Abstract

Strong correlation of photons, particularly in the single-photon regime, has recently been exploited for various applications in quantum information processing. Existing correlation measurements, however, do not fully characterize multi-photon correlation in a relevant context and may pose limitations in practical situations. We propose a conceptually rigorous, but easy-to-implement, criterion for detecting correlated multi-photon emission out of a quantum optical system, drawn from the context of wavefunction collapse. We illustrate the robustness of our approach against experimental limitations by considering an anharmonic optical system.

© 2010 Optical Society of America

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  1. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007)
    [Crossref]
  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,” Nat. Phys. 4, 859–863 (2008).
    [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. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
    [Crossref] [PubMed]
  5. M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
    [Crossref]
  6. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
    [Crossref]
  7. D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
    [Crossref]
  8. N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A 77, 031803(R) (2008).
    [Crossref]
  9. D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
    [Crossref]
  10. A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
    [Crossref]
  11. L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
    [Crossref] [PubMed]
  12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
    [Crossref] [PubMed]
  13. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
    [Crossref]
  14. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
    [Crossref]
  15. L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
    [Crossref]
  16. L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999).
    [Crossref]
  17. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
    [Crossref]
  18. C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 411721–1723 (1990).
    [Crossref] [PubMed]
  19. D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A,  2137–15 (1996).
    [Crossref]
  20. H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Comm. 82, 73–79 (1991).
    [Crossref]
  21. R. Hanbury-Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
    [Crossref]
  22. M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
    [Crossref] [PubMed]
  23. Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000).
    [Crossref]
  24. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
    [Crossref] [PubMed]
  25. H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. 77, 631–634 (1996).
    [Crossref] [PubMed]

2009 (2)

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[Crossref] [PubMed]

2008 (7)

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[Crossref]

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[Crossref]

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,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
[Crossref] [PubMed]

N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A 77, 031803(R) (2008).
[Crossref]

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
[Crossref]

2007 (2)

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

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
[Crossref]

2006 (2)

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[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]

2000 (1)

Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000).
[Crossref]

1999 (1)

L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999).
[Crossref]

1997 (1)

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

1996 (2)

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A,  2137–15 (1996).
[Crossref]

H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. 77, 631–634 (1996).
[Crossref] [PubMed]

1992 (1)

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
[Crossref] [PubMed]

1991 (1)

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Comm. 82, 73–79 (1991).
[Crossref]

1990 (2)

C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 411721–1723 (1990).
[Crossref] [PubMed]

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref] [PubMed]

1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[Crossref]

1956 (1)

R. Hanbury-Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[Crossref]

An, K.

Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000).
[Crossref]

Angelakis, D. G.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
[Crossref]

Aoki, T.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
[Crossref] [PubMed]

Aßmann, M.

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[Crossref] [PubMed]

Baur, M.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[Crossref]

Bayer, M.

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[Crossref] [PubMed]

Bianchetti, R.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[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]

Bishop, L. S.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

Blais, A.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[Crossref]

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]

Bose, S.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
[Crossref]

Brandão, F. G. S. L.

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

Brecha, R. J.

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Comm. 82, 73–79 (1991).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. 77, 631–634 (1996).
[Crossref] [PubMed]

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
[Crossref] [PubMed]

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Comm. 82, 73–79 (1991).
[Crossref]

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref] [PubMed]

Chang, D. E.

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
[Crossref]

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

Chough, Y.-T

Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000).
[Crossref]

Chow, J. M.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

Cole, J. H.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

Dayan, B.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
[Crossref] [PubMed]

Demler, E. A.

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
[Crossref]

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

Deutsch, M.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

Devoret, M. H.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[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,” Nat. 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,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Fink, J. M.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[Crossref]

Fuhrmanek, A.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[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,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Gauthier, D. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref] [PubMed]

Girvin, S. M.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[Crossref]

Göppl, M.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[Crossref]

Greentree, A. D.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

Gritsev, V.

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
[Crossref]

Hanbury-Brown, R.

R. Hanbury-Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[Crossref]

Hartmann, M. J.

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

Hollenberg, L. C. L.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

Horvath, L.

L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999).
[Crossref]

Houck, A. A.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

Hvam, J. M.

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[Crossref] [PubMed]

Imamoglu, A.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

Kimble, H. J.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
[Crossref] [PubMed]

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]

Klyshko, D. N.

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A,  2137–15 (1996).
[Crossref]

Koch, J.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

Koch, M.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

Kochan, P.

H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. 77, 631–634 (1996).
[Crossref] [PubMed]

Kubanek, A.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[Crossref]

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

Lee, C. T.

C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 411721–1723 (1990).
[Crossref] [PubMed]

Leek, P. J.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[Crossref]

Lukin, M. D.

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
[Crossref]

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

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]

Moon, H.-J.

Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000).
[Crossref]

Morigi, G.

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
[Crossref]

Morin, S. E.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref] [PubMed]

Mossberg, T. W.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref] [PubMed]

Murr, K.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[Crossref]

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

Na, N.

N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A 77, 031803(R) (2008).
[Crossref]

Nha, H.

Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000).
[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]

Ostby, E. P.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
[Crossref] [PubMed]

Ourjoumtsev, A.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

Parkins, A. S.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
[Crossref] [PubMed]

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,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Pinkse, P. W. H.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[Crossref]

Plenio, M. B.

