Abstract

We measure second- and third-order temporal coherences, g(2)(τ) and g(3)(τ1,τ2), of an optically excited single-photon source: an InGaAs quantum dot in a microcavity pedestal. Increasing the optical excitation power leads to an increase in the measured count rate, and also an increase in multi-photon emission probability. We show that standard measurements of g(2) provide limited information about this multi-photon probability, and that more information can be gained by simultaneously measuring g(3). Experimental results are compared with a simple theoretical model to show that the observed antibunchings are consistent with an incoherent addition of two sources: 1) an ideal single-photon source that never emits multiple photons and 2) a background cavity emission having Poissonian photon number statistics. Spectrally resolved cross-correlation measurements between quantum-dot and cavity modes show that photons from these two sources are largely uncorrelated, further supporting the model. We also analyze the Hanbury Brown-Twiss interferometer implemented with two or three “click” detectors, and explore the conditions under which it can be used to accurately measure g(2)(τ) and g(3)(τ1,τ2).

© 2014 Optical Society of America

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2013 (1)

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

2012 (2)

M. Yamaguchi, T. Asano, S. Noda, “Third emission mechanism in solid-state nanocavity quantum electrodynamics,” Rep. Prog. Phys. 75(9), 096401 (2012).
[CrossRef] [PubMed]

Y. Zhou, J. Liu, J. Simon, Y. Shih, “Resolution enhancement of third-order thermal light ghost imaging in the photon counting regime,” J. Opt. Soc. Am. B 29(3), 377 (2012).
[CrossRef]

2011 (5)

L. Ma, M. T. Rakher, M. J. Stevens, O. Slattery, K. Srinivasan, X. Tang, “Temporal correlation of photons following frequency up-conversion,” Opt. Express 19(11), 10501–10510 (2011).
[CrossRef] [PubMed]

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, O. Thomas, A. J. Shields, “Probing higher order correlations of the photon field with photon number resolving avalanche photodiodes,” Opt. Express 19(14), 13268–13276 (2011).
[CrossRef] [PubMed]

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011).
[CrossRef] [PubMed]

M. D. Eisaman, J. Fan, A. Migdall, S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[CrossRef] [PubMed]

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

2010 (5)

Y. Zhou, J. Simon, J. Liu, Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81(4), 043831 (2010).
[CrossRef]

T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010).
[CrossRef]

M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010).
[CrossRef] [PubMed]

A. Hayat, A. Nevet, M. Orenstein, “Ultrafast partial measurement of fourth-order coherence by HBT interferometry of upconversion-based autocorrelation,” Opt. Lett. 35(5), 793–795 (2010).
[CrossRef] [PubMed]

S. Reitzenstein, A. Forchel, “Quantum dot micropillars,” J. Phys. D Appl. Phys. 43(3), 033001 (2010).
[CrossRef]

2009 (5)

B. Blauensteiner, I. Herbauts, S. Bettelli, A. Poppe, H. Hübel, “Photon bunching in parametric down-conversion with continuous-wave excitation,” Phys. Rev. A 79(6), 063846 (2009).
[CrossRef]

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56(2-3), 422–431 (2009).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[CrossRef] [PubMed]

M. Assmann, F. Veit, M. Bayer, M. van der Poel, J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009).
[CrossRef] [PubMed]

2007 (2)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

2004 (1)

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).

2002 (1)

E. Waks, C. Santori, Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66(4), 042315 (2002).
[CrossRef]

1999 (1)

1998 (1)

P. R. Tapster, J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998).
[CrossRef]

1996 (1)

Y. Qu, S. Singh, C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. 76(8), 1236–1239 (1996).
[CrossRef] [PubMed]

1987 (2)

A. G. Palmer, N. L. Thompson, “Molecular aggregation characterized by high order autocorrelation in fluorescence correlation spectroscopy,” Biophys. J. 52(2), 257–270 (1987).
[CrossRef] [PubMed]

R. Loudon, P. L. Knight, “Squeezed light,” J. Mod. Opt. 34(6-7), 709–759 (1987).
[CrossRef]

1976 (1)

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14(4), 1475–1478 (1976).
[CrossRef]

1969 (2)

F. Davidson, “Measurements of photon correlations in a laser beam near threshold with time-to-amplitude converter techniques,” Phys. Rev. 185(2), 446–453 (1969).
[CrossRef]

R. F. Chang, V. Korenman, C. O. Alley, R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178(2), 612–621 (1969).
[CrossRef]

Abram, I.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

Agafonov, I. N.

