Abstract

Random number generators are essential to ensuring performance in information technologies, including cryptography, stochastic simulations, and massive data processing. The quality of random numbers ultimately determines the security and privacy that can be achieved, while the speed at which they can be generated poses limits to the utilization of the available resources. In this work we propose and demonstrate a quantum entropy source for random number generation on an indium phosphide photonic integrated circuit made possible by a new design using two-laser interference and heterodyne detection. The resulting device offers high-speed operation with unprecedented security guarantees and reduced form factor. It is also compatible with complementary metal-oxide semiconductor technology, opening the path to its integration in computation and communication electronic cards, which is particularly relevant for the intensive migration of information processing and storage tasks from local premises to cloud data centers.

© 2016 Optical Society of America

1. INTRODUCTION

Random numbers (RNs) are essential to a wide range of applications, including secure communications to protect the transmission and storage of confidential data Shannon [1], massive data processing [2], and stochastic simulations [3,4] for stock market predictions, decision making in engineering processes, and Monte Carlo calculations of physical, chemical, nuclear, and biological events. Pseudo-RNs can be generated through computational algorithms, while true RNs can be generated only through physical processes [5]. Quantum mechanical processes are the best guarantee for offering high performance without compromising security and privacy. So far, several quantum entropy sources (QESs) have been proposed for quantum random number generation (QRNG), including single photon splitting [6], homodyne detection of the vacuum field [7], and phase diffusion (PD) in semiconductor lasers [8,9]. To date, PD-QRNGs have achieved the highest bit rates [1012], up to 68 Gb/s [12], and passed severe random tests [13]. However, so far, PD-QRNGs have been realized with discrete optical components, often leading to devices of large size.

Photonic integrated circuit (PIC) technology [14,15] is a key ingredient for building scalable optical devices [16]. The telecommunication industry is a clear example, and it already accounts for commercial products such as semiconductor lasers, 100 GHz photodetectors, and high-bandwidth optical interconnects and transceivers [17]. Recently, the quantum optics community has been making rapid progress in leveraging PIC technology, offering the possibility to build scalable quantum optics experiments. In the field of quantum computation, PIC technology in combination with additional bulk elements, such as lasers, is allowing for the development of novel experiments otherwise impossible using tabletop components. Some examples include quantum simulation [18] and quantum-enhanced sensing [19]. Quantum key distribution (QKD) functionalities have been also integrated using indium phosphide technology [20] and a monolithically integrated QRNG, composed of a light-emitting diode (LED) and a single-photon avalanche photodetector (SPAD), has been recently demonstrated at 1 Mb/s using silicon (Si) photonics technology [21]. Si photonics is a promising candidate for building scalable optical applications due to its compatibility with the microelectronics industry. However, the impossibility of creating a Si laser source poses serious limitations to the level of integration and performance of the PIC.

In this work, we show a fully integrated QES for random number generation on an InP platform (QES-PIC) using standard fabrication techniques only. The device is made possible by a new design using two-laser interference and heterodyne detection, allowing QRNG rates in the Gb/s regime. We observe high interference visibility during long execution runs, as well as superior temperature stability when compared to the bulk implementation of the same scheme. Also, using the Lang–Kobayashi rate equations model, we study in detail the dynamics of the two integrated lasers. We find conditions for operating the two lasers with a negligible coupling effect, and provide an accurate description and modeling of the strong thermal chirp observed in the InP distributed feedback (DFB) lasers.

2. EXPERIMENT

As illustrated in Fig. 1(a), we introduce a QES scheme that combines two DFB lasers on the same chip. The first laser is operated in gain switching (GS) mode, while the second one is in continuous wave (CW) mode. By modulating continuously the GS laser from below to above threshold, optical pulses with nearly identical waveforms and completely randomized phases are generated. Then, by beating the GS and CW (the local oscillator) lasers through a multimode interference (MMI) coupler, an intensity oscillation forms with a beating frequency equal to the difference of the two lasers’ frequencies, which can be detected by a photodetector; see Fig. 1(b). If i(cw) and i(gs) are the intensities from the CW and the GS lasers, respectively, we can write the total intensity at the output of the MMI (see Supplement 1) as

iT(t)=iS(t)+2iP(t)cos(0tdξΩC(ξ)+Δφ),
where iS(t)i(cw)+i(gs) is the sum of the intensities from the two lasers, iP(t)(i(cw)i(gs))1/2 the geometric mean, Δφ=φ(cw)φ(gs) the phase difference between the two lasers fields, and ΩC(t)=Ωβ(t) the frequency detuning as a function of time. We introduced β(t)=β0t phenomenologically to account for frequency chirp arising from fast thermal effects in the directly modulated laser [22]. Here, Ω represents the initial frequency detuning between the two lasers. As illustrated in Fig. 1(b), the resulting signal corresponds to a train of pulses in which the amplitude of each pulse oscillates at 0tdξΩc(ξ) with a random phase Δφ (for simplicity, β0=0 in the illustration). Finally, after the MMI coupler, a photodetector converts the optical signal into the electrical domain, and random numbers are obtained by taking one sample per period.

 figure: Fig. 1.

Fig. 1. (a) Schematic of the QRNG-PIC based on two-laser interference. The two DFB lasers are biased with a current driver, one of them operating in CW, while the other one is periodically GS using an external RF generator. The temperature of the entire chip is controlled through a Peltier element, while that of the area including one of the lasers is locally changed by a stable current source. The outputs from the two lasers are combined and interfered in a 2×2 MMI coupler and two 40 GHz photodiodes are placed at the output of the coupler. The detected signal is sent to a fast oscilloscope. (b) Principle of operation: optical pulses from a GS laser interfere with a CW laser, generating an interference modulation whose frequency is equal to the difference of the two lasers’ frequencies. The random phase of the GS laser pulse produces a random phase of the interference oscillation that can be properly sampled into a random amplitude. In this way, after digitization, one can extract one sample per GS pulse. (c) Microscope image of the PIC on a 1 Euro cent background. Two QRNG-PICs are printed on each chip.

