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Generation of non-classical light in a -number superposition

J. C. Loredo,1, ∗ C. Ant´on,1, ∗ B. Reznychenko,2 P. Hilaire,1 A. Harouri,1 C. Millet,1 H. Ollivier,1 N. Somaschi,3 L. De Santis,1 A. Lemaˆıtre,1 I. Sagnes,1 L. Lanco,1, 4 A. Auff`eves,2 O. Krebs,1 and P. Senellart1, † 1CNRS Centre for Nanoscience and Nanotechnology, Universit´eParis-Sud, Universit´eParis-Saclay, 91120 Palaiseau, France 2Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel,38000 Grenoble, France 3Quandela SAS, 86 rue de Paris, 91400 Orsay, France 4Universit´eParis Diderot, Paris 7, 75205 Paris CEDEX 13, France The ability to generate light in a pure 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 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 the atomic ground and excited state to the emitted light been a primary task enabling progress in optical quantum field. This has so far been explored in the weak-excitation technologies [1, 2]. Single-, and N-photon sources [3–5] regime to produce quantum light that exhibits coher- are at the heart of future quantum communication net- ence with the driving laser—observed with atoms [32], works [6, 7], sensors [8, 9], as well as optical quantum as well as semiconductor quantum dots [33–36]. This computers [10, 11] and simulators [12–15]. These achieve- regime has been shown to produce squeezed light where ments make use of the interference of indistinguish- an atomic dipole—with vanishing population—elastically able single-photons, allowing the realisation of quantum scatters a coherent superposition of vacuum and a small gates [16, 17], and protocols such as quantum telepor- one-photon term [37]. Generating a photon-number su- tation [18] and entanglement swapping [19]. The one- perposition with large single-photon population requires photon term has been exploited heretofore, and the vac- to create an atomic population—inherently coupled to uum component has been considered detrimental to the its environment—that remains insensitive to any deco- overall protocol efficiency, motivating a quest for de- herence until spontaneous emission takes place. To the terministic sources producing single-photon Fock states best of our knowledge, the generation of photon-number with no vacuum component [20–23]—a challenging task, quantum superpositions under strong coherent driving to say the least. If vacuum is set instead in a quantum has not been reported so far, neither with natural atoms, superposition with the single-photon, one could use it nor with artificial ones. to encode quantum information in the photon-number— becoming a resource for optical quantum information In this work, we report on the on-demand generation processing. For instance, vacuum within a pure quantum of quantum superpositions in the photon-number , state can be exploited in quantum teleportation [24], or in light pulses emitted by a single artificial atom. We quantum random number generators [25]. However, ob- observe superpositions of zero, one, and two photons arXiv:1810.05170v2 [quant-ph] 20 Nov 2018 taining quantum superpositions in the photon-number emitted from semiconductor quantum dots coupled to basis has so far demanded complex quantum state engi- optical microcavities [20, 38]. We use pulsed coherent neering and conditioned state preparation [26, 27]. driving, beyond full inversion of the atomic population, and perform interferometric measurements with a path- The text-book model of a quantum emitter is a two unbalanced Mach-Zehnder interferometer (MZI). As sup- level atom—a system shown to generate quantum light ported by our theoretical calculations, phase-dependent in various excitation regimes. Incoherent non-resonant oscillations at the interferometer output demonstrate the excitation of natural [28] and artificial atoms [29–31] can production of coherent superpositions of vacuum, one, produce optical fields with a large single-photon compo- and two photons. Below π-pulse driving, we obtain su- nent, but without coherence in the photon-number ba- perpositions of vacuum and one-photon Fock states, with sis due to the incoherent creation process of the atomic their relative populations controlled by the driving laser population. In contrast, coherent driving of an atom can intensity. By driving the quantum dot with 2π-pulses, in principle be used to transfer the coherence between we obtain a state with the two-photon component larger 2 than the one-photon population—a state allowing phase tinct and complementary behaviour between π-, and 2π- super-resolving interferometry, and incidentally resem- pulse driving. A pronounced antibunched photon statis- bling a small