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

Puppe, T.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[Crossref]

Rempe, G.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[Crossref]

Rice, P. R.

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Comm. 82, 73–79 (1991).
[Crossref]

Sanders, B. C.

L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999).
[Crossref]

H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. 77, 631–634 (1996).
[Crossref] [PubMed]

Santos, M. F.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
[Crossref]

Schmidt, H.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

Schoelkopf, R. J.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

Schuster, I.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[Crossref]

Sørensen, A. S.

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

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,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Tahan, C.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

Thuneberg, E.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

Tian, L.

N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A 77, 031803(R) (2008).
[Crossref]

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
[Crossref] [PubMed]

Twiss, R.

R. Hanbury-Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[Crossref]

Utsunomiya, S.

N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A 77, 031803(R) (2008).
[Crossref]

Vahala, K. J.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
[Crossref] [PubMed]

van der Poel, M.

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[Crossref] [PubMed]

Veit, F.

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[Crossref] [PubMed]

Vuckovic, J.

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,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Vuletic, V.

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
[Crossref]

Wallraff, A.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[Crossref]

Wielinga, B. F.

L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999).
[Crossref]

Woods, G.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

Wu, Q.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref] [PubMed]

Yamamoto, Y.

N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A 77, 031803(R) (2008).
[Crossref]

Zhu, Y.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref] [PubMed]

J. Opt. B: Quantum Semiclassic. Opt. (1)

L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999).
[Crossref]

Nat. Phys. (7)

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[Crossref]

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008).
[Crossref]

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

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,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009).
[Crossref]

Nature (2)

R. Hanbury-Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[Crossref]

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 (London) (1)

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008).
[Crossref]

Opt. Comm. (1)

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Comm. 82, 73–79 (1991).
[Crossref]

Phys. Lett. A (1)

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A,  2137–15 (1996).
[Crossref]

Phys. Rev. (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[Crossref]

Phys. Rev. A (5)

C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 411721–1723 (1990).
[Crossref] [PubMed]

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
[Crossref] [PubMed]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
[Crossref]

N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A 77, 031803(R) (2008).
[Crossref]

Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000).
[Crossref]

Phys. Rev. Lett. (4)

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref] [PubMed]

H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. 77, 631–634 (1996).
[Crossref] [PubMed]

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
[Crossref] [PubMed]

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

Science (2)

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008).
[Crossref] [PubMed]

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[Crossref] [PubMed]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Energy-level diagram for (i) harmonic and (ii) anharmonic system. (b) Energy level structure for cavity QED system. (c) multiphoton coincidence rates 〈a n an 〉 as a function of δ/g for 2κ/g = γ/g = 0.01 with ��/κ = 0.1. The dotted vertical lines represent the locations of the multiphoton resonances, δ = ±g/√n throughout Figs. 1–3.

Fig. 2.
Fig. 2.

(a)–(c) The conventional correlation functions g (n) (0) by Glauber and C (n) (0) by Kubanek et al. [12]. (d)–(f) The quantitative measure �� n together with conditional relative rates R k,k-1. The truncation numbers used are (d) N tr = 4, (e) and (f) N tr = 5. In all plots, 2κ/g=γ/g=��/K = 0.1.

Fig. 3.
Fig. 3.

Comparison between g (n)(0) (black dotted curve) and our measure �� n (red solid curve) for (a) n = 2 and (b) n = 3, with the truncation numbers (a) N tr - 4 and (b) N tr = 5, respectively. The driving intensity is rather high, ��/κ = 1, with the coupling condition 2κ/g=γ/g=0.1.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

𝓡 n Ψ c ( n 1 ) 𝓔 ̂ 𝓔 ̂ + Ψ c ( n 1 ) Ψ c ( n 1 ) Ψ c ( n 1 ) = 𝓔 ̂ n 𝓔 ̂ + n 𝓔 ̂ n 1 𝓔 ̂ + n 1
R k , k 1 𝓡 k 𝓡 k 1 = 𝓔 ̂ k 𝓔 ̂ + k 𝓔 ̂ k 2 𝓔 ̂ + k 2 𝓔 ̂ k 1 𝓔 ̂ + k 1 2 > 1 , ( k = 2 , , n ) ,
R k , k 1 < 1 , ( k = n + 1, ) .
𝓜 n k = 2 n max { R kk 1 1,0 } k = n + 1 N tr max { R kk 1 1 1,0 } ,
g ( n ) 𝓔 ̂ ( x 1 ) 𝓔 ̂ ( x 2 ) 𝓔 ̂ ( x n ) 𝓔 ̂ + ( x n ) 𝓔 ̂ + ( x 2 ) 𝓔 ̂ + ( x 1 ) 𝓔 ̂ ( x 1 ) 𝓔 ̂ + ( x 1 ) 𝓔 ̂ ( x 2 ) 𝓔 ̂ + ( x 2 ) 𝓔 ̂ ( x n ) 𝓔 ̂ + ( x n )
a k a k a k 2 a k 2 = d 2 α α 2 k P ( α ) d 2 α α 2 k 4 P ( α )
( d 2 α α 2 k 2 P ( α ) ) 2 = a k 1 a k 2 2 .
H = h ¯ ω 0 ( a a + 1 2 σ z ) + i h ¯ g ( a σ + ) ,
n h ¯ ω L = E n , ± = n h ¯ ω 0 ± h ¯ g n .

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