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56(2-3), 422–431 (2009).
[CrossRef]

Alley, C. O.

R. F. Chang, V. Korenman, C. O. Alley, R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178(2), 612–621 (1969).
[CrossRef]

Asano, T.

M. Yamaguchi, T. Asano, S. Noda, “Third emission mechanism in solid-state nanocavity quantum electrodynamics,” Rep. Prog. Phys. 75(9), 096401 (2012).
[CrossRef] [PubMed]

Assmann, M.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[CrossRef] [PubMed]

M. Assmann, F. Veit, M. Bayer, M. van der Poel, J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009).
[CrossRef] [PubMed]

Atatüre, M.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

Badolato, A.

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

Baek, B.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010).
[CrossRef] [PubMed]

Balbach, M.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011).
[CrossRef] [PubMed]

Bayer, M.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[CrossRef] [PubMed]

M. Assmann, F. Veit, M. Bayer, M. van der Poel, J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009).
[CrossRef] [PubMed]

Beaudoin, G.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

Berggren, K. K.

Berstermann, T.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[CrossRef] [PubMed]

Bettelli, S.

B. Blauensteiner, I. Herbauts, S. Bettelli, A. Poppe, H. Hübel, “Photon bunching in parametric down-conversion with continuous-wave excitation,” Phys. Rev. A 79(6), 063846 (2009).
[CrossRef]

Beveratos, A.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

Blauensteiner, B.

B. Blauensteiner, I. Herbauts, S. Bettelli, A. Poppe, H. Hübel, “Photon bunching in parametric down-conversion with continuous-wave excitation,” Phys. Rev. A 79(6), 063846 (2009).
[CrossRef]

Braive, R.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

Brida, G.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

Cantrell, C. D.

Y. Qu, S. Singh, C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. 76(8), 1236–1239 (1996).
[CrossRef] [PubMed]

Chang, R. F.

R. F. Chang, V. Korenman, C. O. Alley, R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178(2), 612–621 (1969).
[CrossRef]

Chekhova, M. V.

I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56(2-3), 422–431 (2009).
[CrossRef]

Chibani, H.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011).
[CrossRef] [PubMed]

Corti, M.

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14(4), 1475–1478 (1976).
[CrossRef]

Dauler, E. A.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010).
[CrossRef] [PubMed]

Davidson, F.

F. Davidson, “Measurements of photon correlations in a laser beam near threshold with time-to-amplitude converter techniques,” Phys. Rev. 185(2), 446–453 (1969).
[CrossRef]

Degiorgio, V.

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14(4), 1475–1478 (1976).
[CrossRef]

Degiovanni, I. P.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

Detenbeck, R. W.

R. F. Chang, V. Korenman, C. O. Alley, R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178(2), 612–621 (1969).
[CrossRef]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).

Durian, D. J.

Dynes, J. F.

Eisaman, M. D.

M. D. Eisaman, J. Fan, A. Migdall, S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[CrossRef] [PubMed]

Elvira, D.

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Fält, S.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
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M. D. Eisaman, J. Fan, A. Migdall, S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
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C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).

Finley, J.

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
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T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010).
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S. Reitzenstein, A. Forchel, “Quantum dot micropillars,” J. Phys. D Appl. Phys. 43(3), 033001 (2010).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
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E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

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J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
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E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

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D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

Hamilton, S. A.

Hayat, A.

Hennessy, K.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

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M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
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B. Blauensteiner, I. Herbauts, S. Bettelli, A. Poppe, H. Hübel, “Photon bunching in parametric down-conversion with continuous-wave excitation,” Phys. Rev. A 79(6), 063846 (2009).
[CrossRef]

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T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010).
[CrossRef]

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J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
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J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
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T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010).
[CrossRef]

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M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

Hübel, H.

