Abstract

We propose a method for quantum noise extraction from the interference of laser pulses with random phase. Our technique is based on the calculation of a parameter, which we called the quantum reduction factor, and which allows for the determination of the contributions of quantum and classical noises with the assumption that classical fluctuations exhibit Gaussian distribution. To the best of our knowledge, the concept of quantum reduction factor is introduced for the first time. We use such an approach to implement the post-processing-free optical quantum random number generator with the random bit generation rate of 2 Gbps.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  5. Q. Zhou, R. Valivarthi, C. John, and W. Tittel, “Practical quantum random-number generation based on sampling vacuum fluctuations,” Quantum Eng. 1(1), e8 (2019).
    [Crossref]
  6. Y. Mansour, N. Nisan, and P. Tiwari, “The computational complexity of universal hashing,” Theor. Comput. Sci. 107(1), 121–133 (1993).
    [Crossref]
  7. X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, “Postprocessing for quantum random-number generators: Entropy evaluation and randomness extraction,” Phys. Rev. A 87(6), 062327 (2013).
    [Crossref]
  8. F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express 20(11), 12366–12377 (2012).
    [Crossref]
  9. F. Raffaelli, P. Sibson, J. E. Kennard, D. H. Mahler, M. G. Thompson, and J. C. F. Matthews, “Generation of random numbers by measuring phase fluctuations from a laser diode with a silicon-on-insulator chip,” Opt. Express 26(16), 19730–19741 (2018).
    [Crossref]
  10. C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
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    [Crossref]
  12. 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(26), 261112 (2014).
    [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(25), 250403 (2015).
    [Crossref]
  14. A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
    [Crossref]
  15. D. E. Knuth, The Art of Computer Programming (Addison-Wesley, 1997).
  16. A. Aspect, “Bell's inequality test: more ideal than ever,” Nature 398(6724), 189–190 (1999).
    [Crossref]
  17. N. Nisan and A. Ta-Shma, “Extracting Randomness: A Survey and New Constructions,” J. Comput. Syst. Sci. 58(1), 148–173 (1999).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  24. I. R. Senitzky, “Phase correlation in cascade spontaneous emission by a multilevel system: atomic memory,” J. Opt. Soc. Am. B 1(6), 879–881 (1984).
    [Crossref]
  25. R. H. Dicke, “Coherence in Spontaneous Radiation Processes,” Phys. Rev. 93(1), 99–110 (1954).
    [Crossref]
  26. K. Nakata, A. Tomita, M. Fujiwara, K.-I. Yoshino, A. Tajima, A. Okamoto, and K. Ogawa, “Intensity fluctuation of a gain-switched semiconductor laser for quantum key distribution systems,” Opt. Express 25(2), 622–634 (2017).
    [Crossref]
  27. V. S. Pugachev, Probability Theory and Mathematical Statistics for Engineers (Oxford University, 1984).
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    [Crossref]
  29. R. Shakhovoy, V. Sharoglazova, A. Udaltsov, A. Duplinsky, V. Kurochkin, and Y. Kurochkin, “Influence of chirp, jitter and relaxation oscillations on laser pulse interference in optical quantum random number generator,” to be published
  30. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

2019 (1)

Q. Zhou, R. Valivarthi, C. John, and W. Tittel, “Practical quantum random-number generation based on sampling vacuum fluctuations,” Quantum Eng. 1(1), e8 (2019).
[Crossref]

2018 (1)

2017 (2)

2016 (1)

2015 (1)

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(25), 250403 (2015).
[Crossref]

2014 (2)

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(26), 261112 (2014).
[Crossref]

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, 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(2), 1645–1654 (2014).
[Crossref]

2013 (1)

X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, “Postprocessing for quantum random-number generators: Entropy evaluation and randomness extraction,” Phys. Rev. A 87(6), 062327 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (1)

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(10), 711–715 (2010).
[Crossref]

2007 (1)

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
[Crossref]

2006 (1)

R. J. Glauber, “Nobel Lecture: One Hundred Years of Light Quanta,” Rev. Mod. Phys. 78(4), 1267–1278 (2006).
[Crossref]

2001 (1)

L. Trevisan, “Extractors and pseudorandom generators,” J. Assoc. Comput. Mach. 48(4), 860–879 (2001).
[Crossref]

1999 (2)

