1. Introduction

Multiplexing and de-multiplexing of single photons are key techniques to enhance the scalability of photonic quantum information processing [1–3]. Near-deterministic single-photon generation is possible by combining, i.e. multiplexing, a large number of intrinsically probabilistic heralded single-photon sources (HSPS) based on spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing (SFWM) [4, 5]. If the output photon state is engineered to be quantum mechanically pure, combination of such multiplexed single-photon sources can boost the multi-photon generation rate, which will be useful for efficient quantum state engineering [6] and quantum information processing with a large number of photons [3, 7]. On the other hand, the multiplexing scheme can be inversely applied to distribute, i.e. de-multiplex, output photons from a single source to multiple output modes [8]. This allows to prepare multiple identical pure-state photons that can be applied to, for example, quantum simulation experiments [9].

Spatial [1, 4], temporal [5, 7, 10, 11], hybrid [12], and spectral [13] multiplexings have been proposed and demonstrated. Those techniques raise the photon generation probability in single output spatio-temporal mode by consuming physical resources or time/frequency bands. Temporal multiplexing particularly offers an advantage of saving the number of source materials, routing elements, and heralding counters. This becomes important when a large number of channels have to be multiplexed to produce a series of pure indistinguishable single photons with multi-photon emission noise being suppressed [5, 10]. Up to 40-channel time multiplexing has been demonstrated to achieve a photon generation probability as high as 67% using one SPDC photon-pair source and one heralding photon counter (a set of four single-photon detectors for exclusion of multi-photon heralding events) [7]. Two strategies can be adopted for temporal multiplexing, namely a storage cavity method and a switchable multiple-path method. The former utilizes an optical cavity in which a fast electro-optic modulator (EOM; typically a Pockels cell) controls capturing, storing, and emitting sequences of heralded photons [5, 10]. This method has an advantage that only one or two EOMs are required only if a high-quality free-space cavity can be implemented. The latter utilizes a series of delay lines usually made of optical fibers connected by two-by-two optical switches (OSs) [2, 11, 12, 14]. Such binary-division scheme is potentially preferable for a large number N of multiplexing channels because its maximum transmission loss due to optical components scales as $1-{\eta }^{{\log }_{2}N}$ instead of $1-{\eta }^N$ of a storage cavity, where η is the transmittance of a unit switching element. However, a larger number of OSs or EOMs, which become usually too bulky to realize low-loss and high-speed operation, are required and longer optical delays have to be implemented between the OSs.

This work demonstrates temporal multiplexing of heralded single-photon sources over 32 time-bins using switchable multiple paths composed of dispersion-compensated single-mode fiber (SMF) delay lines. In addition, relative multiplexing [3, 15–17] of two heralded photons is realized by active synchronization in a fiber-based Hong-Ou-Mandel (HOM) interferometer. For quantum interferometry, the relative multiplexing increases the number of “matched photon pairs” compared to both the non-multiplexed, probabilistic multiple-photon generation scheme and the strictly synchronous scheme using two time-multiplexed single-photon sources [3]. Events that any two photons are generated with the preset time difference range (32 time-bins in this work) contribute to the experiments. The accumulated interference fringe verifies the indistinguishability between photons originating from different time-bins.

 

Fig. 1 Experimental setup for the time-multiplexed HSPS. PPKTP: periodically-poled potassium titanyl phosphate crystal, DM: dichroic mirror, OS: optical switches, FC: fiber coupler, SMF: single-mode fiber: FBS: fiber beam splitter, SPAD: single-photon avalanche photodiode, TAC: time-to-amplitude converter, FPGA: field-programmable gate array.

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2. Structure of the time-multiplexed single-photon source

