Abstract

Solid-state cavity quantum-electrodynamics (QED) has great potential owing to advances such as coupled systems combining a nanocavity and a quantum dot (QD). These systems involve two photon-emission mechanisms: the Purcell effect in the weak coupling regime and vacuum Rabi-splitting in the strong coupling regime. In this paper, we describe a third emission mechanism based on the quantum anti-Zeno effect (AZE) induced by the pure-dephasing in a QD. This is significantly enhanced by the inherent characteristics of the nanocavity. This mechanism explains the origin of strong photon emission at a cavity mode largely detuned from a QD, previously considered a counterintuitive, prima facie non-energy-conserving, light-emission phenomenon. These findings could help in controlling the decay and emission characteristics of solid-state cavity QED, and developing solid-state quantum devices.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
  3. S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
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2007

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

K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system," Nature 450, 862−865 (2007).
[CrossRef] [PubMed]

K. Srinivasan and O. Painter, "Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity," Phys. Rev. A 75, 023814 (2007).
[CrossRef]

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

2006

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81−90 (2006).
[CrossRef]

2005

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, "Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals," Science 308, 1296−1298 (2005).
[CrossRef] [PubMed]

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005).
[CrossRef]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207−210 (2005).
[CrossRef]

2004

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

2003

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944−947 (2003).
[CrossRef] [PubMed]

2002

M. Bayer and A. Forchel, "Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots," Phys. Rev. B 65, 041308 (2002).
[CrossRef]

2001

G. S. Agarwal, M. O. Scully, and H. Walther, "Accelerating decay by multiple 2π pulses," Phys. Rev. A 63, 044101 (2001).
[CrossRef]

A. G. Kofman and G. Kurizki, "Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum," Phys. Rev. Lett. 87, 270405 (2001).
[CrossRef]

2000

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000).
[CrossRef] [PubMed]

A. G. Kofman and G. Kurizki, "Acceleration of quantum decay processes by frequent observations," Nature 405, 546−550 (2000).
[CrossRef] [PubMed]

1996

D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, "Fine structure splitting in the optical spectra of single GaAs quantum dots," Phys. Rev. Lett. 76, 3005−3008 (1996).
[CrossRef] [PubMed]

1992

J. L. Pan, "Reduction of the Auger rate in semiconductor quantum dots," Phys. Rev. B 46, 3977-3998 (1992).
[CrossRef]

1989

H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, "Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators," Phys. Rev. A 40, 5516−5519 (1989).
[CrossRef] [PubMed]

1982

Y. Arakawa and H. Sakai, "Multidimensional quantum well laser and temperature dependence of its threshold current," Appl. Phys. Lett. 40, 939−941 (1982).
[CrossRef]

1981

W. H. Zurek, "Pointer basis of quantum apparatus," Phys. Rev. D 24, 1516−1525 (1981).
[CrossRef]

1946

E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

1937

I. I. Rabi, "Space quantization in a gyrating magnetic field," Phys. Rev. 51, 652 (1937).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, M. O. Scully, and H. Walther, "Accelerating decay by multiple 2π pulses," Phys. Rev. A 63, 044101 (2001).
[CrossRef]

Akahane, Y.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207−210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944−947 (2003).
[CrossRef] [PubMed]

Andreani, L. C.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Arakawa, Y.

Y. Arakawa and H. Sakai, "Multidimensional quantum well laser and temperature dependence of its threshold current," Appl. Phys. Lett. 40, 939−941 (1982).
[CrossRef]

Asano, T.

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005).
[CrossRef]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207−210 (2005).
[CrossRef]

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, "Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals," Science 308, 1296−1298 (2005).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944−947 (2003).
[CrossRef] [PubMed]

Atatüre, M.

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

Badolato, A.

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

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Bayer, M.

M. Bayer and A. Forchel, "Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots," Phys. Rev. B 65, 041308 (2002).
[CrossRef]

Becher, C.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000).
[CrossRef] [PubMed]

Bouwmeester, D.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Brecha, R. J.

