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Withdrawn: Strong coupling between a single quantum dot and whisper gallery mode within high-Q photonic crystal microcavity
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Strong coupling between a single quantum dot and whisper gallery mode within high-Q photonic crystal microcavity Hao-Xiang Jiang a, Ling-Yan Li b, Cheng-ping Ying a, Jing-Feng Liu b,n a Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China b School of Electronic Engineering, South China Agriculture University, Guangzhou 510642, China
art ic l e i nf o
a b s t r a c t
Article history: Received 7 January 2015 Received in revised form 4 April 2015 Accepted 6 April 2015
Photonic crystal (PC) slab cavity is an important candidate structure to achieve strong coupling with quantum dot. An additional hole on PC slab will generally lead to a sharp decrease of the Q-factor. We numerically observe strong coupling between a quantum dot and whisper gallery mode (WGM) in a cavity based on photonic crystal slab with an additional central air hole. Compared with other WGM microcavities, our proposal offers ease for fabrication and integration. In our system, quantum dot can be stuck on the edge of the air holes to achieve strong coupling, instead of being embedded into the slab, which is markedly different from other similar proposals. We believe such a characteristic may contribute to the further development in this field. & 2015 Published by Elsevier B.V.
Keywords: Photonic crystal cavities Whisper gallery mode Vacuum Rabi splitting
1. Introduction Over the last decade, the realization of vacuum Rabi splitting has been an exciting subfield of quantum optics and atomic physics [1–3], amongst others, one of the most effective proposals is the coupling between a high-Q optical microcavity and an atomiclike two-level system. Several strategies have been posed to design high-Q optical microcavities, such as photonic crystal (PC) cavity [4–7], micropost (or micropillar) cavity [8,9] and microdisk (or microtoroid resonator) [10,11], which are frequently used to realize strong coupling in experiments. These systems provide test beds for quantum optics and quantum information science. The PC cavities, because of small mode volume, are suited for designing microcavity laser device [12–15]. Recently, researches on interaction between whisper gallery mode (WGM) and two-level system [16–20] have attracted more and more interests for its broad prospects in applications of quantum information processing and nonlinear optics. In the last few years, strong coupling has been realized successively in microdisk–atom system [10], by Takao Aoki et al., and microdisk–QD system [11], by Srinivasan and Painter. However, strong coupling between a single two-level system and WGM within high-Q PC microcavity has not been reported so far. Considering the advantages in integration and fabrication of PC, this work may offer an alternative to the further development in this field. n
Corresponding author. E-mail address: [email protected] (J.-F. Liu).
In this paper, we demonstrate the possibility to obtain high-Q WGM and realize strong coupling effect based on photonic crystal (PC) slab cavity, by numerical simulation with three-dimension finite difference time-domain (3D-FDTD) approach [21]. According to our calculation, quantum dot can be stuck on the edge of the air holes to achieve strong coupling, instead of being embedded into the slab, which is totally different from other similar proposals. We believe such a characteristic may offer convenience for certain experiments.
2. Design of PC microcavity with a central air hole Our proposal basically follows the design of Ryu's group [22], but adding a central air hole in the center of the microcavity in order to achieve single mode operation. To illustrate the necessity of the central hole, we will firstly begin with a brief discussion of Ryu's design, and then discuss the effect of the size of the central hole. As shown in Fig. 1(a), a seven-hole defect (H2) cavity is based on a triangular lattice photonic crystal slab with circle air holes. The thickness of the slab, the radius of the air holes and the refractive index of the slab are 0.6a , 0.29a and 3.4, respectively, where a stands for lattice constant. All the calculations mentioned below are performed using the 3-D FDTD method, where the cell size is set to be a/20, and the size of PC layer is 31a in the x-direction and 31 3a /2 in the y-direction. The optimization is realized not only by a displacement p = (2 − 3 ) a of 6 nearest
http://dx.doi.org/10.1016/j.optcom.2015.04.016 0030-4018/& 2015 Published by Elsevier B.V.
