Abstract

By solving the master equation in the steady-state limit and calculating the zero-delay-time second-order correlation function, a strong photon antibunching is found in two coupled cavities with weak nonlinearity χ(2). An optimal antibunching condition is derived that is complementary to the result in a recent publication [Phys. Rev. A 89, 031803(R) (2014)]. Numerical simulations confirm the optimal condition. These results are extended from the case with a single drive to the case with two drives. We find that the parameter added to the system due to the second drive can lift the restriction on the detuning for the antibunching to occur. This may open a new door towards the application of the antibunching into single photon sources.

© 2016 Optical Society of America

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References

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  1. H. Imamoglu, A. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
    [Crossref]
  2. A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
    [Crossref]
  3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
    [Crossref] [PubMed]
  4. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
    [Crossref]
  5. A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
    [Crossref] [PubMed]
  6. Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
    [Crossref]
  7. A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order nonlinearity,” Phys. Rev. B 87, 235319 (2013).
    [Crossref]
  8. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
    [Crossref]
  9. D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
    [Crossref]
  10. D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
    [Crossref]
  11. T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
    [Crossref] [PubMed]
  12. H. J. Carmichael, “Photon antibunching and squeezing for a single atom in a resonant cavity,” Phys. Rev. Lett. 55, 2790 (1985).
    [Crossref] [PubMed]
  13. M. Bamba, A. Imamoglu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
    [Crossref]
  14. X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
    [Crossref]
  15. X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
    [Crossref]
  16. S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New. J. Phys. 15, 025012 (2013)
    [Crossref]
  17. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
    [Crossref]
  18. S. Ferretti, L. C. Andreani, H. E. Tureci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010)
    [Crossref]
  19. X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).
  20. A. Majumdar, M. Bajcsy, A. Rundquist, and J. VuČković, “Loss-enabled sub-poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
    [Crossref] [PubMed]
  21. W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
    [Crossref]
  22. Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
    [Crossref]
  23. D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
    [Crossref]
  24. O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
    [Crossref]
  25. O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
    [Crossref]
  26. R. W. Boyd, Nonlinear Optics (Academic, 2008).
  27. S. Bergfeld and W. Daum, “Second-harmonic generation in GaAs: experiment versus theoretical predictions of χxyh(2),” Phys. Rev. Lett. 90, 036801 (2003).
    [Crossref]
  28. J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129 (1995).
    [Crossref]
  29. S. Combrié, A. De Rossi, N.-Q.-V. Tran, and H. Benisty, “GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55μ m,” Opt. Lett. 33, 1908 (2008).
    [Crossref]
  30. H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
    [Crossref]
  31. H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

2015 (2)

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

2014 (8)

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

2013 (5)

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order nonlinearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New. J. Phys. 15, 025012 (2013)
[Crossref]

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

2012 (1)

A. Majumdar, M. Bajcsy, A. Rundquist, and J. VuČković, “Loss-enabled sub-poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

2011 (2)

M. Bamba, A. Imamoglu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

2010 (2)

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

S. Ferretti, L. C. Andreani, H. E. Tureci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010)
[Crossref]

2009 (1)

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

2008 (2)

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
[Crossref]

S. Combrié, A. De Rossi, N.-Q.-V. Tran, and H. Benisty, “GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55μ m,” Opt. Lett. 33, 1908 (2008).
[Crossref]

2007 (1)

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

2005 (1)

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

2003 (1)

S. Bergfeld and W. Daum, “Second-harmonic generation in GaAs: experiment versus theoretical predictions of χxyh(2),” Phys. Rev. Lett. 90, 036801 (2003).
[Crossref]

1997 (1)

H. Imamoglu, A. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

1995 (1)

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129 (1995).
[Crossref]

1985 (1)

H. J. Carmichael, “Photon antibunching and squeezing for a single atom in a resonant cavity,” Phys. Rev. Lett. 55, 2790 (1985).
[Crossref] [PubMed]

Andreani, L. C.

S. Ferretti, L. C. Andreani, H. E. Tureci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010)
[Crossref]

Aumentado, J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Bachor, H.-A.

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

Bajcsy, M.