H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, "Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators," Phys. Rev. A 40, 5516−5519 (1989).
[CrossRef] [PubMed]

Carmichael, H. J.

H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, "Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators," Phys. Rev. A 40, 5516−5519 (1989).
[CrossRef] [PubMed]

Choi, Y.-S.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Deppe, D. G.

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

Ell, C.

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

Fält, S.

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

Forchel, A.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

M. Bayer and A. Forchel, "Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots," Phys. Rev. B 65, 041308 (2002).
[CrossRef]

Fujita, M.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, "Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals," Science 308, 1296−1298 (2005).
[CrossRef] [PubMed]

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005).
[CrossRef]

Gammon, D.

D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, "Fine structure splitting in the optical spectra of single GaAs quantum dots," Phys. Rev. Lett. 76, 3005−3008 (1996).
[CrossRef] [PubMed]

Gerace, D.

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

Gibbs, H. M.

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81−90 (2006).
[CrossRef]

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

Götzinger, S.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

Gulde, S.

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

Hennessy, K.

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

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Hofmann, C.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Hu, E. L.

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

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Imamoglu, A.

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

Kamp, M.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

Katzer, D. S.

D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, "Fine structure splitting in the optical spectra of single GaAs quantum dots," Phys. Rev. Lett. 76, 3005−3008 (1996).
[CrossRef] [PubMed]

Keldysh, L. V.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Khitrova, G.

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81−90 (2006).
[CrossRef]

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

Kimble, H. J.

H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, "Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators," Phys. Rev. A 40, 5516−5519 (1989).
[CrossRef] [PubMed]

Kira, M.

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81−90 (2006).
[CrossRef]

Kiraz, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000).
[CrossRef] [PubMed]

Koch, S. W.

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81−90 (2006).
[CrossRef]

Kofman, A. G.

A. G. Kofman and G. Kurizki, "Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum," Phys. Rev. Lett. 87, 270405 (2001).
[CrossRef]

A. G. Kofman and G. Kurizki, "Acceleration of quantum decay processes by frequent observations," Nature 405, 546−550 (2000).
[CrossRef] [PubMed]

Kounoike, K.

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005).
[CrossRef]

Kuhn, S.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Kulakovskii, V. D.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Kurizki, G.

A. G. Kofman and G. Kurizki, "Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum," Phys. Rev. Lett. 87, 270405 (2001).
[CrossRef]

A. G. Kofman and G. Kurizki, "Acceleration of quantum decay processes by frequent observations," Nature 405, 546−550 (2000).
[CrossRef] [PubMed]

Lidong Zhang, P. M.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000).
[CrossRef] [PubMed]

Löffler, A.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Michler, P.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000).
[CrossRef] [PubMed]

Nakanishi, J.

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005).
[CrossRef]

Noda, S.

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005).
[CrossRef]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207−210 (2005).
[CrossRef]

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, "Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals," Science 308, 1296−1298 (2005).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944−947 (2003).
[CrossRef] [PubMed]

Painter, O.

K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system," Nature 450, 862−865 (2007).
[CrossRef] [PubMed]

K. Srinivasan and O. Painter, "Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity," Phys. Rev. A 75, 023814 (2007).
[CrossRef]

Pan, J. L.

J. L. Pan, "Reduction of the Auger rate in semiconductor quantum dots," Phys. Rev. B 46, 3977-3998 (1992).
[CrossRef]

Park, D.

D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, "Fine structure splitting in the optical spectra of single GaAs quantum dots," Phys. Rev. Lett. 76, 3005−3008 (1996).
[CrossRef] [PubMed]

Petroff, P. M.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000).
[CrossRef] [PubMed]

Press, D.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

Purcell, E.

E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Rabi, I. I.

