Abstract

A two-site Ising model is implemented as an injection-locked laser network consisting of a single master laser and two mutually coupled slave lasers. We observed ferromagnetic and antiferromagnetic orders in the in-phase and out-of-phase couplings between the two slave lasers. Their phase difference is locked to either 0 or π even if the coupling path is continuously modulated. The system automatically selects the oscillation frequency to satisfy the in-phase or out-of-phase coupling condition, when the mutual coupling dominates over the injection-locking by the master laser.

© 2015 Optical Society of America

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  1. F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15, 3241–3253 (1982).
    [Crossref]
  2. H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, 2001).
    [Crossref]
  3. E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
    [Crossref] [PubMed]
  4. A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,”” Phys. Rev. Lett. 104, 020502 (2010).
    [Crossref] [PubMed]
  5. S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).
  6. R. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 (1982).
    [Crossref]
  7. A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, “Simulating a quantum magnet with trapped ions,” Nat. Phys. 4, 757 (2008).
    [Crossref]
  8. K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
    [Crossref] [PubMed]
  9. E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
    [Crossref]
  10. R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
    [Crossref]
  11. S. Utsunomiya, K. Takata, and Y. Yamamoto, “Mapping of Ising models onto injection-locked laser systems,” Opt. Express 19(19), 18091–18108 (2011).
    [Crossref] [PubMed]
  12. K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked laser network,” New J. Phys. 14013052 (2012).
    [Crossref]
  13. Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators,” Phys. Rev. A 88, 063853 (2013).
    [Crossref]
  14. A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
    [Crossref]
  15. P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations,” Stocast. Proc. Appl. 33, 233 (1989)
  16. S. Kobayashi and T. Kimura, “Injection Locking in AIGaAs Semiconductor Laser,” IEEE J. Quantum Electron. 17(5), 689 (1981).
    [Crossref]
  17. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053 (1990).
    [Crossref] [PubMed]
  18. S. Saito, O. Nilsson, and Y. Yamamoto, “Oscillation center frequency tuning, quantum FM noise, and direct frequency characteristics in external grating loaded semiconductor lasers,” IEEE J. Quantum Electron. 18, 6 (1982).
    [Crossref]
  19. F. Rogister and M. Blondel, “Dynamics of two mutually delayed-coupled semiconductor lasers,” Opt. Commun. 239, 173–180 (2004).
    [Crossref]
  20. H. Fujino and J. Ohtsubo, “Synchronization of Chaotic Oscillations in Mutually Coupled Semiconductor Lasers,” Opt. Rev. 8(5), 351–357 (2001).
    [Crossref]

2014 (1)

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

2013 (2)

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators,” Phys. Rev. A 88, 063853 (2013).
[Crossref]

2012 (1)

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked laser network,” New J. Phys. 14013052 (2012).
[Crossref]

2011 (2)

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

S. Utsunomiya, K. Takata, and Y. Yamamoto, “Mapping of Ising models onto injection-locked laser systems,” Opt. Express 19(19), 18091–18108 (2011).
[Crossref] [PubMed]

2010 (3)

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,”” Phys. Rev. Lett. 104, 020502 (2010).
[Crossref] [PubMed]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

2008 (1)

A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, “Simulating a quantum magnet with trapped ions,” Nat. Phys. 4, 757 (2008).
[Crossref]

2004 (1)

F. Rogister and M. Blondel, “Dynamics of two mutually delayed-coupled semiconductor lasers,” Opt. Commun. 239, 173–180 (2004).
[Crossref]

2001 (2)

H. Fujino and J. Ohtsubo, “Synchronization of Chaotic Oscillations in Mutually Coupled Semiconductor Lasers,” Opt. Rev. 8(5), 351–357 (2001).
[Crossref]

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
[Crossref] [PubMed]

1990 (1)

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053 (1990).
[Crossref] [PubMed]

1989 (1)

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations,” Stocast. Proc. Appl. 33, 233 (1989)

1982 (3)

S. Saito, O. Nilsson, and Y. Yamamoto, “Oscillation center frequency tuning, quantum FM noise, and direct frequency characteristics in external grating loaded semiconductor lasers,” IEEE J. Quantum Electron. 18, 6 (1982).
[Crossref]

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15, 3241–3253 (1982).
[Crossref]

R. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 (1982).
[Crossref]

1981 (1)

S. Kobayashi and T. Kimura, “Injection Locking in AIGaAs Semiconductor Laser,” IEEE J. Quantum Electron. 17(5), 689 (1981).
[Crossref]

Barahona, F.

