1. INTRODUCTION

The development of optical quantum technologies allows for quantum-enhanced metrology, secure quantum communication, and quantum computing and simulation [1–3] in highly increased dimensions. Maturing quantum photonics requires efficient generation and detection of single photons, as well as their scalable manipulation [4]. Single-photon detection is a well-advanced technology to date, and has already reached near-optimal values in efficiencies [5]. On the generation side, the efficiency of a single-photon source can be characterized by the probability $p_1$ of providing a single photon per excitation pulse—often called “brightness”—which allows comparing various technologies [6]. Significant advances have been demonstrated using heralded approaches based on parametric conversion [7,8]; however, their single-photon purity unavoidably decreases with $p_1$. Temporal multiplexing schemes have been explored to circumvent this limitation [9,10], but at the expense of overall operation rates. On the other hand, spatial multiplexing schemes of many heralded sources [7,8] imply a dramatic increase of resource overhead.

Recently, scalable technologies for single-photon generation have emerged using quantum dots (QDs) in microcavities [6,11–13], where a single artificial atom emits temporal trains of single photons on demand. The brightness already exceeds by more than one order of magnitude that of heralded sources of the same quality, and near-deterministic sources could be reached with a similar technology and modified excitation scheme [14]. This new generation of sources has allowed multi-photon experiments such as Boson sampling [15,16] involving up to five detected single photons [17].

Photon manipulation can suffer from mechanical instabilities in bulk circuits, which lead to optical phase drifts and induce errors in device performance. A scalable photonic platform should instead provide photon routing and control in low-loss, integrated, and reconfigurable chips. These devices have been developed using various materials, such as silicon [18–20], silicon nitride [21], lithium niobate [22,23], or glass [24–26]. The latter, based on femtosecond laser writing, offers fast and cheap production, and has been used to tackle a variety of complex quantum operations, such as Boson sampling [27–29], quantum Fourier transforms [30], and quantum walks [31,32]. In addition, this technique has shown great versatility in terms of polarization control [33] and 3D patterning [30]; thermally tunable phase shifters can also be conveniently integrated to achieve circuit reconfigurability [34,35]. Despite spectacular progress on both solid-state photon sources and reconfigurable photonic circuits on chip, these two promising platforms have not yet been combined—an approach that is crucial for scaling optical quantum technologies.

In this work, we interconnect both scalable photonic platforms: we observe three-photon coalescence using an efficient solid-state based multi-photon source, and a reconfigurable photonic tritter circuit on glass. This first implementation of joint platforms already shows improved performance in terms of the quantity and quality of the three-photon interference. Furthermore, we estimate that feasible improvements can allow for 10-photon experiments at $\sim 40{\mathrm{s}}^{-1}$ rates in the near future.

The experimental scheme, sketched in Fig. 1, is composed of four modules: (I) single-photon generation at high rates from a QD source, (II) time-to-spatial active demultiplexing to prepare a three-photon source, (III) photonic circuit of the reconfigurable tritter, and (IV) detection of the photonic state via pseudo number-resolving measurements. In the following, we present each module before discussing the performances of the combined system.

 

Fig. 1. Experimental scheme of the efficient three-photon coalescence. (I) Generation of single photons from a QD-micropillar device under resonant fluorescence excitation. (II) Preparation of the three-photon trains (in the scheme depicted with colored spheres, each of them containing a train of 16 single photons temporally distributed every 3 ns) simultaneously arriving to the tritter input via active demultiplexing. (III) Circuit of the tritter providing the three-photon coalescence. (IV) Detection of the quantum state of light at the output of the tritter via pseudo number-resolving measurements.

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2. SINGLE-PHOTON SOURCE

We use a solid-state single-photon source consisting of a single InGaAs/GaAs QD deterministically coupled, with nanometric accuracy, to a micropillar cavity [36]. The micropillar device is gradually doped in the vertical direction to form a p-i-n diode structure embedding the QD in the intrinsic region of the cavity. Electrical contacts are defined in the top and bottom parts of the pillar to gain tunability of the QD energy via the confined Stark effect [11,37]. The device is mounted in a closed-cycle cryostat at $\sim 8\mathrm{K}$, and an optical confocal cross-polarization setup is used to excite the single-photon source with a resonant pulsed laser; see Fig. 1(I). The experiments are performed with a neutral exciton in resonance with the cavity mode, which presents a single-photon lifetime of 160 ps and a wavelength of 925.47 nm. This short single-photon lifetime allows increasing the laser repetition rate (81 MHz) by a factor of 4 using a passive pulse multiplier composed of beam-splitters (BSs) and delay lines [38].