Please cite this article as: H.-X. Jiang, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.016i
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Fig. 1. (a) Illustration of the optimized cavity within a 2D triangular lattice PC slab. The thickness of the slab, the radius of the air holes and the refractive index of the slab are 0.6a , 0.29a and 3.4, respectively. Displacement and radius of the nearest air holes are p = (2 − 3 ) a and rm = 0.225a . (b) Normalized amplitude of the electric field E /max( E ) of the different single modes in the optimized cavity, in X–Y plane.
neighbor holes, but also by a modification of the radius of 12 nearest neighbor holes from r = 0.29a to rm = 0.225a . Such a 12fold symmetry contributes to the high Q-factor of the WGM. Similar to those in the micro-disk structure, the WGM locates along the boundary of the cavity and forms a standing-wave pattern. But the light is not confined by the total internal reflection at disk boundary here, instead, light confinement depends on photonicbandgap effect in the in-plane direction, and total internal refection at the interface between the slab and the air clad in the vertical direction. According to our FDTD calculation, such a design can achieve WGM with the azimuthal mode number of 6, with the 12 lobes
perfectly matched with the 12-fold symmetry of the structure, see Fig. 1(b). The WGM has a high Q-factor above 105 at normalized frequency ωa/2πc = 0.27292, with mode volume V ∼ 1.3(λ /n)3. However, there are a few modes other than the WGM within the nearby frequency band, which may adversely influence single mode operation of strong coupling. We calculate the normalized amplitudes of the electric field E /max( E ) of ωa/2πc = 0.27292, 0.26113, 0.28005 and 0.27399, as shown in Fig. 1(b), from which we can notice that all the single modes other than WGM have certain light distribution in the center of the cavity. Therefore, an additional central air hole will be a breach of the electric field of
Fig. 2. The effect of the size of the central air hole. (a) shows the resonant spectra of the cavities with central air holes of different sizes over photonic bandgap ωa/2πc = 0.25 ∼ 0.33. The structure illustrations on the left hand side are one-to-one corresponding to the resonant spectra, showing the cases of R ¼0, 0.3a and 0.7a . (b) shows the WGM resonant peak over a narrow frequency range, where a single Lorentzian peak is obtained. (c) reveals the trend of Q factor and normalized resonant frequency of the WGM as R increasing.
Please cite this article as: H.-X. Jiang, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.016i
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such modes but will not affect the distribution of WGM. Next, we discuss the effect of the size of the central air hole. For brevity's sake, we simply compare the resonant spectrum of a H2 cavity with those that have a central air hole of R = 0.3a and R = 0.7a , as shown in Fig. 2(a), where the ordinate represents the radiated power intensity, and the abscissa the normalized frequencies ωa/2πc . Obviously, a central air hole with appropriate radius helps to obtain a “cleaner” resonant spectrum, in other words, a single mode operation for a broad range of wavelengths, which is very useful to realize the QD–cavity strong coupling regime in the designed structure. But the size of the central air hole should not be too large so as to avoid affecting the light distribution of WGM and a decrease of its Q factor, see Fig. 2(c). While changing the value of R, the shape of WGM resonant peak remains a single Lorentzian function, as is shown in Fig. 2(b), despite a little frequency shift when R ≥ 0.7a . This suggests that the clockwise (CW) and counterclockwise (CCW) propagating mode, which form the WGM, are degenerate. In other words, frequency splitting between the CW and the CCW mode due to inner surface scattering [23,24] cannot be observed in this structure. Therefore, when we talk about the QD–cavity interaction in this structure, the basic model underlying our analysis is that of a single two-state system coupled to a resonant cavity, namely the Jaynes–Cummings model.
3
Here, it is noticeable that all of the maximum points locate inside the nearest air holes, but not in the dielectric material like that in the micro-disk. Additionally, the maximum is close to the inner edge of the air holes, due to the continuity of electric displacement and the step discontinuity of relative permittivity. This may offer convenience for some experiments, because the QD is unnecessary to be embedded into the slab but only be stuck on the edge of the air holes at the central plane of the slab. Since the maximum of each lobe is approximately equal to each other, for convenience's sake, we place an x-orientated electric dipole at the maximum of Ex, as illustrated in Fig. 3(a). As evidences of the strong-coupling regime of cavity QED, Fig. 3 (b) shows the Rabi splitting of 2g ¼24.5 GHz from power spectrum by ignoring the effects of dephasing under fully resonant circumstance and the eigenvalues of the QD–cavity. Interestingly, although the mode volume of WGM is relatively large, the coupling strength g of the system is still considerable, in opposition to the usual expectation. This results from the extremely high intensity of electric field on the edge of the nearest air holes.
4. Summary 3. Strong coupling of QD–cavity system In this section, we discuss the strong coupling of QD–cavity system based on the H2 cavity with an R = 0.7a central air hole. The parameters used here remain h = 0.6a , r = 0.29a , n ¼3.4, and the 12 nearest neighbor holes are symmetrically tuned, as has mentioned before. But the lattice constant is set to be 420 nm in this section and the life time of QD is set to be 8.7 ns [6]. To observe strong coupling between the WGM mode and a single QD, we need to settle the QD at the field maximum point.