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15, 3241–3253 (1982).
[Crossref]

Bjork, G.

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053 (1990).
[Crossref] [PubMed]

Blondel, M.

F. Rogister and M. Blondel, “Dynamics of two mutually delayed-coupled semiconductor lasers,” Opt. Commun. 239, 173–180 (2004).
[Crossref]

Boixo, S.

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Byer, R. L.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators,” Phys. Rev. A 88, 063853 (2013).
[Crossref]

Carmichael, H.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

Chakrabarti, A.

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations,” Stocast. Proc. Appl. 33, 233 (1989)

Chakrabarti, B. K.

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations,” Stocast. Proc. Appl. 33, 233 (1989)

Chang, M.-S.

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

Duan, L.-M.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

Edwards, E. E.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

Farhi, E.

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
[Crossref] [PubMed]

Feynman, R.

R. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 (1982).
[Crossref]

Freericks, J. K.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

Friedenauer, A.

A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, “Simulating a quantum magnet with trapped ions,” Nat. Phys. 4, 757 (2008).
[Crossref]

Fujino, H.

H. Fujino and J. Ohtsubo, “Synchronization of Chaotic Oscillations in Mutually Coupled Semiconductor Lasers,” Opt. Rev. 8(5), 351–357 (2001).
[Crossref]

Gillner, L.

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053 (1990).
[Crossref] [PubMed]

Glueckert, J. T.

A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, “Simulating a quantum magnet with trapped ions,” Nat. Phys. 4, 757 (2008).
[Crossref]

Goldstone, J.

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
[Crossref] [PubMed]

Gutmann, S.

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
[Crossref] [PubMed]

Isakov, S. V.

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Islam, R.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

Joseph Wang, C.-C.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

Kim, K.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

Kimura, T.

S. Kobayashi and T. Kimura, “Injection Locking in AIGaAs Semiconductor Laser,” IEEE J. Quantum Electron. 17(5), 689 (1981).
[Crossref]

Knysh, S.

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,”” Phys. Rev. Lett. 104, 020502 (2010).
[Crossref] [PubMed]

Kobayashi, S.

S. Kobayashi and T. Kimura, “Injection Locking in AIGaAs Semiconductor Laser,” IEEE J. Quantum Electron. 17(5), 689 (1981).
[Crossref]

Korenblit, S.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

Lapan, J.

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
[Crossref] [PubMed]

Lidar, D. A.

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Lin, G.-D.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

Lundgren, A.

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
[Crossref] [PubMed]

Marandi, A.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators,” Phys. Rev. A 88, 063853 (2013).
[Crossref]

Martinis, J. M.

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Monroe, C.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

Nilsson, O.

S. Saito, O. Nilsson, and Y. Yamamoto, “Oscillation center frequency tuning, quantum FM noise, and direct frequency characteristics in external grating loaded semiconductor lasers,” IEEE J. Quantum Electron. 18, 6 (1982).
[Crossref]

Nishimori, H.

H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, 2001).
[Crossref]

Noh, C.

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

Ohtsubo, J.

H. Fujino and J. Ohtsubo, “Synchronization of Chaotic Oscillations in Mutually Coupled Semiconductor Lasers,” Opt. Rev. 8(5), 351–357 (2001).
[Crossref]

Porras, D.

A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, “Simulating a quantum magnet with trapped ions,” Nat. Phys. 4, 757 (2008).
[Crossref]

Preda, D.

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
[Crossref] [PubMed]

Ray, P.

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations,” Stocast. Proc. Appl. 33, 233 (1989)

Rogister, F.

F. Rogister and M. Blondel, “Dynamics of two mutually delayed-coupled semiconductor lasers,” Opt. Commun. 239, 173–180 (2004).
[Crossref]

Ronnow, T. F.

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Saito, S.