The first-lens brightness of the source (i.e., the probability to collect a single photon before the first lens placed above the structure—each time the source is triggered) is measured to be $p_1=16.0\%$ in line with state-of-the-art performance for the neutral exciton [6]. The fibered brightness is measured to be $p_1=7.0\%$, where this value is limited by a finite numerical aperture $\mathrm{NA}=0.45$ of the first lens. This corresponds to a generation rate of $22\times {10}^6{\mathrm{s}}^{-1}$ of single photons in a single-mode fiber for a 324 MHz pumping rate. Standard photon correlation measurements are used to characterize the source performance under a repetition rate of 81 MHz. The single-photon purity is found to be $g^{(2)}(0)=0.035\pm 0.003$. It is important to note that when the laser repetition rate is 324 MHz, a lower single-photon purity of $g^{(2)}(0)=0.071\pm 0.003$ is measured. In such a case, the single-photon purity is deteriorated by the slow jitter time of our detectors and the size of the temporal windows for three-photon coincidence counting. This effective value of $g^{(2)}(0)$ will be the relevant one for our three-photon interference experiment. Finally, the photon indistinguishability is measured through a standard Hong–Ou–Mandel experiment, making use of an unbalanced Mach–Zehnder interferometer (for details, see Ref. [11] and Supplement 1). The deduced mean-wave packet overlap $M$ corrected (uncorrected) from $g^{(2)}(0)$ is measured to be $M=0.920\pm 0.007$ ($0.850\pm 0.007$) for $\sim 12.5\mathrm{ns}$ between emitted photons, and $0.880\pm 0.009$ ($0.810\pm 0.009$) for $\sim 100\mathrm{ns}$, which is the maximum temporal distance between the emitted photons subsequently used for interference.

3. DEMULTIPLEXER

Multi-photon interference requires the efficient preparation of indistinguishable single photons arriving simultaneously to the input ports of a photonic circuit [16,17,39]. One can build the required multi-photon sources starting from a single-photon source via a demultiplexer—a device that routes its input train of temporal modes into separate and simultaneous spatial modes. In general, a demultiplexer can be characterized by their relative time-varying output signals. The device is a passive demultiplexer if these signals are static in time, and its conversion rate—the ratio between the output $n$-photon event rates and the input single-photon rate—is given by ${\mathcal{C}}_n^{(\mathrm{passive})}=\prod _{k=1}^n{p}_k^{\mathrm{out}}$, where $p_k^{\mathrm{out}}$ is the static probability of the input signal to exit the $k$th output. For instance, in the particular case with $n$ equal output probabilities $p_k^{\mathrm{out}}=1/n$, the conversion rate scales as ${\mathcal{C}}_n^{(\mathrm{passive})}={(1/n)}^n$, showing the non-scalability of passive approaches. On the other hand, an active demultiplexer with time-varying relative output signals ${\mathcal{S}}_k^{\mathrm{out}}(t)$ results in a conversion rate ${\mathcal{C}}_n^{(\mathrm{active})}=\frac{1}{T}{\int }_T[\prod _{k=1}^n{\mathcal{S}}_k^{\mathrm{out}}(t)]\mathrm{d}t$, where the integral is taken over the demultiplexing period $T$. In this active case, the conversion rate is typically polynomial in $n$, thus constituting a scalable approach.

Solid-state based demultiplexed multi-photon sources have been reported with both passive [15] and active schemes [16,17,40]. Thus far, approaches for active demultiplexing either have employed on-chip architectures [40] with fast reconfigurable speeds (20 MHz), but low device throughput transmission, or have combined high-transmission bulk electro-optic modulators (Pockels cells) [16,17], but requiring very high voltages ($\sim 2000\mathrm{V}$) for each modulator, and with relatively slow reconfigurable speeds (0.76 MHz). Here we make use of resonance-enhanced electro-optical modulators (r-EOMs), from QUBIG GmbH, that allow for combining high transmission and fast reconfigurability, while requiring low-voltage control ($\sim 5\mathrm{V}$).

