1. Introduction

Linear-optical quantum computing requires an efficient method of on-demand generation of a definite number of indistinguishable photons, which carry physical qubits encoded in optical modes. The majority of the reported experimental implementations make use of either spontaneous parametric down-conversion (SPDC) in bulk crystals [1,2] or four-wave mixing in waveguides [3,4] in order to generate single photons. Recent advances in semiconductor quantum dot (QD) sources [5–9] and the emergence of commercially available QD source chips (Quandela, Sparrow Quantum, or AegiQ) propelled the spread of this type of source in the field of optical quantum computing [10,11]. Fabrication of a set of identical QD sources is a notoriously difficult task, and hence the experimental challenge of using a single QD as a mutiphoton source has emerged.

The SPDC process is easily adopted to generate several photons at once. A straightforward solution implies assembling $n$ identical SPDC photon pair sources to generate $2n$ photons with probability $p_{\mathrm {SPDC}}^{n}$. The drawback is an extremely low probability $p_{\mathrm {SPDC}}<<1$ of successful scattering. The $p_{\mathrm {SPDC}}$ can be substantially enhanced if the heralding feature of the SPDC photon pair source is used to implement the multiplexing principle. Detection of one photon of a pair heralds the successful generation of a pair, and hence the second photon can be stored in an optical memory cell and released on demand. Kaneda and Kwiat provided the most prominent demonstration of this method [12]. Their implementation includes two Pockels cells (PCs), one of which is used for the storage and release of the heralded photon and the second one filters out the unheralded photons. As a result they show generation of pure and indistinguishable single photons with probability $p_{\mathrm {MUX}}=66.7\left (24\right ){\% }$ per cycle. The scheme can be cloned to produce several photons simultaneously at a cost of using many SPDC crystals, storage cells, single-photon detectors, and fast switches (at least two Pockels cells per each multiplexed photon). The review, Ref. [13], provides the comprehensive viewpoint on single-photon multiplexing techniques.

A state-of-the-art QD source is a single-photon emitter coupled to a microcavity. This concept makes close to deterministic generation of single-photons feasible in principle [5,9]. Multiphoton state preparation relies on a reciprocal idea of demultiplexing—the stream of photons emitted by a dot is split into several independent spatial modes. Photons, that were generated during $N$ consecutive clock cycles (or $N$ consecutive bursts of cycles) of the pump laser, are forwarded toward delay lines which produce $N$ synchronized photon packs at the output [14]. This scheme requires $N-1$ switches (PC, resonant phase modulators. or other). The maximal number of demultiplexed photons, which has been demonstrated to date with this method, is 20 [15].

An ideal demultiplexer should demonstrate low losses in switching and routing of photons, and maintain indistinguishability of photons after demultiplexing. It is highly desirable to design a resource-efficient demultiplexer circuit, which includes the least possible number of switches, because each switch introduces loss mostly due to non-ideal switching extinction. The complexity of assembling, adjusting, and controlling the circuit grows significantly with the number of switches and effectively increases the overall loss per demultiplexer channel. The work, Ref. [16], reports four-channel demultiplexing on a lithium niobate photonic chip. Although an on-chip demultiplexer demonstrates 78% efficiency, the input and output coupling to a photonic chip reduces the total transmittance of the demultiplexer to 30% only. This work demonstrates the prospect of developing fast on-chip modulators capable of individual pulse-picking which corresponds to the shortest possible delay lines. Further development of an integrated demultiplexer design led to a nanomechanical on-chip photon switch [17]. Despite being inferior to its electro-optic counterpart [16], the nanomechanical switch demonstrated feasibility of packing both a source and a demultiplexer in a single technological platform without invoking the use of hybrid integration technologies.