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A microscope image of the two-laser QES-PIC is shown in Fig. 1(c). The chip was placed on top of a Peltier controller and its temperature was maintained at 25° with variations below 0.1°. The first DFB laser, with a bias of 10 mA, was operated in GS mode by superimposing a 100 MHz modulation from an Anritsu MP1800A pulse generator through a bias T port. We chose this relatively low modulation frequency to capture properly the dynamics of the interference pattern within the GS pulse. However, modulation frequencies up to 2 GHz are within immediate reach, allowing for 10 s of Gb/s raw generation rates using current analog-to-digital conversion technologies, these being limited only by the stabilization time of the build-up dynamics of the laser intensity. The CW laser was operated by applying a constant 30 mA current. The beating signal was detected by an on-chip 40 GHz photodetector and digitized with a 20 GHz and 50 GSa/s real-time scope (Digital Phosphor Oscilloscope, Tektronix DPO72004C), providing a temporal resolution of 20 ps to analyze the beat note.

The central frequency of the two lasers could be independently tuned by injecting a constant current from a stable source (Keithley 2401) through a metallic contact on the grating structure. The heating effect changed the average refractive index of the grating and, consequently, the Bragg condition, thus the operating frequency (wavelength) of each laser. By tuning these currents, the detuning frequency between the two lasers could be reduced and brought within the detection bandwidth.

3. RESULTS

A. Modeling

When the frequencies of the two lasers are tuned close to each other, back reflections from the MMI coupler can give rise to phase- (frequency-) locking effects. This phenomenon can be explained on the basis of the general mechanism of Adler’s synchronization of two coupled nonlinear oscillators [2325] and modeled by the Lang–Kobayashi rate equation analysis for two mutually coupled semiconductor lasers [2628]. Indicating by κ the effective coupling rate between the two laser cavities and by Ω the frequency detuning between the two bare longitudinal modes of the uncoupled cavities, and neglecting delay effects, it is known in simple Adler’s theory of synchronization that frequency locking occurs for |Ω|<2κ [23]. A more accurate analysis can be gained from rate equation analysis. We model the coupled laser system with rate equations for the normalized complex slowly varying envelope of the optical fields E1,2 and the normalized inversions N1,2 using standard Lang–Kobayashi rate equations [2628], which in dimensionless form read [28]

E˙1=γ(1+iα)N1E1+κexp(iψ)E2(tτd)+R1ξ1(t),
E˙2=γ(1+iα)N2E2+κexp(iψ)E1(tτd)+i[Ω+β(t)]E2+R2ξ2(t),
τN˙1=P1N1(1+2N1)|E1|2,
τN˙2=P2N2(1+2N2)|E2|2.

In Eqs. (2)–(5), α is the linewidth enhancement factor, γ is the photon decay rate in the two laser cavities, τ is the carrier lifetime, κ is the coupling rate due to spurious optical feedback, τd and ψ are the time and phase delays of optical feedback, respectively, P1,2(t) are the normalized pump parameters of the two lasers, and Ω is the difference between the oscillating frequencies of the two uncoupled lasers. The normalized pump parameter P is given by P=GNN0(x1)/2, where GN is the differential gain, N0 is the current density at threshold, and x=J/Jth is the actual pumping current density normalized to its threshold value. Spontaneous emission is modeled by Langevin forces, describing a Gaussian white noise process ξ(t) with zero mean and correlations given by ξR(t)ξR(t)=ξl(t)ξl(t)=δ(tt) and ξR(t)ξI(t)=0 [29]. For the laser operated with a carrier density not too far from its threshold value, the noise term becomes additive with a variance R=Rsp, where Rsp is the spontaneous emission variance [29].

To simulate the experimental results, we assume that the first laser is pumped with a constant current (P1=P¯1), whereas the second laser is periodically GS from below to above threshold with a current pulse P2(t). In the simulations, the normalized pump parameter for the second laser in each modulation cycle is assumed to be of the form

P2(t)=P¯2{12+32exp[(t/Δτ)2M]},
where P¯2 is the peak of normalized pump current, Δτ the current pulse duration, and M the super-Gaussian parameter.

B. Numerical Simulation and Experimental Parameters

In the experiments, we used two PICs: a high waveguide propagation loss PIC that ensures absence of significant backreflection from the CW into the GS laser, and another one with a similar structure but lower loss. In the lower-loss PIC, backreflections at the MMI interface eventually induce phase-locking effects, preventing the generation of random oscillations. The coupling constant κ can be estimated from the locking region of the two lasers in the low-loss chip (see Fig. 2) and is given by κ=5ns1. With such a high coupling constant, the Langevin forces entering into Eqs. (2)–(5) are too weak, and therefore phase locking is observed in spite of spontaneous emission noise.

 figure: Fig. 2.

Fig. 2. Beat-note frequency at the output of the MMI measured by sweeping one of the integrated lasers, while keeping the other one constant in the low-loss QRNG-PIC. The beat-note frequency can be continuously tuned by current control for large detuning frequencies, whereas for small detuning frequencies phase- (frequency-) locking may occur (gray square), leading eventually to disappearance of the oscillation.

