OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 23486–23497
« Show journal navigation

Vacuum Rabi splitting in a coupled system of single quantum dot and photonic crystal cavity: effect of local and propagation Green’s functions

Yi-Cong Yu, Jing-Feng Liu, Xiao-Lu Zhuo, Gengyan Chen, Chong-Jun Jin, and Xue-Hua Wang  »View Author Affiliations


Optics Express, Vol. 21, Issue 20, pp. 23486-23497 (2013)
http://dx.doi.org/10.1364/OE.21.023486


View Full Text Article

Acrobat PDF (2888 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We investigate the light emission characteristics for single two level quantum dot (QD) in a realistic photonic crystal (PC) L3 cavity based upon the local coupling strength between the QD and cavity together with the Green’s function in which the propagation function related to the position of the detector is taken into account. We find for a PC cavity that the line shape of the propagation function in frequency domain is identical to that of the cavity and independent on the detector's position. We confirm that this identity is not influenced by the horizontal decay of the cavity. Furthermore, it is revealed that the vacuum fluorescence spectrum of the coupled system never give the triplet in strong coupling regime. Our work demonstrates that the experimental spectral-triplet in coupled system of single QD and PC cavity cannot be individually understood by vacuum Rabi splitting without including other physics mechanism.

© 2013 Optical Society of America

Cavity quantum electrodynamics (QED) [1

1. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002). [CrossRef] [PubMed]

,2

2. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]

] has been a hot topic that has been attracting sustaining interest for decades. It not only provides a test bed for quantum physics but also has important applications in quantum information science with atoms and photons [3

3. C. Monroe, “Quantum information processing with atoms and photons,” Nature 416(6877), 238–246 (2002). [CrossRef] [PubMed]

,4

4. J. L. O'Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009). [CrossRef]

]. By tailoring the light-matter interactions between the quantum dot (QD) and microcavity, various striking quantum phenomena have been revealed including vacuum Rabi splitting [5

5. J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432(7014), 197–200 (2004). [CrossRef] [PubMed]

9

9. 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]

], quantum entanglement [10

10. J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001). [CrossRef]

12

12. R. Johne, N. A. Gippius, and G. Malpuech, “Entangled photons from a strongly coupled quantum dot-cavity system,” Phys. Rev. B 79(15), 155317 (2009). [CrossRef]

], laser oscillation [13

13. E. del Valle, F. P. Laussy, and C. Tejedor, “Luminescence spectra of quantum dots in microcavities. II. Fermions,” Phys. Rev. B 79(23), 235326 (2009). [CrossRef]

,14

14. 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]

], single photon source with photon antibunching [15

15. W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006). [CrossRef] [PubMed]

,16

16. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98(11), 117402 (2007). [CrossRef] [PubMed]

], blockade [17

17. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4(11), 859–863 (2008). [CrossRef]

] and spontaneous emission control [18

18. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]

].

Since the first observation of the vacuum Rabi splitting (VRS) in a solid state system composing of QDs and photonic-crystal-cavity (PC cavity) [6

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 432(7014), 200–203 (2004). [CrossRef] [PubMed]

], this system has been considered as a great candidate for realizing strong coupling between quantum dots and a microcavity. In the work of Hennessy et al. [8

8. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]

], an additional middle peak appeared along with the two VRS peaks under on-resonance conditions, forming a spectral triplet. This middle peak preserved exactly the wavelength, line width and polarization of the bare-cavity mode. This phenomenon has attracted intensive attentions and incurred controversial discussions. A view has been echoed that this phenomenon indicates a clear deviation from standard atomic cavity QED theories that treat a single QD as a two level artificial atom, and then a theoretical model was proposed for understanding this phenomenon based on both the exciton complexes in a QD and the quantum anti-Zeno effect caused by pure dephasing [19

19. M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Quantum dot spectroscopy using cavity quantum electrodynamics,” Phys. Rev. Lett. 101(22), 226808 (2008). [CrossRef] [PubMed]

21

21. M. Yamaguchi, T. Asano, and S. Noda, “Third emission mechanism in solid-state nanocavity quantum electrodynamics,” Rep. Prog. Phys. 75(9), 096401 (2012). [CrossRef] [PubMed]

]. In contrast, another theoretical model was proposed by Hughes et al. [22

22. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef] [PubMed]

] in which the vacuum fluorescence spectrum is expressed as the product of a propagation function and the local dipole spectrum [23

23. T. Ochiai, J.-i. Inoue, and K. Sakoda, “Spontaneous emission from a two-level atom in a bisphere microcavity,” Phys. Rev. A 74(6), 063818 (2006). [CrossRef]

]. In this model, they assumed that the line width of the propagation function can be significantly smaller that the line width of the projected local density of states (PLDOS). This assumption is the key to produce a triple spectrum with VRS without considering other physics mechanism.