In conclusion, we offer evidences for the possibility to realize strong coupling between QD and microcavity based on photonic crystal slab. There are at least two advantages of our system: firstly, microcavity based on photonic crystal slab can be fabricated much easier compared to micro-disk while showing similar optical properties; secondly, it may offer convenience for some experiments for the fact that the QD is unnecessary to be embedded into the slab but only be stuck on the edge of the air holes.
Fig. 3. (a) Normalized radiated power spectrum of dipole source. (b) Power spectrum showing vacuum Rabi splitting when the QD is resonant with WGM.
Please cite this article as: H.-X. Jiang, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.016i
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Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 11204089 and 11104086), Young Teachers Scientific Research and Cultivating Fund of South China Normal University (Grant no. 13KJ05) and Natural Science Foundation of Guangdong Province of China (Grant no. S2011040001908).
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67 V. Keldysh, V.D. Kulakovskii, T.L. Reinecke, A. Forchel, Nature 432 (2004) 197. [9] S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauß, S.H. Kwon, C. Schneider, 68 A. Löffler, S. Höfling, M. Kamp, A. Forchel, Appl. Phys. Lett. 90 (2007) 251109. 69 [10] T. Aoki, B. Dayan, E. Wilcut, W.P. Bowen, A.S. Parkins, T.J. Kippenberg, K. 70 J. Vahala, H.J. Kimble, Nature 443 (2006) 671. [11] K. Srinivasan, O. Painter, Nature 450 (2007) 862. 71 [12] R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C.F. Gmachl, D.M. Tennant, 72 A.M. Sergent, D.L. Sivco, A.Y. Cho, F. Capasso, Science 302 (2003) 1374. 73 [13] H. Altug, D. Englund, J. Vuckovic, Nat. Phys. 2 (2006) 484. [14] Y.V. Radeonychev, I.V. Koryukin, O. Kocharovskaya, Laser Phys. 19 (2009) 1207. 74 [15] V.A. Trofimov, M.V. Fedotov, A.G. Volkov, N.V. Tcherniega, V.V. Savranskii, 75 S. Lan, Laser Phys. 20 (2010) 1137. 76 [16] K. Srinivasan, O. Painter, Phys. Rev. A 75 (2007) 023814. [17] B. Dayan, A.S. Parkins, T. Aoki, E.P. Ostby, K.J. Vahala, H.J. Kimble, Science 319 77 (2008) 1062. 78 [18] F.-Y. Hong, S.-J. Xiong, Phys. Rev. A 78 (2008) 013812. 79 [19] T. Aoki, A.S. Parkins, D.J. Alton, C.A. Regal, B. Dayan, E. Ostby, K.J. Vahala, H. J. Kimble, Phys. Rev. Lett. 102 (2009) 083601. 80 [20] J.-s. Jin, C.-s. Yu, P. Pei, H.-s. Song, Phys. Rev. A 81 (2010) 042309. 81 [21] D.M. Sullivan, Electromagnetic Simulation Using the FDTD Method, WileyQ2 82 IEEE Press, 2000. 83 [22] H.-Y. Ryu, M. Notomi, G.-H. Kim, Y.-H. Lee, Opt. Express 12 (2004) 1708. [23] D.S. Weiss, V. Sandoghdar, J. Hare, V. Lefèvre-Seguin, J.M. Raimond, S. Haroche, 84 Opt. Lett. 20 (1995) 1835. 85 [24] X. Yi, Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, Phys. Rev. A 83 86 (2011) 023803.
Please cite this article as: H.-X. Jiang, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.016i
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Strong coupling between a single quantum dot and whisper gallery mode within high-Q photonic crystal microcavity Hao-Xiang Jiang a, Ling-Yan Li b, Cheng-ping Ying a, Jing-Feng Liu b,n a Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China b School of Electronic Engineering, South China Agriculture University, Guangzhou 510642, China
art ic l e i nf o
a b s t r a c t
Article history: Received 7 January 2015 Received in revised form 4 April 2015 Accepted 6 April 2015
Photonic crystal (PC) slab cavity is an important candidate structure to achieve strong coupling with quantum dot. An additional hole on PC slab will generally lead to a sharp decrease of the Q-factor. We numerically observe strong coupling between a quantum dot and whisper gallery mode (WGM) in a cavity based on photonic crystal slab with an additional central air hole. Compared with other WGM microcavities, our proposal offers ease for fabrication and integration. In our system, quantum dot can be stuck on the edge of the air holes to achieve strong coupling, instead of being embedded into the slab, which is markedly different from other similar proposals. We believe such a characteristic may contribute to the further development in this field. & 2015 Published by Elsevier B.V.