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To obtain the QRNG functionality, it is therefore mandatory to reduce the feedback arising from spurious reflections at the MMI coupler. This goal was simply achieved by increasing the optical losses of the bus waveguides (15dB/mm). As a result, the coupling rate κ between the two laser cavities is reduced by 30dB from the previously discussed low-loss PIC. For this high-loss PIC, the coupling, if any, is very weak, and thus phase randomization due to quantum noise prevails over phase locking, as shown in Fig. 3. Parameter values used in the simulation are shown in Table 1. We use the instantaneous coupling approximation [27], i.e., the delay τd can be neglected when solving Eqs. (2) and (3), since the time delay and coupling rate of feedback satisfy the constraint τdκ<1/1+α2. On the other hand, the phase delay ψ is difficult to estimate owing to its strong sensitivity to length changes over one wavelength; as a matter of fact, it can be considered an independent variable [24,25]. Nevertheless, for parameter values that apply to our experimental conditions, a change of the phase delay, e.g., from ψ=0 to ψ=π/2, does not introduce substantial changes in the beating dynamics.

Tables Icon

Table 1. Value of the Chirp Rate β0 has been Chosen to Qualitatively Fit the Experimental Results and is Consistent with the Data Reported in Ref. [22]a

 figure: Fig. 3.

Fig. 3. Temporal dynamics of the beating between the two lasers forming the high-loss QRNG-PIC and comparison with numerical results. (a)–(d) Experimental data with different temperature settings (currents). Chirp due to thermal effects and attenuation of beating amplitude due to the bandwidth limit of the detection electronics are evident. (e)–(h) Numerical results with initial detuning frequencies set to fit the experimental observations in (a).

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C. Measuring the Beating Dynamics

For both the high- and low-loss PICs, the optical pulses of the GS laser were strongly chirped due to thermal effects, yielding a frequency-varying oscillation of the beating pattern, as depicted in Fig. 3. As a result, a nearly zero detuning (NZD) region was observed within the optical pulses when the chirped frequency of the GS laser coincided with the stable frequency of the CW laser. The position of the NZD region depends on the initial frequency separation between the GS and the CW emission lines. When both lasers were initially close (far) in frequency, the NZD region occurred at the beginning (end) of the pulse; see Fig. 3. In the high (15 dB/cm) waveguide loss PIC, the interference amplitude within the NZD region changed from pulse to pulse, a clear signature that phase noise dominated. Instead, in the low-loss PIC (2 dB/cm), backreflection from the CW into the GS laser was not negligible and phase locking between the two lasers was observed. In this case, the interference amplitude in the NZD region did not appreciably change from pulse to pulse.

In the experiment, the NZD region was tuned at the end of the pulse [see Fig. 3(d)], maximizing the detuning frequency between the two lasers so as to reduce residual phase-locking effects, if any.

D. PIC Stability and Performance

From a practical point of view, long-term stability of the scheme is a critical aspect. As we are interfering signals from two independent lasers, intrinsic phase noise and temperature drifts can severely affect the performance. In Fig. 4(a), we plot the histogram for six datasets with 200,000 samples each. The digitized signal was distributed according to the arcsine probability distribution function because of the initial random phase [10,11]. High stability was observed between acquisitions taken over 14 h, confirming the robustness of the two-laser scheme QES-PIC. Instead, a similar implementation with discrete (bulk) components suffered from slow temperature drifts (see Supplement 1 for details on the bulk implementation and corresponding experimental results). The higher stability of the PIC over the bulk design was associated mainly with the fact that the two lasers are closely located in a region with uniform temperature.

 figure: Fig. 4.

Fig. 4. Statistics on the output of the QES-PIC. (a) Histograms on six sets of 200,000 samples each taken during 14 h, confirming stable operation of the QES-PIC device. (b) Autocorrelation function for 107 random samples taken with a 20 GHz scope and 50 GSa/s. Magenta (green) circles correspond to positive (negative) correlation coefficients.

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In Fig. 4(b), we show the autocorrelation function Γx(k)xixi+kxi2 of a sequence of n=107 samples up to a delay of 500 samples. For such sequence length, the statistical uncertainty due to finite size effects is 3.16×104. Except for the d=1 coefficient, which is significantly larger than the statistical noise sensitivity, all the other coefficients fall within the statistical noise level and pass the D’Agostino–Person’s normality test with a p value of 0.18. We attribute the larger correlation at d=1 to limitations in the direct modulation of the DFB laser diode in the experiment, leading to residual photons in the cavity from pulse to pulse. Samples from the higher loss PIC were acquired using a 50 GSa/s resolution and 20 GHz bandwidth real-time scope, followed by a 30 dB RF amplifier. The amplifier introduced noise at several frequency bands, so we employed a 30 MHz high passband digital filter to remove low-frequency components. In addition, we want to assess the quality of the QES, so we want to analyze the correlation of the beat signal only. For doing so, we assume the noise is independent of our signal and calculate Γx=ΓyΓn, where Γy corresponds to the autocorrelation of samples taken within the GS pulse, and Γn to samples taken outside the GS pulse.

4. DISCUSSION

The production of application-ready random numbers from the QES requires a randomness extraction stage [30]. In real QRNG devices, untrusted noise degrades (corrupts) the purity of the randomness associated to quantum processes. The application of proper randomness extractors allows for elimination of corruption of the quantum signal [30]. Random extraction requires (i) the estimation of the amount of available min-entropy from the QES, taking into account electronic noise, memory effects, and digitization noise [31]; and (ii) an appropriate hashing of the data after digitization. Thus, a full QRNG solution that includes both the QES and the electronics (including the digitizer) is required before any meaningful entropy estimate can be derived. However, the statistical data reported in Fig. 4 (correlation and distribution) already confirms the high randomness quality, as it is comparable to the raw data obtained with previously demonstrated bulk architectures, in which large entropy rates have been reported when employing different digitization strategies in complete PD-QRNG solutions [13,31]. With respect to the postprocessing algorithm, field-programmable gate arrays (FPGAs) can be used for real-time randomness extraction above 1 Gb/s [32] for high-performance applications, while for lower-end applications, such as consumer electronics, the central processing unit (CPU) can sustain up to several Mb/s [33].