Schr¨odinger-catstate. tics at π-pulse is observed for both QD1 and QD2, with (2) Coherent driving and photon statistics gπ (0)=0.037±0.002 for QD1, see Fig. 1c. Such observa- We investigate semiconductor devices consisting of a tions show light wavepackets consisting mostly of either single quantum dot (QD) positioned with nanometer- vacuum or one photon. However, bunched statistics is (2) scale accuracy at the centre of a connected-pillar cav- observed at 2π-pulse for QD2, with g2π (0)=2.98±0.11, ity [20, 39, 40]. The QD layer is inserted in a p-i-n see Fig. 1d. This evidences, as recently observed [41], diode structure, and electrical contacts are defined to wavepackets containing two-photon populations. In the control the QD resonance through the confined Stark following, we investigate the nature of light in the photon- effect. We note that the experimental results reported number degree-of-freedom: whether it contains photon here have been observed on various QD-cavity devices. Fock-states emitted in a mixture or in a pure quantum We focus hereafter on two devices: a neutral (QD1) and state. a charged (QD2) exciton coupled to the cavity mode, of zero and one photon see Methods. QD1 (QD2) is excited resonantly with The Hong-Ou-Mandel (HOM) effect [42] describes two linearly-polarised 40 ps (15 ps) laser pulses at 925 nm, single-photons simultaneously impinging on a beamsplit- and its emission is collected using a crossed-polarisation ter. If the photons are polarisation, spatially, and fre- scheme that separates it from the laser, see Fig. 1a. Fig- quency indistinguishable, they bunch at the output of ure 1b shows the detected countrates for QD1 as a func- the beamsplitter—a behaviour exclusively of quantum tion of the excitation pulse area A, evidencing well de- mechanical origin. This requires that the interfering pho- fined Rabi oscillations. The signal is damped by sponta- tons are in the same pure quantum state in these degrees- neous emission due to the relatively long 40 ps excitation of-freedom. pulses [38] as compared to the measured emission decay As discussed now, interference can also be used to time of 166±16 ps. Second-order autocorrelation func- unravel coherences in the Fock-state basis. Consider tions g(2)(∆t) measured along Rabi cycles evidence a dis- a beamsplitter with inputs a, b, and outputs c, d, onto which pure states of photon-number superpositions im- √ √ iα pinge. These are in the form |Ψai= p0|0ai+ p1e |1ai, √ √ i(α+φ) 1/2 1/2 and |Ψbi= p0|0bi+ p1e |1bi, with p0+p1=1, p0,1 Power (nW ) a b 0.0 0.5 1.0 1.5 2.0 the vacuum and one-photon populations, and φ a rel- 2.0 QD 1 ative phase between the states. When p1=1, their 1.5 quantum interference leads to the√ well known two- photon output state (|2c0di−|0c2di) / 2—the HOM ef- 1.0 fect. However, as soon as p1<1, the output state shows 0.5 other photon terms that lead to a mean photon-number Counts (MHz) Nc,d=p1 (1 ± p0 cos φ) at the beamsplitter outputs, see 0.0 0 π 2π 3π 4π Supplementary Information. That is, if states are pure Area 3000 in the photon-number basis, their interference leads to c QD 1 d QD 2 2500 200 oscillations measured at the output of the interferometer 2000 device, with a visibility amplitude equal to the vacuum 150 population p . 1500 0 100 The previous example describes the idealised case 1000 Coincidences 50 of pure states—instances non-existing in the physical 500 world. To account for impurity in the photon-number 0 0 -50 -25 0 25 50 -50 -25 0 25 50 basis, we consider that each light wavepacket imping- ∆t delay (ns) ∆t delay (ns) ing on the beamsplitter is described by a density ma- trix ρ =λρ +(1−λ)ρ , with ρ =|Ψ ihΨ | a pure FIG. 1. Coherent control of an artificial atom. a S pure mixed pure i i state (i=a, b), ρ =diag{p , p } a diagonal matrix, Schematics of the setup. A single semiconductor QD is kept in mixed 0 1 a cryostat at 9 K, and is excited under pulsed resonant excita- and 0≤λ≤1 a parameter tuning the photon-number pu- tion. The QD emitted state |Ψi is separated from the laser in rity. Moreover, limited purity in the frequency domain is a cross-polarisation scheme, by using a polarising beamsplit- taken into account by the non-unity mean wave-packet ter (PBS), a quarter- (Q), and half-wave plate (H). b Rabi overlap M between interfering photons. It can be shown, oscillation of the QD coherent driving. The emission is col- see Supplementary Information, that such interfering in- lected by a single-mode fibre and directly detected with an put states result in APD. c Second-order autocorrelation function g(2)(∆t) mea- sured at π-pulse with QD1, and d at 2π-pulse driving with 1 QD2. n = (1 ± v cos φ) , (1) c,d 2 3

