feedback with Rydberg atoms [START_REF] Sayrin | Real-time quantum feedback prepares and stabilizes photon number states[END_REF], or through synthesized methods using a superconducting phase qubit [START_REF] Hofheinz | Synthesizing arbitrary quantum states in a superconducting resonator[END_REF]. Here, the quantum superposition is directly obtained from the spontaneous emission of a quantum emitter. This is observed not only in the weakexcitation regime [START_REF] Matthiesen | Phase-locked indistinguishable photons with synthesized waveforms from a solid-state source[END_REF]-i.e., elastic scattering-where the atomic population nearly vanishes, but also up to population inversion. As a result, by adjusting the excitation pulse area, we can generate quantum superpositions of zero and one photon with controlled populations. We note that our measurements provide information of the purity of the quantum state at the output of the emitter. Imperfect photon extraction from the device, or losses in the optical setup have no impact in the presented interferometric measurements, see Supplementary Information. In the next part, we study the case of even stronger driving, under 2π-pulse area, and report on the generation of a quantum superposition of zero-, one-, and two-photon Fock-states. Quantum superpositions up to two photons Strong driving of the atom has been proposed as a mean to generate photon-bundles [START_REF] Muñoz | Emitters of n-photon bundles[END_REF], and evidences for twophoton emission from an artificial atom has been reported recently by coherently driving a charged exciton at 2π-pulse [START_REF] Fischer | Signatures of two-photon pulses from a quantum two-level system[END_REF]. The excited state population at 2πpulse drive is expected to be zero, unless some relaxation process takes place during the pulse. In particular, as long as the driving pulse duration is not infinitely short, the atom in its excited state shows a non-zero probability to undergo spontaneous emission during the pulse. In such a case, a first photon is spontaneously emitted, and the probability for a second excitation during the pulse is non-zero, leading to the emission of a second photon at the end of the excitation.

The pronounced photon bunching observed at 2π-pulse excitation with our second device QD2, see Fig. 1d, quantified by g [START_REF] Erhard | Twisted photons: new quantum perspectives in high dimensions[END_REF] 2π (0)=2.98±0.11, shows that its emission at A=2π contains non-zero two-photon terms. The generated light wavepacket is then composed by zero-, one-and two-photon Fock-states (higher-number terms are negligible when the excitation pulse-length is significantly shorter than the spontaneous emission time [START_REF] Fischer | Signatures of two-photon pulses from a quantum two-level system[END_REF]). We now argue that the generated state contains quantum coherences in the photon-number basis. Indeed, the experimental setup depicted in Fig. 2a allows to quantify the photon-number populations-including the two-photon component-and degree of purity in this basis.

We learned, from Eq. 1, that the counts of a single detector at the output of the path-unbalanced MZI interferometer carry information on the quantum coherence between the zero and one-photon Fock states. If we now consider the coincidence counts from the two output detectors as well, we show that we can obtain information on the purity in the number basis up to two photons. In general, we can extend the previous analysis and consider an input state ρ S now containing terms up to the |m Fock-state. We consider a state with the same general form as before, i.e., ρ S =λρ pure +(1-λ)ρ mixed , in a simplified picture where the state impurity is described through a single parameter λ, reducing here all coherences in the same way. Within such framework, it can be shown, see Supplementary Information, that the detected single-click count rates N c,d at the interferometer outputs c and d, see Fig. 2a, read

N c,d = 1 2 [ n ± C 1 cos(φ)] , (2)
where n = m n np n is the system mean photon-number, and

C 1 =λ 2 m n √ np n p n-1
2 is a first-order coherence term, with the summation indices hereafter from 0 to m, and λ accounts for the photon-number purity. The coincidence rate (at zero delay) follows

C(0) = 1 8 [ n(n-1) -C 2 cos(2φ)] , (3)
with n(n-1) = m n n(n-1)p n the nonnormalised second-order correlation function, and

C 2 =λ 2 m n n(n -1)p n p n-2 2
a coherence term of second-order. In virtue of Eqs. 2 and 3, we obtain the normalised output coincidences at zero delay

C(0)=C(0)/ (N c N d ) = 1 2 g (2) (0) (1-v 2 cos 2φ) / 1-v 2 1 cos 2 φ
, where g (2) (0)= n(n-1) / n 2 is the normalised second-order correlation function of the input state, v 1 =C 1 / n is the single detector counts visibility, and v 2 =C 2 / n(n-1) the coincidences visibility. See that Eq. 3 shows oscillations modulated at twice the phasedependence of Eq. 2: coherences in the photon-number allow phase super-resolving interferometry.

The state generated at 2π-pulse driving contains up to two-photon terms.

In which case we obtain v 1 =λ 2 p 1 p 0 +2 √ 2p 0 p 2 +2p 2 / (p 1 +2p 2 ), v 2 =λ 2 p 0 , and g (2) (0)=2p 2 / (p 1 +2p 2 )

2 . Thus, by measuring v 1 , v 2 , g (2) (0), and taking into account normalisation p 0 +p 1 +p 2 =1, we univocally determine p 0 , p 1 , p 2 , and λ. Figures 3a,3b show respectively our measurements for n c =N c / n = (1+v 1 cos φ) /2, and coincidences proportional to C(0)∝ (1-v 2 cos(2φ)) with a rate specific to the losses of our setup. Figure 3c shows our obtained timecorrelated coincidence measurements C(∆t). As predicted, the coincidences at zero delay C(0) oscillate with 2φ-with minima (maxima) of coincidences occurring for φ=0 (φ=π/2). The full phase span for C(0) (modulo π) is shown in Fig. 3d. In the case of fully mixed (pure) states in the photon-number basis, i.e., λ=0 (λ=1), oscillation visibilities in the coincidence-counts fully vanish (maximally oscillate), see dashed grey (dot-dashed blue) curve in Fig. 3d.

We extract v 1 =0.192±0.008, v 2 =0.452±0.038, which together with the measured value of g [START_REF] Erhard | Twisted photons: new quantum perspectives in high dimensions[END_REF] 2π (0)=2.98±0.11, see Fig. 1d, and the normalisation of probabilities, results in the distribution {p n , λ}, with p 0 =0.838±0.012, p 1 =0.051±0.002, p 2 =0.111±0.010, and λ=0.734±0.025. The generation of light states with p 2 >p 1 is observed with charged excitons under a strong 2π-pulse drive of the QD, see Methods. The state ρ 2π

S contains zero, one and two photons, with a quantum state purity of P=0.870±0.024, see Fig. 3(e). Note that the above theoretical analysis for 2π-pulse driving does not account for the effect of limited photon indistinguishability (M <1). Accordingly, the reported purity contains both the photon-number purity and indistinguishability imperfections, thus representing a lower bound for the photon-number purity alone. This state, with p 2 >p1, incidentally resembles other quantum states of interest.

