## True random numbers from amplified quantum vacuum |

Optics Express, Vol. 19, Issue 21, pp. 20665-20672 (2011)

http://dx.doi.org/10.1364/OE.19.020665

Acrobat PDF (872 KB)

### Abstract

Random numbers are essential for applications ranging from secure communications to numerical simulation and quantitative finance. Algorithms can rapidly produce pseudo-random outcomes, series of numbers that mimic most properties of true random numbers while quantum random number generators (QRNGs) exploit intrinsic quantum randomness to produce true random numbers. Single-photon QRNGs are conceptually simple but produce few random bits per detection. In contrast, vacuum fluctuations are a vast resource for QRNGs: they are broad-band and thus can encode many random bits per second. Direct recording of vacuum fluctuations is possible, but requires shot-noise-limited detectors, at the cost of bandwidth. We demonstrate efficient conversion of vacuum fluctuations to true random bits using optical amplification of vacuum and interferometry. Using commercially-available optical components we demonstrate a QRNG at a bit rate of 1.11 Gbps. The proposed scheme has the potential to be extended to 10 Gbps and even up to 100 Gbps by taking advantage of high speed modulation sources and detectors for optical fiber telecommunication devices.

© 2011 OSA

## 1. Introduction

1. F. Galton, “Dice for statistical experiments,” Nature **42**, 13–14 (1890). [CrossRef]

5. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**, 145–195 (2002). [CrossRef]

8. N. Metropolis and S. Ulam, “The Monte Carlo Method,” J. Am. Statist. Assoc. **44**, 335–341 (1949). [CrossRef]

9. S. Banks, P. Beadling, and A. Ferencz, “FPGA Implementation of Pseudo Random Number Generators for Monte Carlo Methods in Quantitative Finance,” in *Proceedings of the 2008 International Conference on Reconfigurable, Computing and FPGAs, RECONFIG’08*, (IEEE, 2008), pp. 271–276. [CrossRef]

10. S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature **464**, 1021–1024 (2010). [CrossRef] [PubMed]

11. T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. **71**, 1675–1680 (2000). [CrossRef]

12. O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Quantum random number generator using photon-number path entanglement,” Appl. Opt. **48**, 1774–1778 (2009). [CrossRef] [PubMed]

13. M. Stipcevic and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum. **78**, 045104 (2007). [CrossRef] [PubMed]

17. M. Wahl, M. Leifgen, M. Berlin, T. Rhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett. **98**, 171105 (2011). [CrossRef]

18. 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]

19. T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett. **98**, 231103 (2011). [CrossRef]

20. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics **2**, 728–732 (2008). [CrossRef]

22. H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E **81**, 051137 (2010). [CrossRef]

## 2. Device operation

*t*

_{loop}≈ 10 ns provides stable single-mode operation of the interferometer, as shown in Fig. 1(b).

## 3. Laser physics analysis

*a*is the field operator for the mode,

*ω*is its angular frequency,

*γ*is the (energy) decay rate and Γ =

*γ*

^{1/2}

*a*

_{res}+ Γ

_{ASE}is a noise operator, with

*a*

_{res}a reservoir mode. The first term is from attenuation [24

24. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A **30**, 1386–1391 (1984). [CrossRef]

*γ*=

*γ*

_{cav}+

*γ*

_{mat}as follows: The cavity contribution is

*γ*

_{cav}= −

*c*ln(

*R*)/(2

*nL*) = 5 × 10

^{10}s

^{−1}, where

*c*is the speed of light in vacuum,

*R*= 0.3 is the out-coupler reflectivity,

*n*= 3.6 is the refractive index, and

*L*= 300

*μ*m is the cavity length. The material contribution

*γ*

_{mat}ranges from

*cα*/

*n*≈ 10

^{11}s

^{−1}at zero current to

*γ*

_{mat}= −

*γ*

_{cav}at threshold. Here

*α*≈ 10

^{4}cm

^{−1}is the intrinsic absorption of GaAs at 852nm [25

25. M. D. Sturge, “Optical absorption of gallium arsenide between 0.6 and 2.75 ev,” Phys. Rev. **127**, 768–773 (1962). [CrossRef]

*γ*≈ 10

^{11}s

^{−1}, or about 400 dB/ns. This renders completely negligible any prior coherence in the cavity, and the remaining field is an equilibrium between ASE and attenuation. The phase of this field is a true quantum random variable, its value determined by ASE which is driven by vacuum fluctuations. When the laser is taken above threshold, the equilibrated field is amplified, limited by gain depletion [26

26. Y. Suematsu and S. Arai, “Single-mode semiconductor lasers for long-wavelength optical fiber communications and dynamics of semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. **6**, 1436–1449 (2000). [CrossRef]

*P*≈ 3.5 mW or 1.5 × 10

^{7}photons/ns, with about

*P*/

*γ*

_{cav}≈ 3 × 10

^{5}photons in the cavity. The amplification is phase-insensitive, and the phase of the cavity field remains truly random.

