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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 5 — Mar. 2, 2009
  • pp: 3322–3330
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Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system

S. Hughes and P. Yao  »View Author Affiliations


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

1. Introduction

Fig. 1. Schematic of a planar semiconductor PC with an embedded QD; for a self-assembled QD, then the spatial position would be near the center of the slab to maximize coupling to the cavity mode. Also shown is a typical spatial profile of a confined cavity mode f c(r,ωc), within the slab, dominated by fyc in this example (ωc is the cavity resonance frequency); the effective mode volume is less than 0.1μm3. The right side of the figure shows a side view of the cavity, indicating the vertical background radiation-leakage (γb), as well as a vertical (κ v) and horizontal (κ h) cavity leakage. We define the cavity decay rates as Γ v/h = 2κ v/h, and the radiation decay rate as Γ b = 2γb.

2. Theory

We consider a medium whose classical electromagnetic properties are described by the photon Green function [13–15

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 (rr′)I/ε(r) = ∑k ω 2kfk(r)[fk(r′)]*/(ω 2ω 2k) [14

14. S. Hughes, “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. 98, 083603 (2007). [CrossRef] [PubMed]

, 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]

], that is defined in terms of the transverse modes of the system, fk; these modes are solutions of the standard eigenvalue problem: [××−ω 2/c 2 ε(r)] fk(r) = 0.

To describe the quantum mechanics of light-matter coupling, we adopt a canonical Hamiltonian approach introduced by Wubs 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]

] and use a “two-level atom” model for the QD. The resulting Hamiltonian of the system includes one QD exciton, a sum over the light modes, and the coupling between the exciton and the light through the electric-dipole approximation:

H=h¯ωxσ̂+σ̂+λh¯ωλâλâλih¯λ(σ̂+σ̂+)(gλâλgλ*âλ),
(1)

from which the time-dependent operator equations are derived for â˙ λ, â˙ λ,σ^. ̇+,σ^. and σ^. ̇z, using the Heisenberg equations of motion. We follow the usual convension, where the operators âλ represent the photon modes, while the operators σ^ +,σ^ and σ^ 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]

], and using Ê(r, t) = iλ [ ω λ/2ε 0]½ âλ (t) f λ(r)+ H.c. and g λ = [ω λ /2 ε 0]½ μ· fλ(r d), one can derive the quantized electric-field operator,

Ê(R,ω)=1ε0K(R,rd;ω)·μ[σ̂(ω)+σ̂+(ω)],
(2)

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

S(R,ω)=0dt20dt1eiω(t2t1)Ê()(R,t2)Ê(+)(R,t1),
(3)

where 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:

S(R,ω)=K(R,rd,ω)·με02(σ̂(ω))σ̂(ω).
(4)

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.

Srad(R,ω)F(R)Γb(ω+ωx)(ω2ωc2+iωΓc)(ω2ωx2+iωΓ)(ω2ωc2+iωΓc)4g2ωcω2,
(5)
Scav(R,ω)F(R)Γv2gωc(ω+ωx)(ω2ωc2+iωΓc)(ω2ωc2+iωΓv)(ω2ωx2+iωΓ)(ω2ωc2+iωΓc)4g2ωcω2,
(6)

where F(R) is a geometrical factor that depends on the detector location [16

16. 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.

], Γ is an effective exciton decay rate, and g = [ωc/2 ε 0]ν μ · f c(r d) is a cavity-exciton coupling constant.

Equations (2–6) contain a rich amount of information about the coupling between a QD and a PC medium. By inspection of 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 4g 2/(ΓbΓ c) (assuming that Γ hΓ v). Thus, for strongly coupled QDs, with g and Γ v ≈ 0.1meV, and Γ b ≈ 0.05μeV [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)).

To the best of our knowledge, the above equations either extend, recover, or correct all known analytical spectral forms of single photon spectra that have been presented in the literature. For example, Carmichael 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]

] and Andreani 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]

] derived a rotating wave version of 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]

] derived a coupled mode solution for cavity emission and radiation emission for a simple geometrical cavity system, which has been extended recently by Auffeves 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]

]. Finally, our general Green function spectrum (Eq. (4)), which is exact, both corrects and extends the Green function spectrum derived by Ochiai 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

22. 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.

].