5. CONCLUSION

We have presented an ultrafast quantum entropy source on a photonic integrated circuit (QES-PIC) for use in quantum random number generation (QRNG). The resulting device shows high performance, including bit rate, degree of randomness (low correlation values), and stability, in a miniaturized geometry that also integrates the receiver. Considering its small footprint and the possibility for hybrid integration with CMOS electronics, the proposed QES-PIC has the potential to become a future functionality in computer and communication cards, especially for cryptography and stochastic simulations. The high level of miniaturization may even make the integration of the QES-PIC device in smartphones and tablets possible.

Funding

European Regional Development Fund (FEDER) (TEC2013-46168-R); Ministerio de Economía y Competitividad (MINECO) Qu-CARD (SRTC1400C002844XV0); Severo Ochoa (SEV-2015-0522); XPLICA (FIS2014-62181-EXP); European Research Council (ERC) (AQUMET 280169, ERIDIAN 713682); European Union QUIC (641122); Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) (2014 LLAV 00078, 2014-SGR-1295, 2014-SGR-1623); Fundación CELLEX .

Acknowledgment

We thank M. Jofre and M. Curty for stimulating discussions on the optical design and R. Baños for laboratory assistance.

 

See Supplement 1 for supporting content.

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References

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  1. C. E. Shannon, “Communication theory of secrecy systems,” Bell Syst. Tech. J. 28, 656–715 (1949).
    [Crossref]
  2. S. Brin and L. Page, “The anatomy of a large-scale hypertextual web search engine,” Comput. Networks ISDN Syst. 30, 107–117 (1998).
    [Crossref]
  3. M. Mascagni, Y. Qiu, and L. Y. Hin, “High performance computing in quantitative finance: a review from the pseudo-random number generator perspective,” Monte Carlo Methods Appl. 20, 101–120 (2014).
    [Crossref]
  4. T. H. Click, A. Liu, and G. A. Kaminski, “Quality of random number generators significantly affects results of Monte Carlo simulations for organic and biological systems,” J. Comput. Chem. 32, 513–524 (2011).
    [Crossref]
  5. J. von Neumann, “Various techniques used in connection with random digits,” Natl. Bur. Stand. Appl. Math Ser. 12, 16–38 (1951).
  6. J. G. Rarity, P. Owens, and P. R. Tapster, “Quantum random-number generation and key sharing,” J. Mod. Opt. 41, 2435–2444 (1994).
    [Crossref]
  7. C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010).
    [Crossref]
  8. B. Qi, Y. M. Chi, H. K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. 35, 312–314 (2010).
    [Crossref]
  9. M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, and V. Pruneri, “True random numbers from amplified quantum vacuum,” Opt. Express 19, 20665–20672 (2011).
    [Crossref]
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  12. Y. Q. Nie, L. Huang, Y. Liu, F. Payne, and J. Zhang, “The generation of 68  Gbps quantum random number by measuring laser phase fluctuations,” Rev. Sci. Instrum. 86, 063105 (2015).
    [Crossref]
  13. C. Abellán, W. Amaya, D. Mitrani, V. Pruneri, and M. W. Mitchell, “Generation of fresh and pure random numbers for loophole-free Bell tests,” Phys. Rev. Lett. 115, 250403 (2015).
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  14. M. J. R. Heck, J. F. Bauters, M. L. Davenport, J. K. Doylend, S. Jain, G. Kurczveil, S. Srinivasan, Y. Tang, and J. E. Bowers, “Hybrid silicon photonic integrated circuit technology,” IEEE J. Sel. Top. Quantum Electron. 19, 6100117 (2012).
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  15. M. Smit, X. Leijtens, H. Ambrosius, E. Bente, J. van der Tol, B. Smalbrugge, T. de Vries, E.-J. Geluk, J. Bolk, R. van Veldhoven, L. Augustin, P. Thijs, D. D’Agostino, H. Rabbani, K. Lawniczuk, S. Stopinski, S. Tahvili, A. Corradi, E. Kleijn, D. Dzibrou, M. Felicetti, E. Bitincka, V. Moskalenko, J. Zhao, R. Santos, G. Gilardi, W. Yao, K. Williams, P. Stabile, P. Kuindersma, J. Pello, S. Bhat, Y. Jiao, D. Heiss, G. Roelkens, M. Wale, P. Firth, F. Soares, N. Grote, M. Schell, H. Debregeas, M. Achouche, J.-L. Gentner, A. Bakker, T. Korthorst, D. Gallagher, A. Dabbs, A. Melloni, F. Morichetti, D. Melati, A. Wonfor, R. Penty, R. Broeke, B. Musk, and D. Robbins, “An introduction to InP-based generic integration technology,” Semicond. Sci. Technol. 29, 083001 (2014).
    [Crossref]
  16. I. A. Walmsley, “Quantum optics: science and technology in a new light,” Science 348, 525–530 (2015).
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  17. A. Alduino, L. Liao, R. Jones, M. Morse, B. Kim, W.-Z. Lo, J. Basak, B. Koch, H.-F. Liu, H. Rong, M. Sysak, C. Krause, R. Saba, D. Lazar, L. Horwitz, R. Bar, S. Litski, A. Liu, K. Sullivan, O. Dosunmu, N. Na, T. Yin, F. Haubensack, I.-W. Hsieh, J. Heck, R. Beatty, H. Park, J. Bovington, S. Lee, H. Nguyen, H. Au, K. Nguyen, P. Merani, M. Hakami, and M. Paniccia, “Demonstration of a high speed 4-channel integrated silicon photonics WDM link with hybrid silicon lasers,” in Integrated Photonics Research, Silicon and Nanophotonics and Photonics in Switching (Optical Society of America, 2010), paper PDIWI5.
  18. M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7, 540–544 (2013).
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  19. J. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
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  20. P. Sibson, M. Godfrey, C. Erven, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. O’Brien, and M. G. Thompson, “Integrated photonic devices for quantum key distribution,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2015), paper FF1A.6.
  21. A. Khanmohammadi, R. Enne, M. Hofbauer, and H. Zimmermann, “A monolithic silicon quantum random number generator based on measurement of photon detection time,” IEEE Photon. J. 7, 1–13 (2015).
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  22. A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photon. Technol. Lett. 10, 1709–1711 (1998).
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  24. D. G. Aronson, E. J. Doedel, and H. G. Othmer, “An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators,” Physica D 25, 20–104 (1987).
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  25. D. G. Aronson, G. B. Ermentrout, and N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990).
    [Crossref]
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  27. S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: instantaneous coupling limit,” Phys. Rev. E 69, 056221 (2004).
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  28. H. Erzgräber, D. Lenstra, B. Krauskopf, E. Wille, M. Peil, I. Fischer, and W. Elsäßer, “Mutually delay-coupled semiconductor lasers: mode bifurcation scenarios,” Opt. Commun. 255, 286–296 (2005).
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2015 (6)

