Abstract

We numerically and experimentally demonstrate the utilization of the synchronization of chaotic lasers for decision making. We perform decision making to solve the multi-armed bandit problem using lag synchronization of chaos in mutually coupled semiconductor lasers. We observe the spontaneous exchanges of the leader-laggard relationship under lag synchronization of chaos, and we find that the leader laser can be controlled by changing the coupling strengths between the two lasers. To solve the multi-armed bandit problem, we select one of the slot machines with unknown hit probabilities based only on the identity of the leader laser while reconfiguring the coupling strength to determine the correct decision. We successfully perform an on-line experimental demonstration of the decision making based on the two-laser coupled architecture. This is the first time that synchronization in chaotic lasers is utilized for decision making, and this study paves the way for novel resources for future photonic intelligence.

© 2019 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]
  26. R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007).
    [Crossref]
  27. E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E 73(6), 066214 (2006).
    [Crossref]
  28. J. F. M. Avila and J. R. R. Leite, “Time delays in the synchronization of chaotic coupled lasers with feedback,” Opt. Express 17(24), 21442–21451 (2009).
    [Crossref]
  29. M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E 76(4), 045201 (2007).
    [Crossref]
  30. I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
    [Crossref]
  31. K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95(5), 052212 (2017).
    [Crossref]
  32. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  33. S.-J. Kim, M. Aono, and M. Hara, “Tug-of-war model for the two-bandit problem: Nonlocally-correlated parallel exploration via resource conservation,” BioSystems 101(1), 29–36 (2010).
    [Crossref]
  34. S.-J. Kim, M. Aono, and E. Nameda, “Efficient decision-making by volume-conserving physical object,” New J. Phys. 17(8), 083023 (2015).
    [Crossref]

2019 (1)

R. Homma, S. Kochi, T. Niiyama, T. Mihana, Y. Mitsui, K. Kanno, A. Uchida, M. Naruse, and S. Sunada, “On-chip photonic decision maker using spontaneous mode switching in a ring laser,” Sci. Rep. 9(1), 9429 (2019).
[Crossref]

2018 (4)

Y. Kuriki, J. Nakayama, K. Takano, and A. Uchida, “Impact of input mask signals on delay-based photonic reservoir computing with semiconductor lasers,” Opt. Express 26(5), 5777–5788 (2018).
[Crossref]

M. Naruse, T. Mihana, H. Hori, H. Saigo, K. Okamura, M. Hasegawa, and A. Uchida, “Scalable photonic reinforcement learning by time-division multiplexing of laser chaos,” Sci. Rep. 8(1), 10890 (2018).
[Crossref]

J. Bueno, S. Maktoobi, L. Froehly, I. Fischer, M. Jacquot, L. Larger, and D. Brunner, “Reinforcement learning in a large-scale photonic recurrent neural network,” Optica 5(6), 756–760 (2018).
[Crossref]

T. Mihana, Y. Terashima, M. Naruse, S.-J. Kim, and A. Uchida, “Memory effect on adaptive decision making with a chaotic semiconductor laser,” Complexity 2018, 4318127 (2018).
[Crossref]

2017 (5)

M. Naruse, Y. Terashima, A. Uchida, and S.-J. Kim, “Ultrafast photonic reinforcement learning based on laser chaos,” Sci. Rep. 7(1), 8772 (2017).
[Crossref]

K. Ugajin, Y. Terashima, K. Iwakawa, A. Uchida, T. Harayama, K. Yoshimura, and M. Inubushi, “Real-time fast physical random number generator with a photonic integrated circuit,” Opt. Express 25(6), 6511 (2017).
[Crossref]

T. Sasaki, I. Kakesu, Y. Mitsui, D. Rontani, A. Uchida, S. Sunada, K. Yoshimura, and M. Inubushi, “Common-signal induced synchronization in photonic integrated circuits and its application to secure key distribution,” Opt. Express 25(21), 26029–26044 (2017).
[Crossref]