S. Saito, O. Nilsson, and Y. Yamamoto, “Oscillation center frequency tuning, quantum FM noise, and direct frequency characteristics in external grating loaded semiconductor lasers,” IEEE J. Quantum Electron. 18, 6 (1982).
[Crossref]

Schaetz, T.

A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, “Simulating a quantum magnet with trapped ions,” Nat. Phys. 4, 757 (2008).
[Crossref]

Schmitz, H.

A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, “Simulating a quantum magnet with trapped ions,” Nat. Phys. 4, 757 (2008).
[Crossref]

Smelyanskiy, V. N.

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,”” Phys. Rev. Lett. 104, 020502 (2010).
[Crossref] [PubMed]

Takata, K.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked laser network,” New J. Phys. 14013052 (2012).
[Crossref]

S. Utsunomiya, K. Takata, and Y. Yamamoto, “Mapping of Ising models onto injection-locked laser systems,” Opt. Express 19(19), 18091–18108 (2011).
[Crossref] [PubMed]

Troyer, M.

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Utsunomiya, S.

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked laser network,” New J. Phys. 14013052 (2012).
[Crossref]

S. Utsunomiya, K. Takata, and Y. Yamamoto, “Mapping of Ising models onto injection-locked laser systems,” Opt. Express 19(19), 18091–18108 (2011).
[Crossref] [PubMed]

Wang, Z.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators,” Phys. Rev. A 88, 063853 (2013).
[Crossref]

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Wecker, D.

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Wen, K.

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators,” Phys. Rev. A 88, 063853 (2013).
[Crossref]

Yamamoto, Y.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators,” Phys. Rev. A 88, 063853 (2013).
[Crossref]

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked laser network,” New J. Phys. 14013052 (2012).
[Crossref]

S. Utsunomiya, K. Takata, and Y. Yamamoto, “Mapping of Ising models onto injection-locked laser systems,” Opt. Express 19(19), 18091–18108 (2011).
[Crossref] [PubMed]

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053 (1990).
[Crossref] [PubMed]

S. Saito, O. Nilsson, and Y. Yamamoto, “Oscillation center frequency tuning, quantum FM noise, and direct frequency characteristics in external grating loaded semiconductor lasers,” IEEE J. Quantum Electron. 18, 6 (1982).
[Crossref]

Young, A. P.

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,”” Phys. Rev. Lett. 104, 020502 (2010).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (2)

S. Kobayashi and T. Kimura, “Injection Locking in AIGaAs Semiconductor Laser,” IEEE J. Quantum Electron. 17(5), 689 (1981).
[Crossref]

S. Saito, O. Nilsson, and Y. Yamamoto, “Oscillation center frequency tuning, quantum FM noise, and direct frequency characteristics in external grating loaded semiconductor lasers,” IEEE J. Quantum Electron. 18, 6 (1982).
[Crossref]

Int. J. Theor. Phys. (1)

R. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 (1982).
[Crossref]

J. Phys. A: Math. Gen. (1)

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15, 3241–3253 (1982).
[Crossref]

Nat. Commun. (1)

R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe, “Onset of a quantum phase transition with a trapped ion quantum simulator,” Nat. Commun. 2, 77 (2011).
[Crossref]

Nat. Photonics (1)

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Nat. Phys. (1)

A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz, “Simulating a quantum magnet with trapped ions,” Nat. Phys. 4, 757 (2008).
[Crossref]

Nature (1)

K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465, 590 (2010).
[Crossref] [PubMed]

New J. Phys. (1)

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked laser network,” New J. Phys. 14013052 (2012).
[Crossref]

Opt. Commun. (1)

F. Rogister and M. Blondel, “Dynamics of two mutually delayed-coupled semiconductor lasers,” Opt. Commun. 239, 173–180 (2004).
[Crossref]

Opt. Express (1)

Opt. Rev. (1)

H. Fujino and J. Ohtsubo, “Synchronization of Chaotic Oscillations in Mutually Coupled Semiconductor Lasers,” Opt. Rev. 8(5), 351–357 (2001).
[Crossref]

Phys. Rev A (1)

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053 (1990).
[Crossref] [PubMed]

Phys. Rev. A (2)

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators,” Phys. Rev. A 88, 063853 (2013).
[Crossref]

S. Boixo, T. F. Ronnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Quantum annealing with more than one hundred qubits,” Phys. Rev. A 88, 063853 (2013).