Keywords: Photonic crystal cavities Whisper gallery mode Vacuum Rabi splitting
1. Introduction Over the last decade, the realization of vacuum Rabi splitting has been an exciting subfield of quantum optics and atomic physics [1–3], amongst others, one of the most effective proposals is the coupling between a high-Q optical microcavity and an atomiclike two-level system. Several strategies have been posed to design high-Q optical microcavities, such as photonic crystal (PC) cavity [4–7], micropost (or micropillar) cavity [8,9] and microdisk (or microtoroid resonator) [10,11], which are frequently used to realize strong coupling in experiments. These systems provide test beds for quantum optics and quantum information science. The PC cavities, because of small mode volume, are suited for designing microcavity laser device [12–15]. Recently, researches on interaction between whisper gallery mode (WGM) and two-level system [16–20] have attracted more and more interests for its broad prospects in applications of quantum information processing and nonlinear optics. In the last few years, strong coupling has been realized successively in microdisk–atom system [10], by Takao Aoki et al., and microdisk–QD system [11], by Srinivasan and Painter. However, strong coupling between a single two-level system and WGM within high-Q PC microcavity has not been reported so far. Considering the advantages in integration and fabrication of PC, this work may offer an alternative to the further development in this field. n
Corresponding author. E-mail address: [email protected] (J.-F. Liu).
In this paper, we demonstrate the possibility to obtain high-Q WGM and realize strong coupling effect based on photonic crystal (PC) slab cavity, by numerical simulation with three-dimension finite difference time-domain (3D-FDTD) approach [21]. According to our calculation, quantum dot can be stuck on the edge of the air holes to achieve strong coupling, instead of being embedded into the slab, which is totally different from other similar proposals. We believe such a characteristic may offer convenience for certain experiments.
2. Design of PC microcavity with a central air hole Our proposal basically follows the design of Ryu's group [22], but adding a central air hole in the center of the microcavity in order to achieve single mode operation. To illustrate the necessity of the central hole, we will firstly begin with a brief discussion of Ryu's design, and then discuss the effect of the size of the central hole. As shown in Fig. 1(a), a seven-hole defect (H2) cavity is based on a triangular lattice photonic crystal slab with circle air holes. The thickness of the slab, the radius of the air holes and the refractive index of the slab are 0.6a , 0.29a and 3.4, respectively, where a stands for lattice constant. All the calculations mentioned below are performed using the 3-D FDTD method, where the cell size is set to be a/20, and the size of PC layer is 31a in the x-direction and 31 3a /2 in the y-direction. The optimization is realized not only by a displacement p = (2 − 3 ) a of 6 nearest
http://dx.doi.org/10.1016/j.optcom.2015.04.016 0030-4018/& 2015 Published by Elsevier B.V.
Please cite this article as: H.-X. Jiang, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.016i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
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H.-X. Jiang et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 1. (a) Illustration of the optimized cavity within a 2D triangular lattice PC slab. The thickness of the slab, the radius of the air holes and the refractive index of the slab are 0.6a , 0.29a and 3.4, respectively. Displacement and radius of the nearest air holes are p = (2 − 3 ) a and rm = 0.225a . (b) Normalized amplitude of the electric field E /max( E ) of the different single modes in the optimized cavity, in X–Y plane.
neighbor holes, but also by a modification of the radius of 12 nearest neighbor holes from r = 0.29a to rm = 0.225a . Such a 12fold symmetry contributes to the high Q-factor of the WGM. Similar to those in the micro-disk structure, the WGM locates along the boundary of the cavity and forms a standing-wave pattern. But the light is not confined by the total internal reflection at disk boundary here, instead, light confinement depends on photonicbandgap effect in the in-plane direction, and total internal refection at the interface between the slab and the air clad in the vertical direction. According to our FDTD calculation, such a design can achieve WGM with the azimuthal mode number of 6, with the 12 lobes
perfectly matched with the 12-fold symmetry of the structure, see Fig. 1(b). The WGM has a high Q-factor above 105 at normalized frequency ωa/2πc = 0.27292, with mode volume V ∼ 1.3(λ /n)3. However, there are a few modes other than the WGM within the nearby frequency band, which may adversely influence single mode operation of strong coupling. We calculate the normalized amplitudes of the electric field E /max( E ) of ωa/2πc = 0.27292, 0.26113, 0.28005 and 0.27399, as shown in Fig. 1(b), from which we can notice that all the single modes other than WGM have certain light distribution in the center of the cavity. Therefore, an additional central air hole will be a breach of the electric field of
Fig. 2. The effect of the size of the central air hole. (a) shows the resonant spectra of the cavities with central air holes of different sizes over photonic bandgap ωa/2πc = 0.25 ∼ 0.33. The structure illustrations on the left hand side are one-to-one corresponding to the resonant spectra, showing the cases of R ¼0, 0.3a and 0.7a . (b) shows the WGM resonant peak over a narrow frequency range, where a single Lorentzian peak is obtained. (c) reveals the trend of Q factor and normalized resonant frequency of the WGM as R increasing.
