Abstract

In 2010 Qi et al. [Opt. Lett. 35(3), 312 (2010)] demonstrated a random number generator based on the drift of the phase of a laser due to spontaneous emission, The out-of-the-lab implementation of this scheme presents two main drawbacks: it requires a long and highly unbalanced interferometer to generate a random phase with uniform probability distribution, or alternatively, a shorter and slightly unbalanced interferometer that notwithstanding requires active stabilization and does not generate a uniform probability distribution without randomness extraction. Here we demonstrate that making use of the random nature of the phase difference between two independent laser sources and two coherent detectors we can overcome these limitations. The two main advantages of the system demonstrated are: i) it generates a probability distribution of quantum origin which is intrinsically uniform and thus in principle needs no randomness extraction for obtaining a uniform distribution, and ii) the phase is measured with telecom equipment routinely used for high capacity coherent optical communications. The speed of random bit generation is determined by the photodetector bandwidth and the linewidth of the lasers. As a by-product of our method, we have obtained images of how phase noise develops with time in a laser. This provides a highly visual alternative way of measuring the coherence time of a laser.

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

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References

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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 (4)

H. Zhou, P. Zeng, M. Razavi, and X. Ma, “Randomness quantification of coherent detection,” Phys. Rev. A 98(4), 042321 (2018).
[Crossref]

M. Avesani, D. G. Marangon, G. Vallone, and P. Villoresi, “Source-device-independent heterodyne-based quantum random number generator at 17 gbps,” Nat. Commun. 9(1), 5365 (2018).
[Crossref]

C. Leung, A. Brown, H. Nguyen, A. S. Friedman, D. I. Kaiser, and J. Gallicchio, “Astronomical random numbers for quantum foundations experiments,” Phys. Rev. A 97(4), 042120 (2018).
[Crossref]

S. Chen, “Random number generators go public,” Science 360(6396), 1383–1384 (2018).
[Crossref]

2017 (2)

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

S.-H. Sun and F. Xu, “Experimental study of a quantum random-number generator based on two independent lasers,” Phys. Rev. A 96(6), 062314 (2017).
[Crossref]

2016 (2)

2015 (2)

M. W. Mitchell, C. Abellan, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91(1), 012314 (2015).
[Crossref]

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

2014 (1)

2010 (2)

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(3), 312–314 (2010).
[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. Photonics 4(10), 711–715 (2010).
[Crossref]

2009 (1)

Q. Zhou, X. Liao, K. wo Wong, Y. Hu, and D. Xiao, “True random number generator based on mouse movement and chaotic hash function,” Inf. Sci. 179(19), 3442–3450 (2009).
[Crossref]

2007 (1)

P. L’Ecuyer and R. Simard, “Testu01: A c library for empirical testing of random number generators,” ACM Trans. Math. Softw. 33(4), 22–es (2007).
[Crossref]

1986 (1)

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

1982 (1)

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

1978 (1)

R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[Crossref]

1975 (1)

K. V. Mardia and P. J. Zemroch, “Algorithm as 86: The von mises distribution function,” J R Stat. Soc C.-Appl 24(2), 268–272 (1975).
[Crossref]

1949 (1)

C. E. Shannon, “Communication theory of secrecy systems,” Bell Syst. Tech. J. 28(4), 656–715 (1949).
[Crossref]

Abellan, C.

Abellán, C.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (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.

Adleman, L.

R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[Crossref]

Alvaro, M.

L. L. Bonilla, M. Alvaro, and M. Carretero, “Chaos-based true random number generators,” J. Math. Ind. 7(1), 1 (2016).
[Crossref]

Amaya, W.

C. Abellan, W. Amaya, D. Domenech, P. Muñoz, J. Capmany, S. Longhi, M. W. Mitchell, and V. Pruneri, “Quantum entropy source on an InP photonic integrated circuit for random number generation,” Optica 3(9), 989–994 (2016).
[Crossref]

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

M. W. Mitchell, C. Abellan, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91(1), 012314 (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]

Avesani, M.

M. Avesani, D. G. Marangon, G. Vallone, and P. Villoresi, “Source-device-independent heterodyne-based quantum random number generator at 17 gbps,” Nat. Commun. 9(1), 5365 (2018).
[Crossref]

Baish, M. L.

M. S. Turan, E. Barker, J. Kelsey, K. A. McKay, M. L. Baish, and M. Boyle, “Recommendation for the entropy sources used for random bit generation,” Tech. rep., NIST, (2018).

Banks, D. L.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Barker, E.

