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We theoretically demonstrate high quality(Q)-factor micropillar cavities at 1.55-μm wavelength based on Si/SiO2-InP hybrid structure. An adiabatic design in distributed Bragg reflectors (DBRs) improves Q-factor for upto 3 orders of magnitude, while reducing the diameter to sub-micrometer. A moderate Q-factor of ~3000 and a Purcell factor of ~200 are realized by only 2 taper segments and fewer conventional DBR pairs, enabling single photon generation at GHz rate. As the taper segment number is increased, Q-factor can be boosted to ~105-106, enabling coherent exchange between the emitter and the optical mode at 1.55 μm, which is applicable in quantum information networks.
© 2015 Optical Society of America
1. Introduction
Optical microcavities are widely studied for their great potential in many fields of research and technology, such as optical communication, nonlinear optics, optoelectronics and quantum information processing [1]. All these applications rely on the temporal and spatial confinement of light, which are characterized by the quality factor Q and the mode volume V, respectively. Microcavities with high Q and small V are recently attracting special attention due to the applications to ultrasmall lasers [2], all-optical switching [3] and quantum information processing [4,5]. In the field of sold-state quantum information processing, microcavities containing semiconductor quantum dots (QDs) have been demonstrated effective as indispensable devices such as efficient [6–8] and indistinguishable single photon sources (SPSs) [9,10], and coherent quantum-control devices [11,12]. Among many cavity types, micropillar cavities are advantageous for solid-state quantum information processing owing to light emission to normal direction, single-lobed Gaussian-like light pattern, simply isolating single QDs [13], and suitability for current injection [14]. To date, most of the QD-containing micropillar cavities are using GaAs-based materials, but they are mainly limited to photon wavelength shorter than 1 μm. For the purpose of quantum information and communication over silica-fiber-based network, 1.55-μm InAs/InP QDs are promising materials for SPSs [15] and thus micropillar cavity containing InP-based QDs is strongly required.
Recently, we proposed a promising micropillar cavity for the 1.55-μm band by combining Si/SiO2 distributed Bragg reflectors (DBRs) with an InP spacer [16]. This type of microcavity with hybrid materials avoids the difficulty in monolithically fabricating (by epitaxy such as molecular beam epitaxy or metalorganic chemical vapor deposition, and processing such as dry-etching) InP-based micropillar cavities, in which epitaxial materials with a very small refractive index contrast have to be used so that the pillar would be ~30 μm high. Taking into account the known parameters for single InAs/InP QDs, we have shown that this cavity could be efficient in increasing the operating frequency and enhancing the photon indistinguishability of 1.55-μm SPSs [16], owing to the high refractive-index contrast of ~2 in the DBRs. However, this cavity needs relatively large pillar diameter of ~2 μm which is not well satisfying a small V. This character limits the capability to separate a single QD resonant with the cavity mode and prevents the quantum devices from miniaturizing and integrating. Moreover, future large scale quantum networks require 1.55-μm SPSs and other quantum devices allowing coherent transfer of quantum states between QDs and single photons via long-distance optical fibers [17], but a Q factor as high as enabling coherent operation seems still difficult in our previously designed micropillar cavity. Very recently, novel structures of micropillar cavities with submicrometer diameter and high Q factor were proposed for GaAs/AlAs [18] and TiO2/SiO2 [19] systems in which the spacer layers were replaced by tapered DBRs. Although it was proved that such structures were very effective to obtain improved Q and V, the mode wavelength was shorter than 1 μm. For practical applications, it is required to investigate whether and how the mode wavelength can be extended to 1.55-μm telecommunication band while exhibiting sufficiently high Q factor and small V. In this work, we present the design of the hybrid micropillar cavity structure consisting of tapered Si/SiO2 DBRs and InP-based materials containing single InAs QD. By numerical simulations, it is shown that a submicrometer-sized Si/SiO2-InP hybrid micropillar cavity with tapered Si/SiO2 DBRs may have Q factor up to a few 106 and small mode volume, which make it superior to the previous cavity and available as a coherent SPS at 1.55-μm telecommunication band.