Fig. 2. Normalized light spectra from single QD photon emission in a planar PC cavity (c.f. Fig. (1)), with cavity decay rates Γ v = 0.1meV and Γ h = 0, and exciton decay rate Γ (see text and labels in the graph). In (a) is shown the spectrum for an initially excited-exciton, with two different exciton decay rates, with a cavity mode off-resonance (from the exciton) by 0.8 meV; the blue dashed curve is the usual radiation-mode decay (S rad), and the red solid curve shows the emitted spectrum from the leaky cavity mode (S cav). In (b), is shown the influence of detuning as a contour plot. (c) Similar to (a) (top frame) but computed from a master equation solution with pure dephasing (Γ′); the insets display the exciton and cavity mode dynamics. In (d) is shown the on-resonance case.

3. Calculations

Fig. 3. Detected spectra for various cavity-exciton detunings, using the same parameters as in Fig. (2), with Γ = 40μeV. All spectra are scaled by the same constant, and the background radiation decay is Γ b = 0.05μeV. (a) Cavity and radiation mode emission for a detuning of 8 meV. (b) Detuning of 4 meV. (c) Detuning of 1 meV showing only the cavity mode, where already S rad plays a negligible role. (d) On-resonance case, which shows both a pronounced enhancement of the emission (Purcell effect) and the strong coupling regime.

In the calculations above, we have implicitly assumed that the effective exciton decay is “radiative,” which is usually required to get analytical results for the spectra; yet this assumption is somewhat artificial, as the typical radiative decay rate for QD excitons is much smaller than the one used above, and the exciton broadening is usually dominated by phase relaxation mechanisms such a spectral diffusion and electron-phonon scattering, both of which yield “pure dephasing” [24

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

, 25

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]

]. To demonstrate that pure dephasing leads to qualitatively the same spectral shape (as implied from Eq. (4)), and to investigate the dynamics of the cavity mode and exciton mode coupling, in Fig. 2(c) we show the computed spectra that is obtained by solving the corresponding master equation [26

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 Γh = 0 (no in-plane decay).

] of a coupled leaky cavity and exciton system [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]

], with a pure dephasing rate of Γ′ = 40μeV; 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 〈σ^ +(t)σ^ (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.

To clarify the relative strengths of the cavity-mode emission versus radiation-mode emission, in Fig. 3 we show the calculated spectrum–in identical normalized units–for four separate exciton-cavity detunings. With the cavity mode 8 meV away, the radiation mode spectrum dominates; but the cavity mode is still visible. For smaller detunings, both the cavity emission and the exciton emission increase in equal weights, eventually resulting in a pronounced Purcell effect and the strong coupling regime. For a strongly coupled cavity - PC system, evidently 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.

Fig. 4. (a) An example cavity-mode spectrum that results from an off-resonant excitation, from a spectrally wandered exciton or an exciton-biexciton pair. (b) The sum of the off-resonant and on-resonance spectra, showing that the total on-resonance doublet survives. (c) The simple one exciton and one cavity-mode spectrum, but in the presence of in-plane loss or decay, showing the onset of a spectral triplet.

One might also wonder about the role of multiple excitons (spectrally off-resonance, but belonging to the same QD), which are usually assumed unimportant. To address this question, we have numerically confirmed that in the presence of several initially-excited excitons, within the spectral vicinity of the cavity mode, then a significantly larger oscillator strength appears in the cavity mode via exactly the same mechanism above; this point is consistent with the measurements of Press 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]

], Hennessy 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]

], and Kaniber 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]

]. Thus the background excitons can certainly play a qualitatively important role, even if far off resonance. Indeed their effect would also spoil the quantum statistics of light emission from the cavity mode, which is in agreement with cavity-mode autocorrelation measurements showing little quantum correlations [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]

]. We will show this connection directly in a forthcoming publication, where the role of additional excitons act in concert to suppress the antibunching behavior of the cavity autocorrelation function. In addition, since part of the dressed leaky-cavity mode “lives” at the exciton resonance, extra care is needed in understanding the subtle role of spectral filtering.

