Heralded single photons based on spectral multiplexing and feed-forward control

TL;DR

This paper introduces a spectral multiplexing and feed-forward control method to enhance heralded single photon sources, demonstrating increased photon rates and potential for deterministic emission through multiple modes.

Contribution

The authors present a novel spectral multiplexing approach with feed-forward control to improve heralded single photon sources, validated through experimental demonstration.

Findings

3-mode SMUX increases heralded photon rate

Maintains high single-photon quality

Potential for deterministic single-photon sources

Abstract

We propose and experimentally demonstrate a novel approach to a heralded single photon source based on spectral multiplexing (SMUX) and feed-forward-based spectral manipulation of photons created by means of spontaneous parametric down-conversion in a periodically-poled LiNbO3 crystal. As a proof-of-principle, we show that our 3-mode SMUX increases the heralded single-photon rate compared to that of the individual modes without compromising the quality of the emitted single-photons. We project that by adding further modes, our approach can lead to a deterministic SPS.

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Photonic quantum information processing promises delivering optimal security for sensitive communication Gisin2007 , solving certain computational problems much faster than classical computers Shor97 ; Monroe2010 , and estimating physical parameters with significantly improved resolution Giovannetti06 . Many of these applications rely on sources of deterministic (on-demand) and near-perfect single photons Knill01 .

The most common realization of a single-photon source (SPS) is based on the generation of correlated pairs of photons (usually coined idler and signal photons) followed by the detection of one member of the pair (henceforth assumed to be the idler), which heralds the presence of the other. In this scheme a crucial step is the pair generation process, which is achieved through spontaneous parametric down-conversion (SPDC) or spontaneous four-wave-mixing (SFWM) in a nonlinear optical medium. The experimental simplicity and versatility of such heralded sources have earned them a role in numerous quantum information applications Pan2012 . In terms of quality, these sources can produce highly indistinguishable photons, but their main limitation lies in the spontaneous nature of the pair generation. This means that single pairs of photons are generated only with a certain probability $p_{n=1}<1$ and, moreover, there is a chance of generating multiple photon pairs, which results in multi-photon emission from the SPS (for $p_{n=1}\ll 1$ the probability for generating multiple pairs is $p_{n\geq 2}\approx p_{n=1}^{2}$) and hence non-pure single photon states. If the collection and detection efficiency of the idler photon is very large, one can use photon-number resolving detectors Lita08 ; Allman15 to increase the purity, but the maximum single-photon emission probability will still be limited to $p_{n=1}=0.25$ (assuming a thermal distribution)Christ12 .

As an alternative to heralded SPSs (HSPSs) based on photon pairs, efforts have been directed to implement SPSs using single emitters. Such conceptually simple sources are capable of deterministically emitting a single photon within the emitter’s excitation lifetime, and have been demonstrated in different physical systems such as diamond colour centers Babinec10 , single molecules Lounis00 and quantum dots Michler00 ; Shields07 . This approach has led to nearly deterministic generation of light with nearly perfect single-photon character. The main challenge with single-emitter sources has been the lack of indistinguishability between photons emitted by different sources, or even by the same source at different times. Though progress in fabrication methods and active-control of the properties of emitters may solve this problem, it currently poses a significant drawback for practical applications.

A promising avenue to overcome the limitations of both of the above sources is based on revisiting the heralded SPSs. The scheme proposed in Migdall02 ; Shapiro07 realizes in principle a deterministic SPS by actively multiplexing many non-deterministic heralded photon sources that emit photons in different modes. In this scheme, the detection of an idler photon in any mode out of the chosen set heralds the presence of a signal photon in a corresponding optical mode. Then, using a feed-forward signal from the idler photon detector, the signal mode is mapped onto a (predetermined) single mode. In doing so, the probability to generate a single pair in at least one of the modes increases linearly with the number of modes $m$ (for small $p_{n=1}$), and the photon emission probability after feed-forward-based mode mapping of the signal mode can hence be made to approach unity. At the same time, the probability for multi-pair emissions into the feed-forward-mapped output mode also increases proportionally to $m$, which means that the ratio $p_{n=1}/p_{n=2}$ after mode-mapping, and hence the single-photon purity, remains constant. Though, initially, this seems unsatisfactory, it turns out to be far superior to the scaling for an individual heralded SPS. In this case, if the same increase of $p_{n=1}$ by a factor of $k=m$ were to be achieved by increasing the SPDC pump-power, it would result in an increase of $p_{n\geq 2}$ by a factor of $k^{2}$. Hence the ratio of $p_{n=1}/p_{n=2}\propto 1/k$. In other words the non-multiplexed HSPS would produced more and more multi-photons as the emission rate is increased.

