## Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system

Optics Express, Vol. 17, Issue 5, pp. 3322-3330 (2009)

http://dx.doi.org/10.1364/OE.17.003322

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### Abstract

We present a rigorous medium-dependent theory for describing the quantum field emitted and detected from a single quantum dot exciton, strongly coupled to a planar photonic crystal nanocavity, from which the exact spectrum is derived. By using simple mode decomposition techniques, this exact spectrum is subsequently reduced to two separate user-friendly forms, in terms of the leaky cavity mode emission and the radiation mode emission. On application to study exciton-cavity coupling in the strong coupling regime, besides a pronounced modification of the usual vacuum Rabi spectral doublet, we predict several new effects associated with the leaky cavity mode emission, including the appearance of an off-resonance cavity mode and a loss-induced on-resonance spectral triplet. The cavity mode emission is shown to completely dominate the emitted spectrum, even for large cavity-exciton detunings, whereby the usual cavity-QED formulas developed for radiation-mode emission drastically fail. These predictions are in qualitative agreement with several “mystery observations” reported in recent experiments, and apply to a wide range of semiconductor cavities.

© 2009 Optical Society of America

## 2. Theory

13. H. T. Dung, L. Knöll, and D-G. Welsch, “Spontaneous decay in the presence of dispersing and absorbing bodies: General theory and application to a spherical cavity,” Phys. Rev. A **62**, 053804 (2000). [CrossRef]

**G**(

**r**,

**r**′;

*ω*), which is the field response at

**r**to a polarization dipole at

**r**′, as a function of frequency. The photon Green function is defined from Maxwell’s equations, solved with a polarization dipole in a medium defined through a spatially-dependent dielectric constant,

*ε*(

**r**). For a lossless inhomogeneous dielectric, it is also useful to introduce a generalized-transverse Green function,

**K**(

**r**,

**r**′;

*ω*) =

**G**(

**r**,

**r**′;

*ω*) −

*δ*(

**r**−

**r**′)

**I**/

*ε*(

**r**) = ∑

_{k}

*ω*

**(**

^{2}_{k}f_{k}**r**)[

**f**(

_{k}**r**′)]

^{*}/(

*ω*

^{2}−

*ω*

**) [14**

^{2}_{k}14. S. Hughes, “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. **98**, 083603 (2007). [CrossRef] [PubMed]

15. M. Wubs, L.G. Suttorp, and A. Lagendijk. “Multiple-scattering approach to interatomic interactions and superra-diance in inhomogeneous dielectrics,” Phys. Rev. A **70**, 53823 (2004). [CrossRef]

**f**; these modes are solutions of the standard eigenvalue problem: [

_{k}**∇**×

**∇**×−

*ω*

^{2}/

*c*

^{2}

*ε*(

**r**)]

**f**(

_{k}**r**) = 0.

*et al*. [15

15. M. Wubs, L.G. Suttorp, and A. Lagendijk. “Multiple-scattering approach to interatomic interactions and superra-diance in inhomogeneous dielectrics,” Phys. Rev. A **70**, 53823 (2004). [CrossRef]

*operator equations*are derived for

_{λ},

^{†}

_{λ},

^{+},

_{z}, using the Heisenberg equations of motion. We follow the usual convension, where the operators â

_{λ}represent the photon modes, while the operators

^{+},

^{−}and

_{z}are the Pauli operators of the electron-hole pair (or exciton). We consider a QD located at position rd, with the detected field at

**R**, and no initial excitation field in the spectral region of interest; this latter assumption is not a model restriction, but is similar to how the semiconductor cavity medium is typically excited, namely through incoherent loading of a higher-lying exciton state. We also assume a single exciton dipole moment

*=*

**μ****n**

*μ*, aligned along

**n**. After carrying out a Laplace transform of the operator equations [15

15. M. Wubs, L.G. Suttorp, and A. Lagendijk. “Multiple-scattering approach to interatomic interactions and superra-diance in inhomogeneous dielectrics,” Phys. Rev. A **70**, 53823 (2004). [CrossRef]

**E**̂(

**r**,

*t*) =

*i*∑

_{λ}[

*h̅*

*ω*

_{λ}/2

*ε*

_{0}]

^{½}â

_{λ}(

*t*)

**f**

_{λ}(

**r**)+

*H*.