2. Cavity model
We here propose a micropillar cavity structure schematically shown in Fig. 1(a). The diagram is a vertical cross section of the circularly shaped pillar with diameter D. The top and bottom parts of the pillar are conventional DBRs composed of alternating Si and SiO2 layers. Each layer in these DBRs is set quarter-wavelength thick, i.e. the layer thickness t1 = λB/(4ne1) for Si, t2 = λB/(4ne2) for SiO2, where λB is the Bragg wavelength here firstly set to be 1.55 μm, and ne1(2) is the effective refractive index of Si(SiO2), which is calculated and known to be dependent on D by using the standard waveguide theory [20]. In between the conventional DBRs, we incorporate more Si/SiO2 segments as tapered DBRs on both the top and bottom sides. These layers are the additional parts compared to our previous Si/SiO2 micropillar cavities [16]. Here “taper” means adiabatically deducing the layer thicknesses as the DBR extends towards the cavity center (spacer) [18,19]. In detail, the tapered DBRs have linearly decreasing layer thicknesses t1i = t1(1-ρ(2i-1)) for Si and t2i = t2(1-2ρi) for SiO2, where i is the taper segment number and ρ the tapering slope of the layer thickness, i.e. the decreased fraction per tapered layer. In between the tapered Si/SiO2 DBRs, an InP layer containing InAs QD as the light source is inserted as the spacer layer with thickness
where ne0 is the effective refractive index of the spacer material, and N the total taper segment number. It is worth noting that the spacer here is much thinner than that in conventional structures such like our previous cavities. The whole micropillar is standing on a semi-infinite Si substrate.
Fig. 1 (a) The schematic cross section of the cavity model. (b) The optimized quality factor Q as a function of the total taper segment number of cavities with pillar diameter D of 0.8 μm, in comparison with the Q factor of the traditional cavities. Colored insets are the profile patterns of the electric field along x direction, Ex, of the mainly x-polarized fundamental mode. The black dashed frames attached on the profile patterns indicate the outline of the cavity structure.
We simulate the system using finite-difference-time-domain method with the optical constants of the materials cited from [21]. The detailed calculation method is mainly the same as before [16], with simulation domain more than 3 μm larger than pillar size, horizontal grid size smaller than 1/20 of the cavity diameter and maximally 0.05 μm, vertical grid size less than 1/10 of the thinnest layer and maximally 0.02 μm, simulation time long enough so that the mode intensity fades below at least half of its initial value.
3. Simulation results
3.1 General result
Our study was started from all-Si/SiO2 tapered micropillar cavities, which are similar to the reported tapered cavities absent of a third material [18,19], by setting the spacer in Fig. 1(a) to be Si . On such cavities, quality factor Q for the fundamental mode is optimized by simply tuning the tapering slope ρ. No surprising, different structure has different ρ for optimizing the Q factor. As an example, a cavity with diameter D = 0.8 μm, DBR of 4/6.5 pairs and taper segment number N = 3 exhibits an optimized Q factor of 1.1 × 105 if ρ = 0.045. Based on the optimized conditions obtained from these all-Si/SiO2 tapered cavities, one might intuitively expect a good hybrid cavity simply by changing the spacer to InP with the appropriate thickness t0 determined by Eq. (1), but this does not really give a high Q factor. It is found that, it is better to tune the InP spacer thickness as
with the tuning parameter σ >1 while using the optimized ρ of the all-Si/SiO2 case. Naturally, different N corresponds to different σ for the optimized Q factor. Shown in Fig. 1(b) is the result of an example structure, the cavity with D = 0.8 μm and 4/6.5 pairs of conventional DBRs. It is seen that the Q factor increases monotonically with the total taper segment number N, by in average one order for every additional taper segment.Compared to traditional counterparts, which have (4 + N)/(6.5 + N) pairs of quarter-wavelength-thick Si/SiO2 DBRs and a wavelength-thick InP spacer and show Q factor of below 100, tapered DBRs increase the Q factor for 1-3 orders of magnitude. Typically, the Q factor is increased to be ~8 × 104 by three segments of tapered Si/SiO2 DBRs. With 4 taper segments, there seems a saturation effect so that the Q factor reaches 1.4 × 105, only about twice that of 3 taper segments. This is because that there are no longer enough conventional DBR pairs to take the role of vertical optical confinement for the would-be higher Q factor. As a matter of fact, when increasing the conventional DBR pairs to 6/9.5, the Q factor can be further increased to be 2.7 × 105, as shown by a solid square symbol in Fig. 1(b). We may note that the above Q factors are a bit lower than those of all-Si/SiO2 tapered micropillar cavities. This is the result of replacing Si with InP as the spacer. The mismatch in the refractive indices between Si and InP (~8.5%) determines that the mode profile matching between the spacer and the DBRs, which is considered to be associated with the Q improvement in tapered cavities [19], is not as perfect as that in all-Si/SiO2 tapered cavities. Fortunately, the index mismatch is not so large that we can still obtain high Q factors in the Si/SiO2-InP hybrid tapered micropillar cavities. It can be confirmed by replacing InP with InGaAsP which has an index mismatch of 3.5% with respect to Si. In this case, optimized Q factor of 3-segment tapered cavity is ~9.5 × 104, much closer to that of all-Si/SiO2 tapered cavities.
