Abstract

In the realization of ultrasmall semiconductor lasers, cavity-QED effects are used to enhance spontaneous emission and enable the lasing threshold to be crossed with gain contributions from only a few solid-state emitters. Operation in this regime fosters correlation effects that leave their fingerprint especially in the emission dynamics of nanolasers. Using time-resolved photon-correlation spectroscopy, we show that in a quantum-dot photonic-crystal nanolaser emitting in the telecom band, second-order coherence associated with lasing is established on a different timescale than the emission itself. By combining measurements with a microscopic semiconductor laser theory, we attribute the origin to carrier-photon correlations that give rise to non-Markovian effects in the emission dynamics that are not captured by laser rate-equation theories. Our results have direct implications with respect to the modulation response, repetition rate, noise characteristics, and coherence properties of nanolasers for device applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
  24. H. A. M. Leymann, A. Foerster, and J. Wiersig, “Expectation value based equation-of-motion approach for open quantum systems: a general formalism,” Phys. Rev. B 89, 085308 (2014).
    [Crossref]
  25. E. B. Flagg, S. V. Polyakov, T. Thomay, and G. S. Solomon, “Dynamics of nonclassical light from a single solid-state quantum emitter,” Phys. Rev. Lett. 109, 163601 (2012).
    [Crossref]
  26. J. Wiersig, C. Gies, F. Jahnke, M. Assman, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Hofling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
    [Crossref]
  27. M. Aßmann, F. Veit, M. Bayer, C. Gies, F. Jahnke, S. Reitzenstein, S. Hofling, L. Worschech, and A. Forchel, “Ultrafast tracking of second-order photon correlation in the emission of quantum-dot microresonator lasers,” Phys. Rev. B 81, 165314 (2010).
    [Crossref]
  28. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
    [Crossref]

2017 (1)

S. Lichtmannecker, M. Florian, T. Reichert, M. Blauth, M. Bichler, F. Jahnke, J. J. Finley, C. Gies, and M. Kaniber, “A few-emitter solid-state multi-exciton laser,” Sci. Rep. 7, 7420 (2017).
[Crossref]

2016 (3)

T. Wang, G. P. Puccioni, and G. L. Lippi, “How mesoscale lasers can answer fundamental questions related to nanolasers,” Proc. SPIE 9884, 98840B (2016).
[Crossref]

F. Jahnke, C. Gies, M. Aßmann, M. Bayer, H. A. M. Leymann, A. Foerster, J. Wiersig, C. Schneider, M. Kamp, and S. Höfling, “Giant photon bunching, superradiant pulse emission and excitation trapping in quantum-dot nanolasers,” Nat. Commun. 7, 11540 (2016).
[Crossref]

P. Tighineanu, R. S. Daveau, T. B. Lehmann, H. E. Beere, D. A. Ritchie, P. Lodahl, and S. Stobbe, “Single-photon superradiance from a quantum dot,” Phys. Rev. Lett. 116, 163604 (2016).
[Crossref]

2015 (2)

I. Prieto, J. M. Llorens, L. E. Munoz-Camunez, A. G. Taboada, J. Canet-Ferrer, J. M. Ripalda, C. Robles, G. Munoz-Matutano, J. P. Martinez-Pastor, and P. A. Postigo, “Near thresholdless laser operation at room temperature,” Optica 2, 66–69 (2015).
[Crossref]

A. Lebreton, I. Abram, R. Braive, N. Belabas, I. Sagnes, F. Marsili, V. B. Verma, S. W. Nam, T. Gerrits, I. Robert-Philip, M. J. Stevens, and A. Beveratos, “Pulse-to-pulse jitter measurement by photon correlation in high-beta lasers,” Appl. Phys. Lett. 106, 031108 (2015).
[Crossref]

2014 (2)

W. W. Chow, F. Jahnke, and C. Gies, “Emission properties of nanolasers during the transition to lasing,” Light Sci. Appl. 3, e201 (2014).
[Crossref]

H. A. M. Leymann, A. Foerster, and J. Wiersig, “Expectation value based equation-of-motion approach for open quantum systems: a general formalism,” Phys. Rev. B 89, 085308 (2014).
[Crossref]