a b 1.0 freely evolving Detectors 0.8 a c

d 0.6 n

b d c n 0.4 A (π) = 0.61 0.2 A (π) = 0.14 0.0 A (π) A (π) = 0.61 0 50 100 150 200 250 300 350 0 0.25 0.5 0.75 1 Time (s) 1.0 c d 1.0 e 0.8 λ=0.965±0.018 A (π) = 0.14 0.8

0.6 , λ=0.8 | 0.6 v

A (π) = 0.42 01 λ=1

0.4 | ρ 0.4 ,

1 λ=0.8

A (π) = 0.76 p 0.2 λ=0.3 0.2 λ=0.3 0.0 0.0 0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Counts (MHz) M A (π)

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 nd (red) for a pulse area A=0.61π. Light blue (light red) traces display nc (nd) 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 2 follow the theoretical model v=λ 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. where n√c,d=Nc,d/ (Nc+Nd) oscillate with a visibility Figure 2b shows our measurements of nc,d for pulse ar- 2 v=λ p0 M. We observe, from Eq. 1, that if the inter- eas A=0.61π, and A=0.14π. The single detector counts fering states are distinguishable (M=0), or if the state undergo clear oscillations with time, as the optical phase is emitted in a statistical mixture of photon-numbers φ freely evolves in time within the interferometer— (λ=0), then v vanishes. Thus, observing v6=0 implies evidencing quantum coherence in the photon-number ba- that neither case is true: the state contains quantum co- sis. As predicted, the amplitude of the oscillations in- herences in the photon-number basis. creases with the vacuum population, controlled here by Coherent driving of a two-level system creates a quan- choosing the driving pulse area. Figure 2c shows the tum superposition of ground and excited state, with a extracted oscillation visibilities, obtained from maxima relative phase governed by the classical phase of the driv- and minima of nc,d with respect to φ, for different val- ing laser. If this coherence is transferred to the emit- ues of single-photon countrates (bottom axis) as the ted light state through spontaneous emission, we obtain pulse area varies within 0

3 0.60 a Δφ = 2π c φ = 0

c φ = π/2

n 0.50 2 0.40 200 b Δφ = 2π 150 1 100

50 Norm. coincidences

Coincid. (Hz) Coincid. 0 0 0 25 50 75 100 125 -50 -25 0 25 50 Time (s) Δt delay (ns) 3.0 d e 1.0 2.5 0.8 2.0 0.6 1.5 0.10 1.0 0.5 0.05 0.0 0.00 0.0 0.2 0.4 0.6 0.8 1.0 p0 p1 p2 Phase φ (π)