Y. Q. Nie, L. Huang, Y. Liu, F. Payne, and J. Zhang, “The generation of 68  Gbps quantum random number by measuring laser phase fluctuations,” Rev. Sci. Instrum. 86, 063105 (2015).
[Crossref]

C. Abellán, W. Amaya, D. Mitrani, V. Pruneri, and M. W. Mitchell, “Generation of fresh and pure random numbers for loophole-free Bell tests,” Phys. Rev. Lett. 115, 250403 (2015).
[Crossref]

I. A. Walmsley, “Quantum optics: science and technology in a new light,” Science 348, 525–530 (2015).
[Crossref]

A. Khanmohammadi, R. Enne, M. Hofbauer, and H. Zimmermann, “A monolithic silicon quantum random number generator based on measurement of photon detection time,” IEEE Photon. J. 7, 1–13 (2015).
[Crossref]

M. W. Mitchell, C. Abellán, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91, 012314 (2015).
[Crossref]

A. Martin, B. Sanguinetti, C. C. W. Lim, R. Houlmann, and H. Zbinden, “Quantum random number generation for 1.25  GHz quantum key distribution systems,” J. Lightwave Technol. 33, 2855–2859 (2015).
[Crossref]

2014 (5)

B. Sanguinetti, A. Martin, H. Zbinden, and N. Gisin, “Quantum random number generation on a mobile phone,” Phys. Rev. X 4, 031056 (2014).
[Crossref]

M. Smit, X. Leijtens, H. Ambrosius, E. Bente, J. van der Tol, B. Smalbrugge, T. de Vries, E.-J. Geluk, J. Bolk, R. van Veldhoven, L. Augustin, P. Thijs, D. D’Agostino, H. Rabbani, K. Lawniczuk, S. Stopinski, S. Tahvili, A. Corradi, E. Kleijn, D. Dzibrou, M. Felicetti, E. Bitincka, V. Moskalenko, J. Zhao, R. Santos, G. Gilardi, W. Yao, K. Williams, P. Stabile, P. Kuindersma, J. Pello, S. Bhat, Y. Jiao, D. Heiss, G. Roelkens, M. Wale, P. Firth, F. Soares, N. Grote, M. Schell, H. Debregeas, M. Achouche, J.-L. Gentner, A. Bakker, T. Korthorst, D. Gallagher, A. Dabbs, A. Melloni, F. Morichetti, D. Melati, A. Wonfor, R. Penty, R. Broeke, B. Musk, and D. Robbins, “An introduction to InP-based generic integration technology,” Semicond. Sci. Technol. 29, 083001 (2014).
[Crossref]

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acn, J. Capmany, V. Pruneri, and M. W. Mitchell, “Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode,” Opt. Express 22, 1645–1654 (2014).
[Crossref]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104, 261112 (2014).
[Crossref]

M. Mascagni, Y. Qiu, and L. Y. Hin, “High performance computing in quantitative finance: a review from the pseudo-random number generator perspective,” Monte Carlo Methods Appl. 20, 101–120 (2014).
[Crossref]

2013 (1)

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7, 540–544 (2013).
[Crossref]

2012 (1)

M. J. R. Heck, J. F. Bauters, M. L. Davenport, J. K. Doylend, S. Jain, G. Kurczveil, S. Srinivasan, Y. Tang, and J. E. Bowers, “Hybrid silicon photonic integrated circuit technology,” IEEE J. Sel. Top. Quantum Electron. 19, 6100117 (2012).
[Crossref]

2011 (2)

T. H. Click, A. Liu, and G. A. Kaminski, “Quality of random number generators significantly affects results of Monte Carlo simulations for organic and biological systems,” J. Comput. Chem. 32, 513–524 (2011).
[Crossref]