I. I. Rabi, "Space quantization in a gyrating magnetic field," Phys. Rev. 51, 652 (1937).
[CrossRef]

Raizen, M. G.

H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, "Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators," Phys. Rev. A 40, 5516−5519 (1989).
[CrossRef] [PubMed]

Rakher, M. T.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Reinecke, T. L.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Reithmaier, J. P.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Reitzenstein, S.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Rice, P. R.

H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, "Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators," Phys. Rev. A 40, 5516−5519 (1989).
[CrossRef] [PubMed]

Rupper, G.

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

Sakai, H.

Y. Arakawa and H. Sakai, "Multidimensional quantum well laser and temperature dependence of its threshold current," Appl. Phys. Lett. 40, 939−941 (1982).
[CrossRef]

Scherer, A.

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81−90 (2006).
[CrossRef]

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

Schoenfeld, W. V.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000).
[CrossRef] [PubMed]

Scully, M. O.

G. S. Agarwal, M. O. Scully, and H. Walther, "Accelerating decay by multiple 2π pulses," Phys. Rev. A 63, 044101 (2001).
[CrossRef]

Sek, G.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Shanabrook, B. V.

D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, "Fine structure splitting in the optical spectra of single GaAs quantum dots," Phys. Rev. Lett. 76, 3005−3008 (1996).
[CrossRef] [PubMed]

Shchekin, O. B.

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

Snow, E. S.

D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, "Fine structure splitting in the optical spectra of single GaAs quantum dots," Phys. Rev. Lett. 76, 3005−3008 (1996).
[CrossRef] [PubMed]

Song, B. S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207−210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944−947 (2003).
[CrossRef] [PubMed]

Srinivasan, K.

K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system," Nature 450, 862−865 (2007).
[CrossRef] [PubMed]

K. Srinivasan and O. Painter, "Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity," Phys. Rev. A 75, 023814 (2007).
[CrossRef]

Strauf, S.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Takahashi, S.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, "Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals," Science 308, 1296−1298 (2005).
[CrossRef] [PubMed]

Tanaka, Y.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, "Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals," Science 308, 1296−1298 (2005).
[CrossRef] [PubMed]

Walther, H.

G. S. Agarwal, M. O. Scully, and H. Walther, "Accelerating decay by multiple 2π pulses," Phys. Rev. A 63, 044101 (2001).
[CrossRef]

Winger, M.

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

Yamaguchi, M.

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005).
[CrossRef]

Yamamoto, Y.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

Yoshie, T.

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

Zurek, W. H.

W. H. Zurek, "Pointer basis of quantum apparatus," Phys. Rev. D 24, 1516−1525 (1981).
[CrossRef]

Appl. Phys. Lett.

Y. Arakawa and H. Sakai, "Multidimensional quantum well laser and temperature dependence of its threshold current," Appl. Phys. Lett. 40, 939−941 (1982).
[CrossRef]

Electron. Lett.

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005).
[CrossRef]

Nat. Mater.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207−210 (2005).
[CrossRef]

Nat. Phys.

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81−90 (2006).
[CrossRef]

Nature

A. G. Kofman and G. Kurizki, "Acceleration of quantum decay processes by frequent observations," Nature 405, 546−550 (2000).
[CrossRef] [PubMed]

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

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004).
[CrossRef] [PubMed]

K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system," Nature 450, 862−865 (2007).
[CrossRef] [PubMed]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944−947 (2003).
[CrossRef] [PubMed]

Phys. Rev.