Multiplexed heralded sources have thus far been realized using spatial Ma2011 ; Collins2013 , temporal Xiong2016 and spatio-temporal modes Mendoza2016 . In some cases, they have shown to outperform non multiplexed sources in terms of throughput and quality. However, scaling up the number of modes in the employed degrees of freedom requires more resources, and generally impacts the overall performance. In the case of spatial multiplexing, each additional mode requires an independent source and an added switching connection that induces some amount of extra loss Bonneau15 . Temporal multiplexing does not necessarily consume more physical resources but does inevitably encroach on the repetition rate of the source and hence limits the single photon throughput.

Here, we propose and demonstrate an SPS based on a novel spectral multiplexing (SMUX) scheme in which the source requirements and the system loss are independent of the number of modes being multiplexed. The scheme is based on defining spectral modes within the broadband spectrum of an SPDC pair source and applying a feed-forward frequency-shift operation on the heralded photon. We experimentally show that the single-photon character is preserved by measuring a heralded auto-correlation function $g^{(2)}_{\mathrm{H},0}\ll 1$ for the heralded photons with and without multiplexing and feed-forward control. Moreover, directly comparing the multiplexed and non-multiplexed output we deduce that, as expected, the heralded single-photon emissions increase linearly with the number of modes. This allows compensating for the additional loss caused by the non-ideal elements used for the feed-forward operation for as few as three modes.

The experimental implementation of our scheme is shown in Fig. 1a. A pulsed laser creating 18 ps-long pulses centred at 523.5 nm wavelength pumps a 2 cm long periodically-poled lithium-niobate (PPLN) crystal to produce 350 GHz wide, frequency non-degenerate photon pairs composed of signal photon centred at 795 nm and the idler at 1532 nm. The spectral distribution of the pairs is conveniently illustrated by their joint spectral amplitude (JSA) shown in Fig. 1b. The JSA represents the probability amplitude to detect a pair of photons with photons of frequencies $\nu_{s}$ and $\nu_{i}$. Since each pair of photons has to satisfy energy conservation, the JSA is confined to the diagonal band (green region), whose cross-sectional width is given by the spectral width of the pump laser $\delta\nu_{p}$. (The phase-matching condition is assumed to be less restrictive than the energy conservation, and is not considered in this pictorial representation. This assumption reflects standard experimental conditions.)

The signal photons are transmitted to an optical delay line while the idler photons are sent to a pair of diffraction gratings (DGs) that map photons with spectra centred at $\nu_{i0}=195.612~{}\mathrm{THz}$ (1532.59 nm), $\nu_{i+}=\nu_{i0}+19~{}\mathrm{GHz}$ (1532.44 nm) and $\nu_{i-}=\nu_{i0}-22~{}\mathrm{GHz}$ (1532.76 nm), each featuring a spectral width of 12 GHz (see Supplemental Material), onto distinct spatial modes. In the JSA of Fig. 1b these idler spectral modes are highlighted as orange (middle), yellow (upper) and red (lower) horizontal bands. The idler photons in each mode are detected using WSi superconducting nanowire single-photon detectors (SNSPDs). When an idler photon is detected in mode $\nu_{i+}$, $\nu_{i0}$ or $\nu_{i-}$, it heralds the presence of a signal photon with a central frequency of $\nu_{s0}=377.059~{}\mathrm{THz}$ (795.08 nm), $\nu_{s+}=\nu_{s0}-19~{}\mathrm{GHz}$ (795.12 nm) or $\nu_{s-}=\nu_{s0}+22~{}\mathrm{GHz}$ (795.03 nm), respectively.

