Hong-Ou-Mandel interference is a cornerstone of optical quantum technologies. We explore both theoretically and experimentally how the nature of unwanted multi-photon components of single photon sources affect the interference visibility. We apply our approach to quantum dot single photon sources in order to access the mean wavepacket overlap of the single-photon componentan important metric to understand the limitations of current sources. We find that the impact of multi-photon events has thus far been underestimated, and that the effect of pure dephasing is even milder than previously expected.

Quantum interference of indistinguishable single photons is a critical element of quantum technologies. It allows the implementation of logical photon-photon gates for quantum computing [1,2] as well as the development of quantum repeaters for secure long distance communications [3,4]. The development of efficient sources of single and indistinguishable photons has become a challenge of the utmost importance in this regard, with two predominant, distinct approaches. The first one is based on non-linear optical photon pair production [5,6], and multiplexing of heralded single photon sources is being explored to overcome an intrinsic inefficiency [7][8][9][10]. The other is based on single quantum emitters such as semiconductor quantum dots [11,12] where ever-growing control of the solid-state emitter has enabled the combination of high efficiency and high indistinguishability [13][14][15][16].

The standard method to quantify the indistinguishability of single-photon wavepackets is to perform Hong-Ou-Mandel (HOM) interference [17]. In perfect HOM interference, two indistinguishable single photons incident at each input of a 50:50 beam-splitter will exit the beamsplitter together, resulting in no two-photon coincidental detection events at both outputs. In practice, however, the two inputs only exhibit partial indistinguishability described by a non-unity mean wave-packet overlap M s (also defined as the single-photon trace purity [18,19]). Partial indistinguishability of the input states leads to coincidental detection events at the outputs and reduces the HOM interference visibility. The interference visibility V HOM can therefore give direct access to the singlephoton indistinguishability M s = V HOM [19].

For non-ideal single-photon sources, for which the photonic wavepackets present a residual multi-photon component, the HOM visibility remains the relevant quantity that determines the quality of the abovementioned quantum operations. However, the visibility of HOM interference is reduced due to multi-photon contributions, even if M s = 1, i.e. for an ideal singlephoton indistinguishability. In most cases, the multiphoton component of the photonic wavepacket, characterized by the second order intensity autocorrelation at zero time delay g (2) (0), depends on the system parameters in a manner that is completely independent of the single photon indistinguishability, and it is critical to have tools to access the latter in order to understand the physics at play and improve the performance of single photon sources.

Here we explore both theoretically and experimentally HOM interference with imperfect single-photon sources. Previously, the impact of multi-photon contributions on HOM interference has been investigated in the limited case where the additional photons are in the same spectral and temporal mode as the predominant ones [20][21][22]. It has been shown that the visibility of HOM interference in this case is given by V HOM = M tot -g (2) (0) [19,23], where M tot is the mean wavepacket overlap of the total input state, i.e. including the multi-photon component. Here we show that the properties of the additional or "noise" photons play a critical role in HOM interference, and that it is crucial to know the origin of the imperfections to be able to correctly extract the intrinsic single-photon indistinguishability M s . We validate our approach by experimentally emulating two types of imperfect sources. Finally, we investigate the case of quantum-dot based single photon sources (QDSPS) based on both neutral and charged excitons. By understanding the physical mechanisms in both cases, we are able to provide a proper way of extracting the single photon indistinguishability that accounts for the nature of the multi-photon events.

We model an imperfect "single-photon" state (g (2) (0) > 0) by mixing a true single photon (g (2) (0) = 0) with separable noise at a beam splitter. We limit our analysis to small g (2) (0) values so that the noise field itself is well-approximated by an optical field with at most one additional photon and a large vacuum contribution. This restriction to a weak, separable noise field remains relevant in practice for many situations as illustrated for QDSPSs later on.

It can be shown (see Supplementary Material), that for separable noise and a small resultant g (2) (typically g (2) < 0.3), the visibility of HOM interference is given by:

V HOM = M s - 1 + M s 1 + M sn g (2) (1)
where M sn is the mean wavepacket overlap between the single photon and an additional noise photon satisfying 0 ≤ M sn ≤ M s . We have defined g (2) ≡ g (2) (0) for simplicity.

It is instructive to consider the two limiting cases of Equation 1. If the additional photons are identical to the single photons, i.e. M sn = M s , then Equation 1 reduces to the simple case that V HOM = M s -g (2) , showing that the total and single photon mean wavepacket overlap coincide, M s = M tot . Alternatively, if the noise has no overlap with the single photons and M sn = 0 , then the visibility is further reduced and given by V HOM = M s -(1 + M s ) g (2) . The degree to which HOM interference is affected by a non-zero g (2) is therefore dependent on the origin of the additional photons.

We experimentally test this model by emulating imperfect single photon sources. We prepare a train of near-optimal single photons and mix them with additional photons to controllably increase g (2) and measure the impact on the HOM interference. We experimentally emulate the two limiting cases outlined above: when the additional photons are completely distinguishable (M sn = 0) from our single photon input, and when they are completely identical (M sn = M s ). In each case, we measure the g (2) and HOM interference visibility of the resultant wavepacket in an unbalanced Mach-Zehnder interferometer (see Supplementary Material for further details).

We use a state-of-the-art single photon source based on a quantum dot (QD) deterministically embedded in an electrically contacted micropillar cavity [13]. The QD acts as an artificial atom which we coherently control via resonant excitation to generate single photons with high single photon purity, g (2) < 0.05, and high indistinguishability, M tot > 0.9. The single photons are separated from the excitation laser using a cross-polarization set-up, as shown in Figure 1(a).

The experimental set-ups that enable a controlled increase of the multi-photon probability are shown in Figure 1(b) and (c) for the two limiting cases. First, to add fully distinguishable photons, we mix the single photons from the QDSPS with attenuated laser pulses at a different wavelength. A 3 ps Ti-Sapph pulsed laser centered at 925 nm is spectrally dispersed using a diffraction grating, and a narrow portion is selected to obtain a 15 ps excitation pulse resonant with the QD transition (here a charged exciton). A second, non-overlapping part of the spectrum is selected to mix with the emitted single photons. By appropriately tuning the time delay we can add synchronous spectrally-distinguishable photons to the single photon emission. The corresponding output field is then considered as an effective source, and we adjust the power of the laser beam to alter the magnitude of the two-photon component. The measured HOM visibility as a function of g (2) of this effective source is shown in Figure 1(d). Since the spectral overlap between the QDSPS photons and the additional laser photons is zero (M sn = 0) our model predicts that (2) , where a single parameter M s accounts for both the origin of the curve at zero g (2) and its slope. The line in Figure 1(d) shows that this model fits the data very well with M s = 0.94 ± 0.02.

V HOM = M s -(1 + M s )g
To create a wavepacket where the additional photons are identical to the predominant single photon component, we build another effective source where we add a small fraction of photons from the same QDSPS generated at a later time. This is obtained by performing an unbalanced quantum interference between two photon pulses produced by the QDSPS with delay τ . The first half waveplate (HWP) and polarizing beam splitter (PBS) in Figure 1(c) allow us to tune the relative intensity of the predominant single photon pulse and the additional photons. Then, a pair of QWP and HWP is used to make the polarizations of both photons identical before the 50:50 beam splitter (BS). Most of the time, only the main photon gets to the second BS and is transmitted with 50% probability. However, when this photon meets a second one generated after a τ delay and temporally overlapped, they will undergo HOM interference and exit the beam splitter in the same output port. Therefore, the output of the second BS has a higher g (2) , since there is a small probability that some of the output pulses now contain two identical photons. By adjusting the splitting ratio at the first beam splitter, the g (2) of the output state can be controlled. Figure 1(d) presents the HOM visibility as we increase the g (2) via addition of identical photons, where a clear difference is observed compared to the previous limiting case. For M sn = M s , the model predicts V HOM = M s -g (2) , a linear dependence with slope of -1. The line in Figure 1(f) again demonstrates that the model gives a very good fit to the data, with an extracted M s = 0.89 ± 0.01.