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. coincidence rate (at zero delay) follows losses of our setup. Figure 3c shows our obtained time- ¯ 1 correlated coincidence measurements C(∆t). As pre- C(0) = [hn(n−1)i − C2 cos(2φ)] , (3) dicted, the coincidences at zero delay C¯(0) oscillate with 8 Pm 2φ—with minima (maxima) of coincidences occurring for with hn(n−1)i= n n(n−1)pn the non- φ=0 (φ=π/2). The full phase span for C¯(0) (modulo π) normalised second-order correlation function, and 2 is shown in Fig. 3d. In the case of fully mixed (pure) 2 Pm p  C2=λ n n(n − 1)pnpn−2 a coherence term states in the photon-number basis, i.e., λ=0 (λ=1), os- of second-order. In virtue of Eqs. 2 and 3, we ob- cillation visibilities in the coincidence-counts fully vanish tain the normalised output coincidences at zero delay (maximally oscillate), see dashed grey (dot-dashed blue) ¯ 1 (2) 2 2  curve in Fig. 3d. C(0)=C(0)/ (NcNd) = 2 g (0) (1−v2 cos 2φ) / 1−v1 cos φ , where g(2)(0)=hn(n−1)i/hni2 is the normalised We extract v1=0.192±0.008, v2=0.452±0.038, which (2) second-order correlation function of the input state, together with the measured value of g2π (0)=2.98±0.11, v1=C1/hni is the single detector counts visibility, and see Fig. 1d, and the normalisation of probabilities, re- v2=C2/hn(n−1)i the coincidences visibility. See that sults in the distribution {pn, λ}, with p0=0.838±0.012, Eq. 3 shows oscillations modulated at twice the phase- p1=0.051±0.002, p2=0.111±0.010, and λ=0.734±0.025. dependence of Eq. 2: coherences in the photon-number The generation of light states with p2>p1 is observed with allow phase super-resolving interferometry. charged excitons under a strong 2π-pulse drive of the QD, 2π The state generated at 2π-pulse driving contains see Methods. The state ρS contains zero, one and two up to two-photon terms. In which case we ob- photons, with a quantum state purity of P=0.870±0.024, 2 √  2 tain v1=λ p1 p0+2 2p0p2+2p2 / (p1+2p2), v2=λ p0, see Fig. 3(e). Note that the above theoretical analysis for (2) 2 and g (0)=2p2/ (p1+2p2) . Thus, by measuring v1, 2π-pulse driving does not account for the effect of limited (2) v2, g (0), and taking into account normalisation photon indistinguishability (M<1). Accordingly, the re- p0+p1+p2=1, we univocally determine p0, p1, p2, and λ. ported purity contains both the photon-number purity Figures 3a, 3b show respectively our measurements for and indistinguishability imperfections, thus representing nc=Nc/hni= (1+v1 cos φ) /2, and coincidences propor- a lower bound for the photon-number purity alone. tional to C(0)∝ (1−v2 cos(2φ)) with a rate specific to the This state, with p2>p1, incidentally resembles other 6 quantum states of interest. The obtained pho- scientific advisor and co-founder, of the single-photon- 2π ton distribution {pn } presents a statistical fidelity source company Quandela. cat P p 2π cat cat F = n pn pn to {pn }, the probability distribu- tion of an even “Schr¨odinger-cat”state |cati∝|αi+|−αi, METHODS with |αi a , of F cat=0.974±0.016 for a Sample The microcavity samples were grown by molec- small cat state with |α|2=0.5. Thus, by simply driving a ular beam epitaxy. A λ-GaAs cavity is surrounded by a charged quantum dot with 2π-pulses, we are able to gen- bottom and a top mirror made of 29 and 14 pairs of erate other photonic states that may find applications in GaAs/Al0.9Ga0.1As, respectively. The mirrors are grad- coherent-state driven quantum computation [48, 49] and ually n-doped and p-doped in order to tune the quantum quantum metrology [50]. dot transition through the confined Stark effect. The Conclusions cavities are centered on the quantum dots using the in- Quantum states with a high degree of purity are essential situ optical lithography technique [39]. Then the sam- in all quantum-enhanced technologies. Optical quantum ple is etched and standard p-contacts are defined on a technologies have so far exploited various degrees-of- large frame (300×300 µm2) connected to the circular freedom, such as time-frequency, angular momentum, frame around the micropillar. A standard n-contact is or polarisation [1, 2]; but not the photon-number due defined on the sample back surface. A neutral exciton to the absence of suitable sources. Our work demon- is coupled to the cavity mode for QD1. For the op- strates that state-of-the-art semiconductor QD emitters tical measurements, the polarisation of the laser is set not only provide high purity in the frequency basis so that the fine-structure splitting results to emission in but also non-classical photon-number superpositions crossed-polarisation, see ref. [38]. A positively-charged on-demand. We are now able to generate highly-pure exciton is coupled to the cavity mode for QD2, and in light wave-packets with tuneable zero- and one-photon this case the circular-polarisation and optical transition components. Other non-classical states can be also rules naturally allows obtaining a signal in the crossed- generated by adjusting the coherent excitation pulse polarisation configuration. The generation of pure co- duration and intensity, as shown here driving the atom herent superpositions of zero and one photon has been at 2π−pulse. We believe that the generation of quantum observed for half a dozen devices, based either on neu- superpositions of photon-numbers opens new exciting tral or charged exciton transitions, when exciting below routes for optical quantum technologies. For instance, π-pulse. The generation of light states with p2>p1, on the we now can exploit the interference of these novel other hand, is only observed with charged excitons in the photonic states, potentially impacting on the complexity present experimental configuration. Indeed, in a crossed- of existing quantum-enhanced protocols, such as in polarisation collection scheme, the neutral exciton spon- quantum computing, or quantum walks. taneous emission is time delayed by the fine structure splitting [38], preventing an efficient re-excitation within Acknowledgements This work was partially sup- the same pulse. ported by the ERC Starting Grant No. 277885 QD- Time-tagged correlation measurements Simulta- CQED, the French Agence Nationale pour la Recherche neous acquisition of single counts and double coinci- (grant ANR SPIQE and USSEPP), the French RENAT- dences are recorded by measuring the photon count-rate ECH network, a public grant overseen by the French Na- and photon event time-tags in the output detectors (Si tional Research Agency (ANR) as part of the Investisse- avalanche photodiodes) of the MZI, which are connected ments d’Avenir programme (Labex NanoSaclay, refer- to a computer-controlled HydraHarp 400 autocorrelator. ence: ANR-10-LABX-0035). J.C.L. and C.A. acknowl- Under free evolution of the phase φ between the two edge support from Marie Sklodowska-Curie Individual arms of the MZI, the total acquisition time per point Fellowships SMUPHOS and SQUAPH, respectively. We has been set to Tacq=310 ms (810 ms) for the results de- thank N. Carlon Zambon for providing technical assis- scribed in Fig. 2 (Fig. 3), with an integration time for the tance throughout the project. photon time-tags of TTT=200 ms (500 ms). Given the rel- Author contributions The experiments were con- atively fast acquisition of experimental points, the phase ducted by J.C.L. and C.A. with help from P.H., C.M. φ remains approximately unchanged during each acquisi- H.O., and L.D.S. The data analysis was done by C.A. tion run. The measurement protocol for each data point and J.C.L. The theoretical modelling was done by A.A., runs as follows: the Hydraharp autocorrelator reads the B.R, O.K., C.A., and J.C.L. The cavity devices were fab- laser clock signal (24.6700 ± 0.0026 ns), a period of time ricated by A.H. and N.S. from samples grown by A.L., which serves as a reference to determine the photon time and the etching was done by I.S. The project was super- tags of the detected events (accumulated during TTT); vised by L.L., A.A., O.K., and P.S. The manuscript was consecutively, during the interval Tacq−TTT, the coun- written by J.C.L., C.A., and P.S., with input from all trates in the APDs are averaged, from where the phase φ authors. is eventually obtained, see Supplementary Information. Competing interests N.S. is co-founder, and P.S. is Data analysis The outcome of the time-tagged mea- 7 surements renders the countrates and two-photon coinci- //science.sciencemag.org/content/318/5856/1567. dences as function of time (the total integration time is in http://science.sciencemag.org/content/318/5856/1567.full.pdf. the order of 10-15 minutes for a given pulse area and rela- [12] Lanyon, B. 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SUPPLEMENTARY INFORMATION with:

† † SIMULATION OF THE EMISSION PROCESS ρ˜(t, τ) = U(t, t + τ)aoutρ(t)aoutU (t, t + τ) (10)

where U(t1, t2) is an evolution , that satisfies † Here we present the model used to compute the quan- ρ(t) = U(t0, t)ρ(t0)U (t0, t). tum state of the emitted field. For the device used in From these parameters we easily deduce the Fock the experiment, the cavity relaxation rate reads κ = states populations: 400 ± 100 µeV while the light-matter coupling constant equals g = 20 ± 5 µeV (Weak coupling regime). In this C(0) case the cavity mode can be adiabatically eliminated and p0 = 1 − Nout + , (11) the medium is modeled as a two-level system (TLS) cou- 2 pled to a waveguide, a 1D atom [44]. The TLS sponta- p1 = Nout − C(0), (12) neous emission rate γ accounts for the Purcell enhance- C(0) p = . (13) ment induced by the weak coupling to the cavity. Due 2 2 to the interaction with the solid-state environment, the The description of the eventual loss of purity induced TLS also undergoes pure dephasing with a typical rate by pure dephasing would require to model the complete γ∗. 1D atom (i.e., the TLS coupled to an infinite amount of The QD is pumped with a pulsed laser, resonant with photonic modes) coupled to a pure dephasing reservoir, the QD transition. The pump is modeled by a classical which is beyond the scope of the present paper. Instead, field containing n photons per pulse on average. The in we make the assumption that the QD emits pure pho- Hamiltonian of the system in the referential rotating at tonic states |ψ i and |ψ i (See main text and below). In the pump frequency writes a b this simplified model, the pure dephasing events mostly impact the time evolution of the TLS population, hence Hˆ = i Ω(t)(σ − σ†) (4) the relative populations of the Fock states. ~ Experimental imperfections are modeled in an effective √ manner. Reduced indistinguishability between the inter- where Ω(t)= ninγξ(t) is the classical Rabi fre- quency and the temporal profile of the pulse fering photons due to pure dephasing and/or differences ξ(t) is a normalized Gaussian function: ξ(t) = in the temporal profiles of the emitted photons give rise 1/4 to a reduction of the overlapping factor M (See main text 4 ln(2) exp(−t2/τ 28 ln(2)). The lowering operator and below). Reduced purity due to decoherence in the πτ 2 basis is taken into account by the factor λ is denoted σ=|gihe|. The dynamics of the QD is ruled (See main text and below). by the Lindbladt Master equation: i h i ˆ ∗ ρ˙ = − H, ρ + Dγ,σ[ρ] + Dγ /2,σz [ρ] (5) MODELLING INTERFERENCES FOR PURE ~ INDISTINGUISHABLE SUPERPOSITIONS OF  †  FOCK-STATES where σz= σ , σ and D are super-operators defined as