Figure 1(II) depicts our demultiplexing scheme for preparing a three-photon source. The system consists of two cascaded high-transmission (95%) and synchronized r-EOMs. The first r-EOM is driven at one eighth of the laser repetition rate, $\sim 10\mathrm{MHz}$, which combined with a polarizing beam-splitter (PBS) distributes 50 ns long time-bins to either output of the PBS alternately. This modulation produces photon train 2 and sends the other part of the signal to the second r-EOM, driven at one sixteenth of the laser repetition rate, $\sim 5\mathrm{MHz}$. This second modulation routes 100 ns long time-bins to either output of the second PBS, yielding photon trains 1 and 3. Fibered delays of appropriate length are added here (100 ns and 50 ns delay for photon trains 1 and 2, respectively) to ensure the simultaneous arrival of the three single photons to the tritter circuit.

Figure 2 illustrates the working principle of the demultiplexer. The synchronized operation of both r-EOMs [see panels in Fig. 2(a)] results in the active distribution of consecutive $\sim 50\mathrm{ns}$long time-bins into different spatial outputs [see the output modulated signals ${\mathcal{S}}_k^{\mathrm{out}}(t)$ in the ideal modulation case as full lines in the panels of Fig. 2(b)]. The dashed lines correspond to the unmodulated output signals (passive demultiplexing). As an example, the panels of Fig. 2(c) show our measured output signals ${\mathcal{S}}_k^{\mathrm{out}}(t)$ for the same demultiplexing period of $T=200\mathrm{ns}$, taken using laser light (under a repetition rate of 81 MHz) and photodiodes. Note that the current demultiplexing scheme is best suited for four photons since the train of photons in output 2 (black full line) located between 100 and 150 ns [see bottom panels in Figs. 2(b) and 2(c)] corresponds to an unused photon train.

 

Fig. 2. Operation of the demultiplexer. (a),(b) The interplay of two modulators, the first running at 10 MHz, and the second at 5 MHz, results in the relative output signals ${\mathcal{S}}_k^{\mathrm{out}}(t)$ (full red, green, and black traces corresponding to photon trains 1, 3, and 2, respectively) after suitable delays. Dashed lines depict the same signals when the modulator is turned off (passive demultiplexing). (c) Example of the measured output signals using laser pulses at a repetition rate of 81 MHz and a slow photodetector, resembling a similar modulation to the ideal active demultiplexing presented in panels (b).

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Given the modulation sequences, we can estimate the three-photon conversion rates ${\mathcal{C}}_3$ for both the ideal active and passive scenarios. For the ideal active case, we obtain ${\mathcal{C}}_3^{(\mathrm{active})}=1/4$; that is, the time-integrated area of the product of all three relative output signals is one fourth of the total demultiplexing area (period) of $T=200\mathrm{ns}$. If the demultiplexer is turned off, thus operated as a passive one, the conversion rate now becomes ${\mathcal{C}}_3^{(\mathrm{passive})}=1/32$—the product of three static transmission probabilities 1/4, 1/4, and 1/2. The active-to-passive ratio $r_n={\mathcal{C}}_n^{(\mathrm{active})}/{\mathcal{C}}_n^{(\mathrm{passive})}$relates the relative $n$-photon production rates between active and passive schemes, and can be used to assess the demultiplexer’s active efficiency ${\eta }_a=(r_n^{(\exp )}-1)/(r_n^{(\mathrm{ideal})}-1)$—a quantity that equals 1 for an ideal active scheme, and vanishes for a passive one.