In such a hybrid micropillar cavity, taper design is not necessarily restricted to Si/SiO2, but we learned that InP/InGaAsP tapered DBRs are far less effective than Si/SiO2 in improving the cavity quality, because of the difficulty in mode-profile matching between InP/InGaAsP and Si/SiO2.
The above high Q factors suggest thus the nice property of the presently proposed micropillar cavities. Their good character also lies at the mode profiles, as shown by the insets of Fig. 1(b). The mode profile of the fundamental mode is single-lobed and highly symmetrical (by the in-plane xy-profile), which makes it beneficial to couple the QD emission from the cavity into a fiber. The mode profile is well confined in the tapered region (by the vertical yz-profile), which could probably be one of the reasons for high Q factors in such tapered micropillar cavities.
3.2 Dependence on vertical structure
In optimizing the Q factor by only tuning the parameter σ as in the above, the mode wavelength may not exactly match a specific target wavelength. We can resolve it by tuning the cavity structure together in parameter σ, tapering slope ρ, and Bragg wavelength λB, all of which describe the vertical structure of the cavities. For the last parameter λB, we label the previous setting as λB0 = 1.55 μm from now on. Using these procedures on 3-segment tapered cavities with D = 0.8 μm and 4/6.5 conventional DBR pairs, we find that the best Q factor of 8 × 104 for an exact mode wavelength λ = 1.550 μm can be obtained by setting σ = 1.18, ρ = 0.05 and λB = 1.02λB0. In detail, conventional Si/SiO2 DBRs have layer thickness of 124.8/353.0 nm; the tapered Si/SiO2 DBR have layer thicknesses of 118.5/317.8, 106.0/282.5, and 93.6/247.0 nm in segments 1, 2, and 3, respectively; and the InP spacer has thickness of 114.8 nm, labeled t0m hereafter as an optimized spacer thickness.
An optimized design needs detailed knowledge of the dependence on various parameters, which is a little complicated. For simplification, we here characterize the effects of the three tuning parameters σ, ρ and λB in terms of the variation in spacer thickness Δt0 = t0-t0m, since they all give rise to changes in the spacer thickness t0. Figure 2(a) shows the mode wavelength and the Q factor depending on Δt0/t0m, the relative change in the spacer thickness, in the three cases of cavity structure tuning. As σ only is tuned, meaning that only the InP spacer thickness is changing, the mode wavelength changes weakly but the Q factor degrades fast. Viewed in a wider range, the variation of Q factor is actually something like a degrading sinusoidal function of the spacer thickness (not shown). As the spacer thickness increases, it is deduced to a minimum of ~100 at Δt0/t0m ~1.1, then comes up again to another maximum of ~500 at Δt0/t0m ~2.5, and goes down once more. As the tapering slope ρ only is tuned, meaning that the tapered DBRs and the spacer are changing together, the mode wavelength changes faster but the Q factor decreases more slowly. As the Bragg wavelength λΒ only is tuned, meaning that the thicknesses of all layers are changing, the mode wavelength changes even faster but the Q factor almost remains high. As a whole, Q factor can be preserved over 104 while the mode wavelength λ is limited within the range of 1.50-1.60 μm, if the layer thickness fluctuation is within ± 5%, as is indicated by a shaded area in Fig. 2(a).
