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Cited by 168Quantum Information and CryptographySemiconductor Lasers and Optical DevicesRandom lasers and scattering media

Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode

Open access
Abellán, C., Amaya, W., Jofre, M., Curty, M., Acín, A., Capmany, J., Pruneri, V., Mitchell, M. W.·2014·Optics Express
DOI 10.1364/oe.22.001645
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Abstract

We demonstrate a high bit-rate quantum random number generator by interferometric detection of phase diffusion in a gain-switched DFB laser diode. Gain switching at few-GHz frequencies produces a train of bright pulses with nearly equal amplitudes and random phases. An unbalanced Mach-Zehnder interferometer is used to interfere subsequent pulses and thereby generate strong random-amplitude pulses, which are detected and digitized to produce a high-rate random bit string. Using established models of semiconductor laser field dynamics, we predict a regime of high visibility interference and nearly complete vacuum-fluctuation-induced phase diffusion between pulses. These are confirmed by measurement of pulse power statistics at the output of the interferometer. Using a 5.825 GHz excitation rate and 14-bit digitization, we observe 43 Gbps quantum randomness generation.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper demonstrates a high-bit-rate quantum random number generator using interferometric detection of phase diffusion in a gain-switched DFB laser diode. It reports 43 Gbps quantum randomness generation at a 5.825 GHz excitation rate with 14-bit digitization.

Key findings

  • Gain-switched pulses at 5.825 GHz with ~85 ps width and ~7.65 mW peak power show nearly complete vacuum-fluctuation-induced phase diffusion, bounded as at least 9.45 radians rms.
  • Interference in an unbalanced Mach-Zehnder interferometer gives visibility |g(tloop)| = 0.90 and broadens the output power distribution; measured branch powers are 0.97 mW and 0.90 mW on average with standard deviation ~45 µW.
  • The estimated min-entropy is 7.33 bits per sample, yielding a quantum randomness generation rate of 43 Gbps; hashing 120 × 10^6 14-bit numbers with reduction factor RF = 14/7.33 ≈ 1.9 produces 125 × 10^6 7-bit numbers.
  • The hashed data pass all 15 NIST tests with a 1.5 Gbit sequence; the smallest PvalueT is 0.0926 for the non-periodic template matching test.

Methods

  • A 1550 nm DFB laser diode with 10 Gbps modulation bandwidth is gain-switched at 5.825 GHz, producing optical pulses that pass through an unbalanced Mach-Zehnder interferometer with 3.55 cm path delay and are detected by a 14-bit, 12.5 GHz oscilloscope.
  • For statistical testing, 120 × 10^6 pulses were recorded over five days, one sample per pulse taken ~13 ps after the pulse peak; autocorrelation, variance, and min-entropy analyses were used to bound the quantum contribution.
  • Randomness extraction uses the Whirlpool hash function with RF ≈ 1.9, followed by NIST 15-test battery evaluation on 1.5 Gbits.

Limitations

  • The paper does not include a dedicated limitations section. It states that other fluctuating contributions—photo-detection noise, laser current fluctuations, and digitization errors—are not guaranteed to be random, but argues the extraction step removes non-random effects if the reduction factor is chosen using the min-entropy of the quantum contribution.
  • The paper states that several laser parameters (ssat, L, n0, and combined spontaneous-emission parameters) are unknown and are chosen by fitting to the observed pulse shape; the fitting is described as conservative because simulated optical power is always larger than measured.
  • Evident from the text: the min-entropy bound relies on modeling assumptions, including Gaussian phase diffusion, uniform phase on [0, π), and uncorrelation among u1, u2, cos θ, and u_noise.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1364/oe.22.001645

Cites

23 of 23 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 6 cited sources checked · 0 supported · 0 flagged

  • Findingnone given
    § 5. Raw data analysis · p. 6

    The visibility achieved for the interferometer is 0.9.

    “The visibility achieved for the interferometer is 0.9.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode · p. 1

    Using a 5.825 GHz excitation rate and 14-bit digitization, the authors observe 43 Gbps quantum randomness generation.