M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, and V. Pruneri, “True random numbers from amplified quantum vacuum,” Opt. Express 19, 20665–20672 (2011).
[Crossref]

2010 (2)

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010).
[Crossref]

B. Qi, Y. M. Chi, H. K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. 35, 312–314 (2010).
[Crossref]

2009 (1)

J. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[Crossref]

2006 (1)

A. Torcini, S. Barland, G. Giacomelli, and F. Marin, “Low-frequency fluctuations in vertical cavity lasers: experiments versus Lang-Kobayashi dynamics,” Phys. Rev. A 74, 063801 (2006).
[Crossref]

2005 (1)

H. Erzgräber, D. Lenstra, B. Krauskopf, E. Wille, M. Peil, I. Fischer, and W. Elsäßer, “Mutually delay-coupled semiconductor lasers: mode bifurcation scenarios,” Opt. Commun. 255, 286–296 (2005).
[Crossref]

2004 (1)

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: instantaneous coupling limit,” Phys. Rev. E 69, 056221 (2004).
[Crossref]

1998 (2)

A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photon. Technol. Lett. 10, 1709–1711 (1998).
[Crossref]

S. Brin and L. Page, “The anatomy of a large-scale hypertextual web search engine,” Comput. Networks ISDN Syst. 30, 107–117 (1998).
[Crossref]

1994 (1)

J. G. Rarity, P. Owens, and P. R. Tapster, “Quantum random-number generation and key sharing,” J. Mod. Opt. 41, 2435–2444 (1994).
[Crossref]

1990 (1)

D. G. Aronson, G. B. Ermentrout, and N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990).
[Crossref]

1987 (1)

D. G. Aronson, E. J. Doedel, and H. G. Othmer, “An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators,” Physica D 25, 20–104 (1987).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

1973 (1)

R. Adler, “A study of locking phenomena in oscillators,” Proc. IEEE 61, 1380–1385 (1973).
[Crossref]

1951 (1)

J. von Neumann, “Various techniques used in connection with random digits,” Natl. Bur. Stand. Appl. Math Ser. 12, 16–38 (1951).

1949 (1)

C. E. Shannon, “Communication theory of secrecy systems,” Bell Syst. Tech. J. 28, 656–715 (1949).
[Crossref]

Abellán, C.

C. Abellán, W. Amaya, D. Mitrani, V. Pruneri, and M. W. Mitchell, “Generation of fresh and pure random numbers for loophole-free Bell tests,” Phys. Rev. Lett. 115, 250403 (2015).
[Crossref]

M. W. Mitchell, C. Abellán, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91, 012314 (2015).
[Crossref]

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acn, J. Capmany, V. Pruneri, and M. W. Mitchell, “Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode,” Opt. Express 22, 1645–1654 (2014).
[Crossref]

Achouche, M.

M. Smit, X. Leijtens, H. Ambrosius, E. Bente, J. van der Tol, B. Smalbrugge, T. de Vries, E.-J. Geluk, J. Bolk, R. van Veldhoven, L. Augustin, P. Thijs, D. D’Agostino, H. Rabbani, K. Lawniczuk, S. Stopinski, S. Tahvili, A. Corradi, E. Kleijn, D. Dzibrou, M. Felicetti, E. Bitincka, V. Moskalenko, J. Zhao, R. Santos, G. Gilardi, W. Yao, K. Williams, P. Stabile, P. Kuindersma, J. Pello, S. Bhat, Y. Jiao, D. Heiss, G. Roelkens, M. Wale, P. Firth, F. Soares, N. Grote, M. Schell, H. Debregeas, M. Achouche, J.-L. Gentner, A. Bakker, T. Korthorst, D. Gallagher, A. Dabbs, A. Melloni, F. Morichetti, D. Melati, A. Wonfor, R. Penty, R. Broeke, B. Musk, and D. Robbins, “An introduction to InP-based generic integration technology,” Semicond. Sci. Technol. 29, 083001 (2014).
[Crossref]

Acn, A.

Adler, R.

R. Adler, “A study of locking phenomena in oscillators,” Proc. IEEE 61, 1380–1385 (1973).
[Crossref]

Alduino, A.

A. Alduino, L. Liao, R. Jones, M. Morse, B. Kim, W.-Z. Lo, J. Basak, B. Koch, H.-F. Liu, H. Rong, M. Sysak, C. Krause, R. Saba, D. Lazar, L. Horwitz, R. Bar, S. Litski, A. Liu, K. Sullivan, O. Dosunmu, N. Na, T. Yin, F. Haubensack, I.-W. Hsieh, J. Heck, R. Beatty, H. Park, J. Bovington, S. Lee, H. Nguyen, H. Au, K. Nguyen, P. Merani, M. Hakami, and M. Paniccia, “Demonstration of a high speed 4-channel integrated silicon photonics WDM link with hybrid silicon lasers,” in Integrated Photonics Research, Silicon and Nanophotonics and Photonics in Switching (Optical Society of America, 2010), paper PDIWI5.

Amaya, W.

C. Abellán, W. Amaya, D. Mitrani, V. Pruneri, and M. W. Mitchell, “Generation of fresh and pure random numbers for loophole-free Bell tests,” Phys. Rev. Lett. 115, 250403 (2015).
[Crossref]

M. W. Mitchell, C. Abellán, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91, 012314 (2015).
[Crossref]

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acn, J. Capmany, V. Pruneri, and M. W. Mitchell, “Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode,” Opt. Express 22, 1645–1654 (2014).
[Crossref]

Ambrosius, H.