Fig. 2 (a) Mode wavelength λ and Q factor as functions of the variation of spacer thickness t0 relative to an optimized value t0m in the three cases of tuning spacer thickness t0, tapering slope ρ and Bragg wavelength λB for the cavities with diameter of 0.8 μm and 3 taper segments. (b) Line mode profiles along y direction of the electric field Ex of the mainly x-polarized fundamental mode in the central plane of the spacer (solid) and of the Si layer closest to the spacer (dashed) for 4 different cavity structures. Lines are vertically shifted for clarity.
The above Q variations versus vertical structure suggest that a local change destroys, while an overall harmonic change keeps the quality of a cavity. To further understand it, we may try viewing the mode-profile matching, which was considered responsible for high Q factors in tapered cavities [19]. Figure 2(b) shows the line-profiles of the fundamental mode along the in-plane y direction for a few cavity structures O, A, B, and C, as marked in Fig. 2(a). The two lines in each set are taken from the central planes of the spacer layer and of the Si layer closest to the spacer. It is clear that the optimized cavity structure O shows good mode-matching. The structure A, whose Q factor is just a bit lower than structure O, has mode-matching a little bit worse than structure O. The structure B, whose Q factor is greatly degraded, has remarkably mismatching mode-profiles. However, the structure C has mode-matching not as bad as structure B but as good as structure A although its Q factor is even worse than structure B. It suggests that mode-profile mismatching only cannot perfectly explain the Q factors of our tapered micropillar cavities. Similar Q improvement by lattice displacement in 2-dimensional photonic crystal cavity, which has likely the same mechanism as in 1-dimensional cases here and before [18,19], had been ascribed to a certain of (e.g. Gaussian-like) gentle confinement of the optical field [22]. If we take the so-called gentle confinement as the common mechanism, it may contain the effect of mode-profile matching. Another aspect, phase matching, has been considered in gentle confinement [22], so phase mismatching [23] probably takes a role in determining the Q factor in our cavities here, which is open for future investigation.
3.3 Dependence on lateral size
After describing the dependence on the vertical structure as above, we examine here how the cavity property varies with the lateral size, the pillar diameter D. As examples, we show the results of three structures as depicted in Fig. 3 by a comparison with the previous cavity, whose vertical structure was determined by planar quarter-wavelength design [16].
Fig. 3 The (a) Q factor, (b) mode wavelength λ, (c) Purcell factor FP and (d) mode volume V as functions of the pillar diameter D for the tapered Si/SiO2-InP micropillar cavities with vertical structures fixed at optimized conditions, with those of the previous cavity as contrasts. The inset in (a) shows the optimized Q factors at ~1.55 μm for different pillar diameters.
The red square symbols in Figs. 3(a) and 3(b) show that, with 2/3.5 pairs of conventional DBRs and 2 tapered segments, an optimized Q factor of more than 3000 for 1.55 μm emission happens with pillar diameter of 0.85 μm; and changing the pillar diameter in 0.60-0.95 μm gives Q factors of over 1000 for wavelength 1.35-1.60 μm. Compared to the previous cavity indicated by crosses, the Q factor is the same, and the Purcell factor FP, shown in Fig. 3(c), is somehow higher because the pillar diameter is greatly decreased to below 1 μm. Besides, the pillar height remains the same as the previous 4.5 μm, suggesting the superiority of this cavity beyond the previous structure in generating indistinguishable single photons and enabling ultrahigh-speed SPSs operating in several GHz clock.
The more typical is the structure with 4/6.5 pairs of conventional DBRs and 3 taper segments, whose vertical structure is fixed to that optimized with D = 0.8 μm. The blue circles in Fig. 3 show that the Q factor of this structure is 1-8 × 104 in the range of D = 0.45 – 1.10 μm, 1-2 orders of magnitude higher than that of the previous cavities, from below 100 for D < 1 μm to a few thousands for D > 2 μm as shown by crosses. Especially, it remains over 6 × 104 in a D range of 0.65 - 0.90 μm, exhibiting the robustness against diameter variation. There is a peak around D = 1.0 μm, appearing somehow abnormal. It is probably a result of oscillation behavior also observed in GaAs/AlAs tapered cavity [18] and in our previous cavities as shown by crosses in Fig. 3, and is probably related to the coupling with higher-order modes, which often occurs in micropillar cavities [16,24]. This good quality is in debt to a ~7.6 μm high micropillar, higher than that of the previous structure, because not only of tapered layers incorporated but also of smaller effective refractive indices, increasing the thickness of each DBR layer.