α † † †  In this section we present a simple model describing Dα,A[ρ] = 2AρA − A Aρ − ρA A (6) 2 the photon interference taking place in the FBSHOM (de- fined in the Fig. 2 of the main text) of the MZI between Finally, the polarization of the collected light is orthog- quantum states of light in a coherent superposition in the onal to the polarization of the the incident laser light, photon-number basis. Here we ignore the temporal struc- such that the output field a solely depend on the TLS out ture of the field emitted by the QD and treat the input dipole: and output ports of the BS as effective modes a, b, c, d, √ respectively. This allows to directly calculate the inter- aout = γσ (7) ference of the two input states, |ψai and |ψbi, according The Lindbladt equation is time-integrated, giving ac- to the beam splitter transformations. cess to the total number of emitted photons Nout during This basic model allows in principle computing Nc,d the process and the second-order autocorrelation func- and C(0) between arbitrary photon-number superposi- tion C(0): tion input states |ψai and |ψbi, directly interfering in a beam-splitter. The general expression of the two dif- Z h i m √ (a†)n N = dtTr ρ(t)a† a (8) ferent pure states is |ψ i= P a pa eiαn √ |0 i and out out out a n=0 n n! a † n mb p (b ) ma,b Z |ψ i= P pb ei(βn+nφ) √ |0 i, with P pa,b=1, h † i b n=0 n n! b n=0 n C(0) = dtdτTr ρ˜(t, τ)aoutaout (9) considering different maximal number of photons ma,b 10 and different relative phases among Fock terms αn and output state resulting from the interference of |ψai and βn for each input state. Note that the |ψbi state acquires |ψbi is explicitly written in the next equation: an additional phase φ, emulating its passage through one of the arm of the MZI. Assuming that the QD is emitting p 1 i(2α1+φ) for each laser pulse the same kind of state |ψi, we take |ψouti = p0|0, 0i + √ e (|2, 0i − |0, 2i) 2 ma = mb and αn=βn.   p i(α + φ ) φ φ Under such conditions, we write the initial input state + 2p p e 1 2 cos |1, 0i − i sin |0, 1i (17) 0 1 2 2 as |ψini=|ψai ⊗ |ψbi and we apply the transformation of a balanced beam splitter on√ the creation operators√ For the sake of clarity, we have removed the explicit † † † † † † † † a , b following a = c +d / 2 and b = c −d / 2. indication of the output modes c and d in the Fock- The final output state |ψ i contains all the information out states, so that |nc, mdi≡|n, mi. It is√ worth mentioning to calculate the Nc,d and C(0). The expressions for single that the well known NOON state, 1/ 2 (|2, 0i − |0, 2i), detector counts, simultaneous two detector coincidence is retrieved when p0 = 0. Instead, if p0 6= 0 a term with counts and visibility of single detector counts read, photon states |1, 0i and |0, 1i 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 X 2 main text. In this case, and following Eq. 14, the visibil- Nc,d = nc,d|hnc,d|ψouti| (14) ity of the normalised single detector counts nc,d is v=p0, nc,d (where obviously C(0)=0) and, as we can see, there is no X 2 C(0) = ncmd|hncmd|ψouti| (15) influence of α1 in the observed oscillation of the single de- nc,md tector counts. For this reason, our measurements of sin- N − N gle detector counts and two detector coincidence counts v = max c d (16) φ Nc + Nd at the output of the MZI do not allow to retrieve infor- mation about the α1 phase value of the emitted states of light. Then, the normalised simultaneous two detector coin- In the particular case of interfering two states of su- cidence counts that we measure after the MZI, C(0), is perposition of vacuum and one photon, but with differ- obtained following C(0)=C(0)/(NcNd). The normalised ent phases α1, β1, then the normalised single detection single counts in detectors c, d will be obtained by simply counts read nc,d=(1±p0 cos(α1−β1−φ))/2. Therefore, a dividing each expression by the total amount of single finite relative phase α1−β1 would be observed as an ad- detector counts Nc+Nd. We now restrict the study to ditional phase in the oscillation of nc,d as function of φ. the N=1, 2 cases describing our experiments, for an ar- bitrary relative phase αn between the Fock terms in each input state. Vacuum, one, and two photons

In a second case regarding up to N=2 pho- tons, we consider two input states given by Vacuum and one-photon √ √ √ √ |ψ i= p + p eiα1 a†+ p eiα2 (a†)2/ 2 |0 i and a √ 0 √ 1 2√ a √ i(α1+φ) † i(α2+2φ) † 2  |ψbi= p0+ p1e b + p2e (b ) / 2 |0bi, In the first case up to N=1 photons, the two input √ √ which have identical {pn} distributions. The resulting iα1 † states take the form |ψai= p0+ p1e a |0ai and state |ψ i of the interference takes a expression given √ √ out i(α1+φ) † |ψbi= p0+ p1e b |0bi. The expression of the by:

φ  φ φ p p i(α1+ ) 1 i(2α1+φ) |ψouti = p0|0, 0i + 2p0p1e 2 |1, 0i cos − i|0, 1i sin + √ e (|2, 0i − |0, 2i) 2 2 2 √  √  i(α2+φ) + p0p2e (|0, 2i + |2, 0i) cos φ − i 2|1, 1i sin φ r  √ √  p1p2 i α +α + 3φ φ φ + e ( 1 2 2 ) ( 3|3, 0i − |1, 2i) cos + i( 3|0, 3i − |2, 1i) sin 2 2 2 p2 √ √  + ei2(α2+φ) 6|0, 4i − 2|2, 2i + 6|4, 0i (18) 4

Let us consider first that α2=2α1; this renders a visi- bility and double coincidence rate of: 11

oscillating as a function of φ) are found proportional † √ to I = hψin|a b|ψini + cc, where cc stands for complex p0 + 2p2 + 2 2p0p2 v1 = p1 , (19) conjugate. p1 + 2p2 The interference term can thus be rewritten as I = p2 a a b † b C(0) = (1 − p0 cos(2φ)) . (20) hψin|a|ψinihψin|b |ψini + cc. In view of modeling the 4 decoherence in the Fock state basis, we now rewrite o o o It is worthy to note that in this case, the oscillation hψin|o|ψini=Tr[ρino], where Tr stands for the trace op- eration, and ρo =|ψo ihψo | is the density matrix repre- of the single detection counts will be observed (v1 6= 0) in in in senting the quantum state in the input port o=a, b: only if p1 > 0. The overall behaviour of v1 shows that the increase of vacuum is transformed in an increase of √ ρa = p |0 ih0 | + p |1 ih1 | + p p λ|0 ih1 | the oscillation visibility. It can be shown that in the par- in 0 a a 1 a a 0 1 a a ticular case of up to two photons, and in the generalised + h.c. (24) b √ iφ case of up to m photons, one can rewrite Nc,d and C(0) ρin = p0|0bih0b| + p1|1bih1b| + p0p1λe |0bih1b| in the form of Eqs. 2,3 in the main text. + h.c, (25) Finally, let us consider that α1 and α2 have nonzero values in the states with up to N=2 photons. Now the where h.c. stands for hermitic conjugate. a visibility v1 takes a modified form from that given in Eq. Let us consider the term Tr[ρina]. It reduces to a a a 19 (in particular, note the new term cos(2α1 − α2)): Tr[ρina]=h1a|ρina|1ai=h1a|ρin|0ai and is damped with respect to the ideal case of a pure superposition by a b √ factor λ. In the same way, the term Tr[ρinb] is damped α p0 + 2p2 + 2 2p0p2 cos(2α1 − α2) by a factor λ. The decoherence in the Fock state basis v1 = p1 (21) p1 + 2p2 thus leads to a damping of the interference term by a factor |λ|2, such as v → |λ|2v. The same reasoning can Equation 21 shows that α 6=2α reduces the visibility. 2 1 be extended to superpositions of 0, 1 and 2 photons. It can be demonstrated that, in this case, C(0) takes the same form as the one shown in Eq. 20 under the condition α2=2α1. Distinguishable photons

EXPERIMENTAL IMPERFECTIONS 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 As mentioned above, two kinds of experimental imper- have a temporal structure and correspond to differ- fections are considered: decoherence in the Fock-state ent wave packets. The interference term now reads basis or in the spectral basis, that are respectively quan- I= R dthψ |a†(t)b(t)|ψ i+cc, where we have introduced tified by the coherence damping factor λ<1 and the finite in in o(t) the annihilation of a photon at time t in the overlap between photonic states M<1. This finite over- 0 0 mode o, verifying [o(t), o†(t )]=δ(t−t ). One photon lap can be due to spectral decoherence (pure dephasing) R ∗ † in the mode o now writes |1oi= dtfo (t)o |0i, where or to the emission of wave-packets that are not identical. R 2 dt|fo(t)| =1. With these notations, the interference R ∗ term can be rewritten I= dtfa(t)fb (t) + cc. Denot- R ∗ 2 0 + 1 photons decohering in the Fock states basis ing M=| dtfa(t)fb (t)| the real number quantifying the overlap between two photons, we simply find that the dis- † tinguishability of the impinging photons leads to a damp- It is useful to rewrite Eqs. 14 as Nc = hψout|c c|ψouti √ as a function of the input state. Introducing the unitarity ing of√ the interference term by a factor M, such as matrix U that transforms the input modes a, b into the v → Mv. The same effect is reached when the spectral † purity of the two photons is reduced by pure dephasing. output modes√ c, d of the BS, we have c :=a ˜ = √U aU = (a + b)/ 2 (resp. d := ˜b = U †bU = (−a + b)/ 2) and In our experiment, we measured a degree of indis- |ψ i = U|ψ i. Finally, tinguishability M=0.903±0.008, see Fig. 4a, for single- out in photons emitted ∼12 ns appart from the QD, and when † Nc = hψin|a˜ a˜|ψini (22) they are indistinguishable in all other degrees-of-freedom ˜†˜ (polarisation, time of arrival) upon impinging at a beam- Nd = hψin|b b|ψini (23) splitter. From this point, we vary their distinguishabil- a √ √ Denoting as |ψini= p0|0ai+ p1|1ai (resp. ity via tuning their relative polarisations. We measured b √ √ iφ |ψini= p0|0bi+ p1e |1bi) the input state in the the visibility v, as well as the indistinguishability M, as a (resp. b) input port, the global input state is a function of the angle θ between the interfering polari- a b product |ψini=|ψini⊗|ψini. By developing, e.g., the sations (θ=0 for parallel, and θ=π/2 for orthogonal po- † producta ˜ a˜, the interference terms in Nc and Nd (i.e., larisations) for various pulse areas, see Fig. 4b,c. These 12

