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G. Carleo, C. Ignacio, C. Kyle, D. Laurent, S. Maria, T. Naftali, V.-M. Leslie, and Z. Lenka, “Machine learning and the physical sciences,” arXiv:1903.10563 (2019).

D. Woods and T. J. Naughton, “Photonic neural networks,” Nat. Phys. 8, 257–259 (2012).

[Crossref]

F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Beguin, A. Vernier, T. Lahaye, and A. Browaeys, “Single-atom trapping in holographic 2D arrays of microtraps with arbitrary geometries,” Phys. Rev. X 4, 021034 (2014).

[Crossref]

P. Raccuglia, K. C. Elbert, P. D. Adler, C. Falk, M. B. Wenny, A. Mollo, M. Zeller, S. A. Friedler, J. Schrier, and A. J. Norquist, “Machine-learning-assisted materials discovery using failed experiments,” Nature 533, 73–76 (2016).

[Crossref]

X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018).

[Crossref]

A. J. Maren, C. T. Harston, and R. M. Pap, Handbook of Neural Computing Applications (Academic, 2014).

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

[Crossref]

M. Miscuglio, A. Mehrabian, Z. Hu, S. I. Azzam, J. George, A. V. Kildishev, M. Pelton, and V. J. Sorger, “All-optical nonlinear activation function for photonic neural networks,” Opt. Mater. Express 8, 3851–3863 (2018).

[Crossref]

J. Feldmann, N. Youngblood, C. D. Wright, H. Bhaskaran, and W. H. P. Pernice, “All-optical spiking neurosynaptic networks with self-learning capabilities,” Nature 569, 208–214 (2019).

[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutiérrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).

[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).

[Crossref]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631 (1987).

[Crossref]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631 (1987).

[Crossref]

J. George, R. Amin, A. Mehrabian, J. Khurgin, T. El-Ghazawi, P. R. Prucnal, and V. J. Sorger, “Electrooptic nonlinear activation functions for vector matrix multiplications in optical neural networks,” in Signal Processing in Photonic Communications (Optical Society of America, 2018), paper SpW4G-3.

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[Crossref]

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[Crossref]

J. Liu, Y. Qi, Z. Y. Meng, and L. Fu, “Self-learning Monte Carlo method,” Phys. Rev. B 95, 041101 (2017).

[Crossref]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631 (1987).

[Crossref]

P. Raccuglia, K. C. Elbert, P. D. Adler, C. Falk, M. B. Wenny, A. Mollo, M. Zeller, S. A. Friedler, J. Schrier, and A. J. Norquist, “Machine-learning-assisted materials discovery using failed experiments,” Nature 533, 73–76 (2016).

[Crossref]

F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Beguin, A. Vernier, T. Lahaye, and A. Browaeys, “Single-atom trapping in holographic 2D arrays of microtraps with arbitrary geometries,” Phys. Rev. X 4, 021034 (2014).

[Crossref]

S. Du, M. B. Squires, Y. Imai, L. Czaia, R. A. Saravanan, V. Bright, J. Reichel, T. W. Hansch, and D. Z. Anderson, “Atom-chip Bose-Einstein condensation in a portable vacuum cell,” Phys. Rev. A 70, 053606 (2004).

[Crossref]

X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018).

[Crossref]

H. J. Caulfield, J. Kinser, and S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1583 (1989).

[Crossref]

K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).

[Crossref]

D. M. Farkas, K. M. Hudek, E. A. Salim, S. R. Segal, M. B. Squires, and D. Z. Anderson, “A compact, transportable, microchip-based system for high repetition rate production of Bose-Einstein condensates,” Appl. Phys. Lett. 96, 093102 (2010).

[Crossref]

S. Du, M. B. Squires, Y. Imai, L. Czaia, R. A. Saravanan, V. Bright, J. Reichel, T. W. Hansch, and D. Z. Anderson, “Atom-chip Bose-Einstein condensation in a portable vacuum cell,” Phys. Rev. A 70, 053606 (2004).

[Crossref]

S. D. Sarma, D.-L. Deng, and L.-M. Duan, “Machine learning meets quantum physics,” Phys. Today 72(3), 48–54 (2019).

[Crossref]

D.-L. Deng, X. Li, and S. D. Sarma, “Machine learning topological states,” Phys. Rev. B 96, 195145 (2017).

[Crossref]

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

[Crossref]

P. Raccuglia, K. C. Elbert, P. D. Adler, C. Falk, M. B. Wenny, A. Mollo, M. Zeller, S. A. Friedler, J. Schrier, and A. J. Norquist, “Machine-learning-assisted materials discovery using failed experiments,” Nature 533, 73–76 (2016).

[Crossref]

D. M. Farkas, K. M. Hudek, E. A. Salim, S. R. Segal, M. B. Squires, and D. Z. Anderson, “A compact, transportable, microchip-based system for high repetition rate production of Bose-Einstein condensates,” Appl. Phys. Lett. 96, 093102 (2010).