We note that the extracted values of M s for these two (2) for distinguishable (green squares) and identical (black circles) noise sources. The lines are the predictions from the theoretical model.

cases represent the upper and lower bound of the intrinsic single photon indistinguishability of the QDSPS used in these measurements. If the non-zero g (2) of the QDSPS was due to distinguishable noise then we could deduce that M s = 0.94±0.02. Similarly if the noise was identical then the QDSPS has a single photon indistinguishability of M s = 0.89 ± 0.01. This demonstrates that it is necessary to know the origin of the unwanted photon emission in order to be able to extrapolate the data back to g (2) = 0.

Our study highlights the importance of determining the origin of the multi-photon component in order to properly extract the single photon indistinguishability. To the extent of our knowledge, this has so far only been done in the indistinguishable case, independent of the physical phenomena of the multi-photon components. In the following, we discuss how to properly estimate the single photon indistinguishability for the current highest performing single photon sources, i.e. QD based sources.

There are two distinct categories of QDSPS, depending on the charge state of the quantum dot: neutral excitons and charged excitons (hereafter referred to as exciton and trion states respectively). The optical selection rules and photon emission processes differ significantly between the excitons and trions [24], leading to a different origin of the multi-photon component.

For an exciton, the system is described by a three level system where the excitation pulse creates a superposition of the two excitonic linear dipoles with an energy difference given by the fine-structure splitting [25]. This results in a time dependent phase between the two exciton eigenstates, so that the single photon emission in cross-polarization beats with a period determined by the fine structure splitting [24,26], as shown in Figure 2(a). (a) Time traces of the single photon wavepacket emitted by exciton (upper) and trion (lower) based sources. The excitation laser pulse is shown in grey. (b) Measured g (2) for trion (purple squares) and exciton (blue circles) based sources as a function of the excitation pulse duration at π-pulse. The error bars are within the size of the plotted points.

These optical selection rules imply that the single photon emission in cross polarisation is delayed with respect to the excitation pulse. For a trion based source, the optical selection rules correspond to a four level-system with four possible linearly polarized transitions [27]. In a crosspolarised set-up, this system behaves like an effective two-level system, and the single photon emission shows a rapid rise time and monoexponential decay as shown in Figure 2(a).

These optical selection rules result in very different origins of the residual two-photon component for the two types of sources. To illustrate this, we measure the g (2) at maximum emitted brightness for two QD sources, one exciton and one trion, whilst increasing the temporal length of the excitation pulse (Figure 2(b)). For the exciton source, the single photon purity remains very high for pulse durations up to 80 ps, whereas the single photon purity rapidly degrades for longer pulses for the trion based source. For the trion, the single photon emission process is fast and can occur during the laser pulse so that there is a probability that the quantum dot returns to the ground state before the end of the laser pulse and gets excited again, leading to the emission of a second photon [28]. For the exciton, the delayed emission in cross polarization means that the probability of collecting two photons via re-excitation is very small, and the measured value of g (2) for the exciton remains small for pulse durations of up to 80 ps. We notice that for both excitons and trions, the g (2) is higher for very short pulses. This is because the power required to reach maximum emitted brightness (π-pulse) increases as the pulse duration decreases [29]. This implies that in the very short pulse regime (< 10 ps) , the g (2) is limited by imperfect suppression of the excitation laser. This remains the dominant source of an imperfect g (2) for exciton sources up to a pulse duration of 80 ps, whereas trion sources are limited by re-excitation for pulses longer than 15 ps.

To correctly extract the single photon indistinguishability for each type of QDSPS it is critical to account for these different origins of the multi-photon component. To do so, we experimentally increase the multi-photon component by adjusting the main parameter that is responsible for multi-photon emission in each type of source, and then measure the impact this has on the visibility of HOM interference. Specifically, we increase the probability of re-excitation for a trion source and the amount of laser photons for the exciton source.

For the exciton based source, we add laser photons to the single photon emission from the QDSPS by turning the quarter waveplate (QWP) of the excitation pulse (see Figure 1(a)). This means that the excitation pulse is no longer aligned along one of the polarisation axes of the cavity, and the light will experience a small amount of polarisation rotation due to the birefringence of the cavity. Therefore, some fraction of the excitation pulse will now be collected in the orthogonal polarisation with the single photons. By adjusting the QWP we can add more laser photons and increase the g (2) of this effective source and measure the corresponding impact on HOM interference, as shown in Figure 3(a). The added noise photons from the laser are separable from the single photons and therefore Equation 1 can be used to model the data. From the time traces in Figure 2(b), we can calculate that there is very little overlap between the laser photons and the single photons emitted by the quantum dot so that M sn ≈ 0. The line in Figure 3(a) corresponds to a fit using V HOM = M s -(1 + M s )g (2) from which we extract as a single parameter the single photon indistinguishability M s = 0.920 ± 0.003.

For the trion based source, since the imperfect g (2) arises from re-excitation, the assumption of separable 2) for a exciton source, as the suppression of the excitation laser is worsened to increase the g (2) . (b) Measured HOM visibility as a function of the g (2) for a trion source as the pulse duration is increased. For longer pulse durations, there is more re-excitation and the g (2) is higher. Each data point was taken at "π-pulse", corresponding to maximum brightness of the source. In both plots, the solid line gives the theoretical prediction for these data. The error bars are within the size of the plotted points.

noise does not hold because the emission of the first and second photon are time correlated. However, the extra photon must be emitted during the laser pulse for re-excitation to occur [28], whereas the main single photon emission typically takes place after the laser pulse with the trion radiative decay time of approximately 170 ps. As a result the noise photon, while emitted by the QD, presents a very similar temporal profile to the laser photon and can be modelled as a temporally separated noise with M sn = 0. The validity of this analysis is verified by increasing the pulse duration to increase the probability of re-excitation. The g (2) and HOM visibility are measured for different pulse durations from 15 ps to 50 ps at the power that maximises the emitted count rate. The results shown in Figure 3(b) are modelled well using V HOM = M s -(1 + M s )g (2) with a single parameter M s = 0.942 ± 0.004.

To summarize, we find that, despite their different physical origins, the multi-photon component of both exciton and trion based QDSPSs can be treated as separable distinguishable noise. In the limit of low g (2) , the single photon indistinguishability can thus be obtained using:

M s = V HOM + g (2)
1 -g (2) .

(

) 2
This correction factor can be applied to any QDSPS with a small g (2) and fast excitation pulse, in order to extract the intrinsic single photon indistinguishability, M s , given a measurement of g (2) and V HOM . The more general case where the HOM beam splitter has an intensity reflectivity R and transmission T is given in the Supplementary Material. Interestingly, Equation 2 results in a higher singlephoton indistinguishability than the one obtained us-ing the identical noise model. It is therefore likely that many of the values of single photon indistinguishability that have been quoted in the literature [12] for QD sources are in fact an underestimate of the true value. Whilst this does not circumvent the fact that it is the overall wavepacket indistinguishability that is crucial for quantum technologies, this deepened understanding of the Hong-Ou-Mandel experiment allows for a better diagnosis regarding imperfect single photon sources. The single-photon indistinguishability, M s , gives the upper bound to the indistinguishability that could be achieved with an ideal experimental set-up, with no laser leakage for example, and therefore it fundamentally quantifies how temporally coherent the source itself is. Finally, we note that our simple theoretical approach is only valid for separable noise, and further studies are needed to understand the effect of a non-separable noise on the interference.