A. Aspect, “Bell's inequality test: more ideal than ever,” Nature 398(6724), 189–190 (1999).
[Crossref]

N. Nisan and A. Ta-Shma, “Extracting Randomness: A Survey and New Constructions,” J. Comput. Syst. Sci. 58(1), 148–173 (1999).
[Crossref]

1994 (1)

A. N. Oraevskii, “Spontaneous emission in a cavity,” Phys.-Usp. 37(4), 393–405 (1994).
[Crossref]

1993 (1)

Y. Mansour, N. Nisan, and P. Tiwari, “The computational complexity of universal hashing,” Theor. Comput. Sci. 107(1), 121–133 (1993).
[Crossref]

1986 (1)

C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4(3), 298–311 (1986).
[Crossref]

1984 (1)

1982 (1)

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

1954 (1)

R. H. Dicke, “Coherence in Spontaneous Radiation Processes,” Phys. Rev. 93(1), 99–110 (1954).
[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(25), 250403 (2015).
[Crossref]

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, 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(2), 1645–1654 (2014).
[Crossref]

Acín, A.

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, 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(2), 1645–1654 (2014).
[Crossref]

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
[Crossref]

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(25), 250403 (2015).
[Crossref]

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, 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(2), 1645–1654 (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(10), 711–715 (2010).
[Crossref]

Anzolin, G.

Aspect, A.

A. Aspect, “Bell's inequality test: more ideal than ever,” Nature 398(6724), 189–190 (1999).
[Crossref]

Banks, D.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Barker, E.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Brunner, N.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
[Crossref]

Capmany, J.

Comandar, L. C.

Curty, M.

Dicke, R. H.

R. H. Dicke, “Coherence in Spontaneous Radiation Processes,” Phys. Rev. 93(1), 99–110 (1954).
[Crossref]

Dong, R.

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(10), 711–715 (2010).
[Crossref]

Dray, J.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Duplinsky, A.

R. Shakhovoy, V. Sharoglazova, A. Udaltsov, A. Duplinsky, V. Kurochkin, and Y. Kurochkin, “Influence of chirp, jitter and relaxation oscillations on laser pulse interference in optical quantum random number generator,” to be published

Dynes, J. F.

L. C. Comandar, M. Lucamarini, B. Fröhlich, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Near perfect mode overlap between independently seeded, gain-switched lasers,” Opt. Express 24(16), 17849–17859 (2016).
[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(26), 261112 (2014).
[Crossref]

Fröhlich, B.

L. C. Comandar, M. Lucamarini, B. Fröhlich, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Near perfect mode overlap between independently seeded, gain-switched lasers,” Opt. Express 24(16), 17849–17859 (2016).
[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(26), 261112 (2014).
[Crossref]

Fujiwara, M.

Gabriel, C.

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(10), 711–715 (2010).
[Crossref]

Garcia-Escartin, J. C.

M. Herrero-Collantes and J. C. Garcia-Escartin, “Quantum random number generators,” Rev. Mod. Phys. 89(1), 015004 (2017).
[Crossref]

Gisin, N.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
[Crossref]

Glauber, R. J.

R. J. Glauber, “Nobel Lecture: One Hundred Years of Light Quanta,” Rev. Mod. Phys. 78(4), 1267–1278 (2006).
[Crossref]

Heckert, A.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Henry, C.

C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4(3), 298–311 (1986).
[Crossref]

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

Herrero-Collantes, M.

M. Herrero-Collantes and J. C. Garcia-Escartin, “Quantum random number generators,” Rev. Mod. Phys. 89(1), 015004 (2017).
[Crossref]

Jofre, M.

John, C.

Q. Zhou, R. Valivarthi, C. John, and W. Tittel, “Practical quantum random-number generation based on sampling vacuum fluctuations,” Quantum Eng. 1(1), e8 (2019).
[Crossref]

Kennard, J. E.

Knuth, D. E.

D. E. Knuth, The Art of Computer Programming (Addison-Wesley, 1997).

Kurochkin, V.

R. Shakhovoy, V. Sharoglazova, A. Udaltsov, A. Duplinsky, V. Kurochkin, and Y. Kurochkin, “Influence of chirp, jitter and relaxation oscillations on laser pulse interference in optical quantum random number generator,” to be published

Kurochkin, Y.