^{−5}photons, or ≈ 15 bits below the vacuum fluctuations. The physics of the process can thus support QRNG rates in excess of 100 Gbps.

## 5. Statistical testing

^{6}output pulses were collected in order to characterize the statistical correlations of the acquired raw data and to determine the number of extractable random bits per pulse. The normalized correlation of successive samples as a function of sample delay of the raw data is computed as the modulo-N circular auto-correlation for finite length sequences and it is normalized to the maximum, shown in Fig. 4(a). The correlation of data samples follows a delta-function like behavior which indicates a random sequence with low impact of drifts in the system. The quantum random bit content of the recorded signal is determined as follows: The pulse distribution of Fig. 3 is divided into 2

*equally-sized bins and the Shannon entropy is calculated. As shown in Fig. 4(b), the entropy increases linearly with*

^{b}*b*, up to the value

*b*= 12, where it saturates to 11.8 bits of entropy. The same procedure, applied to the detection noise, finds the classical noise entropy. Subtracting the noise entropy, the quantum optical noise contribution reaches a level of 11.1 bits per pulse at

*b*= 12. Multiple samples per pulse achieves larger accuracy when used together with higher resolution ADC. This allows to better bound the contribution of the classical noise and thus permits to extract more true random bits per pulse.

28. Y. Peres, “Iterating Von Neumann’s procedure for extracting random bits,” Ann. Stat. **20**, 590–597 (1992). [CrossRef]

29. N. Nisan and A. Ta-Shma, “Extracting randomness: a survey and new constructions,” J. Comput. Syst. Sci. **58**, 148–173 (1999). [CrossRef]

## 6. Conclusions

## Acknowledgments

## References and links

1. | F. Galton, “Dice for statistical experiments,” Nature |

2. | R. Corporation, ed., |

3. | T. Kanai, M. Tarui, and Y. Yamada, “Random number generator,” International patent WO2010090328 (2009). |

4. | G. Ribordy and O. Guinnard, “Method and apparatus for generating true random numbers by way of a quantum optics process,” US patent 2007127718 (2007). |

5. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

6. | N. Cerf and L.-P. Lamooureux, “Network distributed quantum random number generation,” International patent GB2473078 (2009). |

7. | N. Ferguson, B. Schneier, and T. Kohno, |

8. | N. Metropolis and S. Ulam, “The Monte Carlo Method,” J. Am. Statist. Assoc. |

9. | S. Banks, P. Beadling, and A. Ferencz, “FPGA Implementation of Pseudo Random Number Generators for Monte Carlo Methods in Quantitative Finance,” in |

10. | S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature |

11. | T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. |

12. | O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Quantum random number generator using photon-number path entanglement,” Appl. Opt. |

13. | M. Stipcevic and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum. |

14. | P. Bronner, A. Strunz, C. Silberhorn, and J. P. Meyn, “Demonstrating quantum random with single photons,” Eur. J. Phys. |

15. | M. Wayne and P. Kwiat, “Low-bias high-speed quantum number generator via shaped optical pulses,” Opt. Express |

16. | M. Fürst, H. Weier, S. Nauerth, D. Marangon, C. Kurtsiefer, and H. Weinfurter, “High speed optical quantum random number generation,” Opt. Express |

17. | M. Wahl, M. Leifgen, M. Berlin, T. Rhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett. |

18. | 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 |

19. | T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett. |

20. | A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics |

21. | 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. |

22. | H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E |

23. | P. R. Tapster and P. M. Gorman, “Apparatus and Method for Generating Random Numbers,” US patent 2009013019 (2009). |

24. | M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A |

25. | M. D. Sturge, “Optical absorption of gallium arsenide between 0.6 and 2.75 ev,” Phys. Rev. |

26. | Y. Suematsu and S. Arai, “Single-mode semiconductor lasers for long-wavelength optical fiber communications and dynamics of semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. |

27. | P. Barreto and V. Rijmen, “The Whirlpool hashing fFunction,” pheattarchive.emporia.edu (2010). |