2012 (2)

E. B. Flagg, S. V. Polyakov, T. Thomay, and G. S. Solomon, “Dynamics of nonclassical light from a single solid-state quantum emitter,” Phys. Rev. Lett. 109, 163601 (2012).
[Crossref]

G. T. Noe, J.-H. Kim, J. Lee, Y. Wang, A. K. Wójcik, S. A. McGill, D. H. Reitze, A. A. Belyanin, and J. Kono, “Giant superfluorescent bursts from a semiconductor magneto-plasma,” Nat. Phys. 8, 219–224 (2012).
[Crossref]

G. T. Noe, J.-H. Kim, J. Lee, Y. Wang, A. K. Wójcik, S. A. McGill, D. H. Reitze, A. A. Belyanin, and J. Kono, “Giant superfluorescent bursts from a semiconductor magneto-plasma,” Nat. Phys. 8, 219–224 (2012).
[Crossref]

2011 (2)

D. Elvira, R. Hostein, B. Fain, L. Monniello, A. Michon, G. Beaudoin, R. Braive, I. Robert-Philip, I. Abram, I. Sagnes, and A. Beveratos, “Single InAsP/InP quantum dots as telecommunications-band photon sources,” Phys. Rev. B 84, 195302 (2011).
[Crossref]

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, and A. Beveratos, “Higher-order photon correlation in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).
[Crossref]

2010 (2)

M. Aßmann, F. Veit, M. Bayer, C. Gies, F. Jahnke, S. Reitzenstein, S. Hofling, L. Worschech, and A. Forchel, “Ultrafast tracking of second-order photon correlation in the emission of quantum-dot microresonator lasers,” Phys. Rev. B 81, 165314 (2010).
[Crossref]

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot-nanocavity system,” Nat. Phys. 6, 279–283 (2010).
[Crossref]

2009 (3)

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

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

A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
[Crossref]

2008 (3)

C. Gies, J. Wiersig, and F. Jahnke, “Output characteristics of pulsed and continuous-wave-excited quantum-dot microcavity lasers,” Phys. Rev. Lett. 101, 067401 (2008).
[Crossref]

A. Michon, R. Hostein, G. Patriarche, N. Gogneau, G. Beaudoin, A. Beveratos, S. Robert-Philip, S. Sauvage, P. Boucaud, and I. Sagnes, “Metal organic vapor phase epitaxy of InAsP/InP(001) quantum dots for 1.55 μm applications: growth, structural, and optical properties,” J. Appl. Phys. 104, 043504 (2008).
[Crossref]

R. Hostein, A. Michon, G. Beaudoin, N. Gogneau, G. Patriarche, J.-Y. Marzin, I. Robert-Philip, I. Sagnes, and A. Beveratos, “Time-resolved characterization of InAsP/InP quantum dots emitting in the C-band telecommunications window,” Appl. Phys. Lett. 93, 073106 (2008).
[Crossref]

2007 (2)

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Loffler, A. Forchel, F. Jahnke, and P. Michler, “Photon statistics of semiconductor microcavity lasers,” Phys. Rev. Lett. 98, 043906 (2007).
[Crossref]

C. Gies, J. Wiersig, M. Lorke, and F. Jahnke, “Semiconductor model for quantum-dot-based microcavity lasers,” Phys. Rev. A 75, 013803 (2007).
[Crossref]

2006 (1)

H. Altug, D. Englund, and J. Vučkovic, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2, 484–488 (2006).
[Crossref]

2003 (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref]

1994 (1)

P. A. Rice and H. J. Carmichael, “Photon statistics of a cavity-QED laser: a comment on the laser-phase-transition analogy,” Phys. Rev. A 50, 4318–4329 (1994).
[Crossref]

1989 (1)

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66, 4801–4805 (1989).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probability at radio frequencies,” Phys. Rev. 69, 681 (1946).

Abram, I.