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95(5), 052212 (2017).
[Crossref]

2016 (2)

J. Nakayama, K. Kanno, and A. Uchida, “Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal,” Opt. Express 24(8), 8679–8692 (2016).
[Crossref]

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

2015 (3)

M. Naruse, M. Berthel, A. Drezet, S. Huant, M. Aono, H. Hori, and S.-J. Kim, “Single-photon decision maker,” Sci. Rep. 5(1), 13253 (2015).
[Crossref]

R. Sakuraba, K. Iwakawa, K. Kanno, and A. Uchida, “Tb/s physical random bit generation with bandwidth-enhancedchaos in three-cascaded semiconductor lasers,” Opt. Express 23(2), 1470 (2015).
[Crossref]

S.-J. Kim, M. Aono, and E. Nameda, “Efficient decision-making by volume-conserving physical object,” New J. Phys. 17(8), 083023 (2015).
[Crossref]

2014 (1)

M. Naruse, W. Nomura, M. Aono, M. Ohtsu, Y. Sonnefraud, A. Drezet, S. Huant, and S.-J. Kim, “Decision making based on optical excitation transfer via near-field interactions between quantum dots,” J. Appl. Phys. 116(15), 154303 (2014).
[Crossref]

2013 (3)

S.-J. Kim, M. Naruse, M. Aono, M. Ohtsu, and M. Hara, “Decision maker based on nanoscale photo-excitation transfer,” Sci. Rep. 3(1), 2370 (2013).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364–1367 (2013).
[Crossref]

H. Koizumi, S. Morikatsu, H. Aida, T. Nozawa, I. Kakesu, A. Uchida, K. Yoshimura, J. Muramatsu, and P. Davis, “Information-theoretic secure key distribution based on common random-signal induced synchronization in unidirectionally-coupled cascades of semiconductor lasers,” Opt. Express 21(15), 17869–17893 (2013).
[Crossref]

2012 (1)

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

2010 (2)

O. B. Kroemer, R. Detry, J. Piater, and J. Peters, “Combining active learning and reactive control for robot grasping,” Rob. Auton. Syst. 58(9), 1105–1116 (2010).
[Crossref]

S.-J. Kim, M. Aono, and M. Hara, “Tug-of-war model for the two-bandit problem: Nonlocally-correlated parallel exploration via resource conservation,” BioSystems 101(1), 29–36 (2010).
[Crossref]

2009 (1)

2008 (1)

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

2007 (2)

M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E 76(4), 045201 (2007).
[Crossref]

R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007).
[Crossref]

2006 (3)

E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E 73(6), 066214 (2006).
[Crossref]

I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref]

E. A. Rogers-Dakin, J. García-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E 73(4), 045201 (2006).
[Crossref]

2001 (1)

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

1952 (1)

H. Robbins, “Some aspects of the sequential design of experiments,” Bull. Amer. Math. Soc. 58(5), 527–536 (1952).
[Crossref]

Aida, H.

H. Koizumi, S. Morikatsu, H. Aida, T. Nozawa, I. Kakesu, A. Uchida, K. Yoshimura, J. Muramatsu, and P. Davis, “Information-theoretic secure key distribution based on common random-signal induced synchronization in unidirectionally-coupled cascades of semiconductor lasers,” Opt. Express 21(15), 17869–17893 (2013).
[Crossref]

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

Amano, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Antonoglou, I.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Aono, M.

M. Naruse, M. Berthel, A. Drezet, S. Huant, M. Aono, H. Hori, and S.-J. Kim, “Single-photon decision maker,” Sci. Rep. 5(1), 13253 (2015).
[Crossref]

S.-J. Kim, M. Aono, and E. Nameda, “Efficient decision-making by volume-conserving physical object,” New J. Phys. 17(8), 083023 (2015).
[Crossref]

M. Naruse, W. Nomura, M. Aono, M. Ohtsu, Y. Sonnefraud, A. Drezet, S. Huant, and S.-J. Kim, “Decision making based on optical excitation transfer via near-field interactions between quantum dots,” J. Appl. Phys. 116(15), 154303 (2014).
[Crossref]