Figure 1 shows the schematic of our experimental setup for time multiplexing of heralded single-photon sources. A periodically-poled potassium titanyl phosphate (PPKTP; length 10 mm, poling period 31 μm) crystal generates photon pairs via collinear type-II SPDC process pumped by a second-harmonic-generated mode-locked Ytterbium fiber laser (center wavelength 520 nm, repetition 100 MHz, pulse width 150 fs). Signal and idler photons have the center wavelengths of 780 nm and 1560 nm, respectively. This highly non-degenerate configuration can generate near-factorable joint spectrum of photon pairs due to the group velocity matching between the pump wavelength and the signal wavelength [18]. The signal photons are detected by a heralding Si single-photon avalanche photodiode (SPAD; Excellitas SPCM-AQRH-12). The ‘heralded’ idler photons are sent to an adjustable delay line that consists of six 2×2 OSs (Agiltron NanoSpeed) and optical fiber sections. The five pairs of fiber sections between the adjacent OSs apply differential optical time delays of T, 2T, 4T, 8T, and 16T, respectively, where T (= 10 ns) is the length of a unit time-bin given by the period of pump pulses. The OSs are actively controlled by a field-programmable gate array (FPGA) module to add a time delay of $\left(32-k\right)T$ when a heralding photon is detected in the k-th time-bin, where $k=1,2,3,\cdots ,32$. An optical-fiber spool (Corning SMF-28, length 400 m) holds the idler photons before entering the first OS by 2 μs to cope with the finite response times of the SPADs, the electronic circuits, and the OSs. All the electronic control circuits hereafter use the 100-MHz time base synchronized to the laser repetition rate.

The difference in delay between paired fiber sections are matched to multiples of the pump laser period within the coherence length ($\sim 0.6$ mm) of the heralded photons. Each blue-colored section in Fig. 1 includes a micrometer-controlled variable delay using a pair of fiber-to-free-space couplers. The magnitude of the delays are first adjusted based on the measurement of temporal distribution of coincidence counts using a time-to-amplitude converter (TAC; Ortec 567), and then fine tuned based on the HOM dip measurement described below. The fiber sections introducing delays T, 2T, 4T, 8T, and 16T are designed to have no chromatic dispersion by connecting in series a standard SMF (Corning SMF-28; $D=-16.9\text{ ps}/\text{nm}/\text{km}$) section and a dispersion compensating fiber (Thorlabs DCF3; $D=3.0\text{ ps}/\text{nm}/\text{km}$) section. The ambient temperature of the experimental setup is maintained within $\pm {0.1}^{\circ }$C to ensure that fluctuation of the longest delay (fiber length 64 m; 37 $\text{ps}/\text{km}{/}^{\circ }\mathrm{C}$) due to temperature variation is smaller than the coherence time ($\sim 3$ ps). If the detailed structure of an OS is unknown, it has to be also guaranteed that the optical delays are independent of the switching state. We have designed and demonstrated an experimental setup to test the possible group delay change and verified that the current OSs does not introduce a switching-state-dependent variation (see Appendix). Moreover, the switching operation did not significantly affect the polarization states, thereby the output photon flux in the presence of output polarizers did not show fluctuation depending on the delay path combination.

3. Spectral characteristics of the HSPS: Theory

The indistinguishability between the heralded photons is determined by the quantum mechanical state purity and measured as a quantum interference fringe. The joint spectral intensity (JSI) under the current SPDC configuration is calculated and shown in Fig. 2(a). The pump beam has a 1/e² diameter of 230 μm at the beam waist in the SPDC crystal. The signal and the idler have a spatial mode diameters of 136 μm and 121 μm, respectively, which are determined by the output SMFs and the collimating/focusing lenses. The JSI shows a slight tilt that reduces the purity Tr$\left\{{\rho }^2\right\}$ of the output photon to 0.62, where ρ is the density matrix of the heralded photons. In our experiment, an interference filter with 3.0-nm bandwidth (Semrock LL01-780) is inserted to the signal path to raise the theoretically estimated purity to 0.90. The heralding efficiency, the probability to have an output photon under a heralding signal, between the collecting SMFs is 73% after the filtering.

 

Fig. 2 Numerical calculation results of the joint spectrum and the state purity. (a) Joint spectral intensity of the photon pair (vertical dashed lines: half-maximum bandwidth of the interference filter for signal photons). (b) HOM dip depth according to the dispersion mismatch between two interfering photons. (c) HOM interference fringes. The interference filter for signal photons are included in calculation of (b) and (c).

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Since photons heralded in earlier time-bins propagate a longer fiber delay line, photons from different time-bins can gain temporal distinguishability at the output when chromatic dispersion exists in the optical delay. We analyze this effect by calculating the normalized HOM dip depth (minimum coincidence counts divided by maximum coincidence counts) with two photons from different time-bins being coalesced. Assuming all the fiber sections are standard telecom fibers (Corning SMF-28), the largest amount of differential dispersion is 1.05 ps/nm corresponding to the time delay 31T (fiber length 62 m). The expected dip depth according to the dispersion mismatch is shown in Fig. 2(b). The calculated HOM interference fringes with this maximum dispersion and the average dispersion (16T) are shown in Fig. 2(c). When the dispersion of the delay line is zero, the expected dip depth is 0.90, which is equal to the state purity of the heralded photon. This decreases to 0.80 and 0.76 with the dispersion corresponding to 16T and 31T, respectively, as shown in Fig. 2(c).