*c*. and

*g*

_{λ}= [

*ω*

_{λ}/2

*h̅*

*ε*

_{0}]

^{½}

**· f**

*μ**λ*(

**r**

_{d}), one can derive the quantized electric-field operator,

*exact*. The optical spectrum is calculated from a double time integration over the first-order quantum correlation function:

**Ê**

^{(+)}and

**Ê**

^{(−)}are the positive and negative frequency components of the electric field operator. By substituting in the electric-field operator, we obtain the exact spectrum:

*This analytic optical spectrum (Eq. (4)) is valid for any inhomogeneous dielectric*, including waveguides and open systems, and for any number of photons in the system. Importantly, the Green function spectrum clearly highlights the essential role of light propagation, from the dot to the detector, the dynamics of which are completely contained within the propagator

**K**(

**R**,

**r**

_{d};

*ω*). The theory also shows that there is no need to work with a quantized cavity field operator for describing cavity mode emission from a system prepared in vacuum. Rather, any mode projection is implicitly contained within the propagator. For multiple QDs and (or) multiple excitons, an exact analytical spectral form is also possible. For example, in the presence of multiple excitons from the same QD, the formula above would contain a sum over exciton transitions (multiple Pauli operators), but the propagator remains the same.

**Ê**(

**R**,

*ω*) =

*ε*

^{−1}

_{0}

**K**(

**R**,

**r**

_{d};

*ω*) ·

**d̂**(

*ω*)/(1−

**n**·

**K**(

**r**

_{d},

**r**

_{d};

*ω*) ·

**n**

*α*(

*ω*)), where we have introduced a dipole operator term

**d̂**(

*ω*) = −

*i*

**[**

*μ*^{−}(

*t*= 0)/(

*ω*−

*ω*)+

_{x}^{+}(

*t*= 0)/(

*ω*+

*ω*)] and a bare polarizability amplitude

_{x}*α*(

*ω*) = 2

*μ*

^{2}

*ω*

_{x}/[

*h̅*

*ε*

_{0}(

*ω*

^{2}−

*ω*

^{2}

_{x})];

*ω*is the exciton resonance frequency. This is basically identical in form to the quantum electric-field operator first derived byWubs

_{x}*et al*. [15

**70**, 53823 (2004). [CrossRef]

**K**(

**r**,

**r**′;

*ω*) =

**K**

_{cav}(

**r**,

**r**′;

*ω*)+

**K**

_{rad}(

**r**,

**r**′;

*ω*), where the local and nonlocal cavity contributions are, respectively,

**K**

_{cav}(

**r**

*,*

_{d}**r**

_{d};

*ω*) =

*ω*

^{2}

_{c}|

**f**

_{c}(

**r**

_{d})|

^{2}/(

*ω*

^{2}−

*ω*

^{2}

_{c}+

*i*

*ω*

**Γ**

_{c}) and

**K**

_{cav}(

**R**,

**r**

_{d};

*ω*) =

*ω*

^{2}

_{c}

**f**

_{c}(

**R**)

**f**

^{*}

_{c}(

**r**

_{d})/(

*ω*

^{2}−

*ω*

^{2}

_{c}+

*i*

*ω*

**Γ**

_{v}), while the radiation mode contributions represent the sum of radiation modes above the light line. In essence, we are considering a PC system that consists of a well defined cavity mode (with resonance frequency

*ω*), deep inside the photonic bandgap, and a sum of radiation modes above the light line. The contribution from the radiation modes is typically significantly smaller than the resonant contribution, especially in the presence of an in-plane photonic bandgap. Another important point is that the total cavity decay rate-for example, the broadening of the local density of states (LDOS) near a cavity antinode position center of the slab-is given by the sum of two contributions,

