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Abstract
We report the interferometric photodetection of a phase-diffusion quantum entropy source in a silicon photonics chip. The device uses efficient and robust single-laser accelerated phase diffusion methods, and implements the unbalanced Mach-Zehnder interferometer with optimized splitting ratio and photodetection, in a 0.5 mm×1 mm footprint. We demonstrate Gbps raw entropy-generation rates in a technology compatible with conventional CMOS fabrication techniques.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
On-demand generation of random numbers (RNs) is a key ingredient for fields as diverse as Monte Carlo simulations [1, 2], online gambling applications [3], decision making algorithms, cybersecurity [4, 5], and even tests of fundamental physics [6–9]. Although pseudo-RNs can be easily generated using computational algorithms, true RNs can only be created using physical processes [5, 10]. Quantum entropy sources (QESs) make use of the intrinsic randomness of quantum mechanics to create strings of random bits. Several implementations of QESs have been demonstrated, including splitting of single photons [11, 12], photon arrival time [13], vacuum fluctuations [14], laser chaos [15, 16], and phase diffusion (PD) in laser diodes [17–22]. In particular, PD-QESs have been shown to achieve high bit rates and offer strong randomness guarantees [6].
For future devices it is desirable to scale down these bulky technologies into integrated devices. Recently, an integrated QES using a light emitting diode (LED) and a single-photon avalanche photodetector (SPAD) achieved 1 Mbps bit rates [23]. PD-QESs have the potential to achieve several orders of magnitude higher rates as they use conventional photodetectors instead of SPAD. A PD-QES in an indium phosphide (InP) integrated circuit was demonstrated with Gbps rates [24]. Implementation in silicon photonics, which we show here, allows direct integration with conventional complementary metal oxide semiconductor (CMOS) electronics, enabling QES deployment in the most advanced semiconductor industry.
In this work we demonstrate a PD-QES on a Si chip using an integrated unbalanced Mach-Zehnder interferometer (uMZI) scheme. The laser component cannot be directly implemented in Si photonics, although hybrid technologies [24] have shown the potential for full PD-QES integration onto a single chip. Here the device is driven by an external DFB laser operated in gain-switching (GS) mode to generate pulses with equal amplitudes and random initial phases. These pulses are then interfered in the uMZI, thus creating a train of pulses with random amplitudes that are measured in a high bandwidth integrated photodetector (PD). The scheme shows high stability over time and can potentially deliver Gbps bit rates with appropiate digitization components.
2. Experiment
Figure 1 shows a schematic of the experimental set-up and an image of the Si chip containing QES devices. The Si chip implements both the interferometry and photodetection elements of the PD-QES strategy. The laser component is interfaced to the chip by a grating coupler (GC). A single-frequency (λ = 1550 nm) DFB laser is operated in GS mode, with a mean drive current of 14 mA and a sinusoidal modulation at 1 GHz, applied via a bias-tee. As the laser threshold is 10 mA, this takes the laser above and far below threshold on each cycle, producing a train of linearly-polarized optical pulses of duration ∼300 ps.
Fig. 1 Upper: Microscope image of the 4 mm × 7.5 mm Si photonic chip with a one euro cent coin for reference. One such chip can contain up to 20 PD-QESs. A single PD-QES block is less than 0.5 mm × 1 mm. Lower: Schematic of the experimental set-up. The DFB laser is biased using a current source and modulated with a 1 GHz sinusoidal wave. The output pulses with random phases are amplified using an erbium-doped fiber amplifier (EDFA) and coupled into a single mode Si waveguide using an SMF and a GC. The pulses are split in an MMI, with a power splitting ratio of 0.02:0.98, to the two arms of an uMZI, in order to compensate the losses introduced by the longer path. The interference signal is detected using an integrated photodetector and analyzed with a real-time oscilloscope.