The non-unity contrast of modulation shown in Fig. 2(c) is due to both imperfect polarization switching of the r-EOMs and finite polarization extinction ratios of the PBSs. Using the same demultiplexing scheme with the single-photon source operated at the increased pump repetition rate of 324 MHz, we obtain detected and generated—corrected for detector efficiencies of 30%—two- and three-photon rates as shown in Fig. 3 for both active and passive schemes. The generated (detected) three-photon rate amounts to $3.8\times {10}^3{\mathrm{s}}^{-1}$ ($105{\mathrm{s}}^{-1}$) for the active scheme. The active-to-passive ratio is expected to be $r_3^{(\mathrm{ideal})}=8$, and is measured to reach $r_3^{(\exp )}=6.6$. This corresponds to a three-photon active efficiency of ${\eta }_a=0.80$, which is limited here by switching contrasts and non-instantaneous modulation rise- and down-times.

 

Fig. 3. $n$-photon rates after the demultiplexer. Generated and detected $n$-photon rates for the active (red) and passive (blue) demultiplexing schemes.

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4. RECONFIGURABLE PHOTONIC TRITTER CHIP

The output of the demultiplexer is connected to a fiber array precisely coupled to the tritter chip inputs. The optical waveguides are fabricated by femtosecond laser writing in a commercial alumino-borosilicate glass substrate (EagleXG, Corning Inc., USA). A Yb:KYW cavity-dumped mode-locked laser oscillator was employed, producing 300 fs duration pulses at 1 MHz repetition rate. In detail, 220 nJ laser pulses were focused 30 μm below the surface of the glass substrate, by means of a 0.6 N.A. microscope objective, while the sample was translated at the constant speed of 20 mm/s. Such irradiation parameters result in single-mode waveguides at 930 nm wavelength with $\sim 8\mathrm{\mu m}$ $1/e^2$ mode-diameter and $<1\mathrm{dB}/\mathrm{cm}$ propagation loss.

As shown in Fig. 1, the tritter is built of three directional couplers (DCs), with nominal reflectivities of $R_1=1/2$ (first and last DCs) and $R_2=1/3$ (second DC), and one intermediate phase shifter $\varphi$. When the phase shifter is set at $\varphi =\pi /2$ or $3\pi /2$, the theoretical unitary matrix of the circuit corresponds to a symmetric tritter transformation [41–44], given by the matrix elements ${\mathcal{U}}_{jk}^{\mathrm{th}}=\exp [i2\pi (j-1)(k-1)]/\sqrt{3}$, where $j,k=1,2,3$ are the corresponding matrix element indices. At the input and at the output of the chip, the inter-waveguide distance is set to 127 μm to match the pitch of the fiber arrays. The overall footprint of the waveguide circuit is $25\mathrm{mm}\times 0.25\mathrm{mm}$.

The tunable phase shifter $\varphi$ is realized according to the method of Refs. [34,35]. A 50 nm thick gold layer is sputtered on the top surface of the chip, and a resistive heater is laser-patterned just above the relevant waveguide (the obtained resistance value is about 60 Ω). By driving the resistor with a suitable voltage, the waveguide is locally heated and increases its refractive index according to the thermo-optic effect, thus producing a finely adjustable phase delay.

We characterize the experimental matrix ${\mathcal{U}}^{\exp }$ using a continuous-wave laser tuned to the same wavelength as that of the single photons. We exploit the method described in Ref. [45] where amplitudes and phases are deduced by measuring the intensity and by monitoring the interference at the chip outputs when sending the laser directly into one input only or into two inputs simultaneously. For an applied voltage of 3.1 V on the phase shifter, providing the best operation, we obtain a fidelity to the ideal tritter ${\mathcal{U}}^{\mathrm{th}}$ of $\mathcal{F}=0.960$ (see the Supplement 1 for a complete description of the chip characterization).

5. THREE-PHOTON COALESCENCE

Figure 4(a) shows the occupation probabilities calculated for an ideal circuit for the 10 three-photon output states $|ijk⟩$ corresponding to $i$ (resp. $j$, $k$) photons in mode 1 (respectively 2, 3). Two cases are considered: fully distinguishable photons with pair-wise mean-wave-packet overlap $M=0$ (top panel) and three fully indistinguishable photons with $M=1$ (bottom panel). For fully indistinguishable photons, the output distribution is composed by the no-collision term $|111⟩$, with a probability of 1/3, and by the three-photon bunching terms $|300⟩$, $|030⟩$, and $|003⟩$, with probabilities 2/9 each. In this case, the six possible terms of the output state, having exactly two photons in one of the modes and one photon in another one, completely vanish. As a result, the eventual detection of such events indicates the presence of imperfect single-photon indistinguishability. The case of complete distinguishability indeed shows a different distribution with a maximal probability of 1/9 for these states, a reduced probability of 1/27 for the bunching terms, and 2/9 for the non-collision term [46].