S.-J. Kim, M. Naruse, M. Aono, M. Ohtsu, and M. Hara, “Decision maker based on nanoscale photo-excitation transfer,” Sci. Rep. 3(1), 2370 (2013).
[Crossref]

S.-J. Kim, M. Aono, and M. Hara, “Tug-of-war model for the two-bandit problem: Nonlocally-correlated parallel exploration via resource conservation,” BioSystems 101(1), 29–36 (2010).
[Crossref]

Avila, J. F. M.

Baker, L.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

Barto, A. G.

R. S. Sutton and A. G. Barto, Reinforcement Learning (MIT University, 1998).

Berthel, M.

M. Naruse, M. Berthel, A. Drezet, S. Huant, M. Aono, H. Hori, and S.-J. Kim, “Single-photon decision maker,” Sci. Rep. 5(1), 13253 (2015).
[Crossref]

Bolton, A.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

Brunner, D.

J. Bueno, S. Maktoobi, L. Froehly, I. Fischer, M. Jacquot, L. Larger, and D. Brunner, “Reinforcement learning in a large-scale photonic recurrent neural network,” Optica 5(6), 756–760 (2018).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364–1367 (2013).
[Crossref]

Bueno, J.

Buldu, J. M.

I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref]

Bunsen, M.

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95(5), 052212 (2017).
[Crossref]

Chen, Y.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

Davis, P.

H. Koizumi, S. Morikatsu, H. Aida, T. Nozawa, I. Kakesu, A. Uchida, K. Yoshimura, J. Muramatsu, and P. Davis, “Information-theoretic secure key distribution based on common random-signal induced synchronization in unidirectionally-coupled cascades of semiconductor lasers,” Opt. Express 21(15), 17869–17893 (2013).
[Crossref]

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

Deshazer, D. J.

E. A. Rogers-Dakin, J. García-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E 73(4), 045201 (2006).
[Crossref]

Detry, R.

O. B. Kroemer, R. Detry, J. Piater, and J. Peters, “Combining active learning and reactive control for robot grasping,” Rob. Auton. Syst. 58(9), 1105–1116 (2010).
[Crossref]

Dieleman, S.

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Drezet, A.

M. Naruse, M. Berthel, A. Drezet, S. Huant, M. Aono, H. Hori, and S.-J. Kim, “Single-photon decision maker,” Sci. Rep. 5(1), 13253 (2015).
[Crossref]

M. Naruse, W. Nomura, M. Aono, M. Ohtsu, Y. Sonnefraud, A. Drezet, S. Huant, and S.-J. Kim, “Decision making based on optical excitation transfer via near-field interactions between quantum dots,” J. Appl. Phys. 116(15), 154303 (2014).
[Crossref]

Elsässer, W.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref]

Fischer, I.

J. Bueno, S. Maktoobi, L. Froehly, I. Fischer, M. Jacquot, L. Larger, and D. Brunner, “Reinforcement learning in a large-scale photonic recurrent neural network,” Optica 5(6), 756–760 (2018).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364–1367 (2013).
[Crossref]

M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E 76(4), 045201 (2007).
[Crossref]

R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007).
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Nham, J.

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Niiyama, T.

R. Homma, S. Kochi, T. Niiyama, T. Mihana, Y. Mitsui, K. Kanno, A. Uchida, M. Naruse, and S. Sunada, “On-chip photonic decision maker using spontaneous mode switching in a ring laser,” Sci. Rep. 9(1), 9429 (2019).
[Crossref]

Nomura, W.

M. Naruse, W. Nomura, M. Aono, M. Ohtsu, Y. Sonnefraud, A. Drezet, S. Huant, and S.-J. Kim, “Decision making based on optical excitation transfer via near-field interactions between quantum dots,” J. Appl. Phys. 116(15), 154303 (2014).
[Crossref]

Nozawa, T.

