## Quantum random bit generation using stimulated Raman scattering |

Optics Express, Vol. 19, Issue 25, pp. 25173-25180 (2011)

http://dx.doi.org/10.1364/OE.19.025173

Acrobat PDF (824 KB)

### Abstract

Random number sequences are a critical resource in a wide variety of information systems, including applications in cryptography, simulation, and data sampling. We introduce a quantum random number generator based on the phase measurement of Stokes light generated by amplification of zero-point vacuum fluctuations using stimulated Raman scattering. This is an example of quantum noise amplification using the most noise-free process possible: near unitary quantum evolution. The use of phase offers robustness to classical pump noise and the ability to generate multiple bits per measurement. The Stokes light is generated with high intensity and as a result, fast detectors with high signal-to-noise ratios can be used for measurement, eliminating the need for single-photon sensitive devices. The demonstrated implementation uses optical phonons in bulk diamond.

© 2011 OSA

## 1. Introduction

*π*can be used as random numbers [1], but they are deterministically calculable and hence unsuitable for a cryptographic key: if an adversary determines the sequence used, the process is compromised. Therefore deterministic generation systems are unsuitable for high security applications.

2. Y. Shen, L. Tian, and H. Zou, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Phys. Rev. A **81**, 063814 (2010). [CrossRef]

3. H. Schmidt, “Quantum mechanical random number generator,” J. Appl. Phys. **41**, 462–468 (1970). [CrossRef]

4. 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]

5. 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]

6. W. Wei and H. Guo, “Bias-free true random-number generator,” Opt. Lett. **34**, 1876–1878 (2009). [CrossRef] [PubMed]

9. J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. **93**, 031109 (2008). [CrossRef]

10. S. Pironio, A. Acín, S. Massar, A. Boyer 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. U. Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans. Inf. Theory **39**, 733 –742 (1993). [CrossRef]

^{12}bits per second) bit-rates based on phase measurement of Raman scattered light in the macroscopic regime. In a typical Raman scattering process, an input pump photon is annihilated on scattering from a vibrational excitation, and a longer wavelength Stokes photon is created, with the residual energy deposited in the medium as a vibrational quantum, or phonon. The stimulated scattering of pump photons into the Stokes field can be described classically, however the spontaneous initiation of Raman scattering in a Raman generator in the absence of an input Stokes field, or phonons, is a purely quantum phenomenon [12

12. A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” Prog. Quantum Electron. **6**, 55 (1979). [CrossRef]

13. M. G. Raymer and I. A. Walmsley, “Quantum coherence properties of stimulated Raman scattering,” Prog. Opt. **28**, 181 (1990). [CrossRef]

14. M. G. Raymer, K. Rza̦żewski, and J. Mostowski, “Pulse-energy statistics in stimulated Raman scattering,” Opt. Lett. **7**, 71–73 (1982). [CrossRef] [PubMed]

15. I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. **50**, 962–965 (1983). [CrossRef]

16. S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Spatial interference of macroscopic light fields from independent Raman sources,” Phys. Rev. A **43**, 4083–4086 (1991). [CrossRef] [PubMed]

18. M. Belsley, D. T. Smithey, K. Wedding, and M. G. Raymer, “Observation of extreme sensitivity to induced molecular coherence in stimulated Raman scattering,” Phys. Rev. A **48**, 1514 (1993). [CrossRef] [PubMed]

15. I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. **50**, 962–965 (1983). [CrossRef]

## 2. Experiment

^{−1}. At room temperature the Boltzmann population ratio between the optical phonon band |2〉 and the acoustic phonon band |1〉 is 1.7 × 10

^{−3}, and therefore thermal excitation is negligible. A linearly-polarized pump pulse with duration

*τ*= 100ps, energy ℰ

_{p}*=1.6*

_{p}*μ*J, and wavelength

*λ*= 532nm, is focussed into a 3 mm CVD diamond plate oriented along the 〈100〉 axis. The pump generates longitudinal optical phonons at Ω, and a Stokes pulse with mean energy 0.16