M. S. Turan, E. Barker, J. Kelsey, K. A. McKay, M. L. Baish, and M. Boyle, “Recommendation for the entropy sources used for random bit generation,” Tech. rep., NIST, (2018).

Barker, E. B.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Bassham, L. E.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Bauer, D.

R. G. Brown, D. Eddelbuettel, and D. Bauer, “Dieharder: A Random Number Test Suite. Version 3.31.1,”.

Beyer, J.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Bonilla, L. L.

L. L. Bonilla, M. Alvaro, and M. Carretero, “Chaos-based true random number generators,” J. Math. Ind. 7(1), 1 (2016).
[Crossref]

Boyle, M.

M. S. Turan, E. Barker, J. Kelsey, K. A. McKay, M. L. Baish, and M. Boyle, “Recommendation for the entropy sources used for random bit generation,” Tech. rep., NIST, (2018).

Brown, A.

C. Leung, A. Brown, H. Nguyen, A. S. Friedman, D. I. Kaiser, and J. Gallicchio, “Astronomical random numbers for quantum foundations experiments,” Phys. Rev. A 97(4), 042120 (2018).
[Crossref]

Brown, R. G.

R. G. Brown, D. Eddelbuettel, and D. Bauer, “Dieharder: A Random Number Test Suite. Version 3.31.1,”.

Capmany, J.

Carretero, M.

L. L. Bonilla, M. Alvaro, and M. Carretero, “Chaos-based true random number generators,” J. Math. Ind. 7(1), 1 (2016).
[Crossref]

Chen, S.

S. Chen, “Random number generators go public,” Science 360(6396), 1383–1384 (2018).
[Crossref]

Chi, Y.-M.

Curty, M.

Domenech, D.

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. F.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Eddelbuettel, D.

R. G. Brown, D. Eddelbuettel, and D. Bauer, “Dieharder: A Random Number Test Suite. Version 3.31.1,”.

Friedman, A. S.

C. Leung, A. Brown, H. Nguyen, A. S. Friedman, D. I. Kaiser, and J. Gallicchio, “Astronomical random numbers for quantum foundations experiments,” Phys. Rev. A 97(4), 042120 (2018).
[Crossref]

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]

Gallicchio, J.

C. Leung, A. Brown, H. Nguyen, A. S. Friedman, D. I. Kaiser, and J. Gallicchio, “Astronomical random numbers for quantum foundations experiments,” Phys. Rev. A 97(4), 042120 (2018).
[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]

Gentle, J. E.

J. E. Gentle, Random number generation and Monte Carlo methods, Statistics and computing (Springer, 2003), 2nd ed.

Gerrits, T.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Giustina, M.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Handsteiner, J.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Heckert, N. A.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (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]

Hochrainer, A.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Hu, Y.

Q. Zhou, X. Liao, K. wo Wong, Y. Hu, and D. Xiao, “True random number generator based on mouse movement and chaotic hash function,” Inf. Sci. 179(19), 3442–3450 (2009).
[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]

Kaiser, D. I.

C. Leung, A. Brown, H. Nguyen, A. S. Friedman, D. I. Kaiser, and J. Gallicchio, “Astronomical random numbers for quantum foundations experiments,” Phys. Rev. A 97(4), 042120 (2018).
[Crossref]

Kelsey, J.

M. S. Turan, E. Barker, J. Kelsey, K. A. McKay, M. L. Baish, and M. Boyle, “Recommendation for the entropy sources used for random bit generation,” Tech. rep., NIST, (2018).

Kofler, J.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

L’Ecuyer, P.

P. L’Ecuyer and R. Simard, “Testu01: A c library for empirical testing of random number generators,” ACM Trans. Math. Softw. 33(4), 22–es (2007).
[Crossref]

Larsson, J.-Å.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Leigh, S. D.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (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]

Leung, C.

C. Leung, A. Brown, H. Nguyen, A. S. Friedman, D. I. Kaiser, and J. Gallicchio, “Astronomical random numbers for quantum foundations experiments,” Phys. Rev. A 97(4), 042120 (2018).
[Crossref]

Levenson, M.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Liao, X.

Q. Zhou, X. Liao, K. wo Wong, Y. Hu, and D. Xiao, “True random number generator based on mouse movement and chaotic hash function,” Inf. Sci. 179(19), 3442–3450 (2009).
[Crossref]

Lita, A. E.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Lo, H.-K.

Longhi, S.

Ma, X.

H. Zhou, P. Zeng, M. Razavi, and X. Ma, “Randomness quantification of coherent detection,” Phys. Rev. A 98(4), 042321 (2018).
[Crossref]

Marangon, D. G.