R. Shakhovoy, V. Sharoglazova, A. Udaltsov, A. Duplinsky, V. Kurochkin, and Y. Kurochkin, “Influence of chirp, jitter and relaxation oscillations on laser pulse interference in optical quantum random number generator,” to be published

Leigh, S.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Leuchs, G.

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(10), 711–715 (2010).
[Crossref]

Levenson, M.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Lo, H.-K.

X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, “Postprocessing for quantum random-number generators: Entropy evaluation and randomness extraction,” Phys. Rev. A 87(6), 062327 (2013).
[Crossref]

F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express 20(11), 12366–12377 (2012).
[Crossref]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University, 2000).

Lucamarini, M.

L. C. Comandar, M. Lucamarini, B. Fröhlich, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Near perfect mode overlap between independently seeded, gain-switched lasers,” Opt. Express 24(16), 17849–17859 (2016).
[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(26), 261112 (2014).
[Crossref]

Ma, X.

X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, “Postprocessing for quantum random-number generators: Entropy evaluation and randomness extraction,” Phys. Rev. A 87(6), 062327 (2013).
[Crossref]

F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express 20(11), 12366–12377 (2012).
[Crossref]

Mahler, D. H.

Mansour, Y.

Y. Mansour, N. Nisan, and P. Tiwari, “The computational complexity of universal hashing,” Theor. Comput. Sci. 107(1), 121–133 (1993).
[Crossref]

Marquardt, C.

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(10), 711–715 (2010).
[Crossref]

Massar, S.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
[Crossref]

Matthews, J. C. F.

Mauerer, W.

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(10), 711–715 (2010).
[Crossref]

Mitchell, M. W.

Mitrani, D.

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(25), 250403 (2015).
[Crossref]

Nakata, K.

Nechvatal, J.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Nisan, N.

N. Nisan and A. Ta-Shma, “Extracting Randomness: A Survey and New Constructions,” J. Comput. Syst. Sci. 58(1), 148–173 (1999).
[Crossref]

Y. Mansour, N. Nisan, and P. Tiwari, “The computational complexity of universal hashing,” Theor. Comput. Sci. 107(1), 121–133 (1993).
[Crossref]

Ogawa, K.

Okamoto, A.

Oraevskii, A. N.

A. N. Oraevskii, “Spontaneous emission in a cavity,” Phys.-Usp. 37(4), 393–405 (1994).
[Crossref]

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic Publishers, 1988).

Pironio, S.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
[Crossref]

Plews, A.

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(26), 261112 (2014).
[Crossref]

Pruneri, V.

Pugachev, V. S.

V. S. Pugachev, Probability Theory and Mathematical Statistics for Engineers (Oxford University, 1984).

Qi, B.

X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, “Postprocessing for quantum random-number generators: Entropy evaluation and randomness extraction,” Phys. Rev. A 87(6), 062327 (2013).
[Crossref]

F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express 20(11), 12366–12377 (2012).
[Crossref]

Raffaelli, F.

Rukhin, A.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Scarani, V.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
[Crossref]

Senitzky, I. R.

Shakhovoy, R.

R. Shakhovoy, V. Sharoglazova, A. Udaltsov, A. Duplinsky, V. Kurochkin, and Y. Kurochkin, “Influence of chirp, jitter and relaxation oscillations on laser pulse interference in optical quantum random number generator,” to be published

Sharoglazova, V.

R. Shakhovoy, V. Sharoglazova, A. Udaltsov, A. Duplinsky, V. Kurochkin, and Y. Kurochkin, “Influence of chirp, jitter and relaxation oscillations on laser pulse interference in optical quantum random number generator,” to be published

Shields, A. J.

L. C. Comandar, M. Lucamarini, B. Fröhlich, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Near perfect mode overlap between independently seeded, gain-switched lasers,” Opt. Express 24(16), 17849–17859 (2016).
[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(26), 261112 (2014).
[Crossref]

Sibson, P.

Smid, M.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Soto, J.

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Steinlechner, F.

Svelto, O.