_{p}*μ*J is emitted at

*λ*= 573nm; this gives a photon conversion efficiency to the Stokes field of

_{S}*η*= 0.11. The dephasing time for the vibrational excitation is estimated at Γ

^{−1}= 7ps, based on the Raman linewidth and transient coherent ultrafast phonon spectroscopy measurements [20

20. K. C. Lee, B. J. Sussman, J. Nunn, V. O. Lorenz, K. Reim, D. Jaksch, I. A. Walmsley, P. Spizzirri, and S. Prawer, “Comparing phonon dephasing lifetimes in diamond using transient coherent ultrafast phonon spectroscopy,” Diam. Relat. Mater. **19**, 1289 – 1295 (2010). [CrossRef]

*τ*= 14. Using

_{p}*η*and an analytic result for the Stokes pulse energy taken from the fully quantum model [21

21. M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A **24**, 1980–1993 (1981). [CrossRef]

*gL*≈ 29, where

*g*is the steady-state Raman gain coefficient and

*L*is the gain length of the diamond. The experimental parameters therefore satisfy the necessary conditions for transient SRS (

*gL*> Γ

*τ*) in the high-gain limit (

_{p}*gL*Γ

*τ*≫ 1) [13

_{p}13. M. G. Raymer and I. A. Walmsley, “Quantum coherence properties of stimulated Raman scattering,” Prog. Opt. **28**, 181 (1990). [CrossRef]

18. M. Belsley, D. T. Smithey, K. Wedding, and M. G. Raymer, “Observation of extreme sensitivity to induced molecular coherence in stimulated Raman scattering,” Phys. Rev. A **48**, 1514 (1993). [CrossRef] [PubMed]

*k*is introduced and the resulting fringe pattern is recorded on a 2048-pixel line array charge-coupled device (CCD) camera operating at 200 Hz. The fringe measurement is a comparison of the first Stokes field |

*A*|

_{S}*e*

^{−i(Δkx+ϕS)}with the reference Stokes field |

*A*|

_{r}*e*

^{−iϕr}, where

*A*are the field amplitudes, Δ

_{S,r}*k*is the difference in wavevectors in the plane of the detector,

*x*is the position coordinate along the camera array, and

*ϕ*are the field phases. This yields an interferogram given by

_{r,S}*S*∝ |

_{int}*A*||

_{S}*A*| cos(Δ

_{r}*kx*+Δ

*ϕ*) where Δ

*ϕ*= (

*ϕ*–

_{S}*ϕ*) is a phase factor lying randomly on the interval 0 ≤ Δ

_{r}*ϕ*< 2

*π*due to the quantum mechanical origin of

*ϕ*.

_{S}*e.g.*, heterodyne) and high-repetition, or CW, lasers could significantly improve on these practical limitations. Phase-correlations between Stokes fields from serial pump pulses can persist for many dephasing times, with the correlations expected to expire when the coherent excitation level falls below the zero-point quantum level of one phonon per mode [17

17. D. T. Smithey, M. Belsley, K. Wedding, and M. G. Raymer, “Near quantum-limited phase memory in a Raman amplifier,” Phys. Rev. Lett. **67**, 2446–2449 (1991). [CrossRef] [PubMed]

18. M. Belsley, D. T. Smithey, K. Wedding, and M. G. Raymer, “Observation of extreme sensitivity to induced molecular coherence in stimulated Raman scattering,” Phys. Rev. A **48**, 1514 (1993). [CrossRef] [PubMed]

*e*

^{−2Γτ}, where

*τ*is the delay [18

**48**, 1514 (1993). [CrossRef] [PubMed]

^{11}, this gives an average of 0.1 phonons remaining after a delay of 102 ps. The physical limit to measurement rates in diamond is then 9.8 GHz, although higher rates should be possible in Raman media with shorter dephasing times.