A. Majumdar, M. Bajcsy, A. Rundquist, and J. VuČković, “Loss-enabled sub-poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

Bajer, J.

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Bamba, M.

M. Bamba, A. Imamoglu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

Benisty, H.

Bergfeld, S.

S. Bergfeld and W. Daum, “Second-harmonic generation in GaAs: experiment versus theoretical predictions of χxyh(2),” Phys. Rev. Lett. 90, 036801 (2003).
[Crossref]

Birnbaum, K. M.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Boca, A.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Boozer, A. D.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Carmichael, H. J.

H. J. Carmichael, “Photon antibunching and squeezing for a single atom in a resonant cavity,” Phys. Rev. Lett. 55, 2790 (1985).
[Crossref] [PubMed]

Carusotto, I.

M. Bamba, A. Imamoglu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

Chang, D. E.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Chen, J.

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129 (1995).
[Crossref]

Ciuti, C.

M. Bamba, A. Imamoglu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

Combrié, S.

Daum, W.

S. Bergfeld and W. Daum, “Second-harmonic generation in GaAs: experiment versus theoretical predictions of χxyh(2),” Phys. Rev. Lett. 90, 036801 (2003).
[Crossref]

De Rossi, A.

Demler, E. A.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Deutsch, M.

H. Imamoglu, A. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Englund, D.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
[Crossref]

Faraon, A.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
[Crossref]

Fazio, R.

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Ferretti, S.

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New. J. Phys. 15, 025012 (2013)
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Tureci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010)
[Crossref]

Flayac, H.

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

Fushman, I.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
[Crossref]

Gerace, D.

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order nonlinearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New. J. Phys. 15, 025012 (2013)
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Tureci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010)
[Crossref]

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Giovannetti, V.

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Hoffman, A. J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Houck, A. A.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Imamoglu, A.

M. Bamba, A. Imamoglu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Imamoglu, H.

H. Imamoglu, A. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Kimble, H. J.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Kyriienko, O.

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

Levine, Z. H.

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129 (1995).
[Crossref]

Li, Y.

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

Li, Y. J.

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

Liew, T. C. H.

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

Liu, Y. M.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Liu, Y. X.

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Lukin, M. D.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Majumdar, A.

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order nonlinearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. VuČković, “Loss-enabled sub-poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

Miller, R.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Miranowicz, A.

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Nori, F.

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Northup, T. E.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Paprzycka, M.

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Peng, Y. W.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Petroff, P.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
[Crossref]

Ralph, T. C.

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

Rundquist, A.

A. Majumdar, M. Bajcsy, A. Rundquist, and J. VuČković, “Loss-enabled sub-poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

Savona, V.

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New. J. Phys. 15, 025012 (2013)
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

Schmidt, A.

H. Imamoglu, A. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Schmidt, S.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Shelykh, I. A.

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

Shen, H. Z.

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

Sorensen, A. S.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Spietz, L.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Srinivasan, S. J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Stoltz, N.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
[Crossref]

Tran, N.-Q.-V.

Tureci, H. E.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

S. Ferretti, L. C. Andreani, H. E. Tureci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010)
[Crossref]

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

VuCkovic, J.

A. Majumdar, M. Bajcsy, A. Rundquist, and J. VuČković, “Loss-enabled sub-poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
[Crossref]

Wilkins, J. W.

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129 (1995).
[Crossref]

Woods, G.

H. Imamoglu, A. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Xu, X. W.

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

Yi, X. X.

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

Yu, Z. Y.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Zhang, W.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Zhou, Y. H.

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

Appl. Phys. Lett. (1)

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129 (1995).
[Crossref]

J. Opt. B: At. Mol. Opt. Phys. (1)

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

Nat. Phys. (2)

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Nature (1)

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Nature Phys. (1)

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Phys. 4, 859–863 (2008).
[Crossref]

New. J. Phys. (1)

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New. J. Phys. 15, 025012 (2013)
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (14)

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Tureci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010)
[Crossref]

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

M. Bamba, A. Imamoglu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

Phys. Rev. B (1)

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order nonlinearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

Phys. Rev. Lett. (6)

A. Majumdar, M. Bajcsy, A. Rundquist, and J. VuČković, “Loss-enabled sub-poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