M. Smit, X. Leijtens, H. Ambrosius, E. Bente, J. van der Tol, B. Smalbrugge, T. de Vries, E.-J. Geluk, J. Bolk, R. van Veldhoven, L. Augustin, P. Thijs, D. D’Agostino, H. Rabbani, K. Lawniczuk, S. Stopinski, S. Tahvili, A. Corradi, E. Kleijn, D. Dzibrou, M. Felicetti, E. Bitincka, V. Moskalenko, J. Zhao, R. Santos, G. Gilardi, W. Yao, K. Williams, P. Stabile, P. Kuindersma, J. Pello, S. Bhat, Y. Jiao, D. Heiss, G. Roelkens, M. Wale, P. Firth, F. Soares, N. Grote, M. Schell, H. Debregeas, M. Achouche, J.-L. Gentner, A. Bakker, T. Korthorst, D. Gallagher, A. Dabbs, A. Melloni, F. Morichetti, D. Melati, A. Wonfor, R. Penty, R. Broeke, B. Musk, and D. Robbins, “An introduction to InP-based generic integration technology,” Semicond. Sci. Technol. 29, 083001 (2014).
[Crossref]

Andersen, U. L.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010).
[Crossref]

Andonovic, I.

A. Zadok, H. Shalom, M. Tur, W. D. Cornwell, and I. Andonovic, “Spectral shift and broadening of DFB lasers under direct modulation,” IEEE Photon. Technol. Lett. 10, 1709–1711 (1998).
[Crossref]

Anzolin, G.

Aronson, D. G.

D. G. Aronson, G. B. Ermentrout, and N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41, 403–449 (1990).
[Crossref]

D. G. Aronson, E. J. Doedel, and H. G. Othmer, “An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators,” Physica D 25, 20–104 (1987).
[Crossref]

Au, H.

A. Alduino, L. Liao, R. Jones, M. Morse, B. Kim, W.-Z. Lo, J. Basak, B. Koch, H.-F. Liu, H. Rong, M. Sysak, C. Krause, R. Saba, D. Lazar, L. Horwitz, R. Bar, S. Litski, A. Liu, K. Sullivan, O. Dosunmu, N. Na, T. Yin, F. Haubensack, I.-W. Hsieh, J. Heck, R. Beatty, H. Park, J. Bovington, S. Lee, H. Nguyen, H. Au, K. Nguyen, P. Merani, M. Hakami, and M. Paniccia, “Demonstration of a high speed 4-channel integrated silicon photonics WDM link with hybrid silicon lasers,” in Integrated Photonics Research, Silicon and Nanophotonics and Photonics in Switching (Optical Society of America, 2010), paper PDIWI5.

Augustin, L.

M. Smit, X. Leijtens, H. Ambrosius, E. Bente, J. van der Tol, B. Smalbrugge, T. de Vries, E.-J. Geluk, J. Bolk, R. van Veldhoven, L. Augustin, P. Thijs, D. D’Agostino, H. Rabbani, K. Lawniczuk, S. Stopinski, S. Tahvili, A. Corradi, E. Kleijn, D. Dzibrou, M. Felicetti, E. Bitincka, V. Moskalenko, J. Zhao, R. Santos, G. Gilardi, W. Yao, K. Williams, P. Stabile, P. Kuindersma, J. Pello, S. Bhat, Y. Jiao, D. Heiss, G. Roelkens, M. Wale, P. Firth, F. Soares, N. Grote, M. Schell, H. Debregeas, M. Achouche, J.-L. Gentner, A. Bakker, T. Korthorst, D. Gallagher, A. Dabbs, A. Melloni, F. Morichetti, D. Melati, A. Wonfor, R. Penty, R. Broeke, B. Musk, and D. Robbins, “An introduction to InP-based generic integration technology,” Semicond. Sci. Technol. 29, 083001 (2014).
[Crossref]

Bakker, A.

M. Smit, X. Leijtens, H. Ambrosius, E. Bente, J. van der Tol, B. Smalbrugge, T. de Vries, E.-J. Geluk, J. Bolk, R. van Veldhoven, L. Augustin, P. Thijs, D. D’Agostino, H. Rabbani, K. Lawniczuk, S. Stopinski, S. Tahvili, A. Corradi, E. Kleijn, D. Dzibrou, M. Felicetti, E. Bitincka, V. Moskalenko, J. Zhao, R. Santos, G. Gilardi, W. Yao, K. Williams, P. Stabile, P. Kuindersma, J. Pello, S. Bhat, Y. Jiao, D. Heiss, G. Roelkens, M. Wale, P. Firth, F. Soares, N. Grote, M. Schell, H. Debregeas, M. Achouche, J.-L. Gentner, A. Bakker, T. Korthorst, D. Gallagher, A. Dabbs, A. Melloni, F. Morichetti, D. Melati, A. Wonfor, R. Penty, R. Broeke, B. Musk, and D. Robbins, “An introduction to InP-based generic integration technology,” Semicond. Sci. Technol. 29, 083001 (2014).
[Crossref]

Bar, R.

A. Alduino, L. Liao, R. Jones, M. Morse, B. Kim, W.-Z. Lo, J. Basak, B. Koch, H.-F. Liu, H. Rong, M. Sysak, C. Krause, R. Saba, D. Lazar, L. Horwitz, R. Bar, S. Litski, A. Liu, K. Sullivan, O. Dosunmu, N. Na, T. Yin, F. Haubensack, I.-W. Hsieh, J. Heck, R. Beatty, H. Park, J. Bovington, S. Lee, H. Nguyen, H. Au, K. Nguyen, P. Merani, M. Hakami, and M. Paniccia, “Demonstration of a high speed 4-channel integrated silicon photonics WDM link with hybrid silicon lasers,” in Integrated Photonics Research, Silicon and Nanophotonics and Photonics in Switching (Optical Society of America, 2010), paper PDIWI5.