4. Photon number statistics with multiplexing

Temporal multiplexing capability is experimentally verified by comparing the temporal distribution of the output photons as shown in Fig. 3(a), with and without the switching operation. The multiplexing period was 5 μs, in other words, the clock frequency for the output photons was 200 kHz. This period guarantees a sufficient temporal margin for the OS driving circuits (bandwidth 500 kHz). The 200-kHz clock signal is input to the TAC as a timing reference for the measurements in Fig. 3(a). For noise reduction, the TAC produces an output pulse only in the case that a heralding photon is detected among the 32 channels. The pump power was set to 44 mW to alleviate the effects of multiple pair generation and detector saturation by keeping the mean number of of photon pairs per pulse μ as low as 0.01. The enhancement factor, the ratio between the multiplexed output counts and the single-time-bin counts with the OSs being turned off (with photon propagating through the red lines in Fig. 1), is $19.0\pm 0.4$. Its departure from ideal 32 is mainly due to the fiber-free-space coupling losses and fiber splice losses present in the fiber delay sections in Fig. 1 (blue-colored lines).

 

Fig. 3 Experimental results of temporal multiplexing. μ: photon-pair generation probability per pulse by the SPDC crystal. (a) Temporal distribution of the output photons. (b) Photon generation probability p₁ per output time slot. (c) Second-order correlation g⁽²⁾ (0). Calculation of the theoretical curves in (c) takes into account the on-off detection capability and the dark count rate of the SPADs.

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We characterized the multiplexed single-photon generation probability p₁ and the second-order correlation $g^{\left(2\right)}$ for two different mean photon-pair numbers per pulse μ (0.04 and 0.10) while varying the number N of multiplexing channels as shown in Figs. 3(b) and 3(c). The single-photon or two-photon detection probabilities to calculate p₁ and $g^{\left(2\right)}$ are ratios between the photon count rates and the clock frequency (200 kHz). Here the photons are counted at the output time slots only when a heralding signal is successfully received within the preset N time-bins. This post-processing characterizes the output photon number statistics for the case that the output mode is gated by an optical shutter (not used in the current experiment) based on a heralding signal to block the unheralded noise photons [10]. p₁ in Fig. 3(b) is experimentally the probability of the output SPADs (ID Quantique ID220) to click that is afterwards corrected by the dark count rate ($4.0\times {10}^{-5}$ per unit time slot of 10 ns by each SPAD) and the hold-off time (10 μs) of the SPADs, the insertion loss of the FBS (10%), and reflection from the fiber end faces (3.5%). We henceforth retain the expression p₁ because this click probability is almost the same as the probability to have only one photon when it is small as shown in Fig. 3(b). In contrast, the experimental and theoretical $g^{\left(2\right)}$ values in Fig. 3(c) are second-order correlation of the two SPADs for heralded idler photons without correction. The pump power was 160 mW and 380 mW for the $\mu =0.04$ and $\mu =0.10$ cases, respectively. The data collection period was 1 s for the p₁ measurements, and varied from 460 s to 3600 s and from 100 s to 1800 s for the $g^{\left(2\right)}$ measurements with μ =0.04 and μ = 0.10, respectively. p₁ increases approximately proportional to N to reach $2.2\pm 0.1$ % for $\mu =0.10$ and $N=32$. The error corresponds to the standard deviation of the measured counts, $\pm \sqrt{\text{counts}}$. The transmittance of the heralded photons through the variable delays is 10.0 dB (10%) on average, including the average transmission loss 1.33 dB × 6 by the OSs and their lead fibers, and fiber-free-space coupling loss 0.8 dB × 2.5 (average number of passages). The magnitude of p₁ is given by the average photon-pair number per pulse μ (0.10) multiplied by the number of channels (32), the heralding efficiency (0.73), the above transmission (0.10), and the probability q (0.12) of to detect the heralding photon with a heralded photon being detected [19]. The magnitude of q with our current setup is determined by the transmittance of the signal photons including the coupling efficiency to the lead SMF (0.43), the portion of spectrum transmitted by the interference filter (0.43), and the nominal detection efficiency of the Si SPAD (0.65). The residual 22% loss of the output photons is attributed to unidentified losses by the optical components and the fiber connections.