In conclusion, we have theoretically and experimentally revisited the emblematic Hong-Ou-Mandel interference. This experiment is commonly implemented to test the indistinguishability of single particles including single photons, single plasmons, single electrons or single atoms [30][31][32]. We believe that the new insight brought by our study will benefit these fundamental studies as well as the development of single photon sources, allowing a better diagnosis on the current limitations. The total state of the imperfect single photon can then be written as ρt = p 0 |0 0| + p 1 ρt,1 + p 2 ρt,2 , where ρt,j is the density matrix for the transmitted state containing j photons. The total g (2) and µ can be computed directly from the photon number probabilities by µ = p 1 + 2p 2 and g (2) = 2p 2 /µ 2 . For the average photon number, we have µ = p s,1 cos 2 ϑ + p n,1 sin 2 ϑ and g (2) is given by

Single photon

µ 2 g (2) = 2p s,1 p n,1 (1 + M sn ) cos 2 ϑ sin 2 ϑ, (S.7)
where M sn = Re(ξ s (t, t )ξ * n (t, t ))dtdt is the mean wavepacket overlap of the single photon and noise, and ϑ quantifies the amount of noise.

After applying a propagation phase φ i to each of the photon density wavefunctions, the two-time amplitude correlation of ρt is given by â † t (t )â t (t) = p s,1 cos 2 ϑξ s (t, t )e iφs(t-t ) + p n,1 sin 2 ϑξ n (t, t )e iφn(t-t ) (S.8)

and so the total mean wavepacket overlap is given by where M s = |ξ s (t, t )| 2 dtdt = Tr ρ2 s,1 quantifies the intrinsic single-photon indistinguishability, or single-photon trace purity [18,19], of the source. Here we have that M sn = Re ξ s (t, t )ξ * n (t, t )e i(φs-φn)(t-t ) dtdt (S.10)

µ 2 M tot = â † t (
is not necessarily the same as M sn due to the potential relative propagation phase φ s -φ n . We can reparametrize the expressions for µ, g (2) , and M tot by defining η so that cos 2 η = (p s,1 cos 2 ϑ)/µ and sin 2 η = (p n,1 sin 2 ϑ)/µ. The fact that this reparametrization exists stems from the independence of M tot and g (2) from photon loss. It also implies that the fundamental quantity affecting the photon statistics of this imperfect single photon model is η, which depends on both the beam splitter angle ϑ and the relative input intensities through p s,1 and p n,1 . Using the relation for V HOM from the previous section, equations (S.7), and (S.9), the visibility and g (2) in terms of the noise parameter η are V HOM (η) = 4RT 1 + M s cos 4 η + M n sin 4 η -2(1 + M sn -M sn ) cos 2 η sin 2 η -1 g (2) (η) = 2(1 + M sn ) cos 2 η sin 2 η.

(S.11)

The value of η can be modified by changing the intensity of the noise p n,1 as was done for the emulated distinguishable noise source in the main text. It can also be modified by changing the relative proportions of p n,1 and p s,1 using an unbalanced Mach-Zehnder interferometer, as was done for the emulated identical noise source in the main text.

  Figure 1. (a) Schematic for the experimental set-up of a quantum dot single photon source (QDSPS). A quantum dot embedded in a micropillar cavity is placed in a cryostat and cooled to around 8K. A resonant excitation pulse is used to coherently control the quantum dot, and single photons are collected in the orthogonal polarization to extinguish the excitation laser. The half and quarter waveplates allow alignment of the polarization along one of the microcavity axes [13, 24]. (b),(c) Experimental set-up used to emulate an imperfect single photon source with (b) distinguishable (Msn = 0) and (c) identical noise Msn = Ms. The inset in (b) shows the spectrum of the spectrally-distinct noise photons and QD photons. (d) Visibility of HOM interference, VHOM measured as a function of g(2) for distinguishable (green squares) and identical (black circles) noise sources. The lines are the predictions from the theoretical model.

  Figure 2.(a) Time traces of the single photon wavepacket emitted by exciton (upper) and trion (lower) based sources. The excitation laser pulse is shown in grey. (b) Measured g(2) for trion (purple squares) and exciton (blue circles) based sources as a function of the excitation pulse duration at π-pulse. The error bars are within the size of the plotted points.

9 DFigure 3 .

 93 Figure 3. (a) Measured HOM visibility as a function of the g(2) for a exciton source, as the suppression of the excitation laser is worsened to increase the g(2) . (b) Measured HOM visibility as a function of the g(2) for a trion source as the pulse duration is increased. For longer pulse durations, there is more re-excitation and the g(2) is higher. Each data point was taken at "π-pulse", corresponding to maximum brightness of the source. In both plots, the solid line gives the theoretical prediction for these data. The error bars are within the size of the plotted points.

Figure S. 1 .

 1 Figure S.1. An imperfect photon is modelled by adding separable noise to a perfect single photon at a beam splitter

  t )â t (t) M s cos 4 ϑ + p 2 n,1 M n sin 4 ϑ + 2p s,1 p n,1 M sn cos 2 ϑ sin 2 ϑ,

	2
	dt dt s,1 (S.9) = p 2

Acknowledgements This work was partially supported by the ERC PoC PhoW, the French Agence Nationale pour la Recherche (grant ANR 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 dAvenir" programme (Labex NanoSaclay, reference: ANR-10-LABX-0035). J.C.L. and C.A. acknowledge support from Marie SkodowskaCurie Individual Fellowships SMUPHOS and SQUAPH, respectively. H. O. Acknowledges support from the Paris Ile-de-France Rgion in the framework of DIM SIRTEQ. S.C.W. and C.S. acknowledge support from NSERC (the Natural Sciences and Engineering Research Council), AITF (Alberta Innovates Technology Futures), and the SPIE Education Scholarships program.

Supplementary Material

HOM visibility for unentangled states

In this section, we derive the general relationship between the total mean wavepacket overlap of two interfering states, the single-photon purity, and the HOM visibility. Suppose we have a beam splitter with input modes â1 and â2 and output modes â3 and â4 monitored by single-photon detectors. The output modes can be described by the relation â3 (t) â4 (t) = cos θ -e -iφ sin θ e iφ sin θ cos θ

provided that the beam splitter interaction is constant for the relevant frequency range of the input fields. The coincident events of the detectors at the output are determined from the two-time intensity correlation between the output fields G

(2)

, where we restrict τ ≥ 0 so that the detector monitoring mode 4 clicks after the detector monitoring mode 3. The integrated G (2) 34 around zero delay normalized by the total intensity gives the probability p 34 of having a coincident count. The HOM interference visibility is then defined as V HOM = 1 -2p 34 . By following the methods of Ref. [33] to compute p 34 for two unentangled input states, while also keeping terms associated with multi-photon contributions, the HOM visibility is given by

where R = sin 2 θ is the beam splitter reflectivity, T = cos 2 θ is the beam splitter transmittance, g

(2)

is the integrated intensity correlation around zero delay for input i normalized by the time-integrated mean photon number µ i = â † i (t)â i (t) dt and the mean-wavepacket overlap is

For a balanced interferometer where the input intensities of the final beam splitter are equal, we have that µ 1 = µ 2 and so the relation for HOM visibility simplifies to

where g (2) = (g

2 )/2 is the average g (2) of the interfering states. For HOM interference between identical wavepackets, M 12 = M 11 = M 22 = M tot is the total mean wavepacket overlap of the source and g (2) = g

2 = g (2) quantifies the source single-photon purity. In the ideal case where R = T = 1/2, the above relation reduces to the equation V HOM = M tot -g (2) given in the main text.