We consider a QD as two level exciton placed in inhomogeneous dielectrics, the Hamiltonian of the system within the dipole approximation and rotating wave approach can be written as [24

24. X.-H. Wang, B.-Y. Gu, R. Wang, and H.-Q. Xu, “Decay kinetic properties of atoms in photonic crystals with absolute gaps,” Phys. Rev. Lett. 91(11), 113904 (2003). [CrossRef] [PubMed]

]
=ω0|ee|+λωλaλaλ+λgλ(r0)aλσ+gλ(r0)σ+aλ
(1)
where gλ(r0)=iω0(2ε0ωλ)1/2Eλ(r0)d is the coupling coefficient, σ=|ge| and σ+=|eg| are the Pauli operators of the two-level system, ω0 is the transition frequency of the bare emitter, r0 is the location of the QD, ωλand Eλ(r)are the frequency and electric field of the λ-th eigenmode in the nanostructure, and d=dd^ is the transition dipole moment between the two levels. The vacuum fluorescence spectrum is calculated by [22

22. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef] [PubMed]

,23

23. T. Ochiai, J.-i. Inoue, and K. Sakoda, “Spontaneous emission from a two-level atom in a bisphere microcavity,” Phys. Rev. A 74(6), 063818 (2006). [CrossRef]

]
S(ω)=0dt10dt2eiω(t1t2)E()(r,t1)E(+)(r,t2).
(2)
It is straightforward to obtain that

S(ω)=f(rp,r0,ω)P(r0,ω)
(3)

Here

f(rp,r0,ω)=|K(rp,r0,ω)dε0|2
(4)

is the propagation function which describes the electric field intensity of at the probe point rp emitted by a classical dipole located at r0 with the transition frequency ω. The generalized-transverse Green function is introduced as [22

22. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef] [PubMed]

,25

25. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70(5), 053823 (2004). [CrossRef]

]

K(r,r,ω)=λωλ2Eλ(r)Eλ(r)ω2ωλ2=ω2c2G(r,r,ω)Iδ(rr)εr(r,ω),
(5)

where the dyadic Green’s function [26

26. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1993).

] satisfies the equation

[××εr(r,ω)ω2c2]G(r,r,ω)=Iδ(rr),
(6)

εr(r,ω) is the relative dielectric constant and c is the speed of light in vacuum. P(r0,ω)=[σ(ω)]σ(ω) with σ(ω)=0σ(t)eiωtdt is the local dipole or polarization spectrum. The conventional definition of the emission spectrum from a two-level exciton only consider P(r0,ω) [27

27. H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A 40(10), 5516–5519 (1989). [CrossRef] [PubMed]

,28

28. L. C. Andreani, G. Panzarini, and J.-M. Gérard, “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys. Rev. B 60(19), 13276–13279 (1999). [CrossRef]

]. Assuming that the exciton of the QD is initially at the excited state and the field is in vacuum state, P(r0,ω) can be obtained as
P(r0,ω)={|ωω0ω02d2ω2ε0d^K(r0,r0,ω)d^|2}1.
(7)
Equations (3), (4) and (7) indicate that the vacuum fluorescence spectrum depends on the Green function in two separate ways: one for propagation included in the propagation function, and one for the local field effect on the quantum dot included in the local dipole spectrum. According to Eq. (5), K(r,r,ω) is different from (ω/c)2G(r,r,ω)only when r=r, thus in Eq. (4) the calculation of Kfor the propagation function can be directly linked to the calculation of G. The direct calculation of K(r0,r0,ω) could be more difficult. In present work, the identity 1/(ωω+i0+)=[1/(ωω)]iπδ(ωω)is applied in Eq. (5), and then Eq. (7) can be rewritten as