The obtained photon distribution {p 2π n } presents a statistical fidelity F cat = n p 2π n p cat n to {p cat n }, the probability distribution of an even "Schrödinger-cat" state |cat ∝|α +|-α , with |α a coherent state, of F cat =0.974±0.016 for a small cat state with |α| 2 =0.5. Thus, by simply driving a charged quantum dot with 2π-pulses, we are able to generate other photonic states that may find applications in coherent-state driven quantum computation [START_REF] Ralph | Quantum computation with optical coherent states[END_REF][START_REF] Lund | Fault-tolerant linear optical quantum computing with small-amplitude coherent states[END_REF] and quantum metrology [START_REF] Gilchrist | Schrödinger cats and their power for quantum information processing[END_REF]. Conclusions Quantum states with a high degree of purity are essential in all quantum-enhanced technologies. Optical quantum technologies have so far exploited various degrees-offreedom, such as time-frequency, angular momentum, or polarisation [START_REF] O'brien | Photonic quantum technologies[END_REF][START_REF] Erhard | Twisted photons: new quantum perspectives in high dimensions[END_REF]; but not the photon-number due to the absence of suitable sources. Our work demonstrates that state-of-the-art semiconductor QD emitters not only provide high purity in the frequency basis but also non-classical photon-number superpositions on-demand. We are now able to generate highly-pure light wave-packets with tuneable zero-and one-photon components. Other non-classical states can be also generated by adjusting the coherent excitation pulse duration and intensity, as shown here driving the atom at 2π-pulse. We believe that the generation of quantum superpositions of photon-numbers opens new exciting routes for optical quantum technologies. For instance, we now can exploit the interference of these novel photonic states, potentially impacting on the complexity of existing quantum-enhanced protocols, such as in quantum computing, or quantum walks.

METHODS

Sample The microcavity samples were grown by molecular beam epitaxy. A λ-GaAs cavity is surrounded by a bottom and a top mirror made of 29 and 14 pairs of GaAs/Al 0.9 Ga 0.1 As, respectively. The mirrors are gradually n-doped and p-doped in order to tune the quantum dot transition through the confined Stark effect. The cavities are centered on the quantum dots using the insitu optical lithography technique [START_REF] Dousse | Controlled light-matter coupling for a single quantum dot embedded in a pillar microcavity using far-field optical lithography[END_REF]. Then the sample is etched and standard p-contacts are defined on a large frame (300×300 µm 2 ) connected to the circular frame around the micropillar. A standard n-contact is defined on the sample back surface. A neutral exciton is coupled to the cavity mode for QD1. For the optical measurements, the polarisation of the laser is set so that the fine-structure splitting results to emission in crossed-polarisation, see ref. [START_REF] Giesz | Coherent manipulation of a solid-state artificial atom with few photons[END_REF]. A positively-charged exciton is coupled to the cavity mode for QD2, and in this case the circular-polarisation and optical transition rules naturally allows obtaining a signal in the crossedpolarisation configuration. The generation of pure coherent superpositions of zero and one photon has been observed for half a dozen devices, based either on neutral or charged exciton transitions, when exciting below π-pulse. The generation of light states with p 2 >p 1 , on the other hand, is only observed with charged excitons in the present experimental configuration. Indeed, in a crossedpolarisation collection scheme, the neutral exciton spontaneous emission is time delayed by the fine structure splitting [START_REF] Giesz | Coherent manipulation of a solid-state artificial atom with few photons[END_REF], preventing an efficient re-excitation within the same pulse. Time-tagged correlation measurements Simultaneous acquisition of single counts and double coincidences are recorded by measuring the photon count-rate and photon event time-tags in the output detectors (Si avalanche photodiodes) of the MZI, which are connected to a computer-controlled HydraHarp 400 autocorrelator.

Under free evolution of the phase φ between the two arms of the MZI, the total acquisition time per point has been set to T acq =310 ms (810 ms) for the results described in Fig. 2 (Fig. 3), with an integration time for the photon time-tags of T TT =200 ms (500 ms). Given the relatively fast acquisition of experimental points, the phase φ remains approximately unchanged during each acquisition run. The measurement protocol for each data point runs as follows: the Hydraharp autocorrelator reads the laser clock signal (24.6700 ± 0.0026 ns), a period of time which serves as a reference to determine the photon time tags of the detected events (accumulated during T TT ); consecutively, during the interval T acq -T TT , the countrates in the APDs are averaged, from where the phase φ is eventually obtained, see Supplementary Information. Data analysis The outcome of the time-tagged mea-surements renders the countrates and two-photon coincidences as function of time (the total integration time is in the order of 10-15 minutes for a given pulse area and relative photon polarisation in the MZI). From the oscillation of the single counts, an intensity-to-phase mapping organises the phase-dependent two-photon coincidences as function of the relative phase φ. We use the normalised intensity counts, e.g., n c , to assign a corresponding φ value for each given acquisition time-bin. For example, at a given time-bin, the n c values that are maximum, minimum, or equal to n d , are mapped to phases φ equal to 0, π, or π/2, respectively. See Supplementary Information for further explanations.

Here we present the model used to compute the quantum state of the emitted field. For the device used in the experiment, the cavity relaxation rate reads κ = 400 ± 100 µeV while the light-matter coupling constant equals g = 20 ± 5 µeV (Weak coupling regime). In this case the cavity mode can be adiabatically eliminated and the medium is modeled as a two-level system (TLS) coupled to a waveguide, a 1D atom [START_REF] Auffèves-Garnier | Giant optical nonlinearity induced by a single two-level system interacting with a cavity in the purcell regime[END_REF]. The TLS spontaneous emission rate γ accounts for the Purcell enhancement induced by the weak coupling to the cavity. Due to the interaction with the solid-state environment, the TLS also undergoes pure dephasing with a typical rate γ * .

The QD is pumped with a pulsed laser, resonant with the QD transition. The pump is modeled by a classical field containing n in photons per pulse on average. The Hamiltonian of the system in the referential rotating at the pump frequency writes

Ĥ = i Ω(t)(σ -σ † ) (4)
where

Ω(t)= √ n in γξ(t)
= - i Ĥ, ρ + D γ,σ [ρ] + D γ * /2,σz [ρ] (5)
where σ z = σ † , σ and D are super-operators defined as

D α,A [ρ] = α 2 2AρA † -A † Aρ -ρA † A (6)
Finally, the polarization of the collected light is orthogonal to the polarization of the the incident laser light, such that the output field a out solely depend on the TLS dipole:

a out = √ γσ (7)
The Lindbladt equation is time-integrated, giving access to the total number of emitted photons N out during the process and the second-order autocorrelation function C(0):

N out = dtTr ρ(t)a † out a out (8)
C(0) = dtdτ Tr ρ(t, τ )a † out a out (9)
with:

ρ(t, τ ) = U (t, t + τ )a out ρ(t)a † out U † (t, t + τ ) (10)
where U (t 1 , t 2 ) is an evolution operator, that satisfies ρ(t) = U (t 0 , t)ρ(t 0 )U † (t 0 , t).

From these parameters we easily deduce the Fock states populations:

p 0 = 1 -N out + C(0) 2 , (11)
p 1 = N out -C(0), (12)
p 2 = C(0) 2 . ( 13
)
The description of the eventual loss of purity induced by pure dephasing would require to model the complete 1D atom (i.e., the TLS coupled to an infinite amount of photonic modes) coupled to a pure dephasing reservoir, which is beyond the scope of the present paper. Instead, we make the assumption that the QD emits pure photonic states |ψ a and |ψ b (See main text and below). In this simplified model, the pure dephasing events mostly impact the time evolution of the TLS population, hence the relative populations of the Fock states.

Experimental imperfections are modeled in an effective manner. Reduced indistinguishability between the interfering photons due to pure dephasing and/or differences in the temporal profiles of the emitted photons give rise to a reduction of the overlapping factor M (See main text and below). Reduced purity due to decoherence in the Fock space basis is taken into account by the factor λ (See main text and below).