HOM visibility in a separable noise model

We now develop a theoretical model to describe the visibility of HOM interference for a specific type of imperfect source. This imperfect single photon source is modeled by adding noise to an ideal single photon using a beam splitter interaction, as shown in Figure S.1. Here the noise is separable and exhibits no entanglement with the single photon. For simplicity, we also model the noise by another single photon, which is valid in the limit of a weak noise field or, equivalently, when g (2) is small.

The initial state of the signal (s) and noise (n) is given by ρ = ρs ⊗ ρn where ρi = p i,0 |0 0| + p i,1 ρi,1 and

In our study, we are interested in the slope and intercept of the parametric curve formed by {g (2) (η), V HOM (η)}. The solution for the intercept is clear since g (2) (η) = 0 implies η = 0 and V HOM (0) = 4RT (1 + M s ) -1. To solve for the slope at small η, we have lim η→0 dV HOM (η) dg (2)

For the cases we are interested in, either distinguishable noise or if ξ s = ξ n , we have M sn -M sn 0. This case would also be true if M sn and M sn were phase averaged. Under these conditions, the HOM visibility for small g (2) is given by

In the case where the noise is distinguishable so that M sn = 0, the single-photon indistinguishability M s can be determined by rearranging equation (S.13):

For R = T = 1/2, we recover equations ( 1) and ( 2) presented in the main text.

Measuring g (2) and HOM interference

To measure the single photon purity we perform a Hanbury Brown-Twiss experiment and measure the coincidences between the two outputs of a 50:50 beam splitter. The experimental set-up, and a typical coincidence histogram are shown in Figure S.2(a). The second-order autocorrelation is given by g (2) = A 0 /A uncor , where A 0 is the area of the coincidence peak at zero time delay and A uncor is the average area of the uncorrelated peaks.

We perform a Hong-Ou-Mandel interference experiment by splitting the train of single photons at a beam splitter and delaying one arm by the pulse separation time, τ . Two subsequently emitted photons then interfere at a 50:50 beam splitter, as shown in Figure S.2(b). The visibility of HOM interference is given by V HOM = 1 -2A 0 /A uncor .

(a) g (2) (b) HOM (2) and (b) HOM visibility.


S E Thomas

M Billard

N Coste

S C Wein

Priya Priya

H Ollivier

O Krebs

L Tazaïrt

A Harouri

A Lemaître

A Lemaitre

I Sagnes

C Anton

L Lanco

N Somaschi

J C Loredo
email: juan.loredo1@gmail.com

P Senellart +

Bright Polarized Single-Photon Source Based on a Linear Dipole

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Semiconductor quantum dots in cavities have emerged as high-performance singlephoton sources for quantum technologies [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] Wang | Boson sampling with 20 input photons and a 60-mode interferometer in a 10 14dimensional hilbert space[END_REF]. However, they are still far from ideal deterministic operation, which simultaneously requires perfect linear polarization, population inversion and collection efficiency [START_REF] Senellart | Highperformance semiconductor quantum-dot single-photon sources[END_REF]. Thus far, the best performing sources have used an unpolarized emitter together with polarized photon extraction [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] Wang | Towards optimal single-photon sources from polarized microcavities[END_REF][START_REF] Ollivier | Reproducibility of high-performance quantum dot single-photon sources[END_REF].

Here, we explore a new path to deterministic operation: exploiting the natural asymmetry of neutral quantum dots that present linearly-polarized dipoles and allow for the emission of fully polarized light. To benefit from such a property, we propose the use of off-resonant phonon-assisted excitation that has been theoretically predicted to enable near-unity population inversion and quantum purity [START_REF] Barth | Fast and selective phonon-assisted state preparation of a quantum dot by adiabatic undressing[END_REF][START_REF] Cosacchi | Emission-frequency separated high quality singlephoton sources enabled by phonons[END_REF][START_REF] Gustin | Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities[END_REF]. We experimentally study this approach for quantum dots in micropillar cavities and demonstrate single-photon emission with a degree of linear polarization up to 0.994 ± 0.007 and a high population inversion -85% as high as resonant excitation. We demonstrate a single-photon source with a polarized first lens brightness of 0.51 ± 0.01, a single-photon purity of 0.939 ± 0.001 and single-photon indistinguishability of 0.915 ± 0.003.

The path towards an optimal single-photon source requires finding a scheme in which single photons are generated in a well-defined spatial and polarization mode, with near-unity efficiency, purity and indistinguishability. A device with these properties is highly sought-after for the advancement of quantum technologies such as secure long-distance quantum communication [START_REF] Takemoto | Quantum key distribution over 120 km using ultrahigh purity single-photon source and superconducting single-photon detectors[END_REF] and quantum computing [START_REF] Slussarenko | Photonic quantum information processing: A concise review[END_REF][START_REF] Wang | Integrated photonic quantum technologies[END_REF][START_REF] Rudolph | Why i am optimistic about the siliconphotonic route to quantum computing[END_REF]. Several platforms towards optimised single-photon sources are being developed, including spontaneous parametric down-conversion (SPDC) [START_REF] Ramelow | Highly efficient heralding of entangled single photons[END_REF][START_REF] Weston | Efficient and pure femtosecondpulse-length source of polarization-entangled photons[END_REF][START_REF] Kaneda | Heralded single-photon source utilizing highly nondegenerate, spectrally factorable spontaneous parametric downconversion[END_REF] and four-wave mixing (FWM) [START_REF] Spring | Chip-based array of near-identical, pure, heralded single-photon sources[END_REF][START_REF] Francis-Jones | All-fiber multiplexed source of high-purity single photons[END_REF]. These nonlinear optical sources have an intrinsic efficiency limitation, and various multiplexing schemes are currently being explored to overcome this [START_REF] Francis-Jones | All-fiber multiplexed source of high-purity single photons[END_REF][START_REF] Xiong | Active temporal multiplexing of indistinguishable heralded single photons[END_REF][START_REF] Joshi | Frequency multiplexing for quasi-deterministic heralded single-photon sources[END_REF][START_REF] Kaneda | High-efficiency singlephoton generation via large-scale active time multiplexing[END_REF][START_REF] Meyer-Scott | Singlephoton sources: Approaching the ideal through multiplexing[END_REF]. Sources based on semiconductor quantum dots (QDs) in microcavities have recently shown their capability to deliver high purity, indistinguishable single photons with record brightness [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] Wang | Towards optimal single-photon sources from polarized microcavities[END_REF][START_REF] He | Coherently driving a single quantum two-level system with dichromatic laser pulses[END_REF]. With an efficiency per photon more than one order of magnitude higher than sources based on frequency conversion, QD sources have already allowed a substantial scaling up of optical quantum computing [START_REF] Wang | Boson sampling with 20 input photons and a 60-mode interferometer in a 10 14dimensional hilbert space[END_REF].