## 3. Results

^{6}= 64 possible phases). Any possible bias in the phase measurement is removed by post-processing using a fair bit extractor algorithm [22, 23

23. A. Juels, M. Jakobsson, E. Shriver, and B. Hillyer, “How to turn loaded dice into fair coins,” IEEE Trans. Inf. Theory **46**, 911 –921 (2000). [CrossRef]

*ϕ*. When the Raman process is strong, the coherence of the Stokes light approaches that of the pump, a coherent state, and therefore the minimum phase defined depends on the number of photons per pulse

_{min}*n*as approximately Δ

*ϕ*∼ 1/

_{min}*n*[24].

25. G. Marsaglia, “Diehard battery of tests of randomness,” http://www.stat.fsu.edu/pub/diehard/.

## 4. Theoretical framework

*Q̂*(

*z*,0)

*Q̂*

^{†}(

*z*,0)〉 = 0, the Stokes field initiated by spontaneous Raman scattering is given by [21

21. M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A **24**, 1980–1993 (1981). [CrossRef]

## 5. Conclusion

_{3}, for example, decays ten times faster and still has high Raman gain [19]. This suggests that measurement rates of 100 GHz are feasible without compromising randomness. In this demonstration 6 bits were extracted per measurement, although much higher bit-depths should be possible, in principle. Therefore the product of the measurement rate and the bit-depth per measurement could exceed 1Tbps. Importantly, this would be realizable with fast, high signal-to-noise ratio photodetectors, thus eliminating the need to work with challenging single-photon sensitive devices.

## References and links

1. | G. Marsaglia, “On the randomness of pi and other decimal expansions,” InterStat |

2. | Y. Shen, L. Tian, and H. Zou, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Phys. Rev. A |

3. | H. Schmidt, “Quantum mechanical random number generator,” J. Appl. Phys. |

4. | 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. |

5. | 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 |

6. | W. Wei and H. Guo, “Bias-free true random-number generator,” Opt. Lett. |

7. | M. Ren, E. Wu, Y. Liang, Y. Jian, G. Wu, and H. Zeng, “Quantum random-number generator based on a photon-number-resolving detector,” Phys. Rev. A |

8. | M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, 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. |

9. | J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. |

10. | S. Pironio, A. Acín, S. Massar, A. Boyer 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. | U. Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans. Inf. Theory |

12. | A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” Prog. Quantum Electron. |

13. | M. G. Raymer and I. A. Walmsley, “Quantum coherence properties of stimulated Raman scattering,” Prog. Opt. |

14. | M. G. Raymer, K. Rza̦żewski, and J. Mostowski, “Pulse-energy statistics in stimulated Raman scattering,” Opt. Lett. |

15. | I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. |

16. | S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Spatial interference of macroscopic light fields from independent Raman sources,” Phys. Rev. A |

17. | D. T. Smithey, M. Belsley, K. Wedding, and M. G. Raymer, “Near quantum-limited phase memory in a Raman amplifier,” Phys. Rev. Lett. |

18. | M. Belsley, D. T. Smithey, K. Wedding, and M. G. Raymer, “Observation of extreme sensitivity to induced molecular coherence in stimulated Raman scattering,” Phys. Rev. A |

19. | J. Reintjes and M. Bashkansky, |

20. | K. C. Lee, B. J. Sussman, J. Nunn, V. O. Lorenz, K. Reim, D. Jaksch, I. A. Walmsley, P. Spizzirri, and S. Prawer, “Comparing phonon dephasing lifetimes in diamond using transient coherent ultrafast phonon spectroscopy,” Diam. Relat. Mater. |

21. | M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A |

22. | J. von Neumann, “Various techniques used in connection with random digits,” Nat. Bur. Stand., Appl. Math Ser. |

23. | A. Juels, M. Jakobsson, E. Shriver, and B. Hillyer, “How to turn loaded dice into fair coins,” IEEE Trans. Inf. Theory |

24. | C. Gerry and P. Knight, |

25. | G. Marsaglia, “Diehard battery of tests of randomness,” http://www.stat.fsu.edu/pub/diehard/. |

26. | H. Haken, |

27. | W. Louisell, |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(190.5650) Nonlinear optics : Raman effect

(190.5890) Nonlinear optics : Scattering, stimulated

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(290.5910) Scattering : Scattering, stimulated Raman