A. Lebreton, I. Abram, R. Braive, N. Belabas, I. Sagnes, F. Marsili, V. B. Verma, S. W. Nam, T. Gerrits, I. Robert-Philip, M. J. Stevens, and A. Beveratos, “Pulse-to-pulse jitter measurement by photon correlation in high-beta lasers,” Appl. Phys. Lett. 106, 031108 (2015).
[Crossref]

D. Elvira, R. Hostein, B. Fain, L. Monniello, A. Michon, G. Beaudoin, R. Braive, I. Robert-Philip, I. Abram, I. Sagnes, and A. Beveratos, “Single InAsP/InP quantum dots as telecommunications-band photon sources,” Phys. Rev. B 84, 195302 (2011).
[Crossref]

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, and A. Beveratos, “Higher-order photon correlation in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).
[Crossref]

Altug, H.

H. Altug, D. Englund, and J. Vučkovic, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2, 484–488 (2006).
[Crossref]

Arakawa, Y.

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot-nanocavity system,” Nat. Phys. 6, 279–283 (2010).
[Crossref]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref]

Assman, M.

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

Aßmann, M.

F. Jahnke, C. Gies, M. Aßmann, M. Bayer, H. A. M. Leymann, A. Foerster, J. Wiersig, C. Schneider, M. Kamp, and S. Höfling, “Giant photon bunching, superradiant pulse emission and excitation trapping in quantum-dot nanolasers,” Nat. Commun. 7, 11540 (2016).
[Crossref]

M. Aßmann, F. Veit, M. Bayer, C. Gies, F. Jahnke, S. Reitzenstein, S. Hofling, L. Worschech, and A. Forchel, “Ultrafast tracking of second-order photon correlation in the emission of quantum-dot microresonator lasers,” Phys. Rev. B 81, 165314 (2010).
[Crossref]

Ates, S.

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Loffler, A. Forchel, F. Jahnke, and P. Michler, “Photon statistics of semiconductor microcavity lasers,” Phys. Rev. Lett. 98, 043906 (2007).
[Crossref]

Badolato, A.

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

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, and A. Beveratos, “Higher-order photon correlation in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).
[Crossref]

Bayer, M.

F. Jahnke, C. Gies, M. Aßmann, M. Bayer, H. A. M. Leymann, A. Foerster, J. Wiersig, C. Schneider, M. Kamp, and S. Höfling, “Giant photon bunching, superradiant pulse emission and excitation trapping in quantum-dot nanolasers,” Nat. Commun. 7, 11540 (2016).
[Crossref]

M. Aßmann, F. Veit, M. Bayer, C. Gies, F. Jahnke, S. Reitzenstein, S. Hofling, L. Worschech, and A. Forchel, “Ultrafast tracking of second-order photon correlation in the emission of quantum-dot microresonator lasers,” Phys. Rev. B 81, 165314 (2010).
[Crossref]

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

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, and A. Beveratos, “Higher-order photon correlation in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).
[Crossref]

D. Elvira, R. Hostein, B. Fain, L. Monniello, A. Michon, G. Beaudoin, R. Braive, I. Robert-Philip, I. Abram, I. Sagnes, and A. Beveratos, “Single InAsP/InP quantum dots as telecommunications-band photon sources,” Phys. Rev. B 84, 195302 (2011).
[Crossref]

A. Michon, R. Hostein, G. Patriarche, N. Gogneau, G. Beaudoin, A. Beveratos, S. Robert-Philip, S. Sauvage, P. Boucaud, and I. Sagnes, “Metal organic vapor phase epitaxy of InAsP/InP(001) quantum dots for 1.55 μm applications: growth, structural, and optical properties,” J. Appl. Phys. 104, 043504 (2008).
[Crossref]

R. Hostein, A. Michon, G. Beaudoin, N. Gogneau, G. Patriarche, J.-Y. Marzin, I. Robert-Philip, I. Sagnes, and A. Beveratos, “Time-resolved characterization of InAsP/InP quantum dots emitting in the C-band telecommunications window,” Appl. Phys. Lett. 93, 073106 (2008).
[Crossref]

Beere, H. E.