MODELLING INTERFERENCES FOR PURE INDISTINGUISHABLE SUPERPOSITIONS OF FOCK-STATES

In this section we present a simple model describing the photon interference taking place in the FBS HOM (defined in the Fig. 2 of the main text) of the MZI between quantum states of light in a coherent superposition in the photon-number basis. Here we ignore the temporal structure of the field emitted by the QD and treat the input and output ports of the BS as effective modes a, b, c, d, respectively. This allows to directly calculate the interference of the two input states, |ψ a and |ψ b , according to the beam splitter transformations.

This basic model allows in principle computing N c,d and C(0) between arbitrary photon-number superposition input states |ψ a and |ψ b , directly interfering in a beam-splitter. The general expression of the two different pure states is

|ψ a = ma n=0 √ p a n e iαn (a † ) n √ n! |0 a and |ψ b = m b n=0 p b n e i(βn+nφ) (b † ) n √ n! |0 b , with m a,b n=0 p a,b n =1, considering different maximal number of photons m a,b
and different relative phases among Fock terms α n and β n for each input state. Note that the |ψ b state acquires an additional phase φ, emulating its passage through one of the arm of the MZI. Assuming that the QD is emitting for each laser pulse the same kind of state |ψ , we take m a = m b and α n =β n .

Under such conditions, we write the initial input state as |ψ in =|ψ a ⊗ |ψ b and we apply the transformation of a balanced beam splitter on the creation operators

a † , b † following a † = c † +d † / √ 2 and b † = c † -d † / √ 2.
The final output state |ψ out contains all the information to calculate the N c,d and C(0). The expressions for single detector counts, simultaneous two detector coincidence counts and visibility of single detector counts read,

N c,d = n c,d n c,d | n c,d |ψ out | 2 (14)
C(0) = nc,m d n c m d | n c m d |ψ out | 2 (15) v = max φ N c -N d N c + N d (16)
Then, the normalised simultaneous two detector coincidence counts that we measure after the MZI, C(0), is obtained following C(0)=C(0)/(N c N d ). The normalised single counts in detectors c, d will be obtained by simply dividing each expression by the total amount of single detector counts N c +N d . We now restrict the study to the N =1, 2 cases describing our experiments, for an arbitrary relative phase α n between the Fock terms in each input state.

Vacuum and one-photon

In the first case up to N =1 photons, the two input states take the form |ψ a = √ p 0 + √ p 1 e iα1 a † |0 a and

|ψ b = √ p 0 + √ p 1 e i(α1+φ) b † |0 b .
The expression of the output state resulting from the interference of |ψ a and |ψ b is explicitly written in the next equation:

|ψ out = p 0 |0, 0 + p 1 √ 2 e i(2α1+φ) (|2, 0 -|0, 2 ) + 2p 0 p 1 e i(α1+ φ 2 ) cos φ 2 |1, 0 -i sin φ 2 |0, 1 (17)
For the sake of clarity, we have removed the explicit indication of the output modes c and d in the Fockstates, so that |n c , m d ≡|n, m . It is worth mentioning that the well known NOON state, 1/ √ 2 (|2, 0 -|0, 2 ), is retrieved when p 0 = 0. Instead, if p 0 = 0 a term with photon states |1, 0 and |0, 1 appear (see second line in Eq. 17), which produces the oscillation of single counts as function of the MZI phase φ, reported in Fig. 2 of the main text. In this case, and following Eq. 14, the visibility of the normalised single detector counts n c,d is v=p 0 , (where obviously C(0)=0) and, as we can see, there is no influence of α 1 in the observed oscillation of the single detector counts. For this reason, our measurements of single detector counts and two detector coincidence counts at the output of the MZI do not allow to retrieve information about the α 1 phase value of the emitted states of light.

In the particular case of interfering two states of superposition of vacuum and one photon, but with different phases α 1 , β 1 , then the normalised single detection counts read n c,d =(1±p 0 cos(α 1 -β 1 -φ))/2. Therefore, a finite relative phase α 1 -β 1 would be observed as an additional phase in the oscillation of n c,d as function of φ.

Vacuum, one, and two photons

In a second case regarding up to N =2 photons, we consider two input states given by

|ψ a = √ p 0 + √ p 1 e iα1 a † + √ p 2 e iα2 (a † ) 2 / √ 2 |0 a and |ψ b = √ p 0 + √ p 1 e i(α1+φ) b † + √ p 2 e i(α2+2φ) (b † ) 2 / √ 2 |0
b , which have identical {p n } distributions. The resulting state |ψ out of the interference takes a expression given by:

|ψ out = p 0 |0, 0 + 2p 0 p 1 e i(α1+ φ 2 ) |1, 0 cos φ 2 -i|0, 1 sin φ 2 + p 1 √ 2 e i(2α1+φ) (|2, 0 -|0, 2 ) + √ p 0 p 2 e i(α2+φ) (|0, 2 + |2, 0 ) cos φ -i √ 2|1, 1 sin φ + p 1 p 2 2 e i(α1+α2+ 3φ 2 ) ( √ 3|3, 0 -|1, 2 ) cos φ 2 + i( √ 3|0, 3 -|2, 1 ) sin φ 2 + p 2 4 e i2(α2+φ) √ 6|0, 4 -2|2, 2 + √ 6|4, 0 (18)
Let us consider first that α 2 =2α 1 ; this renders a visi-bility and double coincidence rate of:

v 1 = p 1 p 0 + 2p 2 + 2 √ 2p 0 p 2 p 1 + 2p 2 , (19)
C(0) = p 2 4 (1 -p 0 cos(2φ)) . (20)
It is worthy to note that in this case, the oscillation of the single detection counts will be observed (v 1 = 0) only if p 1 > 0. The overall behaviour of v 1 shows that the increase of vacuum is transformed in an increase of the oscillation visibility. It can be shown that in the particular case of up to two photons, and in the generalised case of up to m photons, one can rewrite N c,d and C(0) in the form of Eqs. 2,3 in the main text.

Finally, let us consider that α 1 and α 2 have nonzero values in the states with up to N =2 photons. Now the visibility v 1 takes a modified form from that given in Eq. 19 (in particular, note the new term cos(2α 1 -α 2 )):

v α 1 = p 1 p 0 + 2p 2 + 2 √ 2p 0 p 2 cos(2α 1 -α 2 ) p 1 + 2p 2 (21)
Equation 21 shows that α 2 =2α 1 reduces the visibility.

It can be demonstrated that, in this case, C(0) takes the same form as the one shown in Eq. 20 under the condition α 2 =2α 1 .

EXPERIMENTAL IMPERFECTIONS

As mentioned above, two kinds of experimental imperfections are considered: decoherence in the Fock-state basis or in the spectral basis, that are respectively quantified by the coherence damping factor λ<1 and the finite overlap between photonic states M <1. This finite overlap can be due to spectral decoherence (pure dephasing) or to the emission of wave-packets that are not identical.

N c = ψ in |ã † ã|ψ in (22)
N d = ψ in | b † b|ψ in (23)
ρ a in = p 0 |0 a 0 a | + p 1 |1 a 1 a | + √ p 0 p 1 λ|0 a 1 a | + h.c. (24)
ρ b in = p 0 |0 b 0 b | + p 1 |1 b 1 b | + √ p 0 p 1 λe iφ |0 b 1 b | + h.c, (25)
(t)f * b (t) + cc. Denot- ing M =| dtf a (t)f * b (t)| 2
the real number quantifying the overlap between two photons, we simply find that the distinguishability of the impinging photons leads to a damping of the interference term by a factor √ M , such as v → √ M v. The same effect is reached when the spectral purity of the two photons is reduced by pure dephasing.