Ohtsu, M.

M. Naruse, W. Nomura, M. Aono, M. Ohtsu, Y. Sonnefraud, A. Drezet, S. Huant, and S.-J. Kim, “Decision making based on optical excitation transfer via near-field interactions between quantum dots,” J. Appl. Phys. 116(15), 154303 (2014).
[Crossref]

S.-J. Kim, M. Naruse, M. Aono, M. Ohtsu, and M. Hara, “Decision maker based on nanoscale photo-excitation transfer,” Sci. Rep. 3(1), 2370 (2013).
[Crossref]

Okamura, K.

M. Naruse, T. Mihana, H. Hori, H. Saigo, K. Okamura, M. Hasegawa, and A. Uchida, “Scalable photonic reinforcement learning by time-division multiplexing of laser chaos,” Sci. Rep. 8(1), 10890 (2018).
[Crossref]

Okumura, H.

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

Oowada, I.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Panneershelvam, V.

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Peil, M.

M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E 76(4), 045201 (2007).
[Crossref]

I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref]

Peters, J.

O. B. Kroemer, R. Detry, J. Piater, and J. Peters, “Combining active learning and reactive control for robot grasping,” Rob. Auton. Syst. 58(9), 1105–1116 (2010).
[Crossref]

Piater, J.

O. B. Kroemer, R. Detry, J. Piater, and J. Peters, “Combining active learning and reactive control for robot grasping,” Rob. Auton. Syst. 58(9), 1105–1116 (2010).
[Crossref]

Robbins, H.

H. Robbins, “Some aspects of the sequential design of experiments,” Bull. Amer. Math. Soc. 58(5), 527–536 (1952).
[Crossref]

Rogers-Dakin, E. A.

E. A. Rogers-Dakin, J. García-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E 73(4), 045201 (2006).
[Crossref]

Rontani, D.

Rosenbluh, M.

E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E 73(6), 066214 (2006).
[Crossref]

Roy, R.

E. A. Rogers-Dakin, J. García-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E 73(4), 045201 (2006).
[Crossref]

Saigo, H.

M. Naruse, T. Mihana, H. Hori, H. Saigo, K. Okamura, M. Hasegawa, and A. Uchida, “Scalable photonic reinforcement learning by time-division multiplexing of laser chaos,” Sci. Rep. 8(1), 10890 (2018).
[Crossref]

Sakuraba, R.

Sasaki, T.

Schrittwieser, J.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Shiki, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Sifre, L.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Silver, D.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Simonyan, K.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

Someya, H.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Sonnefraud, Y.

M. Naruse, W. Nomura, M. Aono, M. Ohtsu, Y. Sonnefraud, A. Drezet, S. Huant, and S.-J. Kim, “Decision making based on optical excitation transfer via near-field interactions between quantum dots,” J. Appl. Phys. 116(15), 154303 (2014).
[Crossref]

Soriano, M. C.

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364–1367 (2013).
[Crossref]

Sunada, S.

R. Homma, S. Kochi, T. Niiyama, T. Mihana, Y. Mitsui, K. Kanno, A. Uchida, M. Naruse, and S. Sunada, “On-chip photonic decision maker using spontaneous mode switching in a ring laser,” Sci. Rep. 9(1), 9429 (2019).
[Crossref]

T. Sasaki, I. Kakesu, Y. Mitsui, D. Rontani, A. Uchida, S. Sunada, K. Yoshimura, and M. Inubushi, “Common-signal induced synchronization in photonic integrated circuits and its application to secure key distribution,” Opt. Express 25(21), 26029–26044 (2017).
[Crossref]

Sutskever, I.

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Sutton, R. S.

R. S. Sutton and A. G. Barto, Reinforcement Learning (MIT University, 1998).

Takano, K.

Terashima, Y.