P. Tighineanu, R. S. Daveau, T. B. Lehmann, H. E. Beere, D. A. Ritchie, P. Lodahl, and S. Stobbe, “Single-photon superradiance from a quantum dot,” Phys. Rev. Lett. 116, 163604 (2016).
[Crossref]

Belabas, N.

A. Lebreton, I. Abram, R. Braive, N. Belabas, I. Sagnes, F. Marsili, V. B. Verma, S. W. Nam, T. Gerrits, I. Robert-Philip, M. J. Stevens, and A. Beveratos, “Pulse-to-pulse jitter measurement by photon correlation in high-beta lasers,” Appl. Phys. Lett. 106, 031108 (2015).
[Crossref]

Belyanin, A. A.

G. T. Noe, J.-H. Kim, J. Lee, Y. Wang, A. K. Wójcik, S. A. McGill, D. H. Reitze, A. A. Belyanin, and J. Kono, “Giant superfluorescent bursts from a semiconductor magneto-plasma,” Nat. Phys. 8, 219–224 (2012).
[Crossref]

Berstermann, T.

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

Beveratos, A.

A. Lebreton, I. Abram, R. Braive, N. Belabas, I. Sagnes, F. Marsili, V. B. Verma, S. W. Nam, T. Gerrits, I. Robert-Philip, M. J. Stevens, and A. Beveratos, “Pulse-to-pulse jitter measurement by photon correlation in high-beta lasers,” Appl. Phys. Lett. 106, 031108 (2015).
[Crossref]

D. Elvira, R. Hostein, B. Fain, L. Monniello, A. Michon, G. Beaudoin, R. Braive, I. Robert-Philip, I. Abram, I. Sagnes, and A. Beveratos, “Single InAsP/InP quantum dots as telecommunications-band photon sources,” Phys. Rev. B 84, 195302 (2011).
[Crossref]

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, and A. Beveratos, “Higher-order photon correlation in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).
[Crossref]

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

R. Hostein, A. Michon, G. Beaudoin, N. Gogneau, G. Patriarche, J.-Y. Marzin, I. Robert-Philip, I. Sagnes, and A. Beveratos, “Time-resolved characterization of InAsP/InP quantum dots emitting in the C-band telecommunications window,” Appl. Phys. Lett. 93, 073106 (2008).
[Crossref]

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Nat. Commun. (1)

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Nat. Phys. (3)

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Nature (2)

J. Wiersig, C. Gies, F. Jahnke, M. Assman, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Hofling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
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Optica (1)

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probability at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. A (3)

P. A. Rice and H. J. Carmichael, “Photon statistics of a cavity-QED laser: a comment on the laser-phase-transition analogy,” Phys. Rev. A 50, 4318–4329 (1994).
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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, and A. Beveratos, “Higher-order photon correlation in pulsed photonic crystal nanolasers,” Phys. Rev. A 84, 061802 (2011).
[Crossref]

C. Gies, J. Wiersig, M. Lorke, and F. Jahnke, “Semiconductor model for quantum-dot-based microcavity lasers,” Phys. Rev. A 75, 013803 (2007).
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Phys. Rev. B (4)

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[Crossref]

A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
[Crossref]

H. A. M. Leymann, A. Foerster, and J. Wiersig, “Expectation value based equation-of-motion approach for open quantum systems: a general formalism,” Phys. Rev. B 89, 085308 (2014).
[Crossref]

D. Elvira, R. Hostein, B. Fain, L. Monniello, A. Michon, G. Beaudoin, R. Braive, I. Robert-Philip, I. Abram, I. Sagnes, and A. Beveratos, “Single InAsP/InP quantum dots as telecommunications-band photon sources,” Phys. Rev. B 84, 195302 (2011).
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Phys. Rev. Lett. (5)

C. Gies, J. Wiersig, and F. Jahnke, “Output characteristics of pulsed and continuous-wave-excited quantum-dot microcavity lasers,” Phys. Rev. Lett. 101, 067401 (2008).
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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, and A. Imamoglu, “Explanation of photon correlation in the far-off-resonance optical emission from a quantum-dot-cavity system,” Phys. Rev. Lett. 103, 207403 (2009).
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Proc. SPIE (1)

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[Crossref]

Other (1)

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Supplementary Material (1)

NameDescription
» Supplement 1       Additional details of the HBT setup, the microscopic calculations, and results at additional temperatures.