    “Using a 5.825 GHz excitation rate and 14-bit digitization, we observe 43 Gbps quantum randomness generation.”
    • Also stated in: § 6.2. Min-entropy estimation · p. 8 · plain
      “With the PRF of 5.825 Gpulses/s, this implies a quantum randomness generation rate of 43 Gbps.”
    • Also stated in: § 7. Conclusions · p. 9 · plain
      “Using a 5.825 GHz excitation rate and 14-bit digitization, we observe 43 Gbps quantum randomness generation.”
    Quote in the paperSubstantial claim
  • Findingnone given
    § Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode · p. 1

    Gain switching at few-GHz frequencies produces a train of bright pulses with nearly equal amplitudes and random phases.

    “Gain switching at few-GHz frequencies produces a train of bright pulses with nearly equal amplitudes and random phases.”
    Quote in the paperSubstantial claim
  • Findingown data
    § 6.2. Min-entropy estimation · p. 8

    In the experiment, with A_ADC = 5 mW, b = 14 bits, u_max - u_min = 3.34 mW, the min entropy is 7.33 bits.

    “In the experiment, where A ADC = 5 mW, b = 14 bits, u max -u min = 4 | g ( tloop ) | √ E [ u 1 ] E [ u 2 ] = 3 . 34 mW, the min entropy is 7.33 bits.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 6.2. Min-entropy estimation · p. 8

    Applying a Whirlpool hash function with reduction factor RF = 14/7.33 ≈ 1.9 to the raw data reduces 120×10^6 14-bit numbers to 125×10^6 7-bit numbers.

    “In order to extract the randomness arising from amplification of the vacuum field, a hash function with a reduction factor RF = 14 / 7 . 33 ≈ 1 . 9 is applied to the raw data, reducing 120 × 10 6 14-bit numbers to 125 × 10 6 7-bit numbers. The hashing is performed by the Whirlpool hash function [23].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [23]

      Not checked in the free run.

  • May combine two claims. 15
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 2

  • May combine two claims.

    Findingown data
    § Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode · p. 1

    Using established models of semiconductor laser field dynamics, the authors predict a regime of high visibility interference and nearly complete vacuum-fluctuation-induced phase diffusion between pulses, confirmed by measurement of pulse power statistics at the output of the interferometer.

    “Using established models of semiconductor laser field dynamics, we predict a regime of high visibility interference and nearly complete vacuum-fluctuation-induced phase diffusion between pulses. These are confirmed by measurement of pulse power statistics at the output of the interferometer.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 5. Raw data analysis · p. 5

    The distribution takes on the expected shape after power saturation occurs, i.e. after the pulse peak, presumably due to damping of transients associated with the rapid turn-on.

    “We observe experimentally that the distribution takes on the expected shape after power saturation occurs, i.e. after the pulse peak, presumably due to damping of transients associated with the rapid turn-on.”
    Quote in the paperNot a substantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingown data
    § 5. Raw data analysis · p. 6

    The normalized autocorrelation of 50 subsequent sampled pulses follows a delta-function like behaviour, indicating the random nature of the process; non-shifted correlation values are at the 40 dB level.

    “We evaluate the normalized autocorrelation function to confirm the intuition that the process is intrinsically uncorrelated. As seen in Fig. 3(b), the non-shifted correlation values are at the 40 dB level.”
    • Also stated in: § 5. Raw data analysis · p. 6 · plain
      “As expected, it follows a delta-function like behaviour indicating the random nature of the process.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 5. Raw data analysis · p. 5

    The power from each branch of the U-MZI, as measured at the output, are 0.97 mW and 0.9 mW on average, with a standard deviation of ∼45 μW.

    “The power from each branch of the U-MZI, as measured at the output, are 0 . 97 mW and 0 . 9 mW on average, with a standard deviation of ∼ 45 µ W .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 6.1. Procedure · p. 7

    From the data shown in Fig. 3, var(u(out)) = 1.4 mW^2, var(u1)=2.0×10^-3 mW^2, var(u2)=2.1×10^-3 mW^2, E[√u1]^2=0.97 mW, E[√u2]^2=0.90 mW; from measurements with the laser off, var(u_noise)=1.45×10^-4 mW^2; applying Eq. (12) gives |g(tloop)|=0.90.

    “From the data shown in Fig. 3, we obtain var ( u ( out ) ) = 1 . 4 mW 2 , var ( u 1 ) = 2 . 0 × 10 -3 mW 2 , var ( u 2 ) = 2 . 1 × 10 -3 mW 2 , E [ √ u 1 ] 2 = 0 . 97 mW, and E [ √ u 2 ] 2 = 0 . 90 mW. From measurements with the laser off, we find the contribution of detection and digitization noise var ( u noise ) = 1 . 45 × 10 -4 mW 2 . Applying Eq. (12) we find | g ( tloop ) | = 0 . 90.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: u1, u2, cos θ, and u_noise are independent.