T. Mihana, Y. Terashima, M. Naruse, S.-J. Kim, and A. Uchida, “Memory effect on adaptive decision making with a chaotic semiconductor laser,” Complexity 2018, 4318127 (2018).
[Crossref]

M. Naruse, Y. Terashima, A. Uchida, and S.-J. Kim, “Ultrafast photonic reinforcement learning based on laser chaos,” Sci. Rep. 7(1), 8772 (2017).
[Crossref]

K. Ugajin, Y. Terashima, K. Iwakawa, A. Uchida, T. Harayama, K. Yoshimura, and M. Inubushi, “Real-time fast physical random number generator with a photonic integrated circuit,” Opt. Express 25(6), 6511 (2017).
[Crossref]

Torrent, M. C.

I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref]

Uchida, A.

R. Homma, S. Kochi, T. Niiyama, T. Mihana, Y. Mitsui, K. Kanno, A. Uchida, M. Naruse, and S. Sunada, “On-chip photonic decision maker using spontaneous mode switching in a ring laser,” Sci. Rep. 9(1), 9429 (2019).
[Crossref]

M. Naruse, T. Mihana, H. Hori, H. Saigo, K. Okamura, M. Hasegawa, and A. Uchida, “Scalable photonic reinforcement learning by time-division multiplexing of laser chaos,” Sci. Rep. 8(1), 10890 (2018).
[Crossref]

T. Mihana, Y. Terashima, M. Naruse, S.-J. Kim, and A. Uchida, “Memory effect on adaptive decision making with a chaotic semiconductor laser,” Complexity 2018, 4318127 (2018).
[Crossref]

Y. Kuriki, J. Nakayama, K. Takano, and A. Uchida, “Impact of input mask signals on delay-based photonic reservoir computing with semiconductor lasers,” Opt. Express 26(5), 5777–5788 (2018).
[Crossref]

T. Sasaki, I. Kakesu, Y. Mitsui, D. Rontani, A. Uchida, S. Sunada, K. Yoshimura, and M. Inubushi, “Common-signal induced synchronization in photonic integrated circuits and its application to secure key distribution,” Opt. Express 25(21), 26029–26044 (2017).
[Crossref]

K. Ugajin, Y. Terashima, K. Iwakawa, A. Uchida, T. Harayama, K. Yoshimura, and M. Inubushi, “Real-time fast physical random number generator with a photonic integrated circuit,” Opt. Express 25(6), 6511 (2017).
[Crossref]

M. Naruse, Y. Terashima, A. Uchida, and S.-J. Kim, “Ultrafast photonic reinforcement learning based on laser chaos,” Sci. Rep. 7(1), 8772 (2017).
[Crossref]

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95(5), 052212 (2017).
[Crossref]

J. Nakayama, K. Kanno, and A. Uchida, “Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal,” Opt. Express 24(8), 8679–8692 (2016).
[Crossref]

R. Sakuraba, K. Iwakawa, K. Kanno, and A. Uchida, “Tb/s physical random bit generation with bandwidth-enhancedchaos in three-cascaded semiconductor lasers,” Opt. Express 23(2), 1470 (2015).
[Crossref]

H. Koizumi, S. Morikatsu, H. Aida, T. Nozawa, I. Kakesu, A. Uchida, K. Yoshimura, J. Muramatsu, and P. Davis, “Information-theoretic secure key distribution based on common random-signal induced synchronization in unidirectionally-coupled cascades of semiconductor lasers,” Opt. Express 21(15), 17869–17893 (2013).
[Crossref]

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

Ugajin, K.

Van Den Driessche, G.

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

Vicente, R.

R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007).
[Crossref]

I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref]

Yoshimori, S.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Yoshimura, K.