Despite this impressive progress, current QD sources are still far from the ideal deterministic performance that would provide a single photon in a pure quantum state with unity probability. This ambitious goal requires a full inversion of the QD transition that emits a single photon with near-unity probability and quantum purity, and that the photon is perfectly collected into a well-defined polarized optical mode. Coherent control of a QD has allowed generation of single photons in pure quantum states [START_REF] He | On-demand semiconductor singlephoton source with near-unity indistinguishability[END_REF], and near-unity population inversion has been obtained through more sophisticated techniques such as rapid adiabatic passage [START_REF] Wu | Population inversion in a single ingaas quantum dot using the method of adiabatic rapid passage[END_REF][START_REF] Wei | Deterministic and robust generation of single photons from a single quantum dot with 99.5% indistinguishability using adiabatic rapid passage[END_REF]. Unpolarized collection efficiencies in the 60-78% range have been demonstrated with micropillar cavities [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]. However, obtaining such high performance into a polarized mode remains challenging. So far, the most efficient sources have used resonant excitation of a charged exciton state, which inherently emits unpolarized photons. Using unpolarized cavities, only half of the single photons are collected through polarization filtering [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] Ollivier | Reproducibility of high-performance quantum dot single-photon sources[END_REF]. Very recently, polarized cavities were used to accelerate spontaneous emission into one linear polarization providing a factor of two gain in the source efficiency [START_REF] Wang | Towards optimal single-photon sources from polarized microcavities[END_REF][START_REF] Tomm | A bright and fast source of coherent single photons[END_REF]. However, it is unclear whether such method could provide near-unity polarization degree.

Here, we propose a new path towards deterministic operation -making use of the natural asymmetry of a neutral QD whose eigenstates present linearly-polarized optical transitions. Such an approach allows for fully polarized single photon emission. To benefit from this attractive feature, we implement a near off-resonant phononassisted excitation scheme that has recently been theoretically proposed to reach both high QD inversion probability and high quantum purity [START_REF] Barth | Fast and selective phonon-assisted state preparation of a quantum dot by adiabatic undressing[END_REF][START_REF] Cosacchi | Emission-frequency separated high quality singlephoton sources enabled by phonons[END_REF][START_REF] Gustin | Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities[END_REF]. The unique combination of these ingredients allows us to report on a source of nearly-perfectly polarized single photons with state-of-the-art performance.

FIG. 1. Proposed scheme for high-efficiency linearly polarized single photon source. (a) Energy level structure of a neutrally charged quantum dot, with ground state |g and two exciton eigenstates |X and |Y , separated in energy by the fine structure splitting ∆FSS. The longitudinal acoustic (LA) phonon assisted excitation scheme uses an excitation pulse blue-detuned from the transition by an energy of ∆LA, or equivalently detuned in wavelength by ∆λ, and is aligned in polarization along the X dipole. (b) A schematic of the quantum dot deterministically embedded at the centre of an electrically-contacted micropillar cavity [START_REF] Somaschi | Near-optimal single-photon sources in the solid state[END_REF] (c) Polar plot of the cross-polarized emission intensity measured under phonon-assisted excitation as a function of the incident polarization angle. (d) Spectrum of the emitted photons when exciting only the X dipole of a neutral exciton and collecting in the parallel (blue) or orthogonal (red) basis, plotted on a log scale. The off-resonant LA-phonon-assisted excitation is detuned by ∆λ=0.6 nm and has a pulse duration of 19 ps.

We exploit the optical selection rules of a neutral In-GaAs QD to directly generate linearly polarized single photons. Self-assembled quantum dots present a shape asymmetry, both in the growth direction and transversely, that reduces the system symmetry to C 2v . This leads to a three-level structure in neutral QDs as shown in Fig. 1(a) where two excitonic eigenstates, labelled |X and |Y , are separated by the fine structure splitting, ∆ FSS , which is determined by the valence band mixing and Coulomb exchange interaction [START_REF] Bayer | Fine structure of neutral and charged excitons in self-assembled in(ga)as/(al)gaas quantum dots[END_REF]. The optical transition between the QD ground state |g and one of the exciton states corresponds to a linearly-polarized dipole. While optical quantum technologies require polarized single photons, direct use of such a linear dipole has not been considered so far. Indeed, the generation of single photons with near-unity indistinguishability has only been reached under resonant excitation, a technique which is not applicable to a single linear dipole. In such a case, the single photons present the same wavelength and polarization as the excitation laser and cannot be easily separated.

To overcome this limitation, we propose the use of a phonon-assisted excitation scheme, making use of a slightly spectrally-detuned laser that can be easily separated from the single photons. This excitation scheme was recently introduced [START_REF] Barth | Fast and selective phonon-assisted state preparation of a quantum dot by adiabatic undressing[END_REF][START_REF] Glässl | Proposed robust and high-fidelity preparation of excitons and biexcitons in semiconductor quantum dots making active use of phonons[END_REF] and relies on a detuned strong optical pulse that dresses the ground and excited states of the optical transition. During the pulse duration, a strong occupation of the excited state is obtained through an adiabatic undressing mediated by the emission of longitudinal-acoustic (LA) phonons. After the first observation of such phonon-assisted excitation [START_REF] Quilter | Phonon-assisted population inversion of a single ingaas/gaas quantum dot by pulsed laser excitation[END_REF], further theoretical studies predicted this excitation scheme should also provide near-unity singlephoton purity, indistinguishability and occupation probability [START_REF] Cosacchi | Emission-frequency separated high quality singlephoton sources enabled by phonons[END_REF][START_REF] Gustin | Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities[END_REF]. The combination of this phonon-assisted excitation scheme with the use of a QD linear dipole thus appears as a promising route toward near-deterministic single photon sources.

We study several devices consisting of a QD deterministically located at the center of an electrically connected micropillar cavity (Fig. 1(b)) [START_REF] Somaschi | Near-optimal single-photon sources in the solid state[END_REF]. These devices have previously been shown to provide reproducible, highperformance single-photon generation under strictly resonant excitation, where the excitation laser was separated from the emitted photons via cross-polarisation. This reduced the polarized collection efficiency by a factor of two, and the polarized source brightness B FL , i.e. the probability to collect a polarized single photon per pulse, to around B FL ≈ 25% [START_REF] Somaschi | Near-optimal single-photon sources in the solid state[END_REF][START_REF] Ollivier | Reproducibility of high-performance quantum dot single-photon sources[END_REF]. In the present work, the laser is blue-detuned from the QD transition by approximately 0.6 nm, corresponding to a phonon energy of around 1 meV. The single photons are separated from the pump laser using spectral filters (see Methods).

We can visualize the two linear dipoles of the neutral QD, by measuring the single photon emission in the orthogonal polarization to the excitation laser. The corresponding intensity when varying the excitation polarization direction is shown in Fig. 1(c). The intensity in cross polarisation goes to zero when the polarization is aligned along one of the exciton linear dipoles and increases in between. In the following, only one of the excitonic dipoles is excited, and the system reduces to an effective twolevel system, {|g , |X }. The measured spectra of the single photons when collecting parallel or orthogonal to the X polarization direction, are shown in Fig. 1 Occupation probability via the phononassisted excitation scheme. Measured intensity of singlephoton emission from a charged exciton collected in crosspolarization for resonant excitation and LA-phonon-assisted excitation at various detunings ∆λ. The emission intensity is normalized to the maximum achieved under resonant excitation to allow comparison between the two schemes. The error bars are within the size of the plotted points. The solid lines give the theoretical prediction based on the model in [START_REF] Gustin | Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities[END_REF].

tion degree D LP of 0.981 ± 0.004 and 0.971 ± 0.001. This polarization degree significantly exceeds what is accessible using polarized Purcell effect in asymmetric cavities, with maximal values of 0.93 in optimized structures [START_REF] Wang | Towards optimal single-photon sources from polarized microcavities[END_REF] -a limitation that arises from a compromise between reducing the cavity linewidth (to obtain large cavity splitting to linewidth ratios), and remaining in the weak coupling regime.With the current approach, there is no fundamental limit to the degree of polarization that could be reached. Even if the two linear dipoles present a slight non-orthogonality, which arises when the quantum dot shape and strain anisotropies do not coincide [START_REF] Léger | Valence-band mixing in neutral, charged, and mndoped self-assembled quantum dots[END_REF][START_REF] Kowalik | Optical alignment and polarization conversion of the neutral-exciton spin in individual inas/gaas quantum dots[END_REF], it is always possible to excite a single dipole by aligning the excitation light to be orthogonal to the other dipole axis.