(350.5030) Other areas of optics : Phase

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 26, 2011

Revised Manuscript: October 7, 2011

Manuscript Accepted: October 21, 2011

Published: November 23, 2011

**Citation**

Philip J. Bustard, Doug Moffatt, Rune Lausten, Guorong Wu, Ian A. Walmsley, and Benjamin J. Sussman, "Quantum random bit generation using stimulated Raman scattering," Opt. Express **19**, 25173-25180 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25173

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### References

- G. Marsaglia, “On the randomness of pi and other decimal expansions,” InterStat5 (2005).
- Y. Shen, L. Tian, and H. Zou, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Phys. Rev. A81, 063814 (2010). [CrossRef]
- H. Schmidt, “Quantum mechanical random number generator,” J. Appl. Phys.41, 462–468 (1970). [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]
- W. Wei and H. Guo, “Bias-free true random-number generator,” Opt. Lett.34, 1876–1878 (2009). [CrossRef] [PubMed]
- M. Ren, E. Wu, Y. Liang, Y. Jian, G. Wu, and H. Zeng, “Quantum random-number generator based on a photon-number-resolving detector,” Phys. Rev. A83, 023820 (2011). [CrossRef]
- M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, 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]
- J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett.93, 031109 (2008). [CrossRef]
- S. Pironio, A. Acín, S. Massar, A. Boyer 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]
- U. Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans. Inf. Theory39, 733 –742 (1993). [CrossRef]
- A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” Prog. Quantum Electron.6, 55 (1979). [CrossRef]
- M. G. Raymer and I. A. Walmsley, “Quantum coherence properties of stimulated Raman scattering,” Prog. Opt.28, 181 (1990). [CrossRef]
- M. G. Raymer, K. Rza̦żewski, and J. Mostowski, “Pulse-energy statistics in stimulated Raman scattering,” Opt. Lett.7, 71–73 (1982). [CrossRef] [PubMed]
- I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett.50, 962–965 (1983). [CrossRef]
- S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Spatial interference of macroscopic light fields from independent Raman sources,” Phys. Rev. A43, 4083–4086 (1991). [CrossRef] [PubMed]
- D. T. Smithey, M. Belsley, K. Wedding, and M. G. Raymer, “Near quantum-limited phase memory in a Raman amplifier,” Phys. Rev. Lett.67, 2446–2449 (1991). [CrossRef] [PubMed]
- M. Belsley, D. T. Smithey, K. Wedding, and M. G. Raymer, “Observation of extreme sensitivity to induced molecular coherence in stimulated Raman scattering,” Phys. Rev. A48, 1514 (1993). [CrossRef] [PubMed]
- J. Reintjes and M. Bashkansky, Handbook of Optics, Volume IV: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill Professional, 2010), chap. 15, p. 15.1.
- K. C. Lee, B. J. Sussman, J. Nunn, V. O. Lorenz, K. Reim, D. Jaksch, I. A. Walmsley, P. Spizzirri, and S. Prawer, “Comparing phonon dephasing lifetimes in diamond using transient coherent ultrafast phonon spectroscopy,” Diam. Relat. Mater.19, 1289 – 1295 (2010). [CrossRef]
- M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A24, 1980–1993 (1981). [CrossRef]
- J. von Neumann, “Various techniques used in connection with random digits,” Nat. Bur. Stand., Appl. Math Ser.12, 36–38 (1951).
- A. Juels, M. Jakobsson, E. Shriver, and B. Hillyer, “How to turn loaded dice into fair coins,” IEEE Trans. Inf. Theory46, 911 –921 (2000). [CrossRef]
- C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge Univ Pr, 2005).
- G. Marsaglia, “Diehard battery of tests of randomness,” http://www.stat.fsu.edu/pub/diehard/ .
- H. Haken, Encyclopedia of Physics, vol. 25 (Springer, 1970).
- W. Louisell, Quantum Statistical Properties of Radiation (John Wiley and Sons, Inc., New York, 1973).

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