P(r0,ω)={[ωω0Δ(r0,ω)]2+Γ(r0,ω)2/4}1,
(8)

where

Γ(r0,ω)=2πλ|gλ(r0)|2δ(ωωλ)=2ω02d2ω2ε0d^Im[K(r0,r0,ω)]d^
(9)

is the LCS and

Δ(r0,ω)=2π0Γ(r0,ω)ωωdω=ω02d2ω2ε0d^Re[K(r0,r0,ω)]d^
(10)

is the level shift [24

24. X.-H. Wang, B.-Y. Gu, R. Wang, and H.-Q. Xu, “Decay kinetic properties of atoms in photonic crystals with absolute gaps,” Phys. Rev. Lett. 91(11), 113904 (2003). [CrossRef] [PubMed]

]. Here the dielectrics is lossless and εr(r,ω) is real, according to Eq. (5) the imaginary part of K(r0,r0,ω)is identical to that of (ω/c)2G(r0,r0,ω), and then we have

Γ(r0,ω)=2ω02d2ε0c2d^Im[G(r0,r0,ω)]d=Γ0ωω0M(r0,ω,d^),
(11)

where Γ0 is the spontaneous emission rate of the QD in vacuum, M(r0,ω,d^) is the local multiplication factor of the PLDOS and can be obtained by the ab-initio mapping [29

29. G. Chen, Y.-C. Yu, X.-L. Zhuo, Y.-G. Huang, H. Jiang, J.-F. Liu, C.-J. Jin, and X.-H. Wang, “Ab initio determination of local coupling interaction in arbitrary nanostructures: Application to photonic crystal slabs and cavities,” Phys. Rev. B 87(19), 195138 (2013). [CrossRef]

]. The level shift can be obtained by the principle value integral of the LCS in Eq. (10).

Finally, we have

S(ω)=f(rp,r0,ω){[ωω0Δ(r0,ω)]2+Γ(r0,ω)2/4}1,
(12)

where f(rp,r0,ω)=(ω2/c2ε0)2|G(rp,r0,ω)d|2.

We investigate single QD in Photonic Crystal L3 cavity following the design of [6

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 432(7014), 200–203 (2004). [CrossRef] [PubMed]

], as depicted in Fig. 1(a)
Fig. 1 The schematic diagrams of the PC L3 cavity system. (a) Cross-section on central plane (z = 0 plane) of the PC L3 cavity. The location of the QD is r0 = (0, 0, 0) at the center of the cavity and the orientation d^ is along y direction. (b) The 3D schematic diagram with the locations of 15 probe points.
. The base structure is composed of air holes in GaAs with refractive index n = 3.4. The spontaneous emission lifetime of the QD in GaAs is 1.82 ns.

The local multiplication factor of the PLDOS M(r0,ω,d^) is displayed in Fig. 2(a)
Fig. 2 (a) The multiplication factor of the PLDOS for the Sample#1. The propagation functions on probe points (b) A1 = (0, 0, 1.5) a, (c) B1 = (0, 0, 5.5) a and (d) C1 = (0, 0, 9.5) a.
for the PC slab L3 cavity with a = 300nm, r = 0.27a, s = 0.20a, the thickness d = 0.90a, and 31 air holes in the x-direction and 29 air holes in the y-direction. There is thus 14 layers of air holes that surround the defect. For convenience, this slab PC L3 cavity is denoted as Sample#1. We find that the M(r0,ω,d^) can be very well fitted by Lorentz function [29

29. G. Chen, Y.-C. Yu, X.-L. Zhuo, Y.-G. Huang, H. Jiang, J.-F. Liu, C.-J. Jin, and X.-H. Wang, “Ab initio determination of local coupling interaction in arbitrary nanostructures: Application to photonic crystal slabs and cavities,” Phys. Rev. B 87(19), 195138 (2013). [CrossRef]

]

M(r0,ω,d^)=2|gc(r0)|2Γ0κ/2(ωωc)2+(κ/2)2
(13)

where ωc is the frequency of the cavity mode, gc(r0) is the g factor that characterizes the coupling strength between the QD and the cavity mode, κ=ωc/Q is the decay rate of the cavity mode and Q is the quality factor (Q factor). From Eq. (12) we can determine the characteristic parameters of the cavity QED system, the normalized frequency of cavity mode is found to be 0.2433232, the decay rate in the unit of normalized frequency is κ=1.74321×106, the Q factor is thus 139,583, and the g factor is g = 22.1GHz.