Similarly, we examined cavities with 6/9.5 conventional DBR pairs and 4 taper segments. We fix the vertical structure to an optimized cavity with D = 0.65 μm, which gives a Q factor as high as 3 × 106 at mode wavelength of 1.55 μm. Fixing the vertical structure to this condition, we get the cavity properties as functions of the pillar diameter D, as shown in Fig. 3 by green triangle symbols. At the expense of a pillar height 12 μm, a Q factor of over 1 × 105 can be obtained in a cavity with diameter ranging from 0.5 to 0.9 μm. Especially, Q > 2 × 106 are available for mode wavelength λ = 1.55 ± 0.05 μm with D = 0.60 - 0.70 μm in this cavity. Since the traditional cavities with 6/9.5 DBR pairs have typically just ~20% higher Q factors (not shown) than 4/6.5 DBR pairs, the 4-segment tapered cavity here improves the Q factor for 2-3 orders of magnitude.
Just due to thicker DBR layers, all these three structures show mode wavelength varying with the diameter D faster than that of the previous cavities, as seen in Fig. 3(b). Figure 3(c) does not show FP for 3 and 4 taper segments because most of their nominal values are too high to be physically meaningful due to the property beyond weak coupling regime, which will be discussed later. As another important parameter, the mode volume V of these cavities are shown in Fig. 3(d) in the unit of (λ/n)3, where n is the refractive index of the spacer material. It is found mostly less than 1, which is not so different but definitely smaller than that of the previous cavity in the small D area, although recent study suggests significant reduction of the mode volume as low as ~0.1 with the benefit of the tapered DBRs [19]. In the best D region 0.65-0.90 μm, three cases show similar mode volumes because they have roughly the same mode profiles, well confined within z = ± 2 μm in vertical direction (not shown). It implies also similar confinement factors within the active InP spacer layer in the three different cases. From the view of tapering effect, taper segment number N = 2 gives a significant mode component outside the tapered region ~ ± 0.7 μm, N = 3 distributes most of the mode energy in the tapered region ~ ± 1.3 μm (as can also be seen in Fig. 1(b)), and N = 4 confines nearly the whole mode within the tapered region ~ ± 1.9 μm. It suggests that best tapering is resulting in some common gentle mode confinement [22] as just discussed but not really confining the mode tightly within the tapered region. Anyway, the structure with 3 taper segments can typically have V as small as ~0.8, which is reduced to be half of that of our previous cavities, ~1.6 at D ~2.2 μm.
In the above, the allowed pillar diameter for mode wavelength around 1.55 μm is restricted to a small range spanning within 0.1 μm, due to fixed vertical structures. This, however, does not mean that an effective cavity for ~1.55 μm band can only be obtained in such a narrow size range. By employing vertical structure tuning at various diameters, we can see that a high Q factor for mode wavelength of ~1.55 μm is available in a wider diameter range. The example for 3 taper-segment cavity is shown by the inset of Fig. 3(a). It is seen that Q factors over 6 × 104 can be obtained with pillar diameter of 0.70 - 0.95 μm. The degradation of Q factor at diameter less than 0.6 μm is partly because that lateral confinement of the optical mode is more difficult. There is a peak at D = 0.9 μm, appearing somehow unusual though good for application. Similar to the case of fixed vertical structure, this is probably a result of oscillation behavior likely related to the coupling with higher-order modes [16]. Generally, based on vertical cavity structures of high Q factors determined in a way as for the inset of Fig. 3(a), tapered Si/SiO2-InP micropillar cavities with sub-micrometer diameters can have Q factors of near 105 to a few 106, improved for up to 3 orders of magnitude compared to the previous ones, with mode wavelength of 1.55 ± 0.05 μm.
4. Discussion
With the good properties described above, we shall now discuss our tapered Si/SiO2-InP micropillar cavities on their effectiveness for applications.