BioSystems (1)

S.-J. Kim, M. Aono, and M. Hara, “Tug-of-war model for the two-bandit problem: Nonlocally-correlated parallel exploration via resource conservation,” BioSystems 101(1), 29–36 (2010).
[Crossref]

Bull. Amer. Math. Soc. (1)

H. Robbins, “Some aspects of the sequential design of experiments,” Bull. Amer. Math. Soc. 58(5), 527–536 (1952).
[Crossref]

Complexity (1)

T. Mihana, Y. Terashima, M. Naruse, S.-J. Kim, and A. Uchida, “Memory effect on adaptive decision making with a chaotic semiconductor laser,” Complexity 2018, 4318127 (2018).
[Crossref]

IEEE J. Quantum Electron. (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

J. Appl. Phys. (1)

M. Naruse, W. Nomura, M. Aono, M. Ohtsu, Y. Sonnefraud, A. Drezet, S. Huant, and S.-J. Kim, “Decision making based on optical excitation transfer via near-field interactions between quantum dots,” J. Appl. Phys. 116(15), 154303 (2014).
[Crossref]

Nat. Commun. (1)

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364–1367 (2013).
[Crossref]

Nat. Photonics (1)

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Nature (2)

D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of Go with deep neural networks and tree search,” Nature 529(7587), 484–489 (2016).
[Crossref]

D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature 550(7676), 354–359 (2017).
[Crossref]

New J. Phys. (1)

S.-J. Kim, M. Aono, and E. Nameda, “Efficient decision-making by volume-conserving physical object,” New J. Phys. 17(8), 083023 (2015).
[Crossref]

Opt. Express (7)

J. F. M. Avila and J. R. R. Leite, “Time delays in the synchronization of chaotic coupled lasers with feedback,” Opt. Express 17(24), 21442–21451 (2009).
[Crossref]

J. Nakayama, K. Kanno, and A. Uchida, “Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal,” Opt. Express 24(8), 8679–8692 (2016).
[Crossref]

Y. Kuriki, J. Nakayama, K. Takano, and A. Uchida, “Impact of input mask signals on delay-based photonic reservoir computing with semiconductor lasers,” Opt. Express 26(5), 5777–5788 (2018).
[Crossref]

R. Sakuraba, K. Iwakawa, K. Kanno, and A. Uchida, “Tb/s physical random bit generation with bandwidth-enhancedchaos in three-cascaded semiconductor lasers,” Opt. Express 23(2), 1470 (2015).
[Crossref]

K. Ugajin, Y. Terashima, K. Iwakawa, A. Uchida, T. Harayama, K. Yoshimura, and M. Inubushi, “Real-time fast physical random number generator with a photonic integrated circuit,” Opt. Express 25(6), 6511 (2017).
[Crossref]

H. Koizumi, S. Morikatsu, H. Aida, T. Nozawa, I. Kakesu, A. Uchida, K. Yoshimura, J. Muramatsu, and P. Davis, “Information-theoretic secure key distribution based on common random-signal induced synchronization in unidirectionally-coupled cascades of semiconductor lasers,” Opt. Express 21(15), 17869–17893 (2013).
[Crossref]

T. Sasaki, I. Kakesu, Y. Mitsui, D. Rontani, A. Uchida, S. Sunada, K. Yoshimura, and M. Inubushi, “Common-signal induced synchronization in photonic integrated circuits and its application to secure key distribution,” Opt. Express 25(21), 26029–26044 (2017).
[Crossref]

Opt. Lett. (1)

Optica (1)

Phys. Rev. E (4)

E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E 73(6), 066214 (2006).
[Crossref]

M. Peil, L. Larger, and I. Fischer, “Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling,” Phys. Rev. E 76(4), 045201 (2007).
[Crossref]

E. A. Rogers-Dakin, J. García-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E 73(4), 045201 (2006).
[Crossref]

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95(5), 052212 (2017).
[Crossref]

Phys. Rev. Lett. (3)

I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref]

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

Rob. Auton. Syst. (1)

O. B. Kroemer, R. Detry, J. Piater, and J. Peters, “Combining active learning and reactive control for robot grasping,” Rob. Auton. Syst. 58(9), 1105–1116 (2010).
[Crossref]

Sci. Rep. (5)

M. Naruse, M. Berthel, A. Drezet, S. Huant, M. Aono, H. Hori, and S.-J. Kim, “Single-photon decision maker,” Sci. Rep. 5(1), 13253 (2015).
[Crossref]