We now investigate the QD occupation probabilities p LA QD that can be obtained through LA-phonon-assisted excitation. In practice, the measurement of this absolute value is a significant experimental challenge that would require precise estimation of the many parameters governing the single photon emission intensity (see Supplementary Material). It is more instructive to measure the relative occupation probability reachable with phononassisted excitation, compared to the one achieved under resonant excitation p LA QD /p RF QD . To do so, we compare the single photon emission in a cross-polarisation configuration, to be able to separate single photons from the laser for both excitation schemes. We study a charged exciton on Sample B (see Methods), which corresponds to a four-level system that acts as an effective two-level system when collecting the emission in cross-polarization [START_REF] Ollivier | Reproducibility of high-performance quantum dot single-photon sources[END_REF]. Fig. 2 presents the single-photon counts detected under strictly resonant excitation (black) and using LAphonon-assisted excitation (coloured) for a pulse dura-tion of 19 ps and various laser detunings as a function of the pulse area experienced by the quantum dot. The pulse area is proportional to the square root of the intracavity power, which is obtained by correcting from the cavity transmission when the laser energy is detuned from the optical resonance. The resonant excitation data exhibits well-known Rabi oscillations reflecting the coherent control of the optical transition. Conversely, the LAphonon-assisted excitation scheme does not show oscillations but a single rise and then slow decrease of the signal. Solid lines show the theoretical predictions based on the model presented in [START_REF] Gustin | Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities[END_REF] for the parameters corresponding to our experimental situation. Theory and experiment are in good agreement and show the occupation probability under LA-phonon-assisted excitation is as large as 0.85 ± 0.01 compared to that reached under resonant excitation. Such high occupation probability is also observed on device A3 as shown below. Theory indicates that the QD occupation probability could be brought closer to unity using lower temperature and developing appropriate temporal shaping of the excitation laser [START_REF] Glässl | Proposed robust and high-fidelity preparation of excitons and biexcitons in semiconductor quantum dots making active use of phonons[END_REF].

We next investigate the single-photon purity and indistinguishability of the collected photons. We measure the second-order autocorrelation, g (2) (0), and the Hong-Ou-Mandel interference visibility, V HOM , for various excitation conditions for device A3, and typical results are shown in Fig. 3(a) (see Methods for more details). When using an excitation pulse which is detuned from resonance by ∆λ = 0.6 nm and has a pulse duration of τ = 19 ps, we observe g (2) (0) = 0.057 ± 0.001 and V HOM = 0.810 ± 0.004, from which we extract a single-photon purity of P = 1 -g (2) (0) = 0.943 ± 0.001 and a single-photon mean wavepacket overlap [START_REF] Ollivier | Hong-ou-mandel interference with imperfect single photon sources[END_REF] of M s = 0.920±0.005. As a comparison, the same measurements are performed in the standard resonance fluorescence (RF) excitation scheme, rejecting the laser component via cross polarization. In this case, the polarization of the excitation pulse is aligned along one of the cavity axes and the laser excites a combination of both X and Y optical transitions [START_REF] Ollivier | Reproducibility of high-performance quantum dot single-photon sources[END_REF]. At π-pulse excitation, we measure g (2) (0) = 0.011 ± 0.001, and M s = 0.915 ± 0.001, showing that the indistinguishability under the LA-phononassisted excitation scheme is the same to that obtained from resonant excitation. The higher g (2) (0) in the LAphonon-assisted excitation scheme is partly due to reexcitation of the QD for the pulse duration of 19 ps used here (see Supplementary Material). We find that these values further improve to g (2) (0) = 0.011 ± 0.001 and M s = 0.948±0.001 for the LA-phonon-assisted excitation and to g (2) (0) = 0.009 ± 0.001, and M s = 0.962 ± 0.004 for resonant excitation when adding an additional 10 pm bandwidth etalon in the collection path. This additional spectral filtering reduces the phonon sideband emission, as well as any extra spectrally-broader photons, thus improving the single-photon purity and indistinguishabil-FIG. 3. Source Performance (a) Comparison of the measured g (2) (0) (blue) and HOM interference (red) for the LA-phonon-assisted (upper) and resonant (lower) excitation schemes. The g (2) (0) data is offset horizontally by 2 ns for clarity. For LA-phonon-assisted excitation the laser is detuned by ∆λ = 0.6 nm from resonance. For both schemes, the pulse has a duration of τ = 19 ps, and the power is chosen to maximise the emitted photon rate. (b) The first lens brightness, BFL, as a function of the detuning, ∆λ, and the pulse duration, τ . (c) The single-photon purity, P = 1-g (2) (0), and the single-photon indistinguishability, Ms, as a function of the pulse duration, τ , of LA-phonon-assisted excitation with a detuning of ∆λ = 0.4 nm. In all cases, the values are obtained for a power maximizing the occupation probability.

ity for both excitation schemes [START_REF] Grange | Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics[END_REF]. We note that recently, an indistinguishability of 80% was observed for both phonon-assisted and resonant excitation for GaAs QDs [START_REF] Reindl | Highly indistinguishable single photons from incoherently excited quantum dots[END_REF]. Our work shows for the first time that nearunity indistinguishability can be reached using phononassisted excitation as theoretically predicted.

Finally, to evaluate the overall performance of a singlephoton source based on an excitonic linear dipole under LA-phonon-assisted excitation, we measure the polarized brightness, single-photon purity and indistinguishability of device A3 as a function of the detuning, ∆λ, and pulse duration of the excitation laser, τ . For each set of parameters, we adjust the excitation power to reach maximum brightness. In order to evaluate the first lens brightness, B FL , we precisely calibrate all losses and the detector efficiency of our experimental setup, see Supplementary Material for a detailed loss budget. Fig. 3(b) shows that higher brightness is achieved for smaller detunings, and Fig. 3(c) shows the source performances as a function of the pulse length for a detuning of ∆λ = 0.4 nm. Singlephoton indistinguishabilities above 90%, single-photon purities above 94% and first-lens brightnesses above 45% are reached for all pulse lengths. The use of longer pulses allows for a slightly higher first-lens brightness, but with a slight degradation of the single photon purity due to re-excitation. The extracted single-photon indistinguishability is maintained approximately constant for all pulse lengths. For a pulse duration of 22 ps we obtain a first-lens brightness of B FL = 0.51 ± 0.01, with g (2) (0) = 0.061 ± 0.001, M s = 0.915 ± 0.003. The brightness is more than a factor of two greater than one observed on the same sample using resonant excitation in a cross-polarization configuration [START_REF] Ollivier | Reproducibility of high-performance quantum dot single-photon sources[END_REF]. It corresponds to a detected count rate of 6 MHz using a 69% efficient single photon detector and a laser repetition rate of 81 MHz, which is limited by the low optical transmission of 17% of our experimental set-up. Considering the extraction efficiency of our cavities η ext ≈ 0.65, and the theoretically expected p LA QD ≈ 0.85 (see Supplementary Material), the maximum expected first-lens brightness is B FL 0.55, very close to our experimental observation. We note that by improving the spectral filtering we would be able to operate at even smaller detunings, enabling higher occupation probabilities to be reached.