E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

I. I. Rabi, "Space quantization in a gyrating magnetic field," Phys. Rev. 51, 652 (1937).
[CrossRef]

Phys. Rev. A

K. Srinivasan and O. Painter, "Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity," Phys. Rev. A 75, 023814 (2007).
[CrossRef]

G. S. Agarwal, M. O. Scully, and H. Walther, "Accelerating decay by multiple 2π pulses," Phys. Rev. A 63, 044101 (2001).
[CrossRef]

H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, "Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators," Phys. Rev. A 40, 5516−5519 (1989).
[CrossRef] [PubMed]

Phys. Rev. B

J. L. Pan, "Reduction of the Auger rate in semiconductor quantum dots," Phys. Rev. B 46, 3977-3998 (1992).
[CrossRef]

M. Bayer and A. Forchel, "Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots," Phys. Rev. B 65, 041308 (2002).
[CrossRef]

Phys. Rev. D

W. H. Zurek, "Pointer basis of quantum apparatus," Phys. Rev. D 24, 1516−1525 (1981).
[CrossRef]

Phys. Rev. Lett.

A. G. Kofman and G. Kurizki, "Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum," Phys. Rev. Lett. 87, 270405 (2001).
[CrossRef]

D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, "Fine structure splitting in the optical spectra of single GaAs quantum dots," Phys. Rev. Lett. 76, 3005−3008 (1996).
[CrossRef] [PubMed]

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007).
[CrossRef] [PubMed]

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef] [PubMed]

Science

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000).
[CrossRef] [PubMed]

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, "Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals," Science 308, 1296−1298 (2005).
[CrossRef] [PubMed]

Other

V. B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge University Press, Cambridge, 1992).
[CrossRef]

H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1990).

M. Yamaguchi, T. Asano, and S. Noda, "Origin of unexpected light emission in a coupled system of a photonic-crystal nanocavity and a quantum dot," presented at the 8th International Conference on Physics of Light-Matter Coupling in Nanostructures, Tokyo, Japan, 7-11 April 2008.
[PubMed]

M. Tabuchi, S. Noda, and A. Sasaki, "Mesoscopic structure in lattice-mismatched heteroepitaxial interface layers," in Science and Technology of Mesoscopic Structures (eds Namba, S., Hamaguchi, C. & Ando, T.) 379−384 (Springer Verlag, Tokyo, 1992).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic illustration of the system investigated. (a) Example of a solid-state cavity QED system, where a single QD is embedded in a two-dimensional photonic crystal nanocavity. (b) Schematic illustration of our analysis model, a TLS interacting with a single-mode cavity. Under the initial condition of an excited TLS, photons can be emitted to free space by two pathways: a direct pathway to free space or an indirect pathway through the cavity mode. (c) Schematic illustration of elastic interactions between the TLS and electrons in the cladding layer.

Fig. 2.
Fig. 2.

Decay rate of the excited TLS. (a) Decay rate for pure-dephasing rates ħγ phase of 0 µeV (blue line), 35 µeV (red line), 70 µeV (green line) and 350 µeV (black line). (b) Decay rate for ħγ phase=35 µeV (red line), 70 µeV (green line) and 350 µeV (black line) normalized by the decay rate for ħγ phase=0 µeV in Fig. 2(a). The inset shows the magnified image under the onresonant condition. The gray dashed line denotes that the value along the longitudinal axis equals to one.

Fig. 3.
Fig. 3.

Emission spectra for various values of detuning. (a) No pure-dephasing. (b) With puredephasing rate ħγ phase=35 µeV. (c) With pure-dephasing rate ħγ phase=70 µeV. (d) With pure-dephasing rate ħγ phase=350 µeV. The labels TLS and Cav denote the peaks at the transition energy of the TLS and at the cavity resonance energy, respectively.

Fig. 4.
Fig. 4.

Pure-dephasing dependence of the photon emission rates for individual peaks in the emission spectrum. (a) Photon emission rates for detuning of 4 meV. The labels W TLS and W cav denote the photon emission rates from the peaks at the transition energy of the TLS and at the cavity resonance energy, respectively. (b) Factor of F is defined as W cav/(W TLS+W cav), which represents the ratio of the cavity mode emission rate to the total emitted photon rate. (c) Corresponding emission spectrum calculated for ħγ phase=35 µeV, where the percentage represents the ratio of each peak to the total integral value.