M. Avesani, D. G. Marangon, G. Vallone, and P. Villoresi, “Source-device-independent heterodyne-based quantum random number generator at 17 gbps,” Nat. Commun. 9(1), 5365 (2018).
[Crossref]

Mardia, K. V.

K. V. Mardia and P. J. Zemroch, “Algorithm as 86: The von mises distribution function,” J R Stat. Soc C.-Appl 24(2), 268–272 (1975).
[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]

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]

McKay, K. A.

M. S. Turan, E. Barker, J. Kelsey, K. A. McKay, M. L. Baish, and M. Boyle, “Recommendation for the entropy sources used for random bit generation,” Tech. rep., NIST, (2018).

Milton Abramowitz, I. S.

I. S. Milton Abramowitz, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Martino Fine Books, 2014).

Mitchell, M. W.

C. Abellan, W. Amaya, D. Domenech, P. Muñoz, J. Capmany, S. Longhi, M. W. Mitchell, and V. Pruneri, “Quantum entropy source on an InP photonic integrated circuit for random number generation,” Optica 3(9), 989–994 (2016).
[Crossref]

M. W. Mitchell, C. Abellan, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91(1), 012314 (2015).
[Crossref]

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (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]

Muñoz, P.

Nam, S. W.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Nechvatal, J. R.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Nguyen, H.

C. Leung, A. Brown, H. Nguyen, A. S. Friedman, D. I. Kaiser, and J. Gallicchio, “Astronomical random numbers for quantum foundations experiments,” Phys. Rev. A 97(4), 042120 (2018).
[Crossref]

Park, K. I.

K. I. Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications: Including a Concise Review of Mathematical Pre-requisites of Complex Variables, Matrix and Set Operations (Springer Science+Business Media, 2017).

Phelan, K.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Pruneri, V.

C. Abellan, W. Amaya, D. Domenech, P. Muñoz, J. Capmany, S. Longhi, M. W. Mitchell, and V. Pruneri, “Quantum entropy source on an InP photonic integrated circuit for random number generation,” Optica 3(9), 989–994 (2016).
[Crossref]

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (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]

Qi, B.

Qian, L.

Razavi, M.

H. Zhou, P. Zeng, M. Razavi, and X. Ma, “Randomness quantification of coherent detection,” Phys. Rev. A 98(4), 042321 (2018).
[Crossref]

Rivest, R. L.

R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[Crossref]

Rukhin, A. L.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Scheidl, T.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Shalm, L. K.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Shamir, A.

R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[Crossref]

Shannon, C. E.

C. E. Shannon, “Communication theory of secrecy systems,” Bell Syst. Tech. J. 28(4), 656–715 (1949).
[Crossref]

Simard, R.

P. L’Ecuyer and R. Simard, “Testu01: A c library for empirical testing of random number generators,” ACM Trans. Math. Softw. 33(4), 22–es (2007).
[Crossref]

Smid, M. E.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Soto, J.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Steinlechner, F.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Sun, S.-H.

S.-H. Sun and F. Xu, “Experimental study of a quantum random-number generator based on two independent lasers,” Phys. Rev. A 96(6), 062314 (2017).
[Crossref]

Sych, D.

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]

Tittel, W.

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]

Turan, M. S.

M. S. Turan, E. Barker, J. Kelsey, K. A. McKay, M. L. Baish, and M. Boyle, “Recommendation for the entropy sources used for random bit generation,” Tech. rep., NIST, (2018).

Ursin, R.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Valivarthi, R.

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]

Vallone, G.

M. Avesani, D. G. Marangon, G. Vallone, and P. Villoresi, “Source-device-independent heterodyne-based quantum random number generator at 17 gbps,” Nat. Commun. 9(1), 5365 (2018).
[Crossref]

Vangel, M.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Versteegh, M. A.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Villoresi, P.

M. Avesani, D. G. Marangon, G. Vallone, and P. Villoresi, “Source-device-independent heterodyne-based quantum random number generator at 17 gbps,” Nat. Commun. 9(1), 5365 (2018).
[Crossref]

Vo, S.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

Wengerowsky, S.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Wittmann, B.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Wittmann, 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]

wo Wong, K.

Q. Zhou, X. Liao, K. wo Wong, Y. Hu, and D. Xiao, “True random number generator based on mouse movement and chaotic hash function,” Inf. Sci. 179(19), 3442–3450 (2009).
[Crossref]

Xiao, D.