In conclusion, we have reported on a new approach to bring QD-based single photon sources closer to deterministic operation. Our approach relies on the very general property that neutral excitons in quantum dots naturally present linearly-polarized dipoles. Remarkably, the use of phonon-assisted excitation enables near-unity photon indistinguishability and high occupation probability. We note that the use of the exciton linear dipole could also be applied to the recent proposition of dichromatic excitation scheme [START_REF] He | Coherently driving a single quantum two-level system with dichromatic laser pulses[END_REF][START_REF] Koong | Coherent dynamics in quantum emitters under dichromatic excitation[END_REF]. However the use of phononassisted excitation enables a significant gain in stability and robustness, since the scheme is resilient to drifts in QD-laser detuning and excitation power. All these features are of great importance for practical applications in quantum technologies. Finally, the use of unpolarized cavities and the possibility of exciting and collecting the photons in all polarization directions could also be applied to the four-level structure of a charged exciton, in order to generate photonic cluster states [START_REF] Lindner | Proposal for pulsed on-demand sources of photonic cluster state strings[END_REF].

Acknowledgements. The authors would like to thank Andrew White and Marcelo Pereira de Almeida for experimental help. This work was partially supported by the ERC PoC PhoW, the French Agence Nationale pour la Recherche (grant ANR 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 RENAT-ECH network, a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements dAvenir" programme (Labex NanoSaclay, reference: ANR-10-LABX-0035).

Declarations. Niccolo Somaschi and Pascale Senellart

METHODS

Devices

We study two samples, referred to as Sample A and B, each of which contain several devices consisting of a QD deterministically located at the center of an electrically connected micropillar cavity [START_REF] Somaschi | Near-optimal single-photon sources in the solid state[END_REF]. The quality factor of the cavities on Sample A (B) is around 4000 (10000), and the cavities are almost circular: they exhibit only a small degree of polarization, with two linearly-polarized cavity modes typically separated by ≈ 70 µeV ( ≈ 30 µeV), which is much smaller than the cavity linewidth of ≈ 300 µeV (≈ 150 µeV).

Experimental Set-up. To excite the QD via LA-assisted excitation we require a pulse that is blue-detuned from the QD resonance by 0.3 -0.8 nm. We start with a 3 ps pulsed Ti-Sapphire laser with a central wavelength around 924 nm and a 81 MHz repetition rate. The laser spectrum is shaped using a 4-f filtering system to obtain pulses with tunable temporal length from 3 ps to 22 ps and negligible spectral overlap with the QD emission at 924.82 nm. The co-linearly polarized single photons are collected with a non-polarizing beam-splitter transmitting 90% of the light. Three hightransmission (∼ 95% each), 0.8 nm FWHM, high-extinction band-pass filters (Alluxa) separate the single photons from the excitation laser. A half and quarter waveplate are used to precisely align the polarization along the exciton axis (X or Y ) inside the cavity. Superconducting nanowire single photon detectors (SNSPDs) with 25-30 ps timing jitter and an efficiency of 75% at low count rates, and 69% at the maximum measured count rate, are used to record the time dependence of the emission and the second order intensity correlations.

Measuring Linear Dipoles To measure the emission from the two linear dipoles of the neutral exciton, we add a polarizer to the output and measure the signal in crosspolarisation to the input light. By rotating the HWP on the input, we can change the angle of the linear polarization of the excitation light. When the polarization of the laser is orthogonal to one of the dipoles, say Y, then only the X dipole is excited and the quantum dot only emits X-polarized light. Since we are collecting in cross-polarisation we do not collect any signal, and there is a minimum in the collected light. When the light excites a general linear combination of the two dipoles, then the quantum dot can emit light into the orthogonal polarisation [START_REF] Ollivier | Reproducibility of high-performance quantum dot single-photon sources[END_REF]. As the angle of polarizaiton is rotated, we see mimima in the signal when the light excites only one dipole, and maxima in the signal when we excite an equal superposition of the two dipoles, as shown in Figuer 1(c).

Single-photon purity and indistinguishability. To measure the single-photon purity we use a Hanbury Brown-Twiss set-up, where the collected photons from a source are incident on a 50:50 beam splitter and correlations between the two outputs are measured. The proportion of coincident clicks at zero time delay compared to the uncorrelated peaks gives the second-order autocorrelation, g (2) (0), which is related to the single-photon purity P = 1 -g (2) (0). To measure the indistinguishability of the output we perform Hong-Ou-Mandel interference between subsequently emitted photons from the source. The photon train is separated at a beam splitter and one arm is delayed by the pulse separation time. Two subsequently emitted photons are simultaneously incident at a second beam splitter, and undergo quantum interference. The visibility of this interference, VHOM, together with the secondorder autocorrelation of the input, g (2) (0), gives access to the indistinguishability of the single-photon component of each wavepacket, Ms, via Ms = (VHOM + g (2) (0))/(1 -g (2) (0)) [START_REF] Ollivier | Hong-ou-mandel interference with imperfect single photon sources[END_REF].

FIG. S.2. (a)

The second-order autocorrelation g (2) (0) (upper) and the single-photon indistinguishability, Ms, (lower) as a function of the detuning (∆λ) and pulse duration (τ ) of LA-phonon-assisted excitation. For each point, the values are obtained for a power maximizing the occupation probability. (b) Measured g (2) (0) (blue) and HOM interference (red) for LA-phononassisted excitation (upper) and resonant excitation (lower), when a 10 pm etalon is added to the collection path.

Single-Photon Purity and indistinguishability

To investigate the single-photon purity when using phonon-assisted excitation, we measure the second-order autocorrelation, g (2) (0), as a function of the excitation pulse duration and detuning from resonance. For each set of parameters, the excitation power was adjusted to maximise the single-photon emission. Figure S.2(a) shows that the g (2) (0) increases for longer pulse durations, indicating that there is an increasing probability of re-excitation. For a pulse duration greater than 14 ps we can deduce that re-excitation is a significant cause of the non-zero g (2) (0). However, it is not clear what limits the g (2) (0) to around 4% for shorter pulse durations. The single-photon indistinguishability is constant across these range of parameters, as shown in the lower panel of We note that it is theoretically predicted that phonon-assisted excitation should enhance the single-photon purity as compared to resonant excitation, since the phonon-induced relaxation process leads to a delay in photon emission and hence suppresses the probability of re-excitaiton [START_REF] Cosacchi | Emission-frequency separated high quality singlephoton sources enabled by phonons[END_REF]. This suppression should become more significant when the lifetime of the quantum dot is comparable to the phonon-relaxation timescale. By exploring the large parameter space of available detunings, pulse durations, and Purcell factors to adjust the optimal QD emission lifetimes, it may be possible to achieve significantly higher single-photon purity under LA-phonon-assisted excitation.