In our experiment, we measured a degree of indistinguishability M =0.903±0.008, see Fig. 4a, for singlephotons emitted ∼12 ns appart from the QD, and when they are indistinguishable in all other degrees-of-freedom (polarisation, time of arrival) upon impinging at a beamsplitter. From this point, we vary their distinguishability via tuning their relative polarisations. We measured the visibility v, as well as the indistinguishability M , as function of the angle θ between the interfering polarisations (θ=0 for parallel, and θ=π/2 for orthogonal polarisations) for various pulse areas, see Fig. 4b,c. These measurements are used for displaying the behaviour of v in terms of M shown in Fig. 2d of the main text.

ROLE OF OPTICAL LOSSES

Here we show that optical losses in the experimental setup have no impact on the observations reported in the main text. To illustrate this, we drive the QD with a pulse area of A=0.7π, and compare measurements with different amounts of losses.

Figure 5a shows the single detector counts N c (red) and N d (blue) measured through our MZI interferometer with its nominal optical transmission, leading to ∼300kHz of total single-photons (black). At this point, we introduce extra losses to modify the measured countrates for at least one order-of-magnitude, resulting in about ∼10 times less signal. Figure 5b displays the normalised single detector counts n c before and after the extra losses are introduced, thus evidencing that optical losses have no impact on the observed visibilities.

From this, we deduce that our measurements contain information on the photonic state prior to any element acting as optical loss, thus at the level of the emitter.

TIME-TO-PHASE MAPPING

Here we describe how we obtained the interferometric phase φ from the raw measured countrates freely evolving in time. As an example, we take the measurements of n c,d for a pulse area A=0.39π, see Fig. 6a,b. An intensity binning of the data, see Fig. 6c,d, allows to assign a one-to-one correspondence between intensity bins and phase bins, thus obtaining the corresponding values of φ (modulo π) at each instant of time.

Similarly, the raw measurements taken at 2π-pulse area follow the same analysis, see Fig. 7. Here, we monitor single detector, see Fig. 7a,b, and coincidence counts simultaneously, see Fig. 7c. Each time-bin obtained from the single counts is assigned to one phase-bin. Finally, this information is used to obtain the phase-resolved coincidence measurements, see Fig.

FIG. 1 .

 1 FIG.1. Coherent control of an artificial atom. a Schematics of the setup. A single semiconductor QD is kept in a cryostat at 9 K, and is excited under pulsed resonant excitation. The QD emitted state |Ψ is separated from the laser in a cross-polarisation scheme, by using a polarising beamsplitter (PBS), a quarter-(Q), and half-wave plate (H). b Rabi oscillation of the QD coherent driving. The emission is collected by a single-mode fibre and directly detected with an APD. c Second-order autocorrelation function g(2) (∆t) measured at π-pulse with QD1, and d at 2π-pulse driving with QD2.

FIG. 2 .

 2 FIG. 2. Quantum superposition of vacuum and one photon. a Sketch of the MZI used to probe coherences in the photon-number. The MZI delays one arm by τ =12.34 ns as to allow interference of two consecutive wave-packets in the fibre beamsplitter FBSHOM. The phase φ between the two arms of the MZI is not stabilised and thus it freely evolves in time. A half-wave plate H in one arm tunes the photon distinguishability via their polarisation. b Normalised single countrates nc (blue) and n d (red) for a pulse area A=0.61π. Light blue (light red) traces display nc (n d ) for A=0.14π. Each data point here was accumulated for ∼300 ms. c Measured visibility v (blue squares) as a function of the countrates detected from our first collecting fibre. The blue solid line is a linear fit used to obtain the purity of the generated state, and the dashed blue lines consider lower purity values. d Visibility v in terms of the photon indistinguishability M (varied via polarisation). Blue, green, and red data points are taken for pulse areas A of 0.14π, 0.42π, 0.76π, respectively; and their corresponding curves follow the theoretical model v=λ 2 p0 √ M . e Blue line: theoretical prediction for the probability p1 of the QD to emit one photon. Blue data points: experimental one-photon population. Green full line: theoretical prediction of the photon-number coherence amplitude (|ρ01|=λ √ p0p1) assuming that the emitted state is pure (λ=1). Dashed green lines same: same as before for cases with less purity. Green data points: extracted values for |ρ01| deduced from the measured visibilities. Black data points: extracted values of the purity P.

FIG. 3 .

 3 FIG. 3. Quantum superposition of vacuum, one, and two photons. a Normalised single counts nc as the phase φ freely evolves in time. b Coincidence rate at zero delay evolving in time (full data was taken during 1500 s), cycling twice per unit of interferometric phase cycle ∆φ=2π-i.e., showing phase super-resolution. c Normalised time-correlated coincidences for a phase φ=0 (blue), and φ=π/2 (red). d Normalised coincidences at zero delay C(0) (red circles) as function of φ, with the theoretical prediction (red line) considering the extracted values of {pn, λ} described in e, and the expected predictions for the same Fock-state populations {pn} in the cases of maximally pure, i.e., λ=1 (blue dashed line), and classical statistical mixtures, i.e., λ=0 (gray dashed line). e Populations {pn} and purity P of the emitted state at A=2π.

0 + 1

 1 photons decohering in the Fock states basis It is useful to rewrite Eqs. 14 as N c = ψ out |c † c|ψ out as a function of the input state. Introducing the unitarity matrix U that transforms the input modes a, b into the output modes c, d of the BS, we have c := ã = U † aU = (a + b)/ √ 2 (resp. d := b = U † bU = (-a + b)/ √ 2) and |ψ out = U |ψ in . Finally,

FIG. 4 .

 4 FIG.4. Probing photon distinguishability. a Indistinguishability, M =0.903±0.008, measured in a Hong-Ou-Mandel setup, as explained in the main text, for single-photons emitted with ∼12 ns separation. b Single detector countrate visibility in terms of the polarisation angle θ of interfering photons, measured for pulse areas A=0.14π, 0.39π, 0.61π, 0.76π. c Indistinguishability measured for the same values of θ and pulse areas used in b. Note that, for a given angle θ, the indistinguishability remains the same across all pulse areas. In b, c, the solid lines are fits to the experimental data according to the equations described in the main text.

  7d

  .

FIG. 5 .FIG. 6 .