_{c}**Γ**

_{c}=

**Γ**

_{v}+

**Γ**

_{h}, where

**Γ**

_{v}and

**Γ**

_{h}account for vertical and horizontal (in-plane:

*x*/

*y*−) decay loss, respectively; the horizontal decay, for example, can be caused by not having enough holes surrounding the defect cavity, or it can be caused by material losses that partly destroy the photonic bandgap. We now conveniently separate the total emitted spectrum into a background radiation-mode spectrum (emitted at a rate

**Γ**

_{b}) and the cavity-mode spectrum (emitted at a rate

**Γ**

_{v}):

*F*(

**R**) is a geometrical factor that depends on the detector location [16],

**Γ**is an effective exciton decay rate, and

*g*= [

*ω*/2

_{c}*h̅*

*ε*

_{0}]

^{ν}

**·**

*μ***f**

_{c}(

**r**

_{d}) is a cavity-exciton coupling constant.

*S*

_{cav}and

*S*

_{rad}, several important scaling rules and interference effects immediately become clear. First, near resonance, the ratio of the cavity mode emission and the radiation mode emission scales with 4

*g*

^{2}/(

**Γ**b

**Γ**

_{c}) (assuming that

**Γ**

_{h}≪

**Γ**

_{v}). Thus, for strongly coupled QDs, with

*g*and

**Γ**

_{v}≈ 0.1meV, and

**Γ**

_{b}≈ 0.05

*μ*eV [9

9. K. Hennessy, A. Badolato, M. Winger, A. Atäture, S. Fält, E.L. Hu, and A. Imamoǧlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature **445**, 896 (2007). [CrossRef] [PubMed]

*S*

_{cav}will completely dominate the detected spectra, and by several orders of magnitude; in fact, as we will show later, this is also the case for significant exciton-cavity detunings. Second, from

*S*

_{cav}, the presence of in-plane decay can result in unexpected interference effects, whereby a singlet will remain in the presence of a doublet; this can lead to a spectral triplet (see Eq. (6)).

*et al*. [17

17. H. 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 (1989). [CrossRef] [PubMed]

*et al*. [18

18. L. C. Andreani, G. Panzarini, and J-M. Gérard “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys. Rev. B **60**, 13276 (1999). [CrossRef]

*S*

_{rad}. Cui and Raymer [19

19. G. Cui and M.G. Raymer, “Emission spectra and quantum efficiency of single-photon sources in the cavity-QED strong-coupling regime,” Phys. Rev. A **73**, 053807 (2006). [CrossRef]

*et al*. [20

20. A. Auffeves, B. Besga, J. M. Gérard, and J. P. Poizat, “Spontaneous emission spectrum of a two-level atom in a very-high-Q cavity,” Phys. Rev. A **77**, 063833 (2008). [CrossRef]

*et al*. [21

21. T. Ochiai, J-I. Inoue, and K. Sakoda, “Spontaneous emission from a two-level atom in a bisphere microcavity,” Phys. Rev. A **74**, 063818 (2006). [CrossRef]

22. Note an important correction to the emission spectrum in [21], namely **G**(**R**,**r _{d}**;ω) and not Im[

**G**(

**R**,

**r**;ω)] appears, since a principal value term was neglected in that paper.

_{d}## 3. Calculations

24. T. Takagahara, “Theory of exciton dephasing in semiconductor quantum dots,” Phys. Rev. B **60**, 2638 (1999). [CrossRef]

25. B. Krummheuer, V. M. Axt, and T. Kuhn, “Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots,” Phys. Rev. B **65**, 195313 (2002). [CrossRef]

17. H. 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 (1989). [CrossRef] [PubMed]

**Γ**′ = 40

*μ*eV; again we see a pronounced resonance at the cavity mode, when light is emitted via the cavity mode. The temporal dynamics of the two-operator one-time correlation functions, 〈â

^{†}(

*t*)â(

*t*)〉 and 〈

^{+}(

*t*)