[Crossref]

J. Liu, H. Shen, Y. Qi, Z. Y. Meng, and L. Fu, “Self-learning Monte Carlo method and cumulative update in fermion systems,” Phys. Rev. B 95, 241104 (2017).

[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).

[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).

[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

[Crossref]

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20, 22783–22795(2012).

[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).

[Crossref]

M. Miscuglio, A. Mehrabian, Z. Hu, S. I. Azzam, J. George, A. V. Kildishev, M. Pelton, and V. J. Sorger, “All-optical nonlinear activation function for photonic neural networks,” Opt. Mater. Express 8, 3851–3863 (2018).

[Crossref]

J. George, R. Amin, A. Mehrabian, J. Khurgin, T. El-Ghazawi, P. R. Prucnal, and V. J. Sorger, “Electrooptic nonlinear activation functions for vector matrix multiplications in optical neural networks,” in Signal Processing in Photonic Communications (Optical Society of America, 2018), paper SpW4G-3.

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutiérrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).

[Crossref]

D. M. Farkas, K. M. Hudek, E. A. Salim, S. R. Segal, M. B. Squires, and D. Z. Anderson, “A compact, transportable, microchip-based system for high repetition rate production of Bose-Einstein condensates,” Appl. Phys. Lett. 96, 093102 (2010).

[Crossref]

S. Du, M. B. Squires, Y. Imai, L. Czaia, R. A. Saravanan, V. Bright, J. Reichel, T. W. Hansch, and D. Z. Anderson, “Atom-chip Bose-Einstein condensation in a portable vacuum cell,” Phys. Rev. A 70, 053606 (2004).

[Crossref]

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13, 346–351 (2019).

[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).

[Crossref]

G. Carleo and M. Troyer, “Solving the quantum many-body problem with artificial neural networks,” Science 355, 602–606 (2017).

[Crossref]

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer-Verlag, 1999).

X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018).

[Crossref]

F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Beguin, A. Vernier, T. Lahaye, and A. Browaeys, “Single-atom trapping in holographic 2D arrays of microtraps with arbitrary geometries,” Phys. Rev. X 4, 021034 (2014).

[Crossref]

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[Crossref]

L. Huang and L. Wang, “Accelerated Monte Carlo simulations with restricted Boltzmann machines,” Phys. Rev. B 95, 035105 (2017).

[Crossref]

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13, 346–351 (2019).

[Crossref]

P. Raccuglia, K. C. Elbert, P. D. Adler, C. Falk, M. B. Wenny, A. Mollo, M. Zeller, S. A. Friedler, J. Schrier, and A. J. Norquist, “Machine-learning-assisted materials discovery using failed experiments,” Nature 533, 73–76 (2016).

[Crossref]

S. Zhang, J. F. Chen, C. Liu, S. Zhou, M. M. T. Loy, G. K. L. Wong, and S. Du, “A dark-line two-dimensional magneto-optical trap of 85Rb atoms with high optical depth,” Rev. Sci. Instrum. 83, 073102 (2012).

[Crossref]

D. Woods and T. J. Naughton, “Photonic neural networks,” Nat. Phys. 8, 257–259 (2012).

[Crossref]

J. Feldmann, N. Youngblood, C. D. Wright, H. Bhaskaran, and W. H. P. Pernice, “All-optical spiking neurosynaptic networks with self-learning capabilities,” Nature 569, 208–214 (2019).

[Crossref]

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

[Crossref]

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13, 346–351 (2019).

[Crossref]

X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018).

[Crossref]

J. Feldmann, N. Youngblood, C. D. Wright, H. Bhaskaran, and W. H. P. Pernice, “All-optical spiking neurosynaptic networks with self-learning capabilities,” Nature 569, 208–214 (2019).

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P. Raccuglia, K. C. Elbert, P. D. Adler, C. Falk, M. B. Wenny, A. Mollo, M. Zeller, S. A. Friedler, J. Schrier, and A. J. Norquist, “Machine-learning-assisted materials discovery using failed experiments,” Nature 533, 73–76 (2016).

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Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13, 346–351 (2019).

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S. Zhang, J. F. Chen, C. Liu, S. Zhou, M. M. T. Loy, G. K. L. Wong, and S. Du, “A dark-line two-dimensional magneto-optical trap of 85Rb atoms with high optical depth,” Rev. Sci. Instrum. 83, 073102 (2012).

[Crossref]

Y. Zhang and E.-A. Kim, “Quantum loop topography for machine learning,” Phys. Rev. Lett. 118, 216401 (2017).

[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).

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Y. Zuo, B. Li, Y. Zhao, Y. Jiang, Y.-C. Chen, P. Chen, G.-B. Jo, J. Liu, and S. Du, “All optical neural network with nonlinear activation functions,” arXiv: 1904.10819 (2019).

S. Zhang, J. F. Chen, C. Liu, S. Zhou, M. M. T. Loy, G. K. L. Wong, and S. Du, “A dark-line two-dimensional magneto-optical trap of 85Rb atoms with high optical depth,” Rev. Sci. Instrum. 83, 073102 (2012).

[Crossref]

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13, 346–351 (2019).

[Crossref]

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13, 346–351 (2019).