  • May combine two claims.

    Findingreasoning
    § 5. Raw data analysis · p. 5

    The optical pulses have a predictable power and true random phase, so that when interfering these fields, random input phases of quantum origin are converted into macroscopically measurable random output intensities.

    “The fact that the optical pulses have a predictable power and true random phase implies that when interfering these fields, random input phases of quantum origin are converted into macroscopically measurable random output intensities.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data
    § 2. System configuration · p. 2

    The optical output consists of pulses of ∼85 ps width and ∼7.65 mW peak power.

    “The resulting optical output consists of pulses of ∼ 85 ps width and ∼ 7.65 mW peak power.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode · p. 1

    An unbalanced Mach-Zehnder interferometer interferes subsequent pulses to generate strong random-amplitude pulses, which are detected and digitized to produce a high-rate random bit string.

    “An unbalanced Mach-Zehnder interferometer is used to interfere subsequent pulses and thereby generate strong random-amplitude pulses, which are detected and digitized to produce a high-rate random bit string.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 4. Phase diffusion · p. 5

    Numerical integration of Eq. (4) gives 〈∆θ^2〉 > (9.45 rad)^2, which for all practical purposes describes a full randomization.

    “Finally, Eq. (4) is numerically integrated to find 〈 ∆θ 2 〉 > ( 9 . 45 rad ) 2 . For all practical purposes this value describes a full randomization.”
    • Also stated in: § 7. Conclusions · p. 9 · plain
      “We compute a conservative bound on the phase diffusion between pulses and find diffusion by at least 9.45 radians rms, equivalent to complete phase randomness for all practical purposes.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: Phase diffusion is larger for low optical intensities.

  • May combine two claims.

    Findingown data
    § 6.2. Min-entropy estimation · p. 8

    Correlations within the hashed data are at the -40 dB level, which coincides with the statistical uncertainty given the quantity of output numbers.

    “As seen in Fig. 4, correlations within the resulting data are at the -40 dB level, which coincides with the statistical uncertainty given the quantity of output numbers.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 7. Conclusions · p. 9

    The method uses interferometry rather than single-photon or shot-noise-limited detection to convert microscopic quantum observables to macroscopically detectable signals, and is demonstrated with commercially available telecommunications components.

    “The method uses interferometry rather than single-photon or shot-noise-limited detection to convert microscopic quantum observables to macroscopically detectable signals, and is demonstrated with commercially available telecommunications components.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given
    § 5. Raw data analysis · p. 5

    The data were taken over a five day period as a test of stability and repeatability.

    “These data were taken over a five day period as a test of stability and repeatability.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation
    § 4. Phase diffusion · p. 4

    During gain-switching, the field within the cavity alternately experiences two working regimes: far above threshold and far below threshold; during the former, stimulated emission dominates and coherent optical pulses are emitted; during the latter, the intracavity field is strongly attenuated and significant phase diffusion occurs.

    “The experiment is performed under gain-switching conditions, and the field within the cavity alternately experiences two working regimes: far above threshold and far below threshold. During the former, stimulated emission dominates and coherent optical pulses are emitted. During the latter, the intracavity field is strongly attenuated and significant phase diffusion occurs.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [19] Henry, 1986, J. Lightwave Technol. source

      Not checked in the free run.

    • cites [20] Agrawal, 1990, IEEE J. Quantum Electron. source

      Not checked in the free run.

    • cites [21] Nisan, 1999, J. Comput. Sci. Tech.

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 6.1. Procedure · p. 7

    The intra-cavity field describes a diffusion process and therefore θ is gaussianly distributed; with standard deviation larger than π estimated in Section 4, this justifies considering θ uniformly distributed on [0,π), and thus cos θ described by an arcsine distribution.

    “From laser physics, we know that the intra-cavity field describes a diffusion process and therefore θ is gaussianly distributed. Also, using measures and well-established models of semiconductor DFB lasers, an standard deviation larger than π was estimated in Section 4. As described above, this justifies considering θ to be uniformly distributed on [ 0 , π ) , and thus that cos θ is described by a”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 6.2. Min-entropy estimation · p. 8

    The probability of the first bin decreases exponentially with b and therefore the min-entropy increases linearly with the resolution of the digitization.