Due to phase diffusion, subsequent pulses have random relative phases, while also having the same waveforms. In order to couple the pulses into the Si chip, the laser output is directed via an SMF towards a GC at 10° incidence using a 6-axis micropositioner. A polarization controller (PC) is used to adjust the input polarization to minimize the coupling losses due to the GC. Coupling losses were estimated to be ∼7 dB per grating by measuring the transmission losses through a straight waveguide. Total losses of 15 dB were measured in a 0.34 cm long waveguide, from which we can obtain the coupling losses using the relation L = 2Lgc + lwgt, where L are the total losses, Lgc are the coupling losses per grating, lwg are the waveguide losses (∼3 dB/cm in our case) and t = 0.34 cm is the waveguide length. Improved designs of the grating coupler, for instance using a tapered waveguide could further reduce these coupling losses to values around 3 dB.
In the chip, light travels through a single mode Si waveguide with a cross-section of 450×220 nm2, containing only the fundamental TE mode. A first multimode interference coupler (MMI) splits the input light, with a power-splitting ratio of 2:98, to the two arms of an uMZI, in order to compensate the losses introduced by the longer path. The stronger output experiences the longer path, and the two arms are re-combined at a second MMI, with splitting ratio 50:50. The splitting ratio (tl /ts) of the first MMI, where tl (ts) is the transmission to the long (short) arm of the uMZI, is given by
here κ ∼ 0.56 cm−1 is the attenuation coefficient of the Si waveguide and Δl = 6.9 cm is the relative path difference of the uMZI, given by Δl = τc/ng, where τ = 1 ns is the pulse repetition rate and ng ∼ 4.3 is the effective group refractive index of the Si waveguide. This implies a relative attenuation by a factor of ∼50, compensated by the first MMI and thus equalizing the field strength reaching the detector, while the path length difference introduces a delay of 1 ns that creates the conditions for temporal overlap of subsequent pulses. The strong attenuation is due to the length of the long arm and the waveguide losses (∼3 dB/cm). Thus a decrease in the waveguide losses and/or the path length difference would allow to obtain higher visibility and SNR, while at the same time this would require faster modulation, leading to an increase in the raw entropy generation rate. This is an important point to take into account in the design of future devices.Careful control of the MMI splitting ratios is crucial in order to obtain high interference visibility. This ratio, which is given by the length of the MMI, was experimentally measured in 50:50 and 2:98 test MMIs placed inside the chip. No difference between outputs was found in the 50:50 MMI, while a difference of 17 dB was measured in the 2:98 MMI, close to the design value (16.9 dB). Finally, the interfered pulses are detected by a fast (10 GHz) on-chip Ge photodiode (responsivity ~ 0.7 A/W) and sent to a 4 GHz real-time oscilloscope via a bias-tee.
3. Discussion
3.1. Phase-diffusion measurements
As described in [6], the power detected by the integrated photodiode is given by
where Pl(t) is the instantaneous laser power at time t, τ = 1 ns is the pulse repetition period, is the interference visibility, Δϕ is the relative phase between subsequent pulses, and Δθ is the optical phase acquired in the uMZI. Due to strong phase diffusion in the time below threshold, the statistical description of Δϕ is, to a very good approximation, random, i.e., uniformly distributed on [0, 2π). As a result, the cosine of Δϕ follows a bimodal distribution [6], irrespective of Δθ.By measuring the statistics of the electrical signal when the laser is below threshold (green curve in Fig. 2(a)) one can also obtain information about the overall noise of the system, which ultimately determines the quality of the device. Figure 2(a) shows the observed distribution of output powers. The optical and electronic noises produce a monomodal distribution for the equivalent input power, whereas the interference process produces a strongly bimodal distribution, reflecting the arcsine distribution expected from the phase diffusion process smoothed by convolution with the electronic noise distribution.