At first, for SPS-based quantum devices, it is important to isolate a single QD effectively emitting from the cavity. Supposing a high QD density of the order of 1010 cm−2 and an inhomogeneous width of ~50 meV, there could be less than 1 QD resonant to a cavity mode with Q factor more than 3000 in a cavity with diameter of 1 μm. It guarantees the single photon nature of an InAs/InP QD SPS using this submicrometer micropillar cavity. Since the saturated effective emission rate is proportional to the square of the coupling strength g2∝1/V [25], the small mode volume helps increasing the limit in the quantum key rate of an SPS by a factor of 2 with respect to the previous case. This is quite beneficial to enable ultrahigh-speed SPSs operating in several GHz clock.
Coherent operation of SPS requires strong coupling between the QD and the cavity mode, which can be satisfied if
where g is the coupling strength, κ = 2πc/Qλ the loss rate of the cavity mode, γ the spontaneous emission rate and γ* the pure dephasing rate of the QD. When Q is not higher than a few 104, κ is much larger than γ = 1/T1 (T1 ~1.2ns [15]) and γ* = 1/T2* = 1/T2-1/(2T1) (T2~0.13ns [26]) [27], so the condition in Eq. (3) can be simplified as g>κ/4. Using [18], where ε0 is the vacuum permittivity, e the elementary charge, m0 the free-electron mass, and f = ε0m0cλ2/(2πne2T1) [28] the oscillator strength of the QD, the simplified condition reads , where re = e2/(4πε0m0c2) is the classical radius of electron, and V is the mode volume normalized to (λ/n)3. As Q goes higher than that of the satisfied simplified condition, Eq. (3) is always available since |κ-γ-γ*| is decreasing to zero and then approaching γ + γ* (~30 μeV), which is much less than 4g (~365 μeV). Obviously, this condition can be easily satisfied by our tapered micropillar cavity since its can be ~105-106, ~2-3 orders of magnitude higher than that of the previous cavity, maximally ~2000. It is indicative of the feasibility of constructing a coherent SPS or other quantum devices for 1.55-μm band quantum information processing.As the actual fabrication process is concerned, our cavity design has an advantage in lowering the pillar height as compared with those of GaAs/AlAs and SiO2/TiO2 tapered micropillar cavities. The GaAs/AlAs case gives a pillar height of 9.6 μm for mode wavelength λ ~0.9 μm [18], then it would be ~16 μm high for λ ~1.55 μm. The TiO2/SiO2 tapered cavity gives a pillar height of 6.2 μm for λ ~0.64 μm [19], thus it would be ~15 μm high for λ ~1.55 μm. Our present cavities thus serve as better candidates for λ ∼1.55 μm.
Finally we may argue that, the tapered Si/SiO2-InP hybrid micropillar cavities here proposed are not only promising as 1.55-μm quantum information processing devices based on InAs/InP QDs, their high Q and small V also support the applications in other fields such as ultrasmall lasers, slow-light engineering and optical switching.
5. Conclusion
In summary, numerical simulations are carried out on hybrid micropillar cavities consisting of Si/SiO2 DBRs, tapered Si/SiO2 DBRs and a single InP spacer layer. The adiabatic design of the tapered DBRs improves the Q-factor for up to 3 orders of magnitude compared to traditional Si/SiO2-InP micropillar cavities proposed before, and simultaneously reduces the cavity diameter to submicrometer to guarantee single-emitter operation. A moderate Q-factor of ~3000 and then a Purcell factor more than 200 are realized by only 2 taper segments and fewer conventional DBR pairs, leading to ultrahigh-speed single photon generation operating in several GHz clock. As the taper segment number is increased, the Q-factor can be boosted to ~105-106. Together with the small mode volume, such a high Q-factor enables coherent exchange between the emitter and the optical mode at 1.55 μm, which is applicable to advanced fiber-based quantum information networks.