Refs. [14,18] were the first to demonstrate spatial demultiplexing of a single stream of photons produced by a single quantum dot. Both experiments employ the same layout of the demultiplexer—the tree-like array of electro-optic switches (PC). This layout requires $N-1$ switches to separate an input stream into $N$ ports. The authors achieved high transmission per channel equal to approximately 83%–85%. The work [19] employs a fast acousto-optic deflector (AOD) to divert the input beam toward each output port. After that, the photons could be spatially separated by first-order diffraction. For this purpose, the AOD was driven by three different RF frequencies within time windows $\Delta T$. Each frequency corresponds to deflection of the beam toward a designated output fiber coupler. An optical frequency shift induced by an AOD was much smaller than the bandwidth of the photons and thus did not affect their indistinguishability. The best reported transmission was 65%. Here the time window $\Delta T = 320$ ns is defined by the speed of AOD switching. The longest delay line in this example equals to $960$ ns imposing a stringent requirement on the QD source itself: the photons separated by a $960$ ns interval should still maintain a high degree of indistinguishability. An alternative implementation [20] uses resonantly enhanced electro-optic modulators. The resonant frequency was not tuned to match the repetition rate of the pump laser, and hence some of the photons were lost during the switching process. The work, Ref. [21], demonstrated a system where this problem is solved and the demultiplexer is capable of distributing each photon from the stream to a designated channel.

In this paper we present a resource-efficient four-beam demultiplexer requiring a single fast optical switch (PC) only. Although our demultiplexer does not beat the best reported probability to produce a photon at the output, it drastically reduces the number of fast switches required for splitting the input photon sequence and eliminates the necessity to assemble long fiber delay lines in order to compensate for large delay times between the output photons in each channel.

2. Experimental Method

We exploit a simple idea of storing and releasing a bunch of photons inside an optical loop. Our setup is shown in Fig. 1. We use a semiconductor QD source in the micropillar configuration [22]. We pump the QD in a cross-polarized scheme using resonant picosecond pulses emitted at 82.6 MHz repetition rate at 918.83 nm wavelength. The single photons are coupled to a single-mode optical fiber and fed into the demultiplexer through the output coupler. The demultiplexer is an optical loop with a round-trip time equal to the time difference $\Delta t = 12.1$ ns between the consecutive pump pulses. Each round-trip shifts the beam transversely in a horizontal direction by 3 mm on average. A $1:1$ telescope inside the loop serves two purposes: it minimizes the diffraction spread of the beam passing through the loop multiple times and focuses all the beams through the 3 mm aperture of the PC (Leysop). The currently used geometry of the optical components allows us to store only four photons inside the loop. The half-wave and quarter-wave plates at the input adjust the polarization of the input photons to a horizontal state. After passing through the PC each photon reflects of the polarization beam splitter (PBS) and the right-angle prism and completes the round trip by passing the PC again. Each sixth clock cycle the PC rotates the polarization of the photons in the loop and hence releases an incoming photon and three previously stored ones. Ideally, switching has to be done every fourth cycle because our circuit is designed to collect only four output photons; however, hardware restrictions imposed by the high-voltage driver allow for a minimal interval between consecutive switching events of $70$ ns. The high-voltage driver (BME Bergmann) switches the cell using a 12-ns-long voltage pulse. The driver is controlled by a homebuilt FPGA-based circuit which uses the pump laser as a source of the reference clock signal. The voltage pulse peak is synchronized to the moment when all four photons pass through the Pockels cell and the rise and fall fronts do not affect the neighboring photons.

 

Fig. 1. Schematic illustration of the experiment with the QD single photon source setup, the demultiplexing setup, the detection setup, and control and analysis setup. The demultiplexer optical circuit is fed with a sequence of single photon pulses emitted by a QD in a microresonator source. The circuit stores four optical pulses and outputs them once the Pockels cell PC is turned on and the polarization of each pulse is rotated to the orthogonal state. The detailed description of the setup is given in Sec. 2.

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The photons with rotated polarization pass through the PBS and enter the circuit, where each beam is separated from the other using knife-edge reflective prisms. After that, photons are coupled to the single mode fibers. Each fiber coupler is mounted onto a translation stage in order to compensate for slight differences of the output channel lengths including the lengths of optical fibers. The natural synchronization of the output photons makes it possible to divert each photon of the incoming stream to an assigned output mode. This feature relaxes the requirement for the QD source to produce highly indistinguishable photons with a large time interval between emission events. The output of each fiber coupler is sent either to superconducting single-photon detectors (SSPD) or to a balanced fiber beamsplitter which enables the observation of Hong–Ou–Mandel type quantum interference.