Phys. Rev. B (1)

E. E. Edwards, S. Korenblit, K. Kim, R. Islam, M.-S. Chang, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, ”Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins,” Phys. Rev. B 82, 060412 (2010).
[Crossref]

Phys. Rev. Lett. (1)

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,”” Phys. Rev. Lett. 104, 020502 (2010).
[Crossref] [PubMed]

Science. (1)

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science. 292(5516), 472–475 (2001).
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H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, 2001).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Ising spin representation by using polarization state (a) or phase state (b) of slave lasers.
Fig. 2
Fig. 2 Experimental setup. The blue line shows the mutual coupling between two slave lasers (DMLDs). The red line shows uni-directional injection from the master laser that implements the Zeeman term λiz. The relative phase of DMLD1 and 2 is measured by the Mach-Zehnder interferometer on the yellow lines with the FPI and APD. PD1, PD2, APD and FPI each have a refractive neutral density filter (∼ 10dB attenuators) inserted in front of them as isolators. [Inset(a)]: How to implement the Ising interaction between two slave lasers. In-phase coupling between two slave lasers corresponds to ferromagnetic coupling. [Inset(b)]:Schematic picture of two site Ising model implementation.
Fig. 3
Fig. 3 The blue trace represents the Interference output between the master laser and slave laser 1 as a function of the frequency shift of the master laser. The red trace is the ramp voltage to sweep the master laser frequency.
Fig. 4
Fig. 4 Bifurcation of two slave lasers’ phases in mutually coupled semiconductor lasers: (a)The Ising term is dominant when the mutual coupling between the slave lasers is relatively strong (η = 0.04) (b)The mutual coupling ratio for the Ising term is weaker than the case of (a) (η = 0.01). (c)The master injection term is dominant when the mutual coupling is relatively small (η = 0.005). The other numerical parameters are λ1 = λ2 = 0, ω/Q = 1012(s−1), τsp = 10−9(s), ζ = 0.005 and β = 10−4.
Fig. 5
Fig. 5 Interference signal output to show the relative phase Δϕr12 between two slave lasers (DMLD1 and 2) when the mutually coupling optical path length is modulated: (a) Ising coupling term is dominant, (c) Master injection term is dominant, (b) Transition regime between (a) and (c).
Fig. 6
Fig. 6 Interference signals to show the relative phase Δϕr12 between the two slave lasers (DMLD1 and DMLD2). The green lines show the case that the interference path length is modulated. The blue lines show the case that the injection phase from the master laser to the slave laser 1 is modulated. (a) The Ising coupling term is dominant under a weak master signal injection. (b) The master signal is dominant under a strong master signal injection.
Fig. 7
Fig. 7 Frequency shift of a slave laser due to the mutual coupling path length modulation as a function of time under a strong mutual injection (blue line) and a weak master signal injection (green line).
Fig. 8
Fig. 8 Schematic explanation for the sawtooth frequency modulation and phase jump in two coupled slave lasers, which explain the experimental results shown in Fig. 5a and Fig. 7 simultaneously.
Fig. 9
Fig. 9 Ising machine based on a higher harmonic fiber mode-locked laser with N − 1 delay lines and the deferential phase detection.

Equations (6)

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= i < j M J i j σ i z σ j z + i M λ i σ i z
Δ ω ω ω r 0
= ( sin ϕ 0 + α cos ϕ 0 ) F 0 A 0 ω Q e ,
d d t A i ( t ) = 1 2 [ ω Q E CV i ( t ) ] A i ( t ) + ω Q n M { ζ cos ϕ Vi ( t ) η λ i sin ϕ Vi } ω Q j i 1 2 η J i j A j ( t ) cos ( ϕ j ( t ) ϕ i ( t ) ) + F Ai ,
d d t ϕ i ( t ) = 1 A i ( t ) { ω Q n M [ ζ sin ϕ Vi ( t ) η λ i cos ϕ Vi ( t ) ] ω Q j i 1 2 η J i j A j ( t ) sin ( ϕ j ( t ) ϕ i ( t ) ) + F ϕ i ,
d d t N i ( t ) = P N i ( t ) τ sp { 1 + β [ A i ( t ) 2 + 1 ] } + F N i .

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