H. Imamoglu, A. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Tureci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

H. J. Carmichael, “Photon antibunching and squeezing for a single atom in a resonant cavity,” Phys. Rev. Lett. 55, 2790 (1985).
[Crossref] [PubMed]

S. Bergfeld and W. Daum, “Second-harmonic generation in GaAs: experiment versus theoretical predictions of χxyh(2),” Phys. Rev. Lett. 90, 036801 (2003).
[Crossref]

Other (2)

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

R. W. Boyd, Nonlinear Optics (Academic, 2008).

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Figures (7)

Fig. 1
Fig. 1 The system of two coupled cavities. Modes a and b in the cavities are tunnel-coupled, mode b and mode c are coupled via the second-order nonlinear χ(2).
Fig. 2
Fig. 2 The zero-delay-time second-order correlation functions g(2)(0) versus the nonlinear strength g/κ with driving strengths F/κ = 0.5, J/κ = 7 and the detuning Δ/κ = 0, Δ/κ = 0.05, Δ/κ = 0.1 respectively.
Fig. 3
Fig. 3 Dependence of antibunching on the second-harmonic quality factor Qc. (a) In [23], the parameters F/κ = 1, g = 0.1 and J = 19.45. (b) In our scheme, F/κ = 1, g/κ = 0.1 and J/κ = J = 7.1.
Fig. 4
Fig. 4 (a) Logarithmic plot of the zero-time second-order correlation functions g(2)(0) as functions of detunings Δ/κ and the nonlinear interaction strength g/κ with ratio n given by n1 and n2. (b) Logarithmic plot of the zero-time second-order correlation functions g(2)(0) as functions of Δ/κ and n for the nonlinear interaction strength g/κ given by g1 and g2. In both (a) and (b), the tunnel coupling J/κ = 2 and F/κ = 0.1 are taken, and the dashed lines are analytical results.
Fig. 5
Fig. 5 The Logarithmic plot of g(2)(0) as functions of detunings J/κ for different n. The parameters F/κ = 0.5, g/κ = 0.1.
Fig. 6
Fig. 6 Logarithmic plot of the zero-time second-order correlation functions g(2)(0) as functions of the detunings g/κ and the strength ratio n. (a) The tunnel coupling J/κ = 2 and F/κ = 0.1. (b)The tunnel coupling J/κ = 0.2 and F/κ = 0.001.
Fig. 7
Fig. 7 Energy-level diagram showing the zero-, one-, and two-photon states (horizontal black short lines) and the transition paths leading to the quantum interference responsible for the strong antibunching (black lines with arrows). |nanbnc〉 represents the Fock state, where na, nb and nc denote respectively the photon number in mode a, b and c.

Equations (30)