3. Results

Firstly, we characterized the QD itself. Figure 2(a) presents the auto-correlation function $g^{\left (2\right )}\left (\tau \right )$ for our QD pumped in a resonant regime in a cross-polarized configuration. The observed single-photon purity is $g^{\left (2\right )}\left (0\right )=0.024 \pm 0.001$. The indistinguishability of the photons was asserted using a Hong–Ou–Mandel interferometer with $\Delta t = 12.1$ ns delay introduced into one of the arms. The average photon count rate detected using the SSPD was approximately 5 MHz. The fabrication details and optical schemes for pumping and collecting emitted photons are presented in Supplement 1.

 

Fig. 2. (a) Auto-correlation function measured for the single photons emitted by the QD. (b)–(e) Auto-correlation functions measured for the output signal at each of the four channels of the demultiplexer.

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Then, we measured the auto-correlation functions for each output channel, the result is presented in Figs. 2(b)–2(e). The auto-correlation histograms indicate the quality of splitting the input photon sequence. Small parasitic peaks can be witnessed between the demultiplexed photons. They originate from the imperfect polarization switching and finite PBS extinction ratio. Our configuration implies passing four beams through a single PC, and each beam travels in a slightly different direction due to focusing. For this reason it is impossible to reach optimal performance of the PC for each beam. We decided to use the second output channel as a reference one because it was easier to find the best orientation of the PC. Thus, the second channel auto-correlation function has minimal parasitic peaks. On the other hand, focusing on the second channel of the demultiplexer leads to a decrease in the amplitude difference of neighboring peaks in other channels, but increases the amplitude difference in the second channel. We attribute this to the non-ideal operation of the Pockels cells at the limiting frequency (see Supplement 1 for details). Next we tested the indistinguishability of the photons after they have been split into different channels by measuring Hong–Ou–Mandel (HOM) interference visibility. We connected a balanced fiber beamsplitter to the outputs of a selected pair of channels and measured the second-order cross-correlation function $g^{\left (2\right )} \left (\tau \right )$. Since we used a single-mode fiber which does not preserve the polarization state, we had to place an additional pair of half- and quarter-wave plates to each channel and vary their orientation to compensate the unknown polarization rotation inside the fibers. The measurement results indicate that the indistinguishability of the photons remains on the same level as for the photons tested directly from the QD source. We measured raw indistinguishability values between channels 1 and 2 $HOM_{uncorr}^{12}=0.884$, channels 2 and 3 $HOM_{uncorr}^{23}=0.893$, and channels 3 and 4 $HOM_{uncorr}^{34}=0.752$. The corresponding values corrected for non-zero $g^{\left (2\right )} \left (0\right )$ and interferometer imperfections are $HOM_{corr}^{12}=0.959$, $HOM_{corr}^{23}=0.964$, and $HOM_{corr}^{34}=0.894$. We used a correction formula from Ref. [5] to infer the estimated value of the Hong–Ou–Mandel interference visibility (see Supplement 1). We note that raw HOM values $HOM_{uncorr}^{12}$ and $HOM_{uncorr}^{23}$ exceed the HOM value measured by directly interfering the photons from the source. These overestimated values are possibly due to imbalanced losses in the HOM interferometer arms. The correlation histogram then includes contributions from both the two-photon interference events associated with the HOM measurement and single-photon splitting events associated with the $g^{\left (2\right )}$ measurement. The area of the central peak of the histogram is thus less than in the case of the proper HOM measurement. The HOM values displayed in Figs. 3(c) and 3(d) must not be interpreted as a positive effect of the demultiplexer on the indistinguishability of the source.

 

Fig. 3. Uncorrected and corrected indistinguishability of the single photons from (a) QD, (b) channels 1 and 2, (c) channels 1 and 3, and (d) channels 1 and 4 of the demultiplexer. The indistinguishability value is corrected in order to take account of non-ideal beamsplitter reflection and non-zero $g^{\left (0\right )}$ value of the interfering light. See Supplement 1 for a detailed explanation.