The current level of transmission loss through the switching network is mainly the result of our choice of compact fiber-based optical switches. This can be improved by a free-space polarization-based switching configuration using Pockels cells whose transmission loss is as low as 1% [7, 10]. The overall transmission loss will be lowered to around 4 dB considering increased fiber-free-space coupling losses. This may, however, require a significantly larger occupied space and water-cooling facilities for high-voltage drivers. Further development of low-loss optical switches helped by improvement of fiber coupling optics, fiber splicing techniques, and heralding/detection efficiencies for photons will lead to a practically meaningful higher photon generation efficiency.

$g^{\left(2\right)}$ decreases according to N down to $0.89\pm 0.23$ because the multiplexing raises both the single-photon probability p₁ and the probability to have two idler photons p₂ by the same ratio and $g^{\left(2\right)}$ is approximately $p_2/\left(2p_1^2\right)$. The opposite heralding efficiency q is another important factor for the magnitude of $g^{\left(2\right)}$. When q is small, the probability to successfully generate the heralding signal is proportional to q regardless of whether single photon pair or double photon pairs are generated. Thus $g^{\left(2\right)}$ becomes inversely proportional to q because p₁ and p₂ decrease by the same rate according to q ($g^{\left(2\right)}\approx \left(4/q-2\right)/N$ assuming the Bose-Einstein photon statistics and a small probability of photon pair generation per pulse). Note that $g^{\left(2\right)}$ can be much greater than 2 because p₁ and p₂ are practically the coincidence counting probabilities of one heralding photon plus one output photon and of one heralding photon plus two output photons, respectively, per a unit multiplexing run. In contrast, $g^{\left(2\right)}$ values for unheralded idler photons or only under the successful heralding condition (heralded $g^{\left(2\right)}$) are not greater than 2 and mainly determined by the magnitude of μ in the current condition. $g^{\left(2\right)}$ can be further reduced by optimizing the signal-idler wavelengths to remove the interference filter [7] and adopting a photon-number-resolving heralding detector to suppress heralding of multiple-photon emission events [19]. Rigorous estimation of $g^{\left(2\right)}$ requires consideration of higher-order photon pair generation [20], non-photon-number-resolving detection, and dark counts of the SPADs. The theoretical curves in Fig. 3(c) agree with the experimental data for relatively high pump power and high N. The source of the discrepancy in the low-power and low-N regime is currently unknown.

5. Relative-time-multiplexing Hong-Ou-Mandel interferometry

Figure 4 illustrates our Hong-Ou-Mandel interferometer (HOMI) based on an unbalanced optical-fiber Mach-Zehnder interferometer (MZI). Heralding photons are detected by two Si SPADs connected by a fiber beam splitter. The upper arm of the MZI contains the binary-division variable delay. The lower arm has a motorized optical fiber delay line (MDL; Newport MDL-2-6) that is scanned to obtain HOM interference fringes. The optical path length of the lower arm is equal to the minimum path length of the upper arm with all the OSs being ‘OFF’ state (the red lines in Fig. 4). The output polarizations of all the possible switching states are equalized using the polarization controllers in the figure and finally filtered by two polarizers in front of the two output SPADs.

 

Fig. 4 Experimental setup for relative-time-multiplexing HOM interferometry. OS: optical switch, PC: polarization controller, FBS: fiber beam splitter, MDL: motorized delay line.

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Within the stream of randomly generated photon pairs, the two Si SPADs herald the events that two heralding photons are detected with a time difference of kT, where $k=1,\cdots ,31$. The outputs of the two Si SPADs are connected by an OR gate inside the FPGA module. Use of the two SPADs is required to detect time intervals smaller than the hold-off time of the SPADs. When a signal photon is detected first by any SPAD, a counting process begins until the next photon is detected by either of the two SPADs within the allowed time interval ($=310$ ns). The recorded time interval between the two clicks is converted to a six-digit binary number that sets the OSs to delay photons by the same time interval. If a second heralding photon does not arrive within the 31 time-bins, the next earliest photon becomes the first heralding photon. Coincidence counts of the two heralded photons are recorded. The first and second heralded photons arrive at the SPADs at the same time only when they pass through the upper and lower arms of the HOMI in Fig. 4, respectively. The next heralding session begins after an idle time of 300T (3 μs) considering the switching speed limitation.