M. Naruse, Y. Terashima, A. Uchida, and S.-J. Kim, “Ultrafast photonic reinforcement learning based on laser chaos,” Sci. Rep. 7(1), 8772 (2017).
[Crossref]

S.-J. Kim, M. Naruse, M. Aono, M. Ohtsu, and M. Hara, “Decision maker based on nanoscale photo-excitation transfer,” Sci. Rep. 3(1), 2370 (2013).
[Crossref]

M. Naruse, T. Mihana, H. Hori, H. Saigo, K. Okamura, M. Hasegawa, and A. Uchida, “Scalable photonic reinforcement learning by time-division multiplexing of laser chaos,” Sci. Rep. 8(1), 10890 (2018).
[Crossref]

R. Homma, S. Kochi, T. Niiyama, T. Mihana, Y. Mitsui, K. Kanno, A. Uchida, M. Naruse, and S. Sunada, “On-chip photonic decision maker using spontaneous mode switching in a ring laser,” Sci. Rep. 9(1), 9429 (2019).
[Crossref]

Other (2)

R. S. Sutton and A. G. Barto, Reinforcement Learning (MIT University, 1998).

A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

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Figures (16)

Fig. 1.
Fig. 1. Model of mutually coupled semiconductor lasers.
Fig. 2.
Fig. 2. (a) Temporal waveforms of the laser intensities of laser 1 (red) and laser 2 (blue). (b) Low-pass-filtered temporal waveforms of the laser intensities of laser 1 (red) and laser 2 (blue). The cut-off frequency of the low-pass filter is set to 60 MHz.
Fig. 3.
Fig. 3. (a) Temporal evolution of the short-term cross-correlation values of C1 (red) and C2 (blue). (b) The enlarged view of (upper) the low-pass-filtered temporal waveforms of the laser intensities shown in Fig. 2(b), and (lower) the temporal evolution of the short-term cross-correlation values shown in Fig. 3(a).
Fig. 4.
Fig. 4. Probabilities of becoming the leader laser for laser 1 (red) and laser 2 (blue) (L1 and L2) as a function of the detuning of the coupling strengths Δκ.
Fig. 5.
Fig. 5. Schematic diagram of decision-making using lag synchronization of chaos in mutually coupled semiconductor lasers.
Fig. 6.
Fig. 6. Temporal evolution of (a) short-term cross-correlation values, (b) selection result of the slot machines, and (c) estimated hit probabilities ${\bar{P}_{S1}}$ (red) and ${\bar{P}_{S2}}$ (blue) by decision making. The dotted lines indicate the ideal hit probabilities PS1 and PS2.
Fig. 7.
Fig. 7. Correct decision rate (CDR) as a function of the number of plays. The black, red, and green curves correspond to the hit probabilities {PS1, PS2} = {0.2, 0.8}, {0.3, 0.7}, and {0.4, 0.6}, respectively.
Fig. 8.
Fig. 8. (a) Correct decision rate (CDR) at the play t = 1000 and (b) average hit rate (AHR) as a function of the hit probability of slot machine 1 PS1. The hit probability of slot machine 2 is set to PS2 = 1 − PS1.
Fig. 9.
Fig. 9. Average hit rate (AHR) as a function of (a) the sampling interval τSI and (b) the ratio τ / τSI. The coupling delay times are fixed at τ = 2.0 ns (black), τ = 5.0 ns (red), and τ = 20.0 ns (blue). The ideal AHR is 0.6 for {PS1, PS2} = {0.4, 0.6} (the dotted line).
Fig. 10.
Fig. 10. (a) Experimental setup for lag synchronization of chaos in mutually coupled semiconductor lasers. (b) Experimental configuration of decision-making system.
Fig. 11.
Fig. 11. Experimental result of (a), (b) optical spectra and (c), (d) RF spectra, (a), (c) without optical coupling and (b), (d) with optical coupling for laser 1 (red) and laser 2 (blue).
Fig. 12.
Fig. 12. Experimental result of the temporal waveforms of two laser outputs for different coupling strengths for lasers 1 (red) and 2 (blue). Detuning of the two coupling strengths: (a) Δκ = −0.90, (b) Δκ = −0.15, (c) Δκ = 0.00, (d) Δκ = 0.15, and (e) Δκ = 0.90.
Fig. 13.
Fig. 13. Experimental results of short-term cross-correlation C1 (red) and C2 (blue) between the temporal waveforms of the two laser outputs. Detuning of the two coupling strengths: (a) Δκ = −0.90, (b) Δκ = −0.15, (c) Δκ = 0.00, (d) Δκ = 0.15, and (e) Δκ = 0.90.
Fig. 14.
Fig. 14. Experimental result of probabilities of becoming the leader laser for laser 1 (red) and 2 (blue) (L1and L2) as a function of the detuning of coupling strengths Δκ.
Fig. 15.
Fig. 15. Experimental result of correct decision rate (CDR) as a function of the number of plays. The black and red curves correspond to the hit probabilities {PS1, PS2} = {0.8, 0.2} and {0.7, 0.3}, respectively.
Fig. 16.
Fig. 16. Probabilities of becoming the leader laser for laser 1 (red) and laser 2 (blue) (L1 and L2) as a function of the detuning of the coupling strengths Δκ with the initial optical frequency detuning: (a) Δfini = 4 GHz and (b) Δfini = −4 GHz.