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Figures (4)

Fig. 1.
Fig. 1. (a) Quantum PDF of the interference signal (Eq. (4)) for three different values of the visibility $\eta$ (0.6, 0.8, and 1). (b) Monte-Carlo simulations of the signal PDF in the presence of fluctuation of ${s_1}$ and ${s_2}$ in Eq. (1). (c) Monte-Carlo simulations of the signal PDF in the presence of fluctuation of ${s_1}$ and ${s_2}$ and the photodetector’s noise as well.
Fig. 2.
Fig. 2. Monte-Carlo simulations of the signal PDF taking into account the influence of the “linear chirp + jitter” effect. The values of the jitter rms are shown on the corresponding simulations.
Fig. 3.
Fig. 3. (a) The theoretical dependence of the quantum reduction factor $\Gamma $ on the photodetector noise width ${\sigma _\zeta }$ in case of digitization of the interference signal by the comparator. Monte-Carlo simulations of the PDF of the interference signal $\tilde{S}$ corresponding to the three different values of ${\sigma _\zeta }$ are shown to the right of the curve. (b) The theoretical dependence of the quantum reduction factor $\Gamma $ on the PDF broadening factor $B = {W \mathord{\left/ {\vphantom {W {({{\tilde{S}}_2} - {{\tilde{S}}_1})}}} \right.} {({{\tilde{S}}_2} - {{\tilde{S}}_1})}}$. (c) Experimental PDF of the interference signal.
Fig. 4.
Fig. 4. (a) The schematic diagram of the QRNG: C0, C1, and C2 are high-speed comparators, LD is the laser diode, $V_{th}^1$ and $V_{th}^2$ in the inset stand for the threshold voltages of the comparators C1 and C2, respectively. (b) The result of the NIST statistical suite for one of the obtained raw random bit sequences. To pass the test we imposed the condition:

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

S ~ = s 1 + s 2 + 2 η s 1 s 2 cos Δ Φ ,
f Δ Φ = { 1 π , Δ Φ [ 0 , π ) 0 , Δ Φ [ 0 , π ) .
F S ~ Q ( y ) = S ~ < y f Δ Φ ( x ) d x ,
f S ~ Q ( x ) = [ π ( x S ~ m i n ) ( S ~ m a x x ) ] 1 ,
S ~ m i n = s 1 + s 2 2 η s 1 s 2 , S ~ m a x = s 1 + s 2 + 2 η s 1 s 2 .
S ~ m a x S ~ m i n w Δ Φ = 4 η s 1 s 2 ,
F S ~ ( y ) = S ~ < y f Δ Φ ( x 1 ) f s 1 ( x 2 ) f s 2 ( x 3 ) d x 1 d x 2 d x 3 ,
S ~ S ~ + ζ ,
η = e ( 1 + α 2 ) Δ t 2 8 w 2 ,
H Q = log 2 ( S ~ m i n S ~ m i n + w Δ Φ / w Δ Φ 2 2 f S ~ Q ( x ) d x ) = 1 ,
H = log 2 ( S ~ m i n S ~ m i n + w Δ Φ / w Δ Φ 2 2 f S ~ ( x ) d x ) 1.
Γ = 1 2 H .
p m a x = S ~ m i n S ~ m i n + Δ u f S ~ Q ( x ) d x ,
Γ = n 1 + H Q H ,
H = log 2 ( S ~ m i n S ~ m i n + Δ u f S ~ ( x ) d x )
f S ~ i = | R i R i + 1 | Δ V ( 1 + R i + R i + 1 + R i R i + 1 ) ,
P = V t h Δ V Γ 1 V t h + Δ V Γ 2 f S ~ ( x ) d x ,
V t h V t h + Δ V Γ f S ~ ( x ) d x = Γ 1 2 Γ ,
f Δ Φ ( x ) = 1 σ Δ Φ 2 π exp ( ( x Δ θ ) 2 2 σ Δ Φ 2 ) .
f Δ Φ ( x ) { p = ± 1 j = f Δ Φ ( p x + 2 π j ) , x [ 0 , π ) 0 , x [ 0 , π ) ,
f Δ Φ ( x ) = J ( x 2 Δ θ 2 , e σ Δ φ 2 / σ Δ φ 2 2 2 ) + J ( x 2 + Δ θ 2 , e σ Δ φ 2 / σ Δ φ 2 2 2 ) 2 π ,
J ( u , q ) = 1 + 2 j = 1 q j 2 cos ( 2 j u ) .
J ( u , q ) = 1 + 2 q cos 2 u .
f Δ Φ = { 1 π , Δ Φ [ 0 , π ) 0 , Δ Φ [ 0 , π ) ,

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