B. Blauensteiner, I. Herbauts, S. Bettelli, A. Poppe, H. Hübel, “Photon bunching in parametric down-conversion with continuous-wave excitation,” Phys. Rev. A 79(6), 063846 (2009).
[CrossRef]

Hvam, J. M.

M. Assmann, F. Veit, M. Bayer, M. van der Poel, J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009).
[CrossRef] [PubMed]

Imamoglu, A.

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
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I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, L.-A. Wu, “High-visibility intensity interference and ghost imaging with pseudo-thermal light,” J. Mod. Opt. 56(2-3), 422–431 (2009).
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Kalden, J.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
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Kerman, A. J.

Kistner, C.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
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Koch, M.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011).
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Kok, P.

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J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
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Kubanek, A.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011).
[CrossRef] [PubMed]

Kuck, S.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

Lemieux, P.-A.

Liu, J.

Y. Zhou, J. Liu, J. Simon, Y. Shih, “Resolution enhancement of third-order thermal light ghost imaging in the photon counting regime,” J. Opt. Soc. Am. B 29(3), 377 (2012).
[CrossRef]

Y. Zhou, J. Simon, J. Liu, Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81(4), 043831 (2010).
[CrossRef]

Loudon, R.

R. Loudon, P. L. Knight, “Squeezed light,” J. Mod. Opt. 34(6-7), 709–759 (1987).
[CrossRef]

Ma, L.

Migdall, A.

M. D. Eisaman, J. Fan, A. Migdall, S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[CrossRef] [PubMed]

Migdall, A. L.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

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P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).

Mirin, R. P.

Molnar, R. J.

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).

Murr, K.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011).
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D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, S. W. Nam, “High-order temporal coherences of chaotic and laser light,” Opt. Express 18(2), 1430–1437 (2010).
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P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).

Nevet, A.

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Peters, S.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

Piacentini, F.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

Polyakov, S. V.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

M. D. Eisaman, J. Fan, A. Migdall, S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[CrossRef] [PubMed]

Poppe, A.

B. Blauensteiner, I. Herbauts, S. Bettelli, A. Poppe, H. Hübel, “Photon bunching in parametric down-conversion with continuous-wave excitation,” Phys. Rev. A 79(6), 063846 (2009).
[CrossRef]

Portolan, S.

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

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Y. Qu, S. Singh, C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. 76(8), 1236–1239 (1996).
[CrossRef] [PubMed]

Quattropani, A.

T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010).
[CrossRef]

Rakher, M. T.

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).

Rarity, J. G.

P. R. Tapster, J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998).
[CrossRef]

Reitzenstein, S.

S. Reitzenstein, A. Forchel, “Quantum dot micropillars,” J. Phys. D Appl. Phys. 43(3), 033001 (2010).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[CrossRef] [PubMed]

Rempe, G.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011).
[CrossRef] [PubMed]

Robert-Philip, I.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

Ruo Berchera, I.

E. A. Goldschmidt, F. Piacentini, I. Ruo Berchera, S. V. Polyakov, S. Peters, S. Kuck, G. Brida, I. P. Degiovanni, A. L. Migdall, M. Genovese, “Mode reconstruction of a light field by multi-photon statistics,” Phys. Rev. A 88(1), 013822 (2013).
[CrossRef]

Sagnes, I.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

Sames, C.

M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, G. Rempe, “Three-photon correlations in a strongly driven atom-cavity system,” Phys. Rev. Lett. 107(2), 023601 (2011).
[CrossRef] [PubMed]

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C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).

E. Waks, C. Santori, Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66(4), 042315 (2002).
[CrossRef]

Savona, V.

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

Schneider, C.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[CrossRef] [PubMed]

Schwendimann, P.