Loss Budget

We measure the transmission of the optical components of our set-up using a continuous wave laser centered at the quantum dot emission wavelength, 924.82 nm, and details are given in Table S.1. To measure the transmission of the window of the cryostation and the objective lens, we measure the reflection of the beam off the flat diode surface of the sample, and assume that the reflectivity of the surface is 100% in order to give an upper bound on the transmission of the optical elements. The light passes through the cryostation window, the objective lens and two waveplates twice, and we give the single-pass transmission of these combined elements in Table S.1.

The transmission of the filters is less than the maximum of 95% per filter, since we angle-tune the filters to fully suppress the excitation laser. The filters were aligned here for an excitation laser with a detuning of ∆λ = 0.8 nm and a pulse duration of τ = 20 ps. In order to calculate the first lens brightness at a different detuning or pulse duration, the filters are aligned to suppress the laser and then the relative transmission efficiency is measured with respect to this known point.

The detector efficiency of the SNSPDs at a fixed bias current depends on the count rate because of the dead time of the detectors [START_REF] Zadeh | Single-photon detectors combining high efficiency, high detection rates, and ultra-high timing resolution[END_REF]. We measured the detection efficiency using an attenuated laser pulse with a repetition rate of 81 MHz. The detection efficiency for a low count rate of 500 kHz was found to be 75%. However, when the count rate increases to 6 MHz the detection efficiency decreases to 69%, as shown in Figure S.3. We note that this may not be the optimal detection efficiency for these parameters as we did not adjust the bias current of the SNSPDs. The total efficiency of the set-up for a detuning of ∆λ = 0.8 nm, a pulse duration of τ = 20 ps, and a detected count rate of R det = 6 MHz is η = 0.17 ± 0.01. The repetition rate of the laser RL = 81 MHz, and therefore this corresponds to a first lens brightness of BFL = R det /(RL × η) = 0.44 for these parameters.

Element

  FIG. 2.Occupation probability via the phononassisted excitation scheme. Measured intensity of singlephoton emission from a charged exciton collected in crosspolarization for resonant excitation and LA-phonon-assisted excitation at various detunings ∆λ. The emission intensity is normalized to the maximum achieved under resonant excitation to allow comparison between the two schemes. The error bars are within the size of the plotted points. The solid lines give the theoretical prediction based on the model in[START_REF] Gustin | Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities[END_REF].

  J.C.L. and C.A. acknowledge support from Marie SkodowskaCurie Individual Fellowships SMUPHOS and SQUAPH, respectively. S.C.W. acknowledges support from NSERC (the Natural Sciences and Engineering Research Council), AITF (Alberta Innovates Technology Futures), and the SPIE Education Scholarships program. H. O. and N. C. acknowledge support from the Paris Ile-de-France Rgion in the framework of DIM SIRTEQ.

  Figure S.2(a). When a 10 pm bandwidth etalon is added to the set-up to provide additional spectral filtering of the collected photons, the g (2) (0) and HOM visibility improve significantly as shown in Figure S.2(b).

FIG. S. 3 .

 3 FIG. S.3. Left: Detected counts on an SNSPD as a function of the input power. Right: Detection efficiency of SNSPD as a function of the detected count rate.

TABLE S .

 S TransmissionCryostat Window + Objective Lens + QWP + HWP 0.80 ± 0.02 1. Loss Budget. The transmission of optical elements in the set-up, and efficiency of the superconducting nanowire detectors

	Beam splitter	0.92 ± 0.01
	5 Mirrors	0.95 ± 0.01
	Telescope	0.98 ± 0.01
	2 Waveplates	0.95 ± 0.01
	3 Bandpass filters	0.64 ± 0.02
	Fibre coupling efficiency	0.60 ± 0.02
	Detection efficiency (low count rate)	0.75 ± 0.03
	Detection efficiency (6 MHz detected rate)	0.69 ± 0.03

are CTO and CSO of Quandela, respectively. All other authors declare no conflicts of interest.

Supplementary Material

Parameters controlling the source brightness

The brightness of a QD source is determined by a subtle combination of many parameters. Firstly, pQD, quantifies the occupation probability of the exciton state of the quantum dot at the end of the excitation pulse. Considering the near unity quantum efficiency of quantum dot optical transitions, it also reflects the probability that one photon is emitted by the QD per pulse. In practice, pQD can be reduced by blinking phenomena or imperfect population inversion [START_REF] Hilaire | Deterministic assembly of a charged quantum dot-micropillar cavity device[END_REF]. Secondly, the extraction efficiency ηext is determined by two factors. On one hand, the QD emits a single photon into the cavity mode with probability β = FP /(FP + γ), (where FP is the Purcell factor into the mode, and γ is the emission rate into all other modes normalized to the nominal rate). On the other hand, the out-coupling efficiency ηout then gives the probability that a photon exits the desired cavity mode and reaches the collection lens. Accordingly, ηext = β ηout. Finally, the polarization efficiency η pol describes the fraction of polarized single photon emission that can be collected in a given scheme. The polarized first lens brightness is thus B FL,pol = β ηout pQD η pol . In our system the extraction efficiency ηext = βηout ≈ 0.65, independent of the excitation scheme. For RF excitation in a mostly unpolarized cavity, one obtains a polarization efficiency of η pol < 0.5, whereas for the phonon-assisted excitation, it is η pol = 1 2 (1 + DLP), and can be as high as 0.997 as demonstrated here.

Occupation probability of LA-assisted excitation

A detailed theoretical model for estimating the efficiency of LA-phonon-assisted excitation is given in [START_REF] Gustin | Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities[END_REF]. We apply this model with the relevant experimental parameters in order to understand how the brightness of the polarized single photon source depends on the detuning and pulse duration of the excitation pulse.

The results of the simulation are given in Fig. S.1, and show the maximum occupation probability, pQD, maximised over excitation power, for a given detuning and pulse duration. The maximum occupation probability depends both on the detuning and on pulse length [START_REF] Gustin | Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities[END_REF]. More precisely, it depends on the ratio of the excitation pulse duration to the lifetime of the two-level system, and for long enough pulses compared to the transition lifetime, the maximum occupancy has little dependence on the detuning. For our system, we can see that for pulse durations of around 19 ps and longer, the occupation probability is high and approximately constant for detunings between 0.3 and 1.0 nm. The maximum occupation probability is approximately 0.85 less efficient than for resonant excitation. This was the case for the measurements shown in Fig. 2 that were obtained on Sample B presenting a higher Purcell factor, hence reduced lifetime. In contrast, in Fig. 3 we explored neutral excitons in Sample A which present lower Purcell factors, and we observed a slight dependence with detuning for our limited region of pulse lengths.

We note that the above theoretical approach is valid for a two level system: it applies perfectly to the case of the linear dipole of a neutral exciton under parallel excitation, but also to the case of the charged exciton when excitation and collection are cross-polarized. However, further modeling would be required if exciting a combination of the two exciton dipoles instead. Indeed, this would result in a complex excitation/emission dynamics within the three level system {|g , |X , |Y } [START_REF] Ollivier | Reproducibility of high-performance quantum dot single-photon sources[END_REF] that is expected to significantly modify the phonon-assisted excitation process and cannot be compared to the case of a two-level system [START_REF] Barth | Fast and selective phonon-assisted state preparation of a quantum dot by adiabatic undressing[END_REF]. Simulated occupation probability pQD numerically maximized over excitation power for a fixed Gaussian pulse FWHM duration τ . For the simulation we assumed a quantum dot with a natural lifetime of 1/γ = 1 ns at 8 K coupled to a cavity mode with a lifetime of 1/κ = 32 ps by the cavity coupling given by 1/g = 100 ps corresponding to a Purcell factor of Fp = 12.8.