The heralding signals from the output of the SNSPDs are processed by a logic circuit that triggers the creation of a feed-forward signal in the form of a 700 ps long pulse with suitable, linearly changing voltage. This ramp signal is applied to the electrical input of a lithium-niobate (LiNbO3) phase-modulator (LN-PM) that is optically connected to the output of the 512 ns-duration delay line for the signal-photon. As we will demonstrate below, the LN-PM actively shifts the spectrum of the heralded signal photon to a spectral band — shown as vertical blue area in the JSA — determined by the transmission of a Fabry Perot cavity (FP) with bandwidth $\delta\nu_{s}=6$ GHz. Note that the frequency shifts are applied to the entire spectrum of the signal photons, as indicated by the dotted arrows in Fig. 1b. As a consequence, signal photons not corresponding to the heralded spectral band are shifted out of the cavity resonance and thus rejected. Finally, the signal photons are detected by a silicon avalanche photodiode (APD) and a logic circuit records coincidences with the heralding signals from all SNSPDs.

To show how the frequency-shifting is realized by means of the LN-PM, we consider a pulse of light described in the slowly-varying-envelope approximation by $\mathcal{E}_{\mathrm{in}}(x,t)=|\mathcal{E}(x,t)|\exp\left(i2\pi\nu_{s0}t-ikx\right)$. The electro-optic effect in the LN-PM translates a linearly changing voltage signal $V(t)=\mathcal{A}t$ — applied during the passage of the optical pulse — into a linear phase-ramp $\varphi(t)=\pi V(t)/V_{\pi}=\mathcal{A}t\pi/V_{\pi}$ of the optical pulse. Here, $\mathcal{A}$ is the slope of the voltage-signal and $V_{\pi}$ is the $\pi$-voltage of the LN-PM i.e. the voltage required to obtain a $\pi$ phase-shift. Hence, upon exiting the LN-PM, the pulse is described by $\mathcal{E}_{\mathrm{out}}(x,t)=|\mathcal{E}(x,t)|\exp\left(i[2\pi\nu_{s0}t+\varphi(t)]-ikx\right)=|\mathcal{E}(x,t)|\exp\left(i2\pi[\nu_{s0}+\mathcal{A}/(2V_{\pi})]t-ikx\right)$Brecht15 . This highlights that the output pulse is frequency shifted by $\Delta\nu=\mathcal{A}/(2V_{\pi})$ — which is the ratio of the voltage slope to the $2\pi$-voltage — while the temporal shape is unchanged. In our experiment a heralding signal generated by the detection of an idler photon with frequency $\nu_{i+}$, $\nu_{i-}$, or $\nu_{i0}$ triggers the application of either a negative ramp with $\mathcal{A}_{-}\approx-70$ V/ns, a postive ramp with $\mathcal{A}_{+}\approx 53$ V/ns, or no ramp ($\mathcal{A}_{0}=0$ V/ns), respectively (see Supplemental Material for details).

We first characterize the spectral difference between signal photons depending on which idler photon serves as a herald. Towards this end we measure the heralded single photon rates for each idler frequency mode ($\nu_{i+}$, $\nu_{i0}$ and $\nu_{i-}$) while tuning the resonance frequency of the FP cavity that acts on the signal photon. To centre the spectral transmission of the FP at $\nu_{s0}$, we maximize the coincidence counts when heralding exclusively with idler photons detected in mode $\nu_{i0}$. Note that although the LN-PM is part of the measurement set-up, it is not active. The results are shown in Fig. 2a. For all modes, we measure bandwidths of the heralded photon spectra of around $37$ GHz, which matches the convolution of the pump laser bandwidth $\delta\nu_{p}=24$ GHz with the filter bandwidths $\delta\nu_{s}=6$ GHz and $\delta\nu_{i}=12$ GHz. Furthermore, we find that the maximum coincidence rate for each pair of photons is at the relative frequency differences of $\Delta\nu_{-}=-19$ GHz and $\Delta\nu_{+}=+22$ GHz.