Fig. 2 a) Histogram of the electrically measured interference signal (blue) and noise (green), and simulation of the expected interference signal (black line) and noise (green line). Due to the randomness of the initial phase the accumulated signal follows a bimodal probability distribution. Histograms contain approximately 7 × 105 samples. As confirmed by simulation (black curve), the bimodal distribution corresponds to Eq. 2 with and random Δϕ. b) Accumulateed interference signal and histogram obtained with the real-time oscilloscope by continously measuring the interfered pulses at a fixed sampling point.
We run a Monte Carlo calculation to find the parameters of Eq. (1) that best fit with the observed distribution. The waveguide losses were set to αwg~ 3 dB/cm, as estimated experimentally. The electronic noise is described by a gaussian noise with 3.9 mV mean and rms width σn = 2.0 mV, extracted after fitting a gaussian distribution to the observed noise in Fig. 2(a). By leaving the loss imbalance between the two arms of the interferometer fixed, i.e., including (i) the characterised unbalance in the pre-compensated MMIs and (ii) the propagation loss acquired in the long path with respect to the short path, we sweep the interference visibility and fine tune the overall signal conversion strength, which accounts for uncertainties in the overall signal chain, such as losses acquired by the two paths or the responsivity of the detector. We find that the simulated probability distribution function is consistent with Eq. (1), with and random Δθ. The value of the visibility is due to noise originating from photodetection and the non-perfect compensation between the two arms of the uMZI. The presence of noise smoothens the distribution function, thus reducing the observed interference, and so does the unbalance between the two arms.
The simulated distribution is shown in the black curve of Fig. 2(a). The mean square error between the observed and simulated curves is ~ 4.5 × 10−3, leading to an observed quantum signal to classical noise ratio (QSNR) of ~ 4.
3.2. XY plot
The autocorrelation function is a very useful metric to assess the functioning of a physical quantum entropy source. Typically, computing the autocorrelation Γd = 〈xi xi+d〉−(x)2 requires the storage of a continuous stream of input bits and then a significant post-processing step for long datasets. For situations in which high speed digitization is not available, we introduce here a new strategy that allows to qualitatively verify the correlation using only a real time oscilloscope, without the need of any post-processing. Letting Vn and Vn−d denote the voltage amplitudes of the n-th and (n − d)-th pulses, we record in the oscilloscope the number of times an event (Vn, Vn−d) occurs for different shifts (d) in an XY plot. These results are then compared with a simulation of the expected distribution for different values of d, as shown in Fig. 3.
Fig. 3 Upper: XY plot recorded in the real-time oscilloscope, showing the accumulated occurences of an event (Vn, Vn−d) for four different shifts (d = 0, 1, 4, 5). Lower: Simulation of the expected occurences (normalized) for the same values of d. The distribution is expected to be a straight line for d = 0 and a 2D bimodal distribution for d ≠ 0.
The distribution is expected to be a straight line for d = 0 and a 2D bimodal distribution for d ≠ 0, showing good agreement with the experimental results. The spread in the experimental distributions is due to sampling of the real-time oscilloscope. Two time windows are defined in which the values of Vn and Vn−d are sampled. Due to the non-zero width of these windows the values can take any of the values within them.
3.3. Min-entropy estimation
A quantum entropy source is typically expected to provide a sequence of uncorrelated and uniformly distributed random bits. However, random physical processes tend to describe non-uniform distributions. Additionally, corruption due to the use of real components, such as digitisers or single photon detectors, as well as bandwidth limited components, introduce correlations and predictabilities into the signal [25]. To eliminate these undesired effects and to obtain a uniformly distributed output too, a randomness extractor is applied to the raw data. In order to apply a randomness extractor, the min-entropy of the raw data must be properly bounded.