 56 FIG.5. Role of losses. a Countrates in each detector (red, blue), and their sum (black) as function of time for a pulse area A=0.7π. The countrates evolve in time as they depend on a freely evolving phase φ(t). b Normalised single detector counts nc, with its oscillation visibility remaining unchanged when introducing extra optical losses.

  product ã † ã, the interference terms in N c and N d (i.e., oscillating as a function of φ) are found proportional to I = ψ in |a † b|ψ in + cc, where cc stands for complex conjugate.The interference term can thus be rewritten asI = ψ a in |a|ψ a in ψ b in |b † |ψ b in + cc.In view of modeling the decoherence in the Fock state basis, we now rewrite ψ o in |o|ψ o in =Tr[ρ o in o], where Tr stands for the trace operation, and ρ o in =|ψ o in ψ o in | is the density matrix representing the quantum state in the input port o=a, b:

	Denoting |ψ b in = √ p 0 |0	as	|ψ a in =	√ p 0 |0 a + √ p 1 |1 a	(resp.

b + √ p 1 e iφ |1 b ) the input state in the a (resp. b) input port, the global input state is a product |ψ in =|ψ a in ⊗|ψ b in . By developing, e.g., the

  where h.c. stands for hermitic conjugate.Let us consider the term Tr[ρ a in a]. It reduces to Tr[ρ a in a]= 1 a |ρ a in a|1 a = 1 a |ρ a in |0 a and is damped with respect to the ideal case of a pure superposition by a factor λ. In the same way, the term Tr[ρ b in b] is damped by a factor λ. The decoherence in the Fock state basis thus leads to a damping of the interference term by a factor |λ| 2 , such as v → |λ| 2 v. The same reasoning can be extended to superpositions of 0, 1 and 2 photons.We now model the effect of the distinguishability of the photons impinging on the BS. Supposing that the quantum states sent in a and b are pure but have a temporal structure and correspond to different wave packets. The interference term now reads I= dt ψ in |a † (t)b(t)|ψ in +cc, where we have introduced o(t) the annihilation of a photon at time t in the mode o, verifying [o(t), o † (t )]=δ(t-t ). One photon in the mode o now writes |1 o = dtf * o (t)o † |0 , where dt|f o (t)| 2 =1. With these notations, the interference term can be rewritten I= dtf a

	Distinguishable photons

scientific advisor and co-founder, of the single-photonsource company Quandela.

Acknowledgements This work was partially supported by the ERC Starting Grant No. 277885 QD-CQED, the French Agence Nationale pour la Recherche (grant ANR SPIQE and USSEPP), the French RENAT-ECH network, a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d'Avenir programme (Labex NanoSaclay, reference: ANR-10-LABX-0035). J.C.L. and C.A. acknowledge support from Marie Sk lodowska-Curie Individual Fellowships SMUPHOS and SQUAPH, respectively. We thank N. Carlon Zambon for providing technical assistance throughout the project. Author contributions The experiments were conducted by J.C.L. and C.A. with help from P.H., C.M. H.O., and L.D.S. The data analysis was done by C.A. and J.C.L. The theoretical modelling was done by A.A., B.R, O.K., C.A., and J.C.L. 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 project was supervised by L.L., A.A., O.K., and P.S. The manuscript was written by J.C.L., C.A., and P.S., with input from all authors.


Hélène Ollivier

Ilse Maillette De Buy Wenniger

Sarah Thomas

Stephen Wein

Guillaume Coppola

Abdelmounaim Harouri

Paul Hilaire

Clément Millet

Aristide Lemaître

Isabelle Sagnes

Olivier Krebs

Loïc Lanco

Juan Carlos Loredo

Carlos Antón

Niccolo Somaschi
email: niccolo.somaschi@quandela.com

Pascale Senellart
email: pascale.senellart-mardon@c2n.upsaclay.fr

Reproducibility of high-performance quantum dot single-photon sources

Single-photon sources based on semiconductor quantum dots have emerged as an excellent platform for high efficiency quantum light generation. However, scalability remains a challenge since quantum dots generally present inhomogeneous characteristics. Here we benchmark the performance of fifteen deterministically fabricated single-photon sources. They display an average indistinguishability of 90.6 ± 2.8% with a single-photon purity of 95.4 ± 1.5% and high homogeneity in operation wavelength and temporal profile. Each source also has state-of-the-art brightness with an average first lens brightness value of 13.6 ± 4.4%. Whilst the highest brightness is obtained with a charged quantum dot, the highest quantum purity is obtained with neutral ones. We also introduce various techniques to identify the nature of the emitting state. Our study sets the groundwork for large-scale fabrication of identical sources by identifying the remaining challenges and outlining solutions.

I. INTRODUCTION

Quantum light sources are key building blocks for the development of quantum enhanced technologies. For instance, indistinguishable single photons are needed for multipartite quantum cryptography [1] and for developing quantum networks [2,3]. They also constitute attractive quantum bits for quantum computing [4,5] that do not suffer from decoherence and can be operated at room temperature [6]. Quantum light is also sought after for quantum sensing applications, be it for subshot noise quantum imaging [7] or superresolution [8].

For a long time, single-photon based technologies have relied on heralded single-photon sources, based on probabilistic photon pair generation in non-linear crystals [9]. However, these sources suffer from intrinsic limitations, where the multi-photon probability scales linearly with the source brightness and reaching high efficiencies requires difficult multiplexing schemes [10,11].

Semiconductor quantum dots (QDs) have emerged as an excellent platform to generate single-photons. After two decades of fundamental investigation and technological developments, a clear path has been laid out to obtain highly efficient sources of high quantum purity single photons [12,13]. Quantum dots behave like artifical atoms that can emit single photons on demand. By fabricating a microcavity around a QD, these single photons can be efficiently collected while simultaneously mitigating pure dephasing [14][15][16].

Despite the excellent performance of QDbased sources, the scalability of the technology remains an open challenge because natural QDs grow at random spatial positions [17] and both the natural and site-controlled QDs show inhomogeneous spectral resonances spanning 2 to 10 meV [18][19][20]. While several groups in the QD community have successfully achieved the fabrication of single-photon sources with state-ofthe-art performance [21][22][23][24][25], most demonstrations still rely on fabrication techniques where the spatial and spectral matching of the QD cavity coupling is not fully controlled. Obtaining a good QD-cavity coupling thus relies on testing a large number of devicessometimes in the thousands. Such a low control in the fabrication process is prohibitive for scalability, since the QD-cavity coupling determines not only the source brightness and its spectral bandwidth through the Purcell effect [26,27], but also its degree of indistinguishability [28,29]. More recently, several groups have developed techniques to precisely position the QD in a photonic structure [30][31][32]. Yet, only a few of them report the performance of more than one or two devices [13,33]. How far these technologies are from a large-scale production of identical single-photon sources remains an open question.

In the present work, we address this question by benchmarking fifteen sources consisting of single QDs inside micropillar cavities. The sources are deterministically fabricated using the in-situ cryogenic photolithography tech- nique [34]. They are operated under resonant excitation in order to obtain the highest degree of quantum purity. For each source, we measure the single-photon purity, indistinguishability, brightness, and study the reproducibility in operation wavelength and temporal profiles.Two types of sources are studied based on different optical transitions: one based on a neutral exciton and the other on a trion, which is a charged exciton. We demonstrate new techniques for identifying these optical transitions, and discuss the physics that determines the source characteristics. Finally, we outline the remaining challenges for larger scale fabrication of identical sources.

II. DEVICES AND OPERATION

A scanning electron microscope (SEM) image and an optical microscope image of two samples are presented in fig. 1a andb respectively. The sources that we study are composed of a single semiconductor InGaAs QD embedded in a micropillar λ-cavity with 14 (28) GaAs/AlAs Bragg pairs in the top (bottom) mirror. A 20 nm-thick Ga 0.1 Al 0.9 As barrier, positioned 10 nm above the QD layer, is used to increase the hole capture time inside the QD [35]. By creating an electron-hole pair in the QD with an additional laser, we can trap a hole since the electron quickly escapes whereas the hole can not [36]. This technique allows us to optically control the QD charge state. The fabrication process makes use of the cryogenic in-situ lithography technique that allows to position the pillar center within 50 nm of the QD and to adjust the pillar cavity diameter to ensure the spectral resonance between the QD and the cavity lines [34]. The pillar is connected through bridges to a circular frame and a large mesa structure where electrical contacts are attached [37]. This allows us to fine-tune the QD-cavity resonance via the Stark effect [37,38].