 

Fig. 4. Three-photon coalescence. (a) Theoretical output distributions for an ideal tritter device, and distinguishable (blue) and indistinguishable (red) photons. (b) Experimental output distribution (orange bars), and distribution expected from the modeling of the experiment (light-blue bars). (c) Comparisons among output distributions of tritter experiments from [8] (yellow), [44] (green), this work (orange), and the idealized case (red).

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To experimentally reconstruct the population distribution at the output of the chip, we use a pseudo number-resolving detection scheme; see module (IV) of Fig. 1. Each output is coupled to two cascaded multimode fibered BSs connected to three silicon APDs. An electronic correlation allows one to reconstruct all the triple photon coincidences. We accumulated a total of 3078 detected triple events during $\sim 1.7\mathrm{h}$, collected within a 2 ns coincidence window. The corresponding reconstruction of the output state is shown in Fig. 4(b) in blue bars. There is a strong contribution of the three-photon bunching terms and the non-collision term with an average probability of $P_{\overline{300}}=0.157\pm 0.015$and $P_{\overline{111}}=0.229\pm 0.011$, respectively, and an average probability for the terms associated with distinguishable photons $P_{\overline{210}}=0.050\pm 0.006$.

We compare our experimental results with the theoretical distribution calculated for the measured mean-wave-packet overlap and the non-zero $g^{(2)}(0)$. The effect of non-perfect indistinguishability is accounted for following the model of generalized multi-photon interference by Tillmann and coworkers [46]. However, this model does not consider the generation of more than one photon per pulse. To do so, one needs to identify the origin of non-perfect $g^{(2)}(0)$, since different contributions are expected whether the extra photons are identical to the others or not. Here, we use a neutral exciton under resonant excitation, in which case the non-zero $g^{(2)}(0)$ is due to imperfect suppression of the excitation laser. Indeed, the excitation laser is spectrally broader than the QD emission, and crossed-polarization rejection is strongly wavelength dependent. As a result, some residual laser remains outside the QD resonance. This can be evidenced by adding a narrow band filter in the collection of the single-photon emission and observing a $g^{(2)}(0)$ well below 1% [11]. As a result, we neglect the probabilities $p_{n>1}^{\mathrm{QD}}$ and consider that the additional photons arise from the residual laser signal, and are thus distinguishable from the QD emission. We quantify the relative amount of average photon number coming from the QD emission, ${\mu }_{\mathrm{QD}}=p_1^{\mathrm{QD}}$, and from the scattered laser, ${\mu }_{\mathrm{L}}=\sum n{p}_n^{\mathrm{L}}$, according to the formalism of the probability generating function [47] (see Supplement 1). The second-order autocorrelation function that results from the mixture of the independent photonic distributions of scattered laser and QD single photons is given by $g^{(2)}(0)=\chi (2+\chi )/{(1+\chi )}^2$, where $\chi ={\mu }_{\mathrm{L}}/{\mu }_{\mathrm{QD}}$. Since we measure a high single-photon purity, we approximate the probability of having a single photon from the laser by $p_1^{\mathrm{L}}\simeq (g^{(2)}(0)/2)p_1^{\mathrm{QD}}$, neglecting higher-order Fock terms from the laser $p_{n>1}^{\mathrm{L}}\ll p_1^{\mathrm{L}}$.