The min-entropy estimation in the phase diffusion quantum entropy source scheme, and in particular, using an unbalanced Mach Zehnder interferometer, has been largely studied recently [6, 26]. In this case, the digitized voltage at the output of the photodetector V is described by a smoothed arcsine distribution. In general, the amount of extractable randomness from this signal, a.k.a conditional min-entropy is given by
where N represents the combined effect of all the untrusted noises.The detected signal described in Eq. (2) contains a large random component due to phase-diffusion Δθ. However, other sources, including photodetection noise, laser current fluctutations and digitization errors also contribute to the digitized signal, corrupting the unpredictability of the trusted process. These sources are not guaranteed to be random and should be eliminated. This is achieved in the randomness extraction step provided that the applied compression factor fulfills RF > n/H∞ [19], where n is the number of bits used in the digitization process. Assuming 1-bit digitizer, the maximum probability for this signal can be bounded by (see [6])
where is the voltage variation due to phase diffusion and Vc is the noise contribution from untrusted sources. Assuming a digitisation noise of VADC = 5 mV [6] and a contribution of untrusted noises of Ve = 2.0 mV, extracted from Fig. (2), we find a 1σ combined effect from the untrusted noises of . By taking a similar approach, we take a 1σ conservative bound on ΔVϕ by assuming Pl(t) and Pl (t + τ) have a 0.1 rms fluctuation with respect to their mean intensities, finding 2ΔVϕ = 20.9 mV and Pr[V |n] = 0.57, or in terms of the min-entropy H∞ = 0.82 bits/pulse.The only assumption for this calculation is the digitization noise, for which we have taken a value of VADC = 5 mV, as measured in some of our previous works [6]. All other sources of noise have been measured experimentally. As the digitization scheme is the main ‘extra’ source of noise between a PD-QES and a full PD-QRNG (postrprocessing, i.e., the randomness extraction algorithm, does not add noise to the random bits), we believe that this value of H∞ is pretty realistic and thus the full entropy capacity of our system would be around 820 Mbps.
4. Conclusion
In conclusion, we have demonstrated a PD-QES with Gbps bit rates and compatible with current CMOS technologies. The device is based on an external DFB laser operated in GS mode at 1 GHz coupled to a Si photonic chip that integrates the critical intererometry and detection components. Up to 20 PD-QESs can be integrated on a single chip with a footprint of only 4×7.5 mm2, with a single PD-QES footprint of 0.5×1 mm2. The amplitude of the interfered pulses follows a smoothed arcsine distribution, as expected for PD-QESs, with a visibility of , random Δθ and a min-entropy of H∞ = 0.82 bits/pulse. Also, we have introduced a method to qualitatively verify the correlation in the real-time oscilloscope without any need of offline post-processing.
The scheme could easily achieve tens of Gbps bit rates by using shorter uMZIs and thus faster modulation frequencies. This would in turn increase the SNR due to the decreased attenuation of the pulses inside the uMZI and allow PD-QES integration in chips with a smaller footprint. Athough Si lasers are not available, the use of hybrid technologies, for instance InP on Si, would allow full integration of the PD-QES on a single chip. If hybrid integration is possible, parallelization of the devices would be straightforward, one would just need to place the lasers instead of the grating couplers and add an additional coplanar waveguide to modulate the laser, similar to the one used to measure the PD response. Thus only a small amount of additional space would be required and we believe this would allow to integrate around 10 devices in a chip with the same size (7 mm × 4.5 mm).
Funding
Fundació Privada Cellex; Generalitat de Catalunya (CERCA Programme); Ministerio de Economía, Industria y Competitividad, Gobierno de España (SEV-2015-0522, TEC2016-75080-R, FIS2015-68039-P, FIS2014-62181-EXP); H2020 European Research Council (ERC) (713682, 641122, 820405); Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) (2017-SGR-1634).
Acknowledgements
This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 820405. MIQUEL RUDÉ, CARLOS ABELLÁN, ALBERT CAPDEVILA, MORGAN W. MITCHELL, WALDIMAR AMAYA, AND VALERIO PRUNERI acknowledge financial support from the Spanish Ministry of Economy and Competitiveness through the "Severo Ochoa" program for Centres of Excellence in R&D (SEV-2015-0522), from Fundació Privada Cellex, and from Generalitat de Catalunya through the CERCA program.
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