Next, we assess the performance of the linear ramp frequency shifting (LRFS) by again recording the heralded-photon spectra for each idler frequency-mode, but now with the corresponding feed-forward signal applied to the LN-PM. For these measurements, the resonance frequency of the FP cavity remains fixed at $\nu_{s0}$. As expected, the spectra shown in Fig. 2b now completely overlap. Moreover, by comparing with the results in Fig. 2a, we observe that the detection rates with and without LRFS are essentially equal. This shows that our setup is capable of applying the on-demand frequency shift at the single-photon level with nearly $100\%$ efficiency.

Now we activate the full setup and evaluate the performance of our frequency multiplexed heralded source in view of the requirements of an ideal single photon source. First, we measure the heralded single photon (HSP) rate as a function of the pump power with and without multiplexing. The red, yellow and orange circles in Fig. 3 show the rates of signal photons (after spectral filtering by the FP cavity) heralded by the individual frequency modes at $\nu_{i+}$, $\nu_{i0}$ and $\nu_{i-}$. The multiplexed HSP rate for the SMUX source (green circles) is about 2.7 times larger than that of the average of the individual sources, i.e. a significant improvement over that for the individual modes. Yet, to make a fair comparison of the effect of the SMUX, we compare the multiplexed HSP rate to the rate when heralding with only the $\nu_{i0}$ mode and with the LN-PM removed, which, for the specific modulator used in our experiment, adds 5 dB loss and is not needed in the case of using only a standard HSPS without multiplexing. However, we keep the spectral filtering elements on both idler and signal fields as these are necessary to achieve pure states Mosley08 . The rate obtained with the thus modified source (purple circles) is similar to the SMUX rate, which can be explained by the increased transmission due to having removed the LN-PM compensating for the lack of multiplexing. Hence, in our experiment, which employs 3 modes, spectral multiplexing does not yet create an advantage in view of the HSP rate and in creating a deterministic source. However, we emphasize that increasing the number of modes would neither increase the system loss nor require additional elements in the signal mode. Hence, we can assume that the HSP rate will continue to increase as more spectral modes are added, rapidly surpassing that of the non-multiplexed source.

Next, we verify that multiplexing maintains the single photon character of our light source. One of the most common methods for this is to determine its purity Somaschi16 , which, for a heralded source, is generally quantified in terms of its heralded auto-correlation function. To measure this, we direct the heralded signal photons after the FP cavity through a 50/50 beam splitter (BS), and record the individual counts $C_{\mathrm{H}}^{\mathrm{(i)}}(i\in\{a,b\})$, in as well as coincidences $C_{\mathrm{H}}^{\mathrm{(ab)}}$ between, the detectors placed at its two outputs (see Fig. 1a). Together with the total number of heralding signals, $H$, we define the heralded auto-correlation as $g^{(2)}_{H}(0)=C_{\mathrm{H}}^{\mathrm{(ab)}}H/(C_{\mathrm{H}}^{\mathrm{(a)}}C_{\mathrm{H}}^{\mathrm{(b)}})$ (see Supplemental Material). Since detection in both detectors can only occur if two or more photons are present we expect $C_{\mathrm{H}}^{\mathrm{(ab)}}/H\approx p_{n=2}^{\mathrm{H}}/2$, where $p_{n=2}^{\mathrm{H}}$ is the heralded two-photon probability. On the other hand, the individual detector counts are dominated by single photon events such that e.g $C_{\mathrm{H}}^{\mathrm{(a)}}/H\approx p_{n=1}^{\mathrm{H}}/2$ (assuming $p_{n=1}\gg p_{n=2}$). Clearly these considerations hold for the multiplexed and non-multiplexed sources alike (see Supplemental Material). Hence, the autocorrelation function is approximated by $g^{(2)}_{H}(0)\approx 2p_{n=2}^{\mathrm{H}}/(p_{n=1}^{\mathrm{H}})^{2}$, which provides a direct relation to the photon-statistics. In particular, $g^{(2)}_{H}(0)\sim 0$ would indicate $p_{n=2}^{\mathrm{H}}\sim 0$ and $p_{n=1}^{\mathrm{H}}>0$.