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Figures (7)

Fig. 1.
Fig. 1. (a) For a nanolaser operating below the threshold, spontaneous emission into the cavity mode leads to thermal radiation with g(2)(t1,t1)>1. Above the threshold, the onset of stimulated emission leads to coherent radiation with g(2)(t1,t1)=1. (b) In a cavity-QED QD nanolaser, in which only a few emitting dipoles are coupled to a high cavity-quality-factor mode, the transition from thermal to coherent radiation (points) is delayed by δt with respect to the emission pulse maximum (solid line). This delay arises from non-Markovian effects in the dynamics of carrier–photon correlations.
Fig. 2.
Fig. 2. (a) InAsP/InP QD spontaneous emission and cavity mode emission for three different temperatures. Measurements of the cavity linewidth at all temperatures are limited by the spectrometer resolution, placing a lower bound on the cavity quality-factor Q>50,000. (b) Nanolaser output intensity versus input optical power for nonresonant pulsed excitation at three different temperatures, demonstrating the transition from the LED regime (5 K) to stimulated emission (300 K). The light-in/light-out curves are reproduced by the microscopic laser theory (solid lines). The arrows denote the excitation powers P1, P2, and P3 for the data shown in Fig. 5.
Fig. 3.
Fig. 3. Illustration of the Hanbury Brown and Twiss setup with fiber-coupled superconducting nanowire single photon detectors (SNSPDs) for mapping the dynamics of g(2)(t1,t2).
Fig. 4.
Fig. 4. (a) Measured and (b) calculated two-time maps of g(2)(t1,t2) for 5 K (left) and 100 K (right).
Fig. 5.
Fig. 5. Blue circles: Time-resolved zero-delay autocorrelation function (diagonal cuts of the two-time maps shown in the previous figure for which t1=t2). (a) In the LED regime at 5 K, g(2)(t1,t1)>1 as expected for incoherent thermal radiation. These dynamics are in marked contrast to 100 K for which g(2)(t1,t1) reaches unity during emission, which is shown in (b). The emission pulse intensity is indicated by the solid red line for reference. (c) Calculated dynamics.
Fig. 6.
Fig. 6. Comparison of the correlation dynamics from the full set of coupled laser equations (solid blue curve) and dynamics in which the correlations between carriers and photons are adiabatically eliminated (dashed blue curve). The disappearance of the delay δt in the formation of coherence in the latter case identifies the non-Markovian polarization dynamics as the origin of the observed effect.
Fig. 7.
Fig. 7. Delay δt between emission intensity maximum and the minimum in the zero-delay second-order photon autocorrelation function as a function of the cavity-Q factor. The results are obtained for g, κ, and β as given in the Supplement 1. A dephasing rate of Γ=0.5  meV (0.7 meV) is used for N=20 and 75 (240 and 500). Note that a few-emitter gain material requires a sufficiently high cavity-Q factor to reach lasing. Below that, g(2) remains thermal and an offset δt can no longer be defined.

Equations (7)

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g(2)(t1,t2)=b(t1)b(t2)b(t2)b(t1)b(t1)b(t1)b(t2)b(t2),
(ddt+2κ)bb=2N|g|2Re[bvscs],
ddtfse,h=2|g|2Re[bvscs]+Rnl(β)+Rpse,h(P).
(ddt+κ+Γ)bvscs=fsefsh(1fsefsh)bb+δbbcscsδbbvsvs,
(ddt)δbbbvscs=(3κ+Γ)δbbbvscs+2|g|2bvscs2(1fsefsh)δbbbb+2fshbbcscs2fsebbvsvs+bb(δbbcscsδbbvsvs).
(ddt+2κ)bbcscs=2|g|2[δbbbvscs+(bb+fse)bvscs],
(ddt+2κ)bbvsvs=2|g|2[δbbbvscs+(bb+fsh)bvscs].

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