^{−}(

*t*)〉, are shown as an inset; during the first few ps, already the cavity mode is efficiently excited by the exciton, which is a mechanism that also occurs for larger detunings. Finally, in the case of on-resonance excitation, shown in Fig. 2(d), we obtain the spectral doublet which is characteristic of the strong coupling regime; however, we observe a qualitatively different doublet between the cavity emitted spectrum and the radiation-mode spectrum. Although the spectral doublet appears in both cases, the cavity emission is much sharper at the wings and has a larger oscillator strength near zero detuning. For this strong coupling regime, the role of the chosen

**Γ**is almost negligible, since the system is now dominated by enhanced radiative coupling and cavity leakage.

*only the cavity mode spectrum is needed*and this is the one that is typically observed; yet, most-if not all-semiconductor experiments to date carry out their data analysis using

*S*

_{rad}, which can cause much confusion in interpreting the underlying physics of their data.

*et al*. [10

10. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, Andreas 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]

*et al*. [9

9. K. Hennessy, A. Badolato, M. Winger, A. Atäture, S. Fält, E.L. Hu, and A. Imamoǧlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature **445**, 896 (2007). [CrossRef] [PubMed]

*et al*. [12

12. M. Kaniber, A. Laucht, A. Neumann, J. M. Villas-Bas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 161303(R) (2008). [CrossRef]

9. K. Hennessy, A. Badolato, M. Winger, A. Atäture, S. Fält, E.L. Hu, and A. Imamoǧlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature **445**, 896 (2007). [CrossRef] [PubMed]

**445**, 896 (2007). [CrossRef] [PubMed]

**445**, 896 (2007). [CrossRef] [PubMed]

*S*

_{cav}, in a regime where

**Γ**

_{h}is finite, which to some degree is always the case for planar PC cavities. As shown in Fig. 4(c), a clear triple peak emerges, which becomes more pronounced with larger in-plane decay/loss; experimentally, this effect may occur as a function of pump power or through cavities that allow for in-plane decay. In the case of a pump-induced loss mechanism, nonlinear optical processes could be responsible for this loss; for example, mediated by free carrier absorption. However, we remark that such an effect would be material and PC-design specific, and this feature may or may not occur depending on the details of the fabricated structure, the pump-induced loss processes, and the pump excitation wavelength; specifically, we find a doublet if

**Γ**

_{h}~ < 0.25

**Γ**

_{v}, else a triplet. We also point out that our model Green functions are phenomenological, but physically well motivated and confirmed by numerical calculations; in this regard, it should be noted that exact Green function calculations for planar PC media are possible [27

27. V. S. C. Manga Rao and S. Hughes, See, e.g., “Single quantum dot spontaneous emission in a finite-size photonic crystal waveguide: proposal for an efficient “on chip” single photon gun,” Phys. Rev. Lett. **99**, 193901 (2007). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | A. Einstein, “On the quantum theory of radiation” (English Translation), Z. Phys. Translated into English in Van der Waerden Sources of Quantum Mechanics (North Holland1967) pp. 63–77. English translation by D. ter Haar, “The Old Quantum Theory,” Pergamon Press, New York, p. 167 (1967). |

2. | E. Moreau, I. Robert, J. M. Gérard, I. Abram, L. Manin, and V. Thierry-Mieg, “Single-mode solid-state single photon source based on isolated quantum dots in pillar microcavities,” Appl. Phys. Lett. |

3. | D. Fattal, E. Diamante, K. Inoue, and Y. Yamamoto, “Quantum teleportation with a quantum dot single photon source,” Phys. Rev. Lett. |

4. | W. Yao, R-B Liu, and L. J. Sham, “Theory of control of the spin-photon interface for quantum networks,” Phys. Rev. Lett. |

5. | Y. Akahane, T. Asano, B.S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

6. | T. Yoshie, A. Scherer, J. Hendrickson, 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 |

7. | E. Peter, P. Senellart, D. Martrou, A. Lemaitre, J. Hours, J. M Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett. |

8. | 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-semiconductor microcavity system,” Nature |

9. | K. Hennessy, A. Badolato, M. Winger, A. Atäture, S. Fält, E.L. Hu, and A. Imamoǧlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature |

10. | D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, Andreas 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. |

11. | J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, See, e.g., “Deterministic generation of single photons from one atom trapped in a cavity,” Science |

12. | M. Kaniber, A. Laucht, A. Neumann, J. M. Villas-Bas, M. Bichler, M.-C. Amann, and J. J. Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities,” Phys. Rev. B 161303(R) (2008). [CrossRef] |

13. | H. T. Dung, L. Knöll, and D-G. Welsch, “Spontaneous decay in the presence of dispersing and absorbing bodies: General theory and application to a spherical cavity,” Phys. Rev. A |

14. | S. Hughes, “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. |

15. | M. Wubs, L.G. Suttorp, and A. Lagendijk. “Multiple-scattering approach to interatomic interactions and superra-diance in inhomogeneous dielectrics,” Phys. Rev. A |

16. |
For simplicity we are assuming that |

17. | H. 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 |

18. | L. C. Andreani, G. Panzarini, and J-M. Gérard “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys. Rev. B |

19. | G. Cui and M.G. Raymer, “Emission spectra and quantum efficiency of single-photon sources in the cavity-QED strong-coupling regime,” Phys. Rev. A |

20. | A. Auffeves, B. Besga, J. M. Gérard, and J. P. Poizat, “Spontaneous emission spectrum of a two-level atom in a very-high-Q cavity,” Phys. Rev. A |

21. | T. Ochiai, J-I. Inoue, and K. Sakoda, “Spontaneous emission from a two-level atom in a bisphere microcavity,” Phys. Rev. A |

22. | Note an important correction to the emission spectrum in [21], namely G(R,r;ω)] appears, since a principal value term was neglected in that paper._{d} |

23. | H. J. Carmichael, See, e.g., “Statistical Methods in Quantum Optics 2,” Springer , p. 235. (2008). |

24. | T. Takagahara, “Theory of exciton dephasing in semiconductor quantum dots,” Phys. Rev. B |

25. | B. Krummheuer, V. M. Axt, and T. Kuhn, “Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots,” Phys. Rev. B |

26. |
In the limit of only radiative decay and simple cavity and exciton modes, we confirm complete agreement between our PC-projected Green function spectrum and the master equation solution, which is to be expected for the model cavity structure if |

27. | V. S. C. Manga Rao and S. Hughes, See, e.g., “Single quantum dot spontaneous emission in a finite-size photonic crystal waveguide: proposal for an efficient “on chip” single photon gun,” Phys. Rev. Lett. |

**OCIS Codes**

(270.5580) Quantum optics : Quantum electrodynamics

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: December 18, 2008

Revised Manuscript: February 5, 2009

Manuscript Accepted: February 6, 2009

Published: February 17, 2009

**Citation**

Stephen Hughes and P. Yao, "Theory of quantum light emission from a
strongly-coupled single quantum dot
photonic-crystal cavity system," Opt. Express **17**, 3322-3330 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3322