Following this assumption, we consider that for each input mode there is a certain probability of having vacuum, a single photon from the QD, a single photon from the laser, or, less likely, simultaneous photons from the QD and the laser. The total sum of these probabilities is normalized such that $p_0+p_1^{\mathrm{QD}}+p_1^{\mathrm{L}}+p_1^{\mathrm{QD}}{p}_1^{\mathrm{L}}=1$. In this context and considering that $p_0\gg p_1^{\mathrm{QD}}\gg p_1^{\mathrm{L}}$, the relevant input tritter states involving three photons are $|1_{\mathrm{QD}},1_{\mathrm{QD}},1_{\mathrm{QD}}⟩$ with probability ${(p_1^{\mathrm{QD}})}^3$, the six combinations of $\{|1_{\mathrm{QD}}{1}_{\mathrm{L}},1_{\mathrm{QD}},0⟩\}$ with probability $p_0{(p_1^{\mathrm{QD}})}^2{p}_1^{\mathrm{L}}$, and the three combinations of $\{|1_{\mathrm{QD}},1_{\mathrm{QD}},1_{\mathrm{L}}⟩\}$ with probability (${(p_1^{\mathrm{QD}})}^2{p}_1^{\mathrm{L}}$). The total output distribution is obtained by summing the weighted contributions of all relevant input states and considering the corresponding pair-wise indistinguishabilities between photons generated from the QD, and $M=0$ between QD single photons and laser light.

Considering the effective $g^{(2)}(0)=0.071\pm 0.003$ of our measurement, we found that $\chi ={\mu }_{\mathrm{L}}/{\mu }_{\mathrm{QD}}=0.038\pm 0.003$, which validates the approximation $p_1^{\mathrm{L}}\simeq (g^{(2)}(0)/2)p_1^{\mathrm{QD}}$ since $p_1^{\mathrm{QD}}\gg p_1^{\mathrm{L}}$. Figure 4(b) shows our experimental results, displaying good agreement with the simulation.

6. DISCUSSION AND CONCLUSION

Table 1 shows various figures-of-merit of sources used in implementations of three-photon interference in a tritter device. Previous experiments used an integrated platform for either the source or the photonic circuit, but not both simultaneously. The present implementation is the first combining both integrated platforms, already showing a significant improvement in performance.

Table 1. Comparative Table of the Various Tritter Implementations^a

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The brightness of the QD single-photon source, combined with an active demultiplexer, provides an increase by at least two orders of magnitude of the three-photon generation rate. Moreover, the tritter output three-photon rate is also increased by a factor of $\sim 2$. Finally, the output distribution of our implementation is closer to the ideal one compared to previous works as shown in Fig. 4(c). Such observation is consistent with the improved source performance as compared to Ref. [44] as shown in Table 1. However, the output distributions of Refs. [44] and [8] are very similar, despite the better claimed source performance for the latter. Such observation indicates that other parameters (quality of the tritter network, additional noise) must have contributed to degrading the output distribution for Ref. [44].

These significant advances have been obtained despite non-minimized losses in the present implementation (see Table 2) and using low-efficiency detectors with limited time resolution. The present study thus constitutes a first step toward improving the scalability that can be obtained by merging solid-state photon sources and reconfigurable chips.

Table 2. Efficiency Budget for the Total Architecture and Foreseen Progress in the Near Future (See Details in the Text)

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Given the challenges imposed in this first merging of scalable photonic platforms, not all parts of the system could be optimized at once. Moreover, technological and experimental progress has recently been reported on both the source and chip sides, which could push further the limit of the combined system. As a result, we present in Table 2 a loss-budget showing the expected values that could be reached in the foreseeable future.

On the source side, the first-lens brightness was limited by the use of a resonant excitation scheme that removes more than 50% of the single photons. Recent studies show that such an excitation scheme could be overcome using side excitation [48] or removed by taking advantage of Raman-assisted excitation [49]. We thus anticipate that the first-lens brightness using this new excitation scheme could reach the maximal value resulting from the Purcell acceleration and outcoupling efficiency, typically 65% for the present technology [11]. The fibered brightness was limited here to 7% due to the use of a microscope objective with a relatively small numerical aperture within our collection setup ($\mathrm{NA}=0.45$). Inserting a high numerical aperture collection lens inside the cryostat has been shown to solve this problem [16]. Moreover, recent technological progress on the glass chip technology shows that its throughput transmission could reach much higher values, by adopting proper post-processing with thermal-annealing after the waveguide inscription [50]. The implementation of the next experiments with superconducting nanowire single-photon detectors will also give us a $\times 3$ increase in detection efficiency per photon. Considering all these possible improvements, we anticipate that one could soon reach on-chip 10-photon manipulation at rates as high as $\sim 40\mathrm{Hz}$.