T. Horikiri, P. Schwendimann, A. Quattropani, S. Hing, A. Forchel, Y. Yamamoto, “Higher order coherence of exciton-polariton condensates,” Phys. Rev. B 81(3), 033307 (2010).
[CrossRef]

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Shields, A. J.

Shih, Y.

Y. Zhou, J. Liu, J. Simon, Y. Shih, “Resolution enhancement of third-order thermal light ghost imaging in the photon counting regime,” J. Opt. Soc. Am. B 29(3), 377 (2012).
[CrossRef]

Y. Zhou, J. Simon, J. Liu, Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81(4), 043831 (2010).
[CrossRef]

Simon, J.

Y. Zhou, J. Liu, J. Simon, Y. Shih, “Resolution enhancement of third-order thermal light ghost imaging in the photon counting regime,” J. Opt. Soc. Am. B 29(3), 377 (2012).
[CrossRef]

Y. Zhou, J. Simon, J. Liu, Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81(4), 043831 (2010).
[CrossRef]

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Y. Qu, S. Singh, C. D. Cantrell, “Measurements of higher order photon bunching of light beams,” Phys. Rev. Lett. 76(8), 1236–1239 (1996).
[CrossRef] [PubMed]

Slattery, O.

Solomon, G. S.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).

Srinivasan, K.

Stevens, M. J.

Tang, X.

Tapster, P. R.

P. R. Tapster, J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45(3), 595–604 (1998).
[CrossRef]

Tarel, G.

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

Thomas, O.

Thompson, N. L.

A. G. Palmer, N. L. Thompson, “Molecular aggregation characterized by high order autocorrelation in fluorescence correlation spectroscopy,” Biophys. J. 52(2), 257–270 (1987).
[CrossRef] [PubMed]

van der Poel, M.

M. Assmann, F. Veit, M. Bayer, M. van der Poel, J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009).
[CrossRef] [PubMed]

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M. Assmann, F. Veit, M. Bayer, M. van der Poel, J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325(5938), 297–300 (2009).
[CrossRef] [PubMed]

Verma, V. B.

D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, A. Beveratos, “Higher-order photon correlations in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).

Volz, T.

M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, A. Imamoğlu, “Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103(20), 207403 (2009).
[CrossRef] [PubMed]

Vuckovic, J.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).

Waks, E.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots,” Phys. Rev. B 69(20), 205324 (2004).

E. Waks, C. Santori, Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66(4), 042315 (2002).
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Figures (10)

Fig. 1
Fig. 1

Optical pulse train illustrating the labeling scheme used in the discrete notation. The repetition time of the source, Trep, is set by that of the excitation laser.

Fig. 2
Fig. 2

Experimental geometry. As the dashed box on the left indicates, the “source” is defined to include the pump laser, quantum dot (QD) sample, and all optics for collection and filtering. Obj. is a microscope objective. The tunable filter is a piezo-tunable Fabry-Perot etalon with a bandwidth of ~0.06 nm and free spectral range of ~1.5 nm; this filter is tuned to line up with the peak QD emission line at a wavelength of ~959.5 nm. The fixed-bandpass filter has a nominal center wavelength of 960 nm and bandwidth of ~10 nm. The “measurement” side is a modified Hanbury Brown-Twiss interferometer, consisting of two ~50:50 beamsplitters (BSa and BSb), three single-photon avalanche diodes (SPADs 0, 1 and 2), and multi-channel time-stamping electronics.

Fig. 3
Fig. 3

Idealized experimental configuration considered for the calculations presented in Section 4 for (a) γ(2)[0] and (b) γ(3)[0,0]. Each detector Di (i = 0,1,2) shown here is an ideal “click” detector with unity detection efficiency. Non-unity detection efficiency ηi is modeled by placing a beamsplitter with transmittance ηi in front of each detector. The reflectances (R,Ra,Rb) and transmittances (T,Ta,Tb) of the other beamsplitters are equal to the modulus squared of the corresponding field reflection and transmission coefficients.