Please cite this article as: H.-X. Jiang, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.016i
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such modes but will not affect the distribution of WGM. Next, we discuss the effect of the size of the central air hole. For brevity's sake, we simply compare the resonant spectrum of a H2 cavity with those that have a central air hole of R = 0.3a and R = 0.7a , as shown in Fig. 2(a), where the ordinate represents the radiated power intensity, and the abscissa the normalized frequencies ωa/2πc . Obviously, a central air hole with appropriate radius helps to obtain a “cleaner” resonant spectrum, in other words, a single mode operation for a broad range of wavelengths, which is very useful to realize the QD–cavity strong coupling regime in the designed structure. But the size of the central air hole should not be too large so as to avoid affecting the light distribution of WGM and a decrease of its Q factor, see Fig. 2(c). While changing the value of R, the shape of WGM resonant peak remains a single Lorentzian function, as is shown in Fig. 2(b), despite a little frequency shift when R ≥ 0.7a . This suggests that the clockwise (CW) and counterclockwise (CCW) propagating mode, which form the WGM, are degenerate. In other words, frequency splitting between the CW and the CCW mode due to inner surface scattering [23,24] cannot be observed in this structure. Therefore, when we talk about the QD–cavity interaction in this structure, the basic model underlying our analysis is that of a single two-state system coupled to a resonant cavity, namely the Jaynes–Cummings model.
3
Here, it is noticeable that all of the maximum points locate inside the nearest air holes, but not in the dielectric material like that in the micro-disk. Additionally, the maximum is close to the inner edge of the air holes, due to the continuity of electric displacement and the step discontinuity of relative permittivity. This may offer convenience for some experiments, because the QD is unnecessary to be embedded into the slab but only be stuck on the edge of the air holes at the central plane of the slab. Since the maximum of each lobe is approximately equal to each other, for convenience's sake, we place an x-orientated electric dipole at the maximum of Ex, as illustrated in Fig. 3(a). As evidences of the strong-coupling regime of cavity QED, Fig. 3 (b) shows the Rabi splitting of 2g ¼24.5 GHz from power spectrum by ignoring the effects of dephasing under fully resonant circumstance and the eigenvalues of the QD–cavity. Interestingly, although the mode volume of WGM is relatively large, the coupling strength g of the system is still considerable, in opposition to the usual expectation. This results from the extremely high intensity of electric field on the edge of the nearest air holes.
4. Summary 3. Strong coupling of QD–cavity system In this section, we discuss the strong coupling of QD–cavity system based on the H2 cavity with an R = 0.7a central air hole. The parameters used here remain h = 0.6a , r = 0.29a , n ¼3.4, and the 12 nearest neighbor holes are symmetrically tuned, as has mentioned before. But the lattice constant is set to be 420 nm in this section and the life time of QD is set to be 8.7 ns [6]. To observe strong coupling between the WGM mode and a single QD, we need to settle the QD at the field maximum point.
In conclusion, we offer evidences for the possibility to realize strong coupling between QD and microcavity based on photonic crystal slab. There are at least two advantages of our system: firstly, microcavity based on photonic crystal slab can be fabricated much easier compared to micro-disk while showing similar optical properties; secondly, it may offer convenience for some experiments for the fact that the QD is unnecessary to be embedded into the slab but only be stuck on the edge of the air holes.
Fig. 3. (a) Normalized radiated power spectrum of dipole source. (b) Power spectrum showing vacuum Rabi splitting when the QD is resonant with WGM.
Please cite this article as: H.-X. Jiang, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.016i
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Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 11204089 and 11104086), Young Teachers Scientific Research and Cultivating Fund of South China Normal University (Grant no. 13KJ05) and Natural Science Foundation of Guangdong Province of China (Grant no. S2011040001908).
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Please cite this article as: H.-X. Jiang, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.016i
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