Acknowledgments
This work was supported by Project for Developing Innovation Systems of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
References and links
1. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]
2. H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305(5689), 1444–1447 (2004). [CrossRef] [PubMed]
3. R. Ozaki, H. Moritake, K. Yoshino, and M. Ozaki, “Analysis of defect mode switching response in one-dimensional photonic crystal with a nematic liquid crystal defect layer,” J. Appl. Phys. 101(3), 033503 (2007). [CrossRef]
4. D. Englund, A. Faraon, I. Fushman, N. Stoltz, P. Petroff, and J. Vucković, “Controlling cavity reflectivity with a single quantum dot,” Nature 450(7171), 857–861 (2007). [CrossRef] [PubMed]
5. K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450(7171), 862–865 (2007). [CrossRef] [PubMed]
6. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoğlu, “A quantum dot single-photon turnstile device,” Science 290(5500), 2282–2285 (2000). [CrossRef] [PubMed]
7. 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(18), 2865–2867 (2001). [CrossRef]
8. M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002). [CrossRef] [PubMed]
9. C. Santori, D. Fattal, J. Vucković, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419(6907), 594–597 (2002). [CrossRef] [PubMed]
10. O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013). [CrossRef] [PubMed]
11. J. P. Reithmaier, G. Sęk, 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(7014), 197–200 (2004). [CrossRef] [PubMed]
12. M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot–nanocavity system,” Nat. Phys. 6(4), 279–283 (2010). [CrossRef]
13. M. Pelton, J. Vučković, G. S. Solomon, A. Scherer, and Y. Yamamoto, “Three-dimensionally confined modes in micropost microcavities: quality factors and Purcell factors,” IEEE J. Quantum Electron. 38(2), 170–177 (2002). [CrossRef]
14. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4(11), 859–863 (2008). [CrossRef]
15. K. Takemoto, Y. Nambu, T. Miyazawa, K. Wakui, S. Hirose, T. Usuki, M. Takatsu, N. Yokoyama, K. Yoshino, A. Tomita, S. Yorozu, Y. Sakuma, and Y. Arakawa, “Transmission experiment of quantum keys over 50 km using high-performance quantum-dot single-photon source at 1.5 μm wavelength,” Appl. Phys. Express 3(9), 092802 (2010). [CrossRef]
16. H. Z. Song, K. Takemoto, T. Miyazawa, M. Takatsu, S. Iwamoto, T. Yamamoto, and Y. Arakawa, “Design of Si/SiO2 micropillar cavities for Purcell-enhanced single photon emission at 1.55 μm from InAs/InP quantum dots,” Opt. Lett. 38(17), 3241–3244 (2013). [CrossRef] [PubMed]
17. H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008). [CrossRef] [PubMed]
18. M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Höfling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108(5), 057402 (2012). [CrossRef] [PubMed]
19. Y. Zhang and M. Lončar, “Submicrometer diameter micropillar cavities with high quality factor and ultrasmall mode volume,” Opt. Lett. 34(7), 902–904 (2009). [CrossRef] [PubMed]
20. A. Yariv, Optical Electronics (Saunders College, 1991).
21. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).
22. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]
23. P. Lalanne, S. Mias, and J. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express 12(3), 458–467 (2004). [CrossRef] [PubMed]
24. Ph. Lalanne, J. P. Hugonin, and J. M. Gerard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84(23), 4726–4728 (2004). [CrossRef]
25. A. Auffèves, D. Gerace, J.-M. Gérard, M. França Santos, L. C. Andreani, and J.-P. Poizat, “Controlling the dynamics of a coupled atom-cavity system by pure dephasing,” Phys. Rev. B 81(24), 245419 (2010). [CrossRef]
26. T. Kuroda, Y. Sakuma, K. Sakoda, K. Takemoto, and T. Usuki, “Decoherence of single photons from an InAs/InP quantum dot emitting at a 1.3 μm wavelength,” Phys. Status Solidi 6(4), 944–947 (2009). [CrossRef]
27. A. J. Bennett, R. B. Patel, A. J. Shields, K. Cooper, P. Atkinson, C. A. Nicoll, and D. A. Ritchie, “Indistinguishable photons from a diode,” Appl. Phys. Lett. 92(19), 193503 (2008). [CrossRef]
28. O. Kojima, H. Nakatani, T. Kita, O. Wada, K. Akahane, and M. Tsuchiya, “Photoluminescence characteristics of quantum dots with electronic states interconnected along growth direction,” J. Appl. Phys. 103(11), 113504 (2008). [CrossRef]