    “Using the fact that arcsin x ≈ x for x small, it can easily be seen that the probability of the first bin decreases exponentially with b and therefore the min-entropy, given by Eq. (16), increases linearly with the resolution of the digitization.”
    • Also stated in: § 7. Conclusions · p. 9 · hedged
      “We provide a new method to determine the quantum contribution to the detected random signals, and show that the quantum contribution to min-entropy grows proportional to the digitization resolution in bits, suggesting a straightforward route to even higher QRNG bit rates.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 6.1. Procedure · p. 7

    The four random variables X1, X2, X3, X4 are uncorrelated because cos θ is equally likely to be positive or negative.

    “Nevertheless, the fact that cos θ is equally likely to be positive or negative implies that the four random variables are uncorrelated.”
    Quote in the paperSubstantial claim
16 of 16

  • The authors state this as an assumption.

    u1, u2, cos θ, and u_noise are independent.

    “Assuming that u 1, u 2, cos θ and u noise are independent, does not imply that the four random variables X 1 , X 2 , X 3 , X 4 are independent.”

  • The paper relies on this without support.

    For the same mean power, pulsing accelerates the phase diffusion rate, which is proportional to the spontaneous emission rate over the intra-cavity photon number.

    “For the same mean power, pulsing accelerates the phase diffusion rate, which is proportional to the spontaneous emission rate over the intra-cavity photon number.”

  • The paper relies on this without support.

    Phase diffusion is larger for low optical intensities.

    “Note that phase diffusion is larger for low optical intensities. Hence, the fitting is conservative because the simulated optical power is always larger than the measurement.”
3 of 3

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • To overcome the bandwidth limitation of shot noise-limited homodyne detection, researchers have developed QRNGs based on measuring phase or intensity noise of amplified spontaneous emission (ASE), which is quantum mechanical by nature.

    “To overcome the bandwidth limitation of shot noise-limited homodyne detection, researchers have developed QRNGs based on measuring phase 40 -45 or intensity noise 46,47 of ampli fi ed spontaneous emission(ASE), which is quantum mechanical by nature. 15,48,49”

    From Quantum random number generation (2016) · cites this paper as [44]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • NIST tests require more hashed data than reported

    Major

    The reported hashed output contains 125 million 7-bit numbers, or 875 million bits. That is insufficient for the stated NIST testing of 1.5 billion bits (1,500 sequences of 1 Mbit each).

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. Cai, 2007, IEEE Signal Process. Mag., 24, 27
  2. Nisan, 1999, J. Comput. Sci. Tech., 58, 148

Cited by

Sorted by citations

Related

Authors
Abellán, C., Amaya, W., Jofre, M., Curty, M., Acín, A., Capmany, J., Pruneri, V., & Mitchell, M. W.
Affiliations
  • Institute of Photonic Sciences (ES)
  • Universidade de Vigo (ES)
  • Universitat Politècnica de València (ES)
ORCID
Venue
Optics Express
Publisher
Optica Publishing Group
Published
2014-01-16
Volume
22
Issue
2
Pages
1645
ISSN
1094-4087 (ISSN-L)
DOI
10.1364/oe.22.001645
OpenAlex
W1998868411
Semantic Scholar
2582b3a1a5dfa199190bbd6f8512eba3ad610b53
Cited by
168
References
23
Citations by year
2026: 8 · 2025: 17 · 2024: 12 · 2023: 22 · 2022: 6 · 2021: 13 · 2020: 19 · 2019: 19 · 2018: 11 · 2017: 14 · 2016: 9 · 2015: 14 · 2014: 4
FWCI
13.61
Type
journal article
Language
en
Open access
Licence
Locations
  • Optics Express (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • UPCommons institutional repository (Universitat Politècnica de Catalunya) (submittedVersion, public-domain)
  • arXiv (Cornell University)
Funders
Atlantic Research Center for Information and Communication Technologies
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Semiconductor Lasers and Optical Devices
  • Physical Sciences › Physics and Astronomy › Acoustics and Ultrasonics › Random lasers and scattering media
Keywords
quantum random number generation, phase diffusion, gain-switched laser diode, distributed feedback lasers, pulsed laser, Mach-Zehnder interferometer, unbalanced interferometer, vacuum fluctuations

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