In the following, we report on sources from five samples (labelled from A to E) fabricated from the same 2-inch wafer grown by molecular beam epitaxy (MBE). Each sample contains 15 to 30 sources. Only a few sources were investigated on samples A through C and all sources were investigated on samples D and E. Among the studied sources, we selected those giving a first lens brightness greater than 5%. For samples D and E, it was the case for 6 and 4 sources respectively, corresponding to 20 to 25% of sources per sample. The labels in fig. 1b show the corresponding sources for sample D.

Highly indistinguishable photons are obtained from QDs when using resonant excitation [39,40]. A typical setup for this purpose is presented in fig. 1c. A laser providing 15 ps pulses at a repetition rate of 81 MHz is set to be resonant with the QD transition. The excitation beam enters the setup through a single-mode fiber; then a first telescope (not shown in the figure) optimizes the spatial overlap of the beam with the fundamental mode of the micropillar [34,41]. An input polarizer sets the polarization along the reflection axis of a polarizing beam splitter (PBS). The beam is sent through a 0.45 NA objective to a low vibration cryostat where the sample is cooled down to 7 K. The QD emission is collected in the transmission mode of the PBS, orthogonal to the input polarization. A half-wave plate (HWP) and a quarter-wave plate (QWP) are used together to control the polarization with respect to the cavity axes and to correct for polarization ellipticities induced by the setup. In the collection mode of the PBS, a second polarizer improves laser extinction and a second telescope adjusts the single-photon beam diameter to match the collection fiber mode.

Fig. 1d and 1e present typical secondorder correlation histograms characterizing the single-photon purity 1-g (2) (0) and the photon indistinguishability, here measured on device S7. The g (2) (0) is given by the integrated coincidences around zero delay normalized by the area of the side-band peaks. The indistinguishability is measured via twophoton Hong-Ou-Mandel (HOM) interference in a path-unbalanced Mach-Zehnder interferometer, where two consecutive single photons initially separated by about 12 ns interfere at a beam splitter [12,15]. The raw indistinguishability is given by V HOM = 1 -2A 0 where A 0 is the ratio of coincidences at zero delay to the averaged coincidences of the uncorrelated peaks. The examples in fig. 1 correspond to g (2) (0) = 0.0237 ± 0.0004 and V HOM = 0.895 ± 0.002.

III. TWO TYPES OF SOURCES

Using the cross-polarization setup depicted in fig. 1c, the source operation relies on the ability of the QD optical transition to generate light in a polarization orthogonal to the excitation. The emission process differs substantially depending on whether the optical transition under consideration is an exciton or a trion, which, as we explain in this section, leads to significant differences in the source performance.

A. Exciton based sources

We first analyze the exciton, which corresponds to a three-level system, as depicted in fig. 2a. It comprises a single ground state |g and the two intrinsic exciton eigenstateshereafter labelled |V and |H -leading to an emission in the corresponding linear polarization V and H . The two exciton states present a fine-structure splitting (FSS) ∆ FSS = E V -E H which is an energy difference arising from the Coulomb exchange interaction in a spatially anisotropic QD [42].

The micropillar is not perfectly circular and therefore the cavity itself presents a small anisotropy leading to two nearly-degenerate linearly polarized fundamental cavity modes. The energy difference typically amounts to 30-70 µeV which is smaller than the cavity spectral linewidth of 150-300 µeV. These two modes, which we label V and H, cause an effective birefringence: when the excitation polarization is not aligned to H nor V , the portion of the excitation beam that is coupled to the cavity sees its polarization rotated. Such polarization rotation, recently used to measure the cavity coupling accurately [41], leads to laser signal in the collection path. By choosing the excitation to be parallel with an axis of the cavity, this cavity rotated light is suppressed in the collection mode. In the following, we assume the excitation polarization is parallel to V and the single-photon collection mode is H. The quantum states of the exciton that have a dipole parallel to V and H are given by

|V = cos(θ) |V + sin(θ) |H |H = -sin(θ) |V + cos(θ) |H , ( 1
)
where θ, presented on fig. 3a, is the angle between the optical polarization axis of V and the dipole orientation of the exciton eigenstate |V .

In a simplified framework where the Purcell effect dominates and pure dephasing effects are negligible, the cavity amplitudes can be adiabatically eliminated to obtain an effective nonhermitian Hamiltonian that describes the emission process [43]:

H eff = E V - i 2τ |V V | + E H - i 2τ |H H | , ( 2
)
where τ is the total Purcell-enhanced emission lifetime, which for small cavity asymmetry is assumed to be the same for both excitonic states. Starting from the excited state |ψ(t = 0) = |V and following a pi-pulse excitation, the time evolution of the exciton is described by

|ψ(t) = cos(θ)e -i E V t e -t 2τ |V + sin(θ)e -i E H t e -t 2τ |H .
(

) 3
The single-photon emission collected from the H mode is then proportional to

| H|ψ(t) | 2 = e -t τ sin 2 t∆ FSS 2 sin 2 (2θ).
(4) This simple model explains important features. A V -polarized excitation creates an exciton state that has no overlap with the H mode at t = 0. The single-photon emission along H is thus time delayed from the excitation, with a timescale inversely proportional to the FSS. While both components of the excited state decay with the total decay rate 1/τ , the emission in the H polarization is modulated by the timedependent oscillation induced by ∆ FSS . In the The black points are the experimental data, the red curves are the fits to the theoretical models for total intensity, which include the contribution from the laser (orange curves). The blue curve is the emission arising from the exciton (eq. ( 4)) after taking into account the timing jitter of the detector.

limit where ∆ FSS → 0, no emission from an exciton in cross-polarized collection mode is expected. In addition, note that if the exciton axes are aligned along the cavity axes so that θ = 0, no cross-polarized emission takes place either. On the other hand, the collection is most efficient when θ = π/4.

The measured decay dynamics of an exciton is shown in fig. 2b for source S7. For visualization purposes, the light of the laser was not perfectly extinguished for these data so that we detect both the laser pulse and the exciton emission dynamics to compare the relative timescales. The maximum of the single-photon emission is delayed by approximately 200 ps from the laser excitation pulse. The overall exponential decay is governed by the Purcellenhanced spontaneous emission rate, and is also modulated by the phase dependence of the frequency components H and V at the rate ∆ FSS . The experimental observations are accurately reproduced by eq. ( 4) using the following parameters: τ = 252±3 ps and ∆ FSS = 8.58±0.03 µeV. The damping of the oscillations observed in the lifetime curve were found to be consistent with a finite gaussian detector jitter time with a full width at half maximum (FWHM) of 53 ps, which also dominated the observed width of the 15 ps Gaussian excitation pulse.

All the different excitons studied in this work show a FSS value ranging roughly from 5 to 10 µeV . Such emission process, governed by the QD and cavity asymmetries, adds extra challenges to the fabrication of identical sources. These complex features are circumvented when one considers sources based on a trion, as discussed in the following.

B. Trion based sources

When the QD contains a single charge, here a hole, the energy levels and optical selection rules are different. In the absence of an in-plane magnetic field, no linear polarization direction is favored and the optical transition rules can be written in the H-V cavity axes basis. The system ground state is composed of two degenerate energy levels of the hole spin state |⇑ and |⇓ , which we define with a quantization axis parallel to H. The excited trion states |X + ⇑ and |X + ⇓ correspond to two holes with opposite spin states and one electron. The optical selection rules governing these transitions are summarized in fig. 2c: each excited state is connected to both ground states through two optical transitions with linear orthogonal polarizations.