Fig. 5.
Fig. 5.

Emission channels through density operator states. (A)–(C) represent diagonal states and (D)–(F) represent off-diagonal (correlation) states. Gray arrows denote couplings between states within first-order perturbation. For simplicity, the Hermitian conjugate states and their related couplings are not shown.

Equations (44)

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W = 2 π 0 d ω ξ ( ω ) 2 D ( ω ) F ( ω ) .
d ρ S d t = i 1 [ H S , ρ S ] + Σ m L m ρ m ,
W Γ cav + Γ spon + R ( Γ cav Γ spon ) 2 + R 2 ,
R 2 g 2 Γ total δ ω TLS , cav 2 + Γ total 2 .
S ( ω ) 2 π Λ + Λ Re [ 1 γ + γ { f ( γ + ) i ω + γ + f ( γ ) i ω + γ } ] .
2 γ ± [ Γ total + i ( ω TLS + ω cav ) ] ± ( Γ cav Γ spon γ phase i δ ω TLS , cav ) 2 4 g 2 .
F W cav ( W TLS + W cav ) ,
W TLS 2 Γ spon + 2 g 2 Γ cav δ ω TLS , cav 2 ,
W cav 2 g 2 ( Γ spon + γ phase ) δ ω TLS , cav 2 ,
F Γ spon + γ phase Γ spon ( δ ω TLS , cav g ) 2 + Γ spon + ( Γ spon + γ phase ) ,
d ρ T d t = i 1 [ H T , ρ T ] ,
H T = H S + H R + H int ,
d ρ S ( t ) d t = i 1 [ H S ( t ) , ρ S ( t ) ] i 1 τ 1 t t + τ d t 1 T r R [ H int ( t 1 ) , ρ S ( t ) ρ R ( 0 ) ]
2 τ 1 t t + τ d t 1 t t 1 d t 2 T r R [ H int ( t 1 ) , [ H int ( t 2 ) , ρ S ( t ) ρ R ( 0 ) ] ] ,
H Coulomb = 1 2 d 3 r d 3 r ψ ̂ ( r ) ψ ̂ ( r ) e 2 r r ψ ̂ ( r ) ψ ̂ ( r ) ,
ψ ̂ ( r ) = c ̂ 1 φ 1 ( r ) + c ̂ 2 φ 2 ( r ) + Σ m c ̂ m φ m ( r ) ,
H Coulomb H 12 , El + H 11 , El + H 22 , El ,
H ii , El = c ̂ i c ̂ i Σ m c ̂ m c ̂ m G ( i , i , m , m ) + c ̂ i c ̂ i Σ n , k c ̂ n c ̂ k G ( i , i , n , k ) , ( n k , i = 1 or 2 )
H 12 , El = c ̂ 1 c ̂ 1 c ̂ 2 c ̂ 2 G ( 1 , 1 , 2 , 2 ) ,
G ( i , j , k , l ) d 3 r d 3 r e 2 r r [ φ i * ( r ) φ j ( r ) φ k * ( r ) φ l ( r ) φ i ( r ) φ j * ( r ) φ l ( r ) φ k * ( r ) ] .
L i , phase X = i Δ i [ c ̂ i c ̂ i , X ] γ phase , 1 ( c ̂ i c ̂ i X + X c ̂ i c ̂ i 2 c ̂ i c ̂ i X c ̂ i c ̂ i ) , ( i = 1 , or 2 )
L cav X = Γ cav ( a ̂ cav a ̂ cav X + X a ̂ cav a ̂ cav 2 a ̂ cav X a ̂ cav ) ,