Fig. 4
Fig. 4

Measured coherence of the single-photon source with CW excitation. g(2)(τ) is shown as green circles in the upper panel. Third-order measurements in the upper panel are cross-sections of the full data set in the lower panel: blue triangles (red squares) lie along the diagonal running from the upper left to lower right (lower left to upper right) in the lower figure. The histogram time bin width (Δτ) is 100 ps for g(2) and 500 ps for g(3). The weak, ~1.8 ns-period oscillations visible g(2) are due to periodic intensity fluctuations in the pump, which is a two-mode HeNe laser.

Fig. 5
Fig. 5

Measured second-order correlation histograms for the source at four different excitation powers, illustrating the degradation of antibunching with increasing pump power. The values in red are the time-averaged pump power for each measurement.

Fig. 6
Fig. 6

Measured coherences at zero delay for a pulsed pump laser at three average excitation powers: (a) 800 nW, (b) 1.4 μW and (c) 3.3 μW.

Fig. 7
Fig. 7

(a) Measured zero-delay coherences (left axis) and total detected count rate (right axis) as a function of time-averaged pump power. The count rate is the sum of the detected rate on the three individual SPADs. (b) Probabilites of the source emitting 1, 2 and 3 photons per pulse, computed from the data in (a).

Fig. 8
Fig. 8

Zero-delay third-order coherence plotted as a function of zero-delay second-order coherence. Data in (a) are for the same quantum dot as in Figs. 47. Green triangles and black squares in (b) denote measurements for two quantum dots located in other mesas on the same wafer. For the open magenta circles in (b), the excitation spot is spatially centered on QD1, but the spectral filter is tuned to be resonant with the cavity mode (magenta arrow in Fig. 9). The blue curves show the prediction of the model described in the text. The gray × ’s in (a) denote the predicted values for one- and two-photon Fock states and a coherent state, |α .

Fig. 9
Fig. 9

(Solid curves, left axis) Unfiltered photoluminescence (PL) spectra of the quantum-dot source for three different pulsed pump laser powers. The black arrow labeled “QD” denotes the approximate central wavelength (λ0 = 959.5 nm) of the ~0.06 nm-wide tunable bandpass filter used for coherence measurements of the quantum dot in Figs. 47. The magenta arrow at 959.2 nm denotes the filter tuning for the “cavity” g(2) and g(3) measurements in Fig. 8(b). The green squares (right axis) show the results of second-order cross-correlation measurements, with one detector measuring photons at the fixed wavelength λ0 = 959.5 nm, and the wavelength of the other detector (λ1) swept across the spectrum.

Fig. 10
Fig. 10

Experimental geometry for spectrally resolved cross-correlation measurements between photons at two different wavelengths. λ0 is fixed at the QD emission wavelength, while λ1 is scanned across the spectrum of cavity and QD emission. BS is a nonpolarizing beamsplitter, and the tunable and fixed filters are the same as in Fig. 2.

Equations (28)