Operating the source at maximum power with the LN-PM (and hence the multiplexing) removed we obtain a value of $g^{(2)}_{H}(0)=0.05\pm 0.01$. With the SMUX activated and using the same pump power, we measure $g^{(2)}_{H}(0)=0.06\pm 0.01$, which equals the value without multiplexing within experimental uncertainty. If the LN-PM loss were lower we would thus achieve a larger HSP rate without increasing the $g^{(2)}_{H}(0)$ (see Supplemental Material). Both experimentally measured $g^{(2)}_{H}(0)\ll 1$, which indicates a highly pure single photon character of the output mode. Hence, we establish that the SMUX maintains the single photon nature of the source. This conclusion is supported by additional measurements of the so-called coincidence to accidental ratio (CAR) measured with and without the SMUX Zhou17 .

We did not directly measure the degree of distinguishability of single photons from our source, but based on the experimental setup, we conjecture they must be nearly indistinguishable. The reason for this is that the photons emitted by our SMUX source pass through a FP-cavity and are coupled into single mode fibre, which removes any distinguishability in the frequency and spatial degrees of freedom. Finally, the fact that all signal photons travel along the same path, from the point where they are created to the point where they are detected, means that their arrival times and polarization are the same (irrespective of which spectral mode they belong to), provided that chromatic dispersion and polarization mode dispersion can be ignored.

In conclusion, we have generalized the idea of multiplexing supplemented by feed-forward control with the goal of creating a high-quality heralded SPS from the spatial and temporal degrees-of-freedom to the spectral domain. We have demonstrated that multiplexing of three spectral modes leads to the expected increase in the heralded single photon rate while keeping the purity – the noise caused by multiple-pair emission – constant. To achieve these results we have implemented on-demand frequency shifting of single photons over approximately $\pm 20$ GHz with nearly 100$\%$ efficiency by driving a commercially available LiNbO3 phase modulator with a linear voltage ramp. Unlike in the case of temporal or spatial multiplexing Bonneau15 , the required resources – with the exception of the single-photon detectors – do not depend on the number of modes, thus making this approach appealing for further development towards an ideal SPS.

There is a number of avenues to increase the heralded single-photon rate of our source, the simplest being to employ a phase-modulator with reduced loss. For instance, the use of a readily available modulator with 1.5 dB loss – 3.5 dB less than in our current modulator — would result in an improvement of the rate by about a factor of 2.5. At a more fundamental level, more spectral modes must be multiplexed. This can be achieved both by increasing the spectral mode density (e.g. by taking advantage of commercially available ultra dense wavelength division multiplexers with 6 GHz bandwidth), as well as by improving the total bandwidth – the latter necessitating improvement of the maximum frequency shifs using higher-bandwidth electronics Wright16 ; Li16 ; Fan16 ; Albrecht14 . Assuming 6 GHz wide spectral channels and an 8-fold increase of the shifting range to 350 GHz (the spectral width of the photons created by our source), it seems feasible to reach 60 spectral channels. This will suffice for creating at least one heralding event per pump pulse and for making the source as close to deterministic as loss in the signal path allows Christ12 . And as the SPDC spectrum is determined by the length of the non-linear crystal and phase-matching conditions, even larger bandwidths and thus more modes may be feasible.

To finish this paper, let us note that spectral multiplexing has also been proposed to improve the rate of entanglement distribution in a quantum repeater architecture Sinclair14 . However, the proof-of-principle demonstration in Sinclair14 was still lacking the feed-forward control. Our current demonstration therefore also establishes an important element in this architecture.