Tables (4)

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Table 1. Parameter values in numerical simulations.

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Table 2. Value of X(t).

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Table 3. Parameter values for decision making.

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Table 4. Parameter values for the calculation of the correct decision rate.

Equations (16)

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d E 1 ( t ) d t = 1 + i α 2 [ G N [ N 1 ( t ) N 0 ] 1 + ε | E 1 ( t ) | 2 1 τ p ] E 1 ( t ) + κ 2 E 2 ( t τ ) exp [ i θ 1 ( t ) ]
d N 1 ( t ) d t = J N 1 ( t ) τ s G N [ N 1 ( t ) N 0 ] 1 + ε | E 1 ( t ) | 2 | E 1 ( t ) | 2
θ 1 = ( ω 2 ω 1 ) t ω 2 τ
d E 2 ( t ) d t = 1 + i α 2 [ G N [ N 2 ( t ) N 0 ] 1 + ε | E 2 ( t ) | 2 1 τ p ] E 2 ( t ) + κ 1 E 1 ( t τ ) exp [ i θ 2 ( t ) ]
d N 2 ( t ) d t = J N 2 ( t ) τ s G N [ N 2 ( t ) N 0 ] 1 + ε | E 2 ( t ) | 2 | E 2 ( t ) | 2
θ 2 = ( ω 1 ω 2 ) t ω 1 τ
C 1 ( t ) = [ I 1 ( t τ ) I ¯ 1 ] [ I 2 ( t ) I ¯ 2 ] τ σ 1 σ 2
C 2 ( t ) = [ I 1 ( t ) I ¯ 1 ] [ I 2 ( t τ ) I ¯ 2 ] τ σ 1 σ 2
κ 1 = { κ max ( Δ κ 0 ) κ {max} + Δ κ ( Δ κ < 0 )
κ 2 = { κ {max} Δ κ ( Δ κ 0 ) κ {max} ( Δ κ < 0 )
Δ κ ( t ) = { k N ( ( int ) T A ( t ) > N ) k T A ( t ) ( | ( int ) T A ( t ) | N ) k N ( ( int ) T A ( t ) < N )
T A ( t + 1 ) = X ( t ) + a T A ( t )
Δ = 2 ( P ¯ S 1 + P ¯ S 2 )
Ω = P ¯ S 1 + P ¯ S 2
CDR ( t ) = 1 n i = 1 n C ( i , t )
A H R = 1 m n i = 1 n t = 1 m H i ( t )

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