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g ( 2 ) ( τ )= a ^ ( t ) a ^ ( t+τ ) a ^ ( t+τ ) a ^ ( t ) a ^ ( t ) a ^ ( t ) 2 ,
g ( 3 ) ( τ 1 , τ 2 )= a ^ ( t ) a ^ ( t+ τ 1 ) a ^ ( t+ τ 2 ) a ^ ( t+ τ 2 ) a ^ ( t+ τ 1 ) a ^ ( t ) a ^ ( t ) a ^ ( t ) 3 ,
g ( 2 ) [ l ]= a ^ [ k ] a ^ [ k+l ] a ^ [ k+l ] a ^ [ k ] a ^ [ k ] a ^ [ k ] a ^ [ k+l ] a ^ [ k+l ] ,
g ( 3 ) [ l,m ]= a ^ [ k ] a ^ [ k+l ] a ^ [ k+m ] a ^ [ k+m ] a ^ [ k+l ] a ^ [ k ] a ^ [ k ] a ^ [ k ] a ^ [ k+l ] a ^ [ k+l ] a ^ [ k+m ] a ^ [ k+m ] ,
g ( 2 ) [ 0 ]= n ^ k ( n ^ k 1 ) n ^ k 2 ,
g ( 2 ) [ 0 ]= Tr{ ρ ^ n ^ ( n ^ 1 ) } ( Tr{ ρ ^ n ^ } ) 2 .
g ( 2 ) [ 0 ]= n=0 n( n1 )P( n ) [ n=0 nP( n ) ] 2 = 2P( 2 )+6P( 3 )+ μ 2 = 2P( 2 )+6P( 3 )+ [ P( 1 )+2P( 2 )+ ] 2 ,
P( n>1 )= n=2 P( n ) 1 2 n=0 n( n1 )P( n ) .
P( n>1 ) 1 2 μ 2 g ( 2 ) [ 0 ].
g ( 2 ) [ 0 ] 2P( 2 ) [ P( 1 ) ] 2 .
g ( 3 ) [ 0,0 ]= n ^ k ( n ^ k 1 )( n ^ k 2 ) n ^ k 3 ,
g ( 3 ) [ 0,0 ]= 6P( 3 )+24P( 4 )+ μ 3 .
g ( 3 ) [ 0,0 ] 6P( 3 ) [ P( 1 ) ] 3 .
γ ( 2 ) [ 0 ] P 01 ( click,click ) P 0 ( click ) P 1 ( click ) .
P( n 0 , n 1 )= m 0 = n 0 m 1 = n 1 ( m 0 n 0 ) η 0 n 0 ( 1 η 0 ) m 0 n 0 ( m 1 n 1 ) η 1 n 1 ( 1 η 1 ) m 1 n 1 ( m 0 + m 1 m 0 ) × R m 0 T m 1 P( m 0 + m 1 ).
γ ( 2 ) [ 0 ]= n 0 =1 n 1 =1 P( n 0 , n 1 ) [ n 0 =1 n 1 =0 P( n 0 , n 1 ) ][ n 0 =0 n 1 =1 P( n 0 , n 1 ) ] .
γ ( 2 ) [ 0 ]= n=2 [ 1 ( 1 η 0 R ) n ( 1 η 1 T ) n + ( 1 η 0 R η 1 T ) n ]P( n ) [ n=1 [ 1 ( 1 η 0 R ) n ]P( n ) ][ n=1 [ 1 ( 1 η 1 T ) n ]P( n ) ] .
γ ( 2 ) [ 0 ]= 2P( 2 )+6P( 3 )( 1 1 2 η 0 R 1 2 η 1 T )+ [ P( 1 )+2P( 2 )( 1 1 2 η 0 R )+ ][ P( 1 )+2P( 2 )( 1 1 2 η 1 T )+ ] ,
lim η 0 , η 1 0 γ ( 2 ) [ 0 ]= g ( 2 ) [ 0 ].
γ ( 2 ) [ 0 ] 2P( 2 ) [ P( 1 ) ] 2 g ( 2 ) [ 0 ].
γ ( 3 ) [ 0,0 ] P 012 ( click,click,click ) P 0 ( click ) P 1 ( click ) P 2 ( click ) ,
γ ( 3 ) [ 0,0 ]= 6P( 3 )+24P( 4 )( 1 1 2 η 0 R a 1 2 η 1 T a R b 1 2 η 2 T a T b )+ [ P( 1 )+2P( 2 )( 1 1 2 η 0 R a )+ ][ P( 1 )+2P( 2 )( 1 1 2 η 1 T a R b )+ ][ P( 1 )+2P( 2 )( 1 1 2 η 2 T a T b )+ ] .
γ ( 3 ) [ 0,0 ] 6P( 3 ) [ P( 1 ) ] 3 g ( 3 ) [ 0,0 ],
P( m )= k=0 P s ( m ) P b ( mk ) ,
P( m )=[ μ s μ b m1 ( m1 )! +( 1 μ s ) μ b m m! ] e μ b .
g ( 2 ) [ 0 ]= 1+2r ( 1+r ) 2
g ( 3 ) [ 0,0 ]= 1+3r ( 1+r ) 3 ,
g ( 3 ) [ 0,0 ]= 1+3r ( 1+r )( 1+2r ) g ( 2 ) [ 0 ].

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