Finally, we turn our attention to the strange observation of a strongly-coupled “triple peak” [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]

]. It has been speculated that spectral diffusion may result in a triple peak profile in the strong coupling spectrum [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]

], via a similar mechanism that we have explained above, namely a channeling of energy into the cavity mode for off-resonant exciton resonances; another possibility could be biexcitonic loading, where the first generated photon (which is off-resonance) feeds the cavity mode, while the second generated photon (on resonance) creates the doublet; thus, one might imagine that the statistical mixture of an off-resonant excitation and an on-resonant excitation could give rise to a triple peak. To investigate this hypothesis, in a simple qualitative way, we show in Fig. 4(a) an example of a transition that is only 1 meV away from the cavity (to look for a best case off-resonant feeding mechanism); in Fig. 4(b) is shown the sum of this contribution and the doublet contribution, which demonstrates that there is no noticeable effect on the doublet. We conclude that spectral wandering or biexciton loading is unlikely responsible for the observed middle peak, since the off-resonant coupling has significantly less oscillator strength than the on-resonant contribution.

We now introduce a slightly different–but more realistic–model for the PC slab cavity, that has been implicit from the beginning, and highlight some unusual differences that may occur for 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 ~ < 0.25Γ 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]

], even with loss, and future work will explore and report on these in more detail.

Of course, one should not rule out the possible influence of spectral diffusion, nor any other well known semiconductor mechanisms such as deep-state emissions and free carrier absorption; any semiconductor QD experiment must be carefully scrutinized and analyzed. Rather, we point out that, regardless of the precise origin of the observed spectral triplet, one could deterministically engineer a situation where observations of the spectral triplet is “guaranteed,” and this alone should motivate new experiments in this exciting and growing field. With properly designed high-Q cavities, and in structures where material loss is small, then a spectral doublet should be the dominant feature, which is certainly the usual case experimentally. If a triplet is obtained, then turning down the power of the pump laser, or spectrally exciting closer to the target exciton resonance could help revert back to a doublet. As mentioned above, a strongly-coupled doublet to triplet investigation could also be carried out by designing cavities with different in-plane decays. In addition, the investigation of free carrier absorption and nonlinear pumping on the Purcell effect and emitted spectrum would be an interesting experiment to try.

4. Conclusions

We have presented and exploited an exact quantized medium-dependent theory to derive “userfriendly” analytical spectra, suitable to study the quantum emission from a strongly coupled single QD - PC cavity system. Our spectral formulas clarify why the cavity-mode emission completely dominates the radiation-mode emission, which manifests in an off-resonant cavity feeding mechanism over large exciton-cavity detunings. Moreover, we have demonstrated that in-plane decay can yield a spectral triplet in the strong coupling regime, caused by a quantum interference effect between the various decay channels that a single photon can take within the medium. Both of these effects are consistent with recent experiments, and, contrary to earlier reports, can be adequately explained by simple atomlike models for the QD. These predicted effects can also be described classically using standard electromagnetic theory and an oscillator model, though naturally these would have a different physical interpretation.

Acknowledgments

We thank Jeff Young, David Press, Tetsuyuki Ochiai, Michael Raymer, Guoqiang Cui, Martijn Wubs, Howard Carmichael, Takehiko Tawara and Hidehiko Kamada for useful discussions. This work was supported by the National Sciences and Engineering Research Council of Canada and the Canadian Foundation for Innovation.

References and links

1.

A. Einstein, “On the quantum theory of radiation” (English Translation), Z. Phys. 18, 121 (1917).

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. 79, 2865 (2001). [CrossRef]

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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]

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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]

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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]

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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]

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. 95, 067401 (2005). [CrossRef] [PubMed]

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 432, 197 (2004). [CrossRef] [PubMed]

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]

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]

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 303, 1992 (2004). [CrossRef] [PubMed]

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 62, 053804 (2000). [CrossRef]

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]

16.

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.

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]

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]

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]

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]

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,rd;ω) and not Im[G(R,rd;ω)] appears, since a principal value term was neglected in that paper.

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 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]

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 Γh = 0 (no in-plane decay).

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]

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

  1. 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).
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  9. 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]
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  11. 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]
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  16. 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.
  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]
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  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]
  20. 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]
  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,rd;ω) and not Im[G(R,rd;ω)] appears, since a principal value term was neglected in that paper.
  23. See, e.g., "Statistical Methods in Quantum Optics 2," H. J. Carmichael, Springer, p. 235. (2008).
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  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 Γh = 0 (no in-plane decay).
  27. 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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