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Finally, Fig. 4 illustrates the detection rates of multiphoton events at the output of the demultiplexer. We used the data to estimate the efficiency across all output channels. The probabilities of detecting $n$-photon events were fitted with an exponential function $p^{n}$, where $p$ is the probability of detecting a photon at the output of the demultiplexer. The value $p$ is related to the source brightness $\nu / r$, where $\nu$ is the detected single-photon count rate at the output of the source, and $r$ is the repetition rate of the pump laser. The ratio $e= p / B$ expresses the efficiency of each demultiplexer channel. Then for the described scheme $e = 0.225$, which does not include the efficiency of the detectors (0.85 on average), and photon losses due to the maximum possible frequency of the Pockels cell (4/6). With this in mind, the efficiency increases to $e=0.397$. This value is due to losses in the polarizing beam splitters, other optical elements of the circuit, the efficiency of optical couplers, and multiple passages through the Pockels cell.

 

Fig. 4. Count rates of different configurations of photon distributions among channels. The red dots are the average count rate values for the given number of photons. Red line is the approximation by an exponential function $p^n$

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4. Discussion

The characteristic feature of the demonstrated device is the optical circuit equipped with a single Pockels cell only. To the best of our knowledge, the majority of the demultiplexers reported to date include $N-1$ electro-optic active elements in order to split the input stream into $N$ channels (excluding probably the AOD-based design [19] which uses a single AOD). The number of channels which can be implemented using the demonstrated principle is limited by two parameters: the repetition rate $r$ of the source pump laser and the switching rate $r_s$ at which the Pockels cell or other electro-optic device can operate continuously. Typically the switching rate of the PC drivers is far below the repetition rate of pump lasers which can reach up to 10 GHz. The ratio $N_{min}\approx \lfloor r / r_s \rfloor$ defines the minimum number of channels of a loop-type demultiplexer. Although in our case the Pockels cell provides maximum switching rate $r_s$ of approximately 13 MHz (such type of cells are capable to produce $r_s$ well up to 20 MHz in burst mode) and in combination with $r=82.6$ MHz it gives $N_{min}=6$. For this reason the use of slow PC drivers is impossible in this type of circuit because the geometry of an optical delay line required to store a large number of photons will quickly become impractical. In our case we can only fit four optical paths inside clear aperture of optical elements, and hence we always lose two out of six photons. The demultiplexer can be easily upgraded to deliver a doubled number of output streams if the repetition rate is also doubled. In this case each input pulse travels exactly half of the round-trip and when the PC is enabled two sets of beams pass the PC in different directions. One needs to add a PBS on the other side of PC to let both sets of streams out of the loop (see Supplement 1).

The loop-type demultiplexer benefits from self-synchronization of photons inside the loop. Once the PC rotates the polarization and releases the photons, they are already synchronous up to a few centimeters difference in the length of the output path to the fiber coupler. This means that there is no need for long delay lines which were used in the previous demonstrations [15,19]. If the application demands the processing of the emitted photons with a photonic circuit, then the input fibers of the fiber array can be directly connected to the fiber couplers thus eliminating any lossy fiber mating sleeves.

The disadvantage of the loop-type demultiplexer circuit is mostly related to the complexity of tuning the optical circuit and tailoring the selection of the elements in order to reach higher efficiency. We struggled with optimizing the position and orientation of the PC because two goals have to be achieved at once. The PC aperture must not cut the beams travelling inside the loop, and the PC has to provide a high switching contrast at the same time. PCs with a larger clear aperture (our PCs have clear aperture equal to 3 mm) might simplify this task, but this also leads to the increase of the half-wave voltage level which in turn makes it harder to reach fast switching rates. The second problem is the beam divergence due to intrinsic diffraction of the Gaussian beam. The photons travel a large distance inside the loop, and hence the coupling efficiency drops from the first channel to the fourth one. The solution would be to use a large focal length aspheric lens which delivers a larger collimated beam; however, this again will require a larger PC aperture and hence a higher half-wave voltage. Nevertheless, we believe that careful optical engineering may significantly boost the demultiplexer efficiency.

Finally, we would like to review the results and discuss the scalability of the demonstrated solution. We have demonstrated the single-photon demultiplexer with four output-channels which requires a single fast Pockels cell and is capable of picking each pulse from a stream photons. The updates of the optical circuit (see Supplement 1) can potentially increase the number of output channels to the order of several tens. The multiphoton source of this scale is believed to outperform classical simulators of the boson sampling problem [23,24]. Our implementation is yet beyond the up-to-date quantum advantage thresholds. The design improvements (see Supplement 1) and custom optics are believed to boost the performance of the demultiplexer.