Q. Zhou, X. Liao, K. wo Wong, Y. Hu, and D. Xiao, “True random number generator based on mouse movement and chaotic hash function,” Inf. Sci. 179(19), 3442–3450 (2009).
[Crossref]

Xu, F.

S.-H. Sun and F. Xu, “Experimental study of a quantum random-number generator based on two independent lasers,” Phys. Rev. A 96(6), 062314 (2017).
[Crossref]

Zeilinger, A.

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Zemroch, P. J.

K. V. Mardia and P. J. Zemroch, “Algorithm as 86: The von mises distribution function,” J R Stat. Soc C.-Appl 24(2), 268–272 (1975).
[Crossref]

Zeng, P.

H. Zhou, P. Zeng, M. Razavi, and X. Ma, “Randomness quantification of coherent detection,” Phys. Rev. A 98(4), 042321 (2018).
[Crossref]

Zhou, H.

H. Zhou, P. Zeng, M. Razavi, and X. Ma, “Randomness quantification of coherent detection,” Phys. Rev. A 98(4), 042321 (2018).
[Crossref]

Zhou, Q.

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]

Q. Zhou, X. Liao, K. wo Wong, Y. Hu, and D. Xiao, “True random number generator based on mouse movement and chaotic hash function,” Inf. Sci. 179(19), 3442–3450 (2009).
[Crossref]

ACM Trans. Math. Softw. (1)

P. L’Ecuyer and R. Simard, “Testu01: A c library for empirical testing of random number generators,” ACM Trans. Math. Softw. 33(4), 22–es (2007).
[Crossref]

Bell Syst. Tech. J. (1)

C. E. Shannon, “Communication theory of secrecy systems,” Bell Syst. Tech. J. 28(4), 656–715 (1949).
[Crossref]

Commun. ACM (1)

R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[Crossref]

IEEE J. Quantum Electron. (1)

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

Inf. Sci. (1)

Q. Zhou, X. Liao, K. wo Wong, Y. Hu, and D. Xiao, “True random number generator based on mouse movement and chaotic hash function,” Inf. Sci. 179(19), 3442–3450 (2009).
[Crossref]

J R Stat. Soc C.-Appl (1)

K. V. Mardia and P. J. Zemroch, “Algorithm as 86: The von mises distribution function,” J R Stat. Soc C.-Appl 24(2), 268–272 (1975).
[Crossref]

J. Lightwave Technol. (1)

C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4(3), 298–311 (1986).
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J. Math. Ind. (1)

L. L. Bonilla, M. Alvaro, and M. Carretero, “Chaos-based true random number generators,” J. Math. Ind. 7(1), 1 (2016).
[Crossref]

Nat. Commun. (1)

M. Avesani, D. G. Marangon, G. Vallone, and P. Villoresi, “Source-device-independent heterodyne-based quantum random number generator at 17 gbps,” Nat. Commun. 9(1), 5365 (2018).
[Crossref]

Nat. Photonics (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]

Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (4)

S.-H. Sun and F. Xu, “Experimental study of a quantum random-number generator based on two independent lasers,” Phys. Rev. A 96(6), 062314 (2017).
[Crossref]

M. W. Mitchell, C. Abellan, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91(1), 012314 (2015).
[Crossref]

H. Zhou, P. Zeng, M. Razavi, and X. Ma, “Randomness quantification of coherent detection,” Phys. Rev. A 98(4), 042321 (2018).
[Crossref]

C. Leung, A. Brown, H. Nguyen, A. S. Friedman, D. I. Kaiser, and J. Gallicchio, “Astronomical random numbers for quantum foundations experiments,” Phys. Rev. A 97(4), 042120 (2018).
[Crossref]

Phys. Rev. Lett. (1)

M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of bell’s theorem with entangled photons,” Phys. Rev. Lett. 115(25), 250401 (2015).
[Crossref]

Quantum Eng. (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]

Rev. Mod. Phys. (1)

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

Science (1)

S. Chen, “Random number generators go public,” Science 360(6396), 1383–1384 (2018).
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Other (6)

M. S. Turan, E. Barker, J. Kelsey, K. A. McKay, M. L. Baish, and M. Boyle, “Recommendation for the entropy sources used for random bit generation,” Tech. rep., NIST, (2018).

J. E. Gentle, Random number generation and Monte Carlo methods, Statistics and computing (Springer, 2003), 2nd ed.

I. S. Milton Abramowitz, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Martino Fine Books, 2014).

K. I. Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications: Including a Concise Review of Mathematical Pre-requisites of Complex Variables, Matrix and Set Operations (Springer Science+Business Media, 2017).