L spon X = Γ spon ( c ̂ 2 c ̂ 1 c ̂ 1 c ̂ 2 X + X c ̂ 2 c ̂ 1 c ̂ 1 c ̂ 2 2 c ̂ 1 c ̂ 2 X c ̂ 2 c ̂ 1 ) ,
H S ( t ) = ( c ̂ 2 c ̂ 1 a ̂ cav g exp ( i δ ω TLS , cav t ) + H . c . ) + H 12 , El ,
d ρ S ( t ) d t = L ρ S ( t ) ,
L L S + L cav + L spon + L 1 , phase + L 2 , phase ,
L S X i 1 [ H S , X ] .
d < c ̂ 2 c ̂ 2 ( t ) > d t = Re [ i g < c ̂ 2 c ̂ 1 a ̂ cav ( t ) > ] 2 Γ spon [ < c ̂ 2 c ̂ 2 ( t ) > < c ̂ 2 c ̂ 1 c ̂ 1 c ̂ 2 ( t ) > ] ,
d < a ̂ cav a ̂ cav ( t ) > d t = Re [ i g < c ̂ 2 c ̂ 1 a ̂ cav ( t ) > ] 2 Γ cav < a ̂ cav a ̂ cav ( t ) > ,
d < c ̂ 2 c ̂ 1 a ̂ cav ( t ) > d t = i g * < c ̂ 2 c ̂ 2 ( t ) > i g * < a ̂ cav a ̂ cav ( t ) > + ( i δ ω TLS , cav Γ total ) < c ̂ 2 c ̂ 1 a ̂ cav ( t ) > ,
d < c ̂ 2 c ̂ 1 c ̂ 1 c ̂ 2 ( t ) > d t = 0 ,
d < c ̂ 2 c ̂ 2 ( t ) > d t = R < c ̂ 2 c ̂ 2 ( t ) > + R < a ̂ cav a ̂ cav ( t ) > 2 Γ spon < c ̂ 2 c ̂ 2 ( t ) > ,
d < a ̂ cav a ̂ cav ( t ) > d t = R < c ̂ 2 c ̂ 2 ( t ) > R < a ̂ cav a ̂ cav ( t ) > 2 Γ cav < a ̂ cav a ̂ cav ( t ) > ,
R + i K 2 g 2 1 i δ ω TLS , cav + Γ total ,
Λ ± ( Γ cav + Γ spon + R ) ± ( Γ cav Γ spon ) 2 R 2 .
S ( ω ) = 2 ε 0 c 0 ( 2 π ) 1 0 d τ 0 d t < E ( t + τ ) · E + ( t ) > exp ( i ω τ ) ,
2 ε 0 c 0 < E ( t + τ ) · E + ( t ) > 2 Γ cav < a ̂ cav ( t + τ ) a ̂ cav ( t ) > + 2 Γ spon < c ̂ 2 c ̂ 1 ( t + τ ) c ̂ 1 c ̂ 2 ( t ) > ,
S ( ω ) S spon ( ω ) + S cav ( ω ) ,
S cav ( ω ) = Γ cav π Re [ 0 d τ 0 d t < a ̂ cav ( t ) a ̂ cav ( t + τ ) > exp ( i ω τ ) ] ,
S spon ( ω ) = Γ spon π Re [ 0 d τ 0 d t < c ̂ 2 ( t ) c ̂ 1 ( t ) c ̂ 1 ( t + τ ) c ̂ 2 ( t + τ ) > exp ( i ω τ ) ] .
S α ( ω ) 2 π Λ + Λ Re [ 1 γ + γ { f α ( γ + ) i ω + γ + f α ( γ ) i ω + γ } ] , ( α = spon , or cav )
2 γ ± [ Γ total + i ( ω TLS + ω cav ) ] ± ( Γ cav Γ spon γ phase i δ ω TLS , cav ) 2 4 g 2 ,
f cav ( γ ± ) Γ cav [ R ( γ ± + i ω TLS + Γ total ) + i K Γ cav ] ,
f spon ( γ ± ) Γ spon [ Γ cav ( i K 2 Γ cav ) ( γ ± + i ω cav ) ( R + 2 Γ cav ) ] .

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