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D i ( r , t ) = ε 0 ε i j E j ( r , t ) + ε 0 [ χ i j k ( 2 ) ( r ) E j ( r , t ) E k ( r , t ) ] +
E ^ ( r , t ) = j = a , b , c h ¯ ω j 2 0 [ j ^ ϕ j ( r ) ε ( r ) e i ω j t + H . C . ] ,
H ^ em = h ¯ ω a a ^ a + h ¯ ω b b ^ b ^ + h ¯ ω c c ^ c ^ + h ¯ J ( a ^ b ^ + b ^ a ^ ) + h ¯ g [ ( b ^ ) 2 c ^ + c ^ b ^ 2 ] + h ¯ v [ ( a ^ ) 2 c ^ + c ^ a ^ 2 ) ] + h ¯ u [ a ^ b ^ c ^ + c ^ b ^ a ^ ] ,
h ¯ J = h ¯ 2 ω a ω b ε i j ( r ) ε ( r ) ϕ a ( r ) ϕ b ( r ) d 3 r , h ¯ g = h ¯ ω b 2 2 ω c 2 0 χ ( 2 ) ( r ) [ ε ( r ) ] 3 / 2 ϕ b ( r ) ϕ b ( r ) ϕ c ( r ) d 3 r , h ¯ v = h ¯ ω a 2 2 ω c 2 0 χ ( 2 ) ( r ) [ ε ( r ) ] 3 / 2 ϕ a ( r ) ϕ a ( r ) ϕ c ( r ) d 3 r , h ¯ u = 2 ε 0 [ h ¯ 2 ε 0 ] 3 / 2 ω a ω b ω c χ ( 2 ) ( r ) [ ε ( r ) ] 3 / 2 ϕ a ( r ) ϕ b ( r ) ϕ c ( r ) d 3 r .
H ^ 0 = h ¯ ω a a ^ a ^ + h ¯ ω b b ^ b ^ + h ¯ ω c c ^ c ^ + h ¯ J ( a ^ b ^ + b ^ a ^ ) + h ¯ g [ ( b ^ ) 2 c ^ + c ^ b ^ 2 ] .
H ^ = H ^ 0 + h ¯ F ( a ^ e i ω L t + a ^ e i ω L t ) .
H ^ d = h ¯ Δ a a ^ a ^ + h ¯ Δ b b ^ b ^ + h ¯ Δ c c ^ c ^ + h ¯ J ( a ^ b ^ + b ^ a ^ ) + h ¯ g [ ( b ^ ) 2 c ^ + c ^ b ^ 2 ] + h ¯ F ( a ^ + a ^ ) ,
ρ ^ t = i h ¯ [ H ^ d , ρ ^ ] + κ a 2 ( 2 a ^ ρ ^ a ^ + a ^ a ^ ρ ^ ρ a ^ a ^ ) + κ b 2 ( 2 b ^ ρ ^ b ^ b ^ b ^ ρ ^ ρ b ^ b ^ ) + κ c 2 ( 2 c ρ ^ c ^ c ^ c ^ ρ ^ ρ c ^ c ^ ) ,
g ( 2 ) ( τ ) = a ^ ( 0 ) a ^ ( τ ) a ^ ( τ ) a ^ ( 0 ) a ^ ( 0 ) a ^ ( 0 ) 2 ,
g ( 2 ) ( 0 ) = Tr [ a ^ a ^ a ^ a ^ ρ s ] Tr [ a ^ a ^ ρ s ] 2 ,
| ψ = C 000 | 000 + C 010 | 010 + C 100 | 100 + C 020 | 020 + C 200 | 200 + C 110 | 110 + C 001 | 001 .
i h ¯ t C 000 0 , i h ¯ t C 010 = i κ 2 C 010 + J C 100 + F C 110 , i h ¯ t C 100 = F C 000 + J C 010 i κ 2 C 100 + 2 F C 200 , i h ¯ t C 020 = i κ C 020 + 2 J C 110 + 2 g C 001 , i h ¯ t C 200 = 2 F C 100 i κ C 200 + 2 J C 110 , i h ¯ t C 110 = F C 010 + 2 J C 020 + 2 J C 200 i κ C 110 , i h ¯ t C 001 = 2 g C 020 2 i κ C 001 .
i κ 2 C 010 + J C 100 = 0 , F + J C 010 i κ 2 C 100 = 0 ,
i κ C 020 + 2 J C 110 + 2 g C 001 = 0 , 2 F C 100 i κ C 200 + 2 J C 110 = 0 , F C 010 + 2 J C 020 + 2 J C 200 i κ C 110 = 0 , 2 g C 020 2 i κ C 001 = 0 .