 

Fig. 5 Total output coincidence counts of the HOMI (period: 1500 s, μ = 0.10). Orange solid line: theoretical fit. Blue dotted line: background noises due to redundant photons and dark counts.

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The coincidence counts with scanning the MDL are shown in Fig. 5. The HOM dip shows a raw depth of $0.46\pm 0.01$ before background subtraction. The data in Fig. 5 is accumulation of the 31 cases in which the heralding time intervals are respectively T, 2T, $\cdots$, and 31T. The HOM dips for all the cases are shown in Fig. 6. The non-monotonous behavior of the data according to the delay is due to the fiber coupling losses present in the longer (blue)delay sections in Fig. 1 and Fig. 4. Photons with a delay of T, 2T, 4T, 8T, and 16T generate most clear fringes because they experience a minimum level of losses by respectively passing through only one lossier delay section. Other cases lead to lower signal levels depending on the number of passes through the fiber couplings. In contrast, the accidental coincidence levels are almost constant because their largest parts (97%) are due to two photons propagating through the interferometer arm that does not contain the switchable delays. The solid red line in Fig. 5 is the numerical fit using the blue curve corresponding to zero dispersion in Fig. 2(c) to the experimental data. The overall amplitude and the magnitude of constant background were used as free fitting parameters. To account for constant background noises due to redundant photons and dark counts, we disconnected in turn each one of the two arms of the MZI, and added up the two coincidence counts. This method can measure the coincidences due to higher-order ($>2$) photon pair generation and dark counts of the SPADs [11]. The effect of detector hold-off time is compensated for by comparing the single-count rates during the accidental coincidence measurement and the HOMI experiment. The HOM dip depth increases to $0.88\pm 0.03$ after subtracting the accidental counts shown as a dotted line in Fig. 5. Good agreement between the theory (0.9) and the experimental result ($0.88\pm 0.03$) of the HOM dip depth verifies the optical path length matching between the 32 different delays within the coherence length of the photons.

 

Fig. 6 Coincidence counts corresponding to the 31 time intervals between the two heralded photons (period: 1500 s).

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The non-ideal HOM dip depth less than unity is mainly caused by imperfect quantum-state purity of the heralded photons due to the slight non-factorability of the JSI in Fig. 2(a). The impurity led to the use of a wavelength bandpass filter for heralding photons at 780 nm wavelength, and thereby further increased the noise due to multi-photon components that are manifested by increased $g^{\left(2\right)}$. Such spectral purity can be improved by using (i) a better optimized wavelength set, (ii) a thinner SPDC crystal, and (iii) a smaller pump bandwidth down to the optimum value [18]. Reducing the overall transmission losses for heralding photons and heralded photons improves the photon generation probability and the photon-number purity. Therefore highly efficient superconducting nanowire photon detectors for both signal and idler photons will improve the overall qualities together with lower noise due to dark counts and hold-off time. For faster optical switching operation with lower insertion loss, bulk Pockels cells can be a better choice as long as cell heads and controllers can be implemented within a manageable space.

6. Conclusion

In conclusion, we have experimentally demonstrated temporal multiplexing of 32-time-bin heralded single-photon sources at the telecom band and its application to relative-multiplexing Hong-Ou-Mandel interferometry. The binary-division variable optical delay based on optical fiber devices was operated at the frequency of 200 kHz, with average transmission of 10% due to the insertion losses of the OSs and the fiber-free-space coupling losses. The HOM interference fringe was measured by accumulating all the cases that two photons are heralded with a time difference of 1 to 31 time-bins (10 ns each), and showed a good agreement with a theoretical calculation for two independent heralded photons. The HOM interferometry results verified that photons from different time-bins were reasonably indistinguishable at the output with the influence of fiber dispersion or temperature fluctuation being effectively suppressed. We believe that this method can be extended to realize higher-performance single-photon sources based on a greater number of multiplexing channels. They will contribute to the scale-up of photonic quantum information processing and quantum metrology, together with development of photon pair sources, detectors, and electro-optic devices.

 

Fig. 7 Low-coherence interferometry setup for testing an optical switch. OS: optical switch, SLD: superluminescent diode, PD: photodiode, MDL: motorized delay line.

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Fig. 8 Output time trace of the low-coherence Mach-Zehnder interferometer.

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