[Crossref]

Y. Zuo, B. Li, Y. Zhao, Y. Jiang, Y.-C. Chen, P. Chen, G.-B. Jo, J. Liu, and S. Du, “All optical neural network with nonlinear activation functions,” arXiv: 1904.10819 (2019).

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D. M. Farkas, K. M. Hudek, E. A. Salim, S. R. Segal, M. B. Squires, and D. Z. Anderson, “A compact, transportable, microchip-based system for high repetition rate production of Bose-Einstein condensates,” Appl. Phys. Lett. 96, 093102 (2010).

[Crossref]

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

[Crossref]

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13, 346–351 (2019).

[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).

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D. Woods and T. J. Naughton, “Photonic neural networks,” Nat. Phys. 8, 257–259 (2012).

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P. Raccuglia, K. C. Elbert, P. D. Adler, C. Falk, M. B. Wenny, A. Mollo, M. Zeller, S. A. Friedler, J. Schrier, and A. J. Norquist, “Machine-learning-assisted materials discovery using failed experiments,” Nature 533, 73–76 (2016).

[Crossref]

K. T. Butler, D. W. Davies, H. Cartwright, O. Isayev, and A. Walsh, “Machine learning for molecular and materials science,” Nature 559, 547–555 (2018).

[Crossref]

J. Feldmann, N. Youngblood, C. D. Wright, H. Bhaskaran, and W. H. P. Pernice, “All-optical spiking neurosynaptic networks with self-learning capabilities,” Nature 569, 208–214 (2019).

[Crossref]

K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).

[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutiérrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20, 3241–3249 (2012).

[Crossref]

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[Crossref]

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[Crossref]

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S. Du, M. B. Squires, Y. Imai, L. Czaia, R. A. Saravanan, V. Bright, J. Reichel, T. W. Hansch, and D. Z. Anderson, “Atom-chip Bose-Einstein condensation in a portable vacuum cell,” Phys. Rev. A 70, 053606 (2004).

[Crossref]

C. Wang and H. Zhai, “Machine learning of frustrated classical spin models. I. Principal component analysis,” Phys. Rev. B 96, 144432 (2017).

[Crossref]

J. Liu, Y. Qi, Z. Y. Meng, and L. Fu, “Self-learning Monte Carlo method,” Phys. Rev. B 95, 041101 (2017).

[Crossref]

L. Huang and L. Wang, “Accelerated Monte Carlo simulations with restricted Boltzmann machines,” Phys. Rev. B 95, 035105 (2017).

[Crossref]

D.-L. Deng, X. Li, and S. D. Sarma, “Machine learning topological states,” Phys. Rev. B 96, 195145 (2017).

[Crossref]

J. Liu, H. Shen, Y. Qi, Z. Y. Meng, and L. Fu, “Self-learning Monte Carlo method and cumulative update in fermion systems,” Phys. Rev. B 95, 241104 (2017).

[Crossref]

Y. Zhang and E.-A. Kim, “Quantum loop topography for machine learning,” Phys. Rev. Lett. 118, 216401 (2017).

[Crossref]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631 (1987).

[Crossref]

F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Beguin, A. Vernier, T. Lahaye, and A. Browaeys, “Single-atom trapping in holographic 2D arrays of microtraps with arbitrary geometries,” Phys. Rev. X 4, 021034 (2014).

[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

S. Zhang, J. F. Chen, C. Liu, S. Zhou, M. M. T. Loy, G. K. L. Wong, and S. Du, “A dark-line two-dimensional magneto-optical trap of 85Rb atoms with high optical depth,” Rev. Sci. Instrum. 83, 073102 (2012).

[Crossref]

Y. Abu-Mostafa and D. Psaltis, “Optical neural computers,” Sci. Am. 256, 88–95 (1987).

[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2, 287 (2012).

[Crossref]

X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018).

[Crossref]

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G. Carleo, C. Ignacio, C. Kyle, D. Laurent, S. Maria, T. Naftali, V.-M. Leslie, and Z. Lenka, “Machine learning and the physical sciences,” arXiv:1903.10563 (2019).

See the supplemental material and Refs. [33–39] for (S1) the technical information of SLM; (S2) the Gerchberg–Saxton algorithm and feedback iteration process; (S3) the principle of linear optical power summation; (S4) the two matrices used for testing the linear operation; (S5) the operation of 2D MOT; (S6) the two-layer AONN implementation; (S7) the training of two-layer AONN; and (S8) Ising model related data processing.

J. George, R. Amin, A. Mehrabian, J. Khurgin, T. El-Ghazawi, P. R. Prucnal, and V. J. Sorger, “Electrooptic nonlinear activation functions for vector matrix multiplications in optical neural networks,” in Signal Processing in Photonic Communications (Optical Society of America, 2018), paper SpW4G-3.

Y. Zuo, B. Li, Y. Zhao, Y. Jiang, Y.-C. Chen, P. Chen, G.-B. Jo, J. Liu, and S. Du, “All optical neural network with nonlinear activation functions,” arXiv: 1904.10819 (2019).

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer-Verlag, 1999).