C 010 = 4 F J 4 J 2 + κ 2 , C 100 = 2 i F κ 4 J 2 + κ 2 , C 020 = 8 2 F 2 J 2 κ 2 ( 4 J 2 + κ 2 ) ( 4 J 2 κ 2 + κ 4 + g 2 ( 2 J 2 + κ 2 ) ) , C 200 = 2 2 F 2 ( κ 4 + g 2 ( 2 J 2 + κ 2 ) ) ( 4 J 2 + κ 2 ) ( 4 J 2 κ 2 + κ 4 + g 2 ( 2 J 2 + κ 2 ) ) , C 110 = 8 i F 2 J κ ( g 2 + κ 2 ) ( 4 J 2 + κ 2 ) ( 4 J 2 κ 2 + κ 4 + g 2 ( 2 J 2 + κ 2 ) ) , C 001 = 8 i F 2 g J 2 κ ( 4 J 2 + κ 2 ) ( 4 J 2 κ 2 + κ 4 + g 2 ( 2 J 2 + κ 2 ) ) .
κ 4 2 g 2 J 2 + g 2 κ 2 = 0 .
g / κ = ± 1 ( 2 ( J / κ ) 2 1 ) .
Δ [ 2 g J 2 n + F ( 4 g 2 + 7 κ 2 20 Δ 2 ) ] = 0 , 4 F g 2 J 2 2 F g 2 κ 2 2 F κ 4 + 4 g J 4 n + g J 2 κ 2 n + 8 F g 2 Δ 2 + 36 F κ 2 Δ 2 4 g J 2 n Δ 2 16 F Δ 4 = 0 ,
{ g 1 = 20 J 2 κ 2 3 κ 4 + 112 J 2 Δ 2 24 κ 2 Δ 2 48 Δ 4 4 3 J , n 1 = F [ 3 ( κ 2 + 4 Δ 2 ) 2 + J 2 ( 10 κ 2 + 56 Δ 2 ) ] 2 3 J 3 4 J 2 ( 5 κ 2 28 Δ 2 ) 3 ( κ 2 + 4 Δ 2 ) 2 ,
{ g 2 = 20 J 2 κ 2 3 κ 4 + 112 J 2 Δ 2 24 κ 2 Δ 2 48 Δ 4 4 3 J , n 2 = F [ 3 ( κ 2 + 4 Δ 2 ) 2 + J 2 ( 10 κ 2 + 56 Δ 2 ) ] 2 3 J 3 4 J 2 ( 5 κ 2 28 Δ 2 ) 3 ( κ 2 + 4 Δ 2 ) 2 .
g / κ = ± 16 [ 5 28 ( Δ / κ ) 2 ] 3 [ 1 + 4 ( Δ / κ ) 2 ] 2 8 3 .
n = ± 0.0036 { 3 [ 1 + 4 ( Δ / κ ) 2 ] 2 + 4 [ 10 + 56 ( Δ 2 / κ ) ] } 16 [ 5 28 ( Δ / κ ) 2 ] 3 [ 1 + 4 ( Δ / κ ) 2 ] 2 ,
4 F g 2 J 2 2 F g 2 κ 2 2 F κ 4 + 4 g J 4 n + g J 2 κ 2 n = 0 .
g / κ = ( J / κ ) 2 n 4 ( J / κ ) 4 n ± 8 ( F / κ ) [ 2 ( F / κ ) + 4 ( F / κ ) ( J / κ ) 2 ] + [ ( J / κ ) 2 n + 4 ( J / κ ) 4 n ] 2 4 [ F / κ + 2 ( F / κ ) ( J / κ ) 2 ] .
4 g 2 + 7 κ 2 20 Δ 2 = 0 , 2 g 2 J 2 g 2 κ 2 κ 4 + 4 g 2 Δ 2 + 18 κ 2 Δ 2 8 Δ 4 = 0 ,
{ Δ 1 = 5 J 2 3 κ 2 + J 2 ( 25 J 2 + 72 κ 2 ) 2 3 , g 1 = 25 J 2 36 κ 2 + 5 J 2 ( 25 J 2 + 72 κ 2 ) 2 3 ,
{ Δ 2 = 5 J 2 3 κ 2 + J 2 ( 25 J 2 + 72 κ 2 ) 2 3 , g 2 = 25 J 2 36 κ 2 + 5 J 2 ( 25 J 2 + 72 κ 2 ) 2 3 ,
{ Δ 3 = 5 J 2 3 κ 2 + J 2 ( 25 J 2 + 72 κ 2 ) 2 3 , g 3 = 25 J 2 36 κ 2 + 5 J 2 ( 25 J 2 + 72 κ 2 ) 2 3 ,
{ Δ 4 = 5 J 2 3 κ 2 + J 2 ( 25 J 2 + 72 κ 2 ) 2 3 , g 4 = 25 J 2 36 κ 2 + 5 J 2 ( 25 J 2 + 72 κ 2 ) 2 3 .
25 J 2 36 κ 2 + 5 J 2 ( 25 J 2 + 72 κ 2 ) 0 .

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