R. G. Brown, D. Eddelbuettel, and D. Bauer, “Dieharder: A Random Number Test Suite. Version 3.31.1,”.

L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, E. B. Barker, S. D. Leigh, M. Levenson, M. Vangel, D. L. Banks, N. A. Heckert, J. F. Dray, and S. Vo, “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., Gaithersburg, MD, United States, (2010).

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Figures (5)

Fig. 1.
Fig. 1. (a) Depiction of an experimental setup for a phase noise interference Quantum Random Number generator, in which a laser interferes with itself at points where the phase has changed. (b) Schematic of a balanced detector, in which a phase $\phi$ is introduced in one of the lasers. A balanced detector with a phase delay $\phi$ can measure the quadrature $\hat {x}\left (\phi \right )$ of the interfering field. (c) Schematic of the setup used for a balanced coherent detector QRNG, which measures the quadratures of the interference between two lasers and thus obtains the phase noise of such interference.
Fig. 2.
Fig. 2. Maximum difference between the discretized values of a von Mises distribution of variance $t/\tau _{c}$ and a uniform distribution. Here, the discretization is performed in $2^{8}$ values.
Fig. 3.
Fig. 3. Depiction of the processing methods used to measure the phase noise difference of the lasers. (a) is measured at sample rates of 25 GSamples/s, whereas (b)-(e) are measured at sample rates of 625 kSamples/s. (a) Presents an experimental measurement of the components $I$ (dashed) and $Q$ (continuous) as they are read out. It is noticeable that these two measurements are offset by $\pi /2$, and indeed they draw a circle when plotted against one another, as can be seen in (b). (b) also shows a two-dimensional representation of the measured electrical noise, normalized to the maximum amplitude of the quadrature signals. Both $I$ and $Q$ are digitalized to 8 bits of depth, and are measured over several cycles, resulting in a thicker unit circle. A phase random walk is obtained by following the method of Eqs. 5 and 6. (d) Normalized histogram of measurements of the quadrature signal $I=\langle \hat {x}\left ( 0 \right ) \rangle $ after detection. Blue: arcsine distribution corresponding to the interference of the signals from the lasers. Orange: electrical noise, which corresponds to a narrow (typical deviation of 7.5 mV) normal distribution. The signal $I$ and noise are normalized to the maximum value of the signal (285 mV). The standard deviation of the electrical noise accounts for less than 3 % of the measured signal amplitude. (e) Here we show the histogram of the different phase kicks that are recovered from the phase random walk in (c).
Fig. 4.
Fig. 4. Histograms (with corresponding fits) and phase space trajectories (wrapped in the phase space and unwrapped tail to tip) for three different sample rates, originating from 5 million point sample measurements on a balanced coherent detector sampled at (a,b,c) 7.81MSamples/s, (d,e,f) 500kSamples/s, and (g,h,i) 156.25kSamples/s. Only the first 300 points of each trajectory in the unit circle are shown, but 30 000 points are shown for the unwrapped walks. The loss of correlation between successive kicks starts to become noticeable once the sampling frequency is smaller than the laser linewidths, i.e., at about 200kHz. All histograms are fit with a von Mises distribution (Eq. (3)) with variance (a) $t/\overline {\tau _{c}}=0.172$, (d) $t/\overline {\tau _{c}}=1.78$, and (g) $t/\overline {\tau _{c}}=7.50\times 10^{5}$. (g) was additionally fit with a uniform distribution, with a Goodness of fit Kolmogorov-Smirnov test p-value of 0.56.
Fig. 5.
Fig. 5. Autocorrelation coefficients when the phase is being measured at rates of 156.25 kSamples/s.

Tables (1)

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Table 1. Statistical test results.

Equations (7)

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x ( ϕ ) I 1 I 2 cos ( Δ ω t + ξ ( t ) + ϕ ) .
var ( ξ ( t ) ) = 2 t ( 1 τ c , 1 + 1 τ c , 2 ) = t τ ¯ c ,
P ( ξ ( t ) = θ ) = 1 2 π I 0 ( t / τ ¯ c ) exp [ τ ¯ c t cos ( θ ) ] .
p i = π + i δ π + i ( δ + 1 ) P ( ξ ( t ) = θ ) d θ
Θ ( t ) := unwrap ( arg ( I + i Q ) ) = Δ ω t + ξ ( t ) .
ξ ( t ) = ( d Θ d t Δ ω ) d t .
K Θ Θ ( d ) = i = 1 N d ( Θ i mean ( Θ ) ) ( Θ i + d mean ( Θ ) ) var ( Θ ) ,

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