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### References

- A. Einstein, "On the quantum theory of radiation" (English Translation), Z. Phys. 18, 121 (1917). Translated into English in Van derWaerden Sources of Quantum Mechanics (North Holland 1967) pp. 63-77. English translation by D. ter Haar, "The Old Quantum Theory," Pergamon Press, New York, p. 167 (1967).
- E. Moreau, I. Robert, J. M. Gerard, I. Abram, L. Manin, and V. Thierry-Mieg, "Single-mode solid-state single photon source based on isolated quantum dots in pillar microcavities," Appl. Phys. Lett. 79, 2865 (2001). [CrossRef]
- D. Fattal, E. Diamante, K. Inoue, and Y. Yamamoto, "Quantum teleportation with a quantum dot single photon source," Phys. Rev. Lett. 92, 7904 (2004). [CrossRef]
- W. Yao, R-B Liu, and L. J. Sham, "Theory of control of the spin-photon interface for quantum networks," Phys. Rev. Lett. 95, 030504 (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 (2003). [CrossRef] [PubMed]
- T. Yoshie, A. Scherer, J. Hendrickson, 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 (2004). [CrossRef] [PubMed]
- E. Peter, P. Senellart, D. Martrou, A. Lemaitre, J. Hours, J. M Gerard, and J. Bloch, "Exciton-photon strongcoupling regime for a single quantum dot embedded in a microcavity," Phys. Rev. Lett. 95, 067401 (2005). [CrossRef] [PubMed]
- J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum-semiconductor microcavity system," Nature 432, 197 (2004). [CrossRef] [PubMed]
- K. Hennessy, A. Badolato, M. Winger, A. Atature, S. Falt, E. L. Hu, A. Imamoglu, "Quantum nature of a strongly coupled single quantum dot-cavity system," Nature 445, 896 (2007). [CrossRef] [PubMed]
- D. Press, S. Gotzinger, S. Reitzenstein, C. Hofmann, A. Loffler, 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]
- See, e.g., J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, H. J. Kimble, "Deterministic generation of single photons from one atom trapped in a cavity," Science 303, 1992 (2004). [CrossRef] [PubMed]
- M. Kaniber, A. Laucht, A. Neumann, J. M. Villas-Bas, M. Bichler, M.-C. Amann, and J. J. Finley, "Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities," Phys. Rev. B161303(R) (2008). [CrossRef]
- H. T. Dung, L. Knoll and D-G. Welsch, "Spontaneous decay in the presence of dispersing and absorbing bodies: General theory and application to a spherical cavity," Phys. Rev. A 62, 053804 (2000). [CrossRef]
- S. Hughes, "Coupled-cavity QED using planar photonic crystals," Phys. Rev. Lett. 98, 083603 (2007). [CrossRef] [PubMed]
- M. Wubs, L.G. Suttorp and A. Lagendijk. "Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics," Phys. Rev. A 70, 53823 (2004). [CrossRef]
- For simplicity we are assuming that F(R) is the same for both cavity and radiation leakage, but in reality this will depend on a number of factors, including the specific collection geometry of the detector.
- H. 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 (1989). [CrossRef] [PubMed]
- L. C. Andreani, G. Panzarini, and J-M. Gerard, "Strong-coupling regime for quantum boxes in pillar microcavities: Theory," Phys. Rev. B 60, 13276 (1999). [CrossRef]
- G. Cui and M. G. Raymer, "Emission spectra and quantum efficiency of single-photon sources in the cavity-QED strong-coupling regime," Phys. Rev. A 73, 053807 (2006). [CrossRef]
- A. Auffeves, B. Besga, J. M. Gerard, and J. P. Poizat, "Spontaneous emission spectrum of a two-level atom in a very-high-Q cavity," Phys. Rev. A 77, 063833 (2008). [CrossRef]
- T. Ochiai, J-I. Inoue, and K. Sakoda, "Spontaneous emission from a two-level atom in a bisphere microcavity," Phys. Rev. A 74, 063818 (2006). [CrossRef]
- Note an important correction to the emission spectrum in [21], namely G(R,rd;ω) and not Im[G(R,rd;ω)] appears, since a principal value term was neglected in that paper.
- See, e.g., "Statistical Methods in Quantum Optics 2," H. J. Carmichael, Springer, p. 235. (2008).
- T. Takagahara, "Theory of exciton dephasing in semiconductor quantum dots," Phys. Rev. B 60, 2638 (1999). [CrossRef]
- B. Krummheuer, V. M. Axt, and T. Kuhn, "Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots," Phys. Rev. B 65, 195313 (2002). [CrossRef]
- In the limit of only radiative decay and simple cavity and exciton modes, we confirm complete agreement between our PC-projected Green function spectrum and the master equation solution, which is to be expected for the model cavity structure if Γh = 0 (no in-plane decay).
- See, e.g., V. S. C. Manga Rao and S. Hughes, "Single quantum dot spontaneous emission in a finite-size photonic crystal waveguide: proposal for an efficient "on chip" single photon gun," Phys. Rev. Lett. 99, 193901 (2007). [CrossRef]

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