28. | Y. Peres, “Iterating Von Neumann’s procedure for extracting random bits,” Ann. Stat. |

29. | N. Nisan and A. Ta-Shma, “Extracting randomness: a survey and new constructions,” J. Comput. Syst. Sci. |

30. | P. L’Ecuyer and R. Simard, “TestU01: AC library for empirical testing of random number generators,” ACM Trans. Math. Softw. |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(230.0230) Optical devices : Optical devices

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: July 8, 2011

Revised Manuscript: August 8, 2011

Manuscript Accepted: September 1, 2011

Published: October 3, 2011

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20665

Sort: Year | Journal | Reset

### References

- F. Galton, “Dice for statistical experiments,” Nature42, 13–14 (1890). [CrossRef]
- R. Corporation, ed., A Million Random Digits with 100,000 Normal Deviates (The Free Press, 1955).
- T. Kanai, M. Tarui, and Y. Yamada, “Random number generator,” International patent WO2010090328 (2009).
- G. Ribordy and O. Guinnard, “Method and apparatus for generating true random numbers by way of a quantum optics process,” US patent 2007127718 (2007).
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys.74, 145–195 (2002). [CrossRef]
- N. Cerf and L.-P. Lamooureux, “Network distributed quantum random number generation,” International patent GB2473078 (2009).
- N. Ferguson, B. Schneier, and T. Kohno, Cryptography Engineering: Design Principles and Practical Applications (Wiley Publishing, Inc., 2010).
- N. Metropolis and S. Ulam, “The Monte Carlo Method,” J. Am. Statist. Assoc.44, 335–341 (1949). [CrossRef]
- S. Banks, P. Beadling, and A. Ferencz, “FPGA Implementation of Pseudo Random Number Generators for Monte Carlo Methods in Quantitative Finance,” in Proceedings of the 2008 International Conference on Reconfigurable, Computing and FPGAs, RECONFIG’08, (IEEE, 2008), pp. 271–276. [CrossRef]
- S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010). [CrossRef] [PubMed]
- T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum.71, 1675–1680 (2000). [CrossRef]
- O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Quantum random number generator using photon-number path entanglement,” Appl. Opt.48, 1774–1778 (2009). [CrossRef] [PubMed]
- M. Stipcevic and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum.78, 045104 (2007). [CrossRef] [PubMed]
- P. Bronner, A. Strunz, C. Silberhorn, and J. P. Meyn, “Demonstrating quantum random with single photons,” Eur. J. Phys.30, 1189–1200 (2009). [CrossRef]
- M. Wayne and P. Kwiat, “Low-bias high-speed quantum number generator via shaped optical pulses,” Opt. Express18, 9351–9357 (2010). [CrossRef] [PubMed]
- M. Fürst, H. Weier, S. Nauerth, D. Marangon, C. Kurtsiefer, and H. Weinfurter, “High speed optical quantum random number generation,” Opt. Express18, 13029–13037 (2010). [CrossRef] [PubMed]
- M. Wahl, M. Leifgen, M. Berlin, T. Rhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett.98, 171105 (2011). [CrossRef]
- 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. Photonics4, 711–715 (2010). [CrossRef]
- T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett.98, 231103 (2011). [CrossRef]
- A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008). [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] [PubMed]
- H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E81, 051137 (2010). [CrossRef]
- P. R. Tapster and P. M. Gorman, “Apparatus and Method for Generating Random Numbers,” US patent 2009013019 (2009).
- M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A30, 1386–1391 (1984). [CrossRef]
- M. D. Sturge, “Optical absorption of gallium arsenide between 0.6 and 2.75 ev,” Phys. Rev.127, 768–773 (1962). [CrossRef]
- Y. Suematsu and S. Arai, “Single-mode semiconductor lasers for long-wavelength optical fiber communications and dynamics of semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron.6, 1436–1449 (2000). [CrossRef]
- P. Barreto and V. Rijmen, “The Whirlpool hashing fFunction,” pheattarchive.emporia.edu (2010).
- Y. Peres, “Iterating Von Neumann’s procedure for extracting random bits,” Ann. Stat.20, 590–597 (1992). [CrossRef]
- N. Nisan and A. Ta-Shma, “Extracting randomness: a survey and new constructions,” J. Comput. Syst. Sci.58, 148–173 (1999). [CrossRef]
- P. L’Ecuyer and R. Simard, “TestU01: AC library for empirical testing of random number generators,” ACM Trans. Math. Softw.33, 1–40 (2007).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.