From this picture, it is straightforward to note that in the absence of spin initialization, namely when the hole spin is a mixture of spin up and down states, a V -polarized optical excitation leads to a population of both trion states, that each radiate with 50% probability along H. This happens on the timescale of the excitation pulse, as opposed to the previous case of the exciton.

The illustrated selection rules lead to the generation of single-photon wavepackets with a mono-exponential decay, where the rise time is governed by the excitation pulse length and the decay time by the Purcell-enhanced spontaneous emission rate. Fig. 2d shows the emission dynamics of the trion-based single-photon source S11. The emission intensity now shows a rapid rise time followed by a mono-exponential decay with a lifetime of 164.9 ± 0.9 ps. Here, the finite detector response of roughly 50 ps is consistent with the observed rise time.

As shown in the following, the nature of the optical transition does not only control the temporal profile but it also determines the brightness and single-photon purity of the sources. It is therefore important to develop tools to identify the nature of the transition.

Advanced equipment is required to observe the exciton splitting, either for high resolution spectral analysis or, as shown in fig 2b, high temporal resolution of the emission dynamics. In the next section, we demonstrate another simple identification tool based on the polarization-dependent optical selection rules.

C. Rotated emission identification method

The performance of each source is determined both by the nature of the transition and, in the case of excitons, the orientation of the cavity axes with respect to the QD dipoles. In this context, it is useful to analyze the emission collected in cross-polarization when turning the excitation polarization V exc by an angle φ with respect to the cavity axes, see fig. 3a. In the experimental configuration described in fig. 1c, the collection polarization is set orthogonal to the excitation polarization by the PBS, hence in the H coll polarization direction (see fig. 3a). For φ = 0, the excitation is parallel to the V cavity axis and only the spectrally narrow emission arising from the QD is collected in the H-mode, which corresponds to the configuration used for the source operation. However, when φ is different from 0 deg or 90 deg, an additional broader emission line is visible as shown in fig. 3b for the two devices S5 and S13. This broader emission arises from the birefringence of the cavity, as explained in subsection III A. For better visualization of the cavity-induced light rotation, we used 3 ps laser pulses, which are spectrally broader than the cavity linewidth. The emission of the cavity rotated light is maximal for around φ = 45 • and 135 • [41].

Fig. 3c shows the emission spectra measured for the exciton S5, as a function of φ. Fig. 3e presents the intensity of the cavity rotated light and the QD emission as a function of φ. The emission arising from the exciton has a sinusoidal dependance with φ. A laser polarized along one of the exciton axes V or H (θ = 0 or π/2) excites an eigenstate of the system and no emission takes place in the orthogonal polarization. The emission of the exciton thus depends on the angle between the incident polarization and the exciton axes (θ-φ). This measurement also allows to determine θ, the angle between the cavity polarization and the exciton axes, The excitons present an average g (2) (0) of 2.89 ± 0.74%, an average indistinguishability of 92.8 ± 1.1% and an average first lens brightness of 11.5 ± 3.7%. The trions present an average g (2) (0) of 5.42 ± 0.92%, an average indistinguishability of 89.5 ± 2.8% and an average first lens brightness of 14.7 ± 4.6%. (e) Table gathering the measured single-photon rates using a ∼ 30% efficient single-photon detector and their corresponding fibered single-photon rates at the output of a single-mode fiber. (f) Temporal profiles of trion-based sources showing an average exponential decay lifetime of 180 ps with a standard deviation of 17 ps.

an important parameter for the source brightness (see eq. ( 4)). Fig. 3d and3f show the same experiment and analysis on a trion-based source S13. The rotated light arising from the cavity has the same sinusoidal dependence but the trion emission is now independent of φ. Indeed, the selection rules illustrated in fig. 2c can be rewritten in any two-orthogonal linear polarization basis and the amplitude of the QD emission in the polarization orthogonal to the excitation is independent of the orientation of the excitation.

IV. SOURCE BENCHMARKING

We now discuss the benchmarking of fifteen devices, whose main quantum properties are presented in fig.

4. We consider not only the figures of merit defining each source performance (single-photon purity, indistinguishability and brightness), but also the emission wavelength and temporal profile, that are crit-ical characteristics for large-scale fabrication of identical sources. All figures of merit are reported using π-pulse excitation [44] with 15 ps pulses at a repetition rate of 81 MHz.

Fig. 4a shows the g (2) (0) values of the sources, with an average of 4.6 ± 1.5%. The single-photon purity (1-g (2) (0)) is systematically higher for excitons than for trion sources. This can be understood by comparing the different emission processes for each source type in cross-polarization (shown in fig. 2b and fig. 2d). Since the emission from the exciton through the H mode of the cavity is delayed compared to the excitation pulse, the probability that the quantum dot gets re-excited afterwards is low because the excitation pulse is over. In the case of a trion, the emission process in cross-polarization begins on the same timescale as the pulse and so there is a higher probability of re-excitation of the transition within the same excitation pulse, leading to the emission of a second photon [45,46]. Fig. 4b shows the mean wavepacket overlap between two consecutive single photons emitted with a 12 ns delay. These values are obtained from the HOM visibility histogram and corrected from the corresponding non-zero g (2) (0) [47]. The mean wavepacket overlap over the various sources is quite homogeneous, with an average value of 90.6 ± 2.8%. Note that this indistinguishability is obtained without spectral filtering and thus it includes the contribution from the phonon sideband, which is strongly suppressed by the cavity funneling effect [16,29]. The slightly lower indistinguishability values of the sources from sample E are likely due to a higher temperature on that chip (around 10-11 K). These observations also show that, despite a very different temporal structure of the single-photon wavepacket in the case of exciton based sources, the coherence is highly preserved in the frequency domain over the emission process.

The source brightness corresponds to the probability to obtain a single photon for a given excitation pulse. It is defined as the collected single-photon rate divided by the excitation repetition rate [48]. This probability is measured at the output of the collection fiber and then deduced at the first lens (fig. 4c). Both values depend on the global efficiency of our setup (transmission and output coupling), which is approximately 40%. The average first lens brightness is 13.6 ± 4.4%, which is on par with state-of-the-art values using a 0.45 NA collection objective.

In a cross-polarization setup using cavities with a small birefringence, both the first lens and fibered brightness are limited to at most 50% due to the rejection of photons orthogonally polarized to the collection mode. However, this limit can be overcome using polarized cavities [13]. In addition, the brightness is limited by the extraction efficiency, which is intrinsic to the cavity and is very similar for all pillars (∼ 65%), and also by the occupation probability of the QD charge state. Indeed, the rather large variation in brightness among the trion sources could be reduced by a better control on the occupation probability of the hole in the QD [36]. The brightness of an exciton based source further depends on both θ and the relative values of ∆ FSS and the emission rate 1/τ . Because the values of θ and FSS are not controlled during fabrication, excitons tend to be slightly less bright than trions.