Barland, S.

A. Torcini, S. Barland, G. Giacomelli, and F. Marin, “Low-frequency fluctuations in vertical cavity lasers: experiments versus Lang-Kobayashi dynamics,” Phys. Rev. A 74, 063801 (2006).
[Crossref]

Basak, J.

A. Alduino, L. Liao, R. Jones, M. Morse, B. Kim, W.-Z. Lo, J. Basak, B. Koch, H.-F. Liu, H. Rong, M. Sysak, C. Krause, R. Saba, D. Lazar, L. Horwitz, R. Bar, S. Litski, A. Liu, K. Sullivan, O. Dosunmu, N. Na, T. Yin, F. Haubensack, I.-W. Hsieh, J. Heck, R. Beatty, H. Park, J. Bovington, S. Lee, H. Nguyen, H. Au, K. Nguyen, P. Merani, M. Hakami, and M. Paniccia, “Demonstration of a high speed 4-channel integrated silicon photonics WDM link with hybrid silicon lasers,” in Integrated Photonics Research, Silicon and Nanophotonics and Photonics in Switching (Optical Society of America, 2010), paper PDIWI5.

Bauters, J. F.

M. J. R. Heck, J. F. Bauters, M. L. Davenport, J. K. Doylend, S. Jain, G. Kurczveil, S. Srinivasan, Y. Tang, and J. E. Bowers, “Hybrid silicon photonic integrated circuit technology,” IEEE J. Sel. Top. Quantum Electron. 19, 6100117 (2012).
[Crossref]

Beatty, R.

A. Alduino, L. Liao, R. Jones, M. Morse, B. Kim, W.-Z. Lo, J. Basak, B. Koch, H.-F. Liu, H. Rong, M. Sysak, C. Krause, R. Saba, D. Lazar, L. Horwitz, R. Bar, S. Litski, A. Liu, K. Sullivan, O. Dosunmu, N. Na, T. Yin, F. Haubensack, I.-W. Hsieh, J. Heck, R. Beatty, H. Park, J. Bovington, S. Lee, H. Nguyen, H. Au, K. Nguyen, P. Merani, M. Hakami, and M. Paniccia, “Demonstration of a high speed 4-channel integrated silicon photonics WDM link with hybrid silicon lasers,” in Integrated Photonics Research, Silicon and Nanophotonics and Photonics in Switching (Optical Society of America, 2010), paper PDIWI5.

Bente, E.

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Supplementary Material (1)

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» Supplement 1: PDF (1275 KB)      Supplementary material

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Figures (4)

Fig. 1.
Fig. 1. (a) Schematic of the QRNG-PIC based on two-laser interference. The two DFB lasers are biased with a current driver, one of them operating in CW, while the other one is periodically GS using an external RF generator. The temperature of the entire chip is controlled through a Peltier element, while that of the area including one of the lasers is locally changed by a stable current source. The outputs from the two lasers are combined and interfered in a 2×2 MMI coupler and two 40 GHz photodiodes are placed at the output of the coupler. The detected signal is sent to a fast oscilloscope. (b) Principle of operation: optical pulses from a GS laser interfere with a CW laser, generating an interference modulation whose frequency is equal to the difference of the two lasers’ frequencies. The random phase of the GS laser pulse produces a random phase of the interference oscillation that can be properly sampled into a random amplitude. In this way, after digitization, one can extract one sample per GS pulse. (c) Microscope image of the PIC on a 1 Euro cent background. Two QRNG-PICs are printed on each chip.
Fig. 2.
Fig. 2. Beat-note frequency at the output of the MMI measured by sweeping one of the integrated lasers, while keeping the other one constant in the low-loss QRNG-PIC. The beat-note frequency can be continuously tuned by current control for large detuning frequencies, whereas for small detuning frequencies phase- (frequency-) locking may occur (gray square), leading eventually to disappearance of the oscillation.
Fig. 3.
Fig. 3. Temporal dynamics of the beating between the two lasers forming the high-loss QRNG-PIC and comparison with numerical results. (a)–(d) Experimental data with different temperature settings (currents). Chirp due to thermal effects and attenuation of beating amplitude due to the bandwidth limit of the detection electronics are evident. (e)–(h) Numerical results with initial detuning frequencies set to fit the experimental observations in (a).
Fig. 4.
Fig. 4. Statistics on the output of the QES-PIC. (a) Histograms on six sets of 200,000 samples each taken during 14 h, confirming stable operation of the QES-PIC device. (b) Autocorrelation function for 107 random samples taken with a 20 GHz scope and 50 GSa/s. Magenta (green) circles correspond to positive (negative) correlation coefficients.

Tables (1)

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Table 1. Value of the Chirp Rate β0 has been Chosen to Qualitatively Fit the Experimental Results and is Consistent with the Data Reported in Ref. [22]a

Equations (6)

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iT(t)=iS(t)+2iP(t)cos(0tdξΩC(ξ)+Δφ),
E˙1=γ(1+iα)N1E1+κexp(iψ)E2(tτd)+R1ξ1(t),
E˙2=γ(1+iα)N2E2+κexp(iψ)E1(tτd)+i[Ω+β(t)]E2+R2ξ2(t),
τN˙1=P1N1(1+2N1)|E1|2,
τN˙2=P2N2(1+2N2)|E2|2.
P2(t)=P¯2{12+32exp[(t/Δτ)2M]},

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