3 120 x10 a 80 40 Coincid. 0 -75 -50 -25 0 25 50 75 ∆t delay (ns) 1.0 1.0 Increasing area 0.8 b 0.8 c 0.6 0.6 v 0.4 M 0.4 0.2 0.2 0.0 0.0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 Angle θ (rad) Angle θ (rad)

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. measurements are used for displaying the behaviour of v From this, we deduce that our measurements contain in terms of M shown in Fig. 2d of the main text. information on the photonic state prior to any element acting as optical loss, thus at the level of the emitter. TIME-TO-PHASE MAPPING ROLE OF OPTICAL LOSSES

Here we show that optical losses in the experimental Here we describe how we obtained the interferometric setup have no impact on the observations reported in phase φ from the raw measured countrates freely evolv- the main text. To illustrate this, we drive the QD with a ing in time. As an example, we take the measurements pulse area of A=0.7π, and compare measurements with of nc,d for a pulse area A=0.39π, see Fig. 6a,b. An inten- different amounts of losses. sity binning of the data, see Fig. 6c, d, allows to assign a one-to-one correspondence between intensity bins and Figure 5a shows the single detector counts Nc (red) and phase bins, thus obtaining the corresponding values of φ Nd (blue) measured through our MZI interferometer with its nominal optical transmission, leading to ∼300kHz of (modulo π) at each instant of time. total single-photons (black). At this point, we intro- Similarly, the raw measurements taken at 2π-pulse duce extra losses to modify the measured countrates for area follow the same analysis, see Fig. 7. Here, we moni- at least one order-of-magnitude, resulting in about ∼10 tor single detector, see Fig. 7a,b, and coincidence counts times less signal. Figure 5b displays the normalised sin- simultaneously, see Fig. 7c. Each time-bin obtained from gle detector counts nc before and after the extra losses the single counts is assigned to one phase-bin. Finally, are introduced, thus evidencing that optical losses have this information is used to obtain the phase-resolved co- no impact on the observed visibilities. incidence measurements, see Fig. 7d. 13

3 400 x10 a Increasing losses at 300 the entrance of the MZI 200 100

Counts (Hz) 0 0.7 b 0.6

c

n 0.5 0.4 Constant visibility of n 0.3 c 100 200 300 400 Time (s) 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.

3 1.0 200 x10 a c 0.8

150 d

n 0.6

c

100 n 0.4 50 0.2

Counts (Hz) 0 0.0 0 100 200 300 400 500 0 10 20 30 40 50 Time (s) Time (s) 1.0 1.0 b φ = 0 d

0.8 c 0.8 n d

n 0.6 0.6 φ = π/2 c

n 0.4 0.4

0.2 Bins in 0.2 φ = π 0.0 0.0 0 100 200 300 400 500 0 50 100 150 Time (s) Acquisitions per bin

FIG. 6. Retrieving phase values. a Single detector countrates Nc (blue), Nd (red), and their sum (black) for a pulse area A=0.39π, used to obtain the normalised counts nc,d shown in b. We select 20 intensity bins (c), and assign to each a phase value between φ=0 and φ=π (d).

FIG. 7. Phase-resolved coincidence detection. a Single detector countrates (blue, red), and their sum (black) for a pulse area A=2π. b The corresponding normalised ηc. c Two-photon countrates accumulated in a 2 ns window for multiple τ delays between detected events, evolving during the same time-lapse in a. d Phase-resolved normalised coincidences obtained from the time-to-phase mapping described in Fig. 6.