The table in fig. 4e presents the fibered single-photon rate obtained by dividing the detected count rates measured at the output of a single-mode fiber by the efficiency of our silicon based avalanche photodiode (30% detection efficiency at ∼925 nm). It is also worth noting that we use a long-distance microscope objective with a numerical aperture of 0.45 that is placed outside of the cryostat. This limits the brightness because not all of the emission is collected by the objective. Also, this spatial truncation of the beam leads to a deformation of the wavefront which reduces the coupling into the single-mode fiber of typically 50 to 60%. Such collection efficiency could be strongly improved by inserting a high numerical aperture lens into the cryostat itself.

V. PERSPECTIVES FOR SCALABILITY

Our systematic study of a large number of single-photon sources allows identification of the remaining challenges for the fabrication of remote sources generating highly identical photons. In this context, a key parameter is the source operation wavelength. The typical inhomogeneous broadening of the InGaAs QDs spectum is around 30 nm. However, the insitu lithography allows us to select QDs in a small spectral range and to fabricate pillars with the correct diameter to match the QD resonance [37]. Fig. 4d shows the optimal operation wavelength of the sources. Overall, a small deviation of the operation wavelength is observed with an average wavelength of 924.7 nm with a standard deviation of only 0.5 nm. When considering sources fabricated on the same part of the original wafer, this deviation is substantially reduced with 0.06 nm for D and 0.12 nm for E, showing the high degree of control provided by the in-situ lithography. Thus, we conclude that a higher homogeneity in the growth process across the wafer, as typically obtained from industrial epitaxy growth systems, would increase the yield of sources operating at the same wavelength. Note that the residual 0.06 nm of deviation can be compensated via Stark tuning with minimal reduction to the brightness considering that the typical linewidth of our micropillar cavities is 0.2 nm.

The temporal profile of the single-photon wavepackets is also another important feature to consider. We show that the challenges are different for sources based on neutral or charged QDs in a cross-polarized setup. Controlling the temporal profile of a exciton based source is challenging because it requires control of the fine-structure splitting and the relative axes orientation of the cavity and QD. Such complete control was not achieved here, but many tools have been developed in the last few years to control both the exciton fine-structure splitting [49] and the cavity birefringence [13]. On the other hand, we find that the temporal profiles of the trion-based sources are very consistent across all samples, with an average decay time of 180 ps and a standard deviation of only 17 ps (see fig. 4f). The most promising path to scalability thus lies in the development of trion based sources where higher single-photon purity can be obtained by using shorter excitation pulses to reduce the probability for re-excitation. Such short pulses have not been used here due to the chromatic dependence of the polarization rejection apparatus. Using a specially designed highly achromatic polarization control or a side excitation scheme [50] would improve of the sources performance. Finally, we note that the variation in the brightness of the trion-based sources is rather large. This is due to an imperfect control of the QD charge state. This could be improved by applying an electric field during the in-situ lithography process, which would allow us to explore the different available charge states and their characteristics prior to the QD selection and cavity definition.

VI. CONCLUSION

We have benchmarked a large number of single-photon sources based on semiconductor QDs. The present technology is based on the deliberate choice of using naturally grown QDs, which currently have a higher quantum purity than site controlled QDs [51,52]. Despite the QD random distribution, in both spatial and spectral degrees of freedom, our technology based on selecting the QDs during an in-situlithography step and building a tailored cavity around them, allows us to obtain a large number of sources with highly homogeneous properties. The first-lens brightness is shown to reach state-of-the-art values for resonant excitation with a highly reproducible indistinguishability of 90.6 ± 2.8%. Moreover, we discussed the physics of the sources behaviour in a cross-polarized resonant excitation scheme, and identified the parameters controlling the wavepacket temporal profile, the source brightness and the single-photon purity. The present study shows a clear path for scaling the fabrication process where highly identical remote sources could be obtained using a trion.

Figure 1 .

 1 Figure 1. Single-photon source devices and operating setup. (a) Scanning electron microscope image of one of the samples under study: each wheel-shaped structure represents a single microcavity coupled to a single QD transition. (b) Optical microscope image of another sample under study. Labels refer to the source numbering used hereafter in the benchmarking. (c) Schematic of the optical setup used for resonant cross-polarization excitation of the QD devices. (d) Typical secondorder correlation histogram g (2) (t) as a function of the delay between two detectors and (e) typical second-order correlation in a Hong-Ou-Mandel two-photon interference experiment.

Figure 2 .

 2 Figure 2. Two types of sources under study. (a) and (c) Schematics of the energy levels and optical selection rules for a single-photon source based on an exciton (a) and a trion (charged exciton) (c). (b) Time evolution of the emission for a source based on an exciton (S7), plotted linearly in the upper panel and logarithmically in the lower one. (d) Time evolution of the emission for a source based on a trion (S11).The black points are the experimental data, the red curves are the fits to the theoretical models for total intensity, which include the contribution from the laser (orange curves). The blue curve is the emission arising from the exciton (eq. (4)) after taking into account the timing jitter of the detector.

Figure 3 .

 3 Figure 3. Line identification through cavityinduced birefringence. (a) Schematic of the various polarization directions involved in the system: cavity axes V and H, exciton axes V and H and polarization of the excitation and collection, Vexc and H coll respectively. (b) Spectra obtained for different values of the polarization angle of the excitation laser φ, illustrating the cavity birefringence contribution. The upper panel corresponds to S5, with the narrower signal from the QD highlighted in green, and the lower panel corresponds to S13, with the narrower signal from the QD highlighted in pink. In both panels, the broader blue curve is the cavity rotated light. (c,d) Emission spectra measured as a function of φ for two devices S5 (c) and S13 (d). The broader emission lines correspond to the signal arising from the cavity birefringence, the narrower lines to the QD emission. (e) and (f) Intensity of the line arising from the cavity birefringence in blue and from the QD emission line in green (red) as a function of the angle φ, for the exciton (trion). The cavity rotated light curve in panels (e,f) are the maximum of the broader peak in panels (c) and (d) respectively. The exciton and trion curves are the maximum of the narrower peaks, minus the linearly interpolated value of the broader peak (cavity rotated light).

Figure 4 .

 4 Figure 4. Performance of fifteen single-photon sources. (a) Second-order correlation values g (2) (0), characterizing the single-photon purity, (b) mean wavepacket overlap, (c) first lens brightness and (d) operation wavelength. The green squares represent sources based on excitons and the orange diamonds represent sources based on trions. The horizontal solid lines show the mean value for each type of sources.The excitons present an average g(2) (0) of 2.89 ± 0.74%, an average indistinguishability of 92.8 ± 1.1% and an average first lens brightness of 11.5 ± 3.7%. The trions present an average g(2) (0) of 5.42 ± 0.92%, an average indistinguishability of 89.5 ± 2.8% and an average first lens brightness of 14.7 ± 4.6%. (e) Table gathering the measured single-photon rates using a ∼ 30% efficient single-photon detector and their corresponding fibered single-photon rates at the output of a single-mode fiber. (f) Temporal profiles of trion-based sources showing an average exponential decay lifetime of 180 ps with a standard deviation of 17 ps.

ACKNOWLEDGMENTS

This work was partially supported by the ERC PoC PhoW, the French Agence Nationale pour la Recherche (grant ANR SPIQE, USSEPP, QuDICE), the IAD -ANR support ASTRID program Projet ANR-18-ASTR-0024 LIGHT, the QuantERA ERA-NET 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: ANR-10-LABX-0035). J.C.L. and C.A. acknowledge support from Marie Skłodowska-Curie Individual Fellowships SMUPHOS and SQUAPH, respectively. H. O. Acknowledges support from the Paris Ile-de-France Région in the framework of DIM SIRTEQ.