We now simulate the propagation function f(rp,r0,ω) for the Sample#1 and determine its line width. 15 different probe points outside the cavity in air with different distances and angles to the QD are chosen, as shown in Fig. 1(b), and the propagation function f(rp,r0,ω) are obtained by the direct numerical simulations of the Green’s function. The propagation functions f(rp,r0,ω) for three probe points along the z axis, A1 = (0, 0, 1.5) a, B1 = (0, 0, 5.5) a and C1 = (0, 0, 9.5) a are also plotted in Figs. 2(b)-2(d). We can see that the propagation functions really have Lorentz line shape. Furthermore, the line widths κf of the propagation function f(rp,r0,ω) for these points are identical to the total decay rate κ of the cavity found from the M(r0,ω,d^), regardless how far these probe points are away from the surface of the PC L3 cavity in this direction.

By fitting the simulation data for all 15 probe points with the Lorentz function, we find that the central frequency in the propagation functions of every point remains unchanged as 0.2433232 identical to the cavity mode frequency. Table 1

Table 1. κf factors obtained by fitting f(rp,r0,ω) in Eq. (13) on every probe point.

table-icon
View This Table
lists the line width κf obtained by the fitting for all 15 probe points, and they are all identical to the total decay rate κ of the cavity. We note that the probe points have cover different vertical distances from the surface of the PC cavity and different azimuth angles, which should be enough to reflect the characteristics of the propagation functions for spatial locations in near, intermediate and far field. It is thus a reliable demonstration that, in a typical high Q PC L3 cavity structure, the line width of the propagation function is identical to that of the PLDOS, i.e., κf=κ, and that this identity is independent on the spatial location, or, in realistic experiments, the position of the detector.

With the simulation data of the PLDOS and the propagation function, we are able to calculate the vacuum fluorescence spectrum S(ω) in Eq. (12). We tune the transition wavelength of the QD across the resonant wavelength of the high Q PC L3 cavity, the detuning between these two wavelengths is set to be −0.06 nm, 0 nm and 0.06 nm, respectively. Figure 3(a)
Fig. 3 (a) The local dipole spectra P(r0,ω) of the Sample#1. (b) The vacuum fluorescence spectra S(ω)for the probe point C1 = (0, 0, 9.5)a. The wavelength detuning between transition wavelength of the QD and the resonant wavelength of the cavity (1232.92 nm) increases from −0.06 nm to 0.06 nm (from top to bottom).
shows that, when the transition wavelength of the QD is resonant with the cavity mode, a symmetric Rabi splitting appears in the local dipole spectrum P(r0,ω), the space of the splitting is 44.1GHz. When the transition frequency of the QD is detuned with the cavity mode, the spectra exhibit asymmetric vacuum Rabi splitting. The space of the splitting shows a typical anti-crossing. These all indicate that the strong coupling between the QD and the cavity mode is achieved. From the above fact that the line shape of the propagation function is independence on the spatial location of the probe point in this structure, we can choose a certain probe point to calculate the vacuum fluorescence spectrum S(ω) that represents the result for arbitrary detector location, here we choose the point C1 = (0, 0, 9.5) a. It can be seen clearly in Fig. 3(b) that the vacuum fluorescence spectrum is still double under on-resonance condition, the spectral triplet observed in [8

8. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]

] doesn’t appear. Furthermore, the frequencies of the two peaks in the vacuum fluorescence spectra remain nearly the same as those in the dipole spectra, and the vacuum Rabi splitting obtained under the on-resonance condition also remains 44.1GHz, the only difference is that the relative heights of the Rabi splitting peaks in the detuned cases are changed slightly. These results demonstrate that the VRS only yields double peaks in vacuum fluorescence spectra.

Although our results in Figs. 2 and 3 support κf=κ, this high Q L3 cavity may be a special structure with horizontal decay κh=0 so that it also satisfies κf=κv. In order to make a further justification, we then consider reducing the number of air hole layers surrounding the defect in Sample#1 to increase the horizontal decay.

Figure 4
Fig. 4 The decay rate κfrom the multiplication factor M(r0,ω,d^) with the corresponding Q factor, and the line width κf of the propagation functions f(rp,r0,ω), for different numbers of air hole layers in the L3 cavity.
displays the decay rate κ obtained from the multiplication factor M(r0,ω,d^), and the corresponding Q factor for L3 cavities with different number of air hole layers surrounding the defect. The sample with 14 layers of air holes is the original Sample#1. When the number of air hole layers decreases from 14 to 9, the decay rate κ only increases slightly, the Q factor is still above 120,000; while if the number of air hole layers is decreased further, the decay rate κ is dramatically increased and the Q factor is reduced. This varying of the Q factor is similar with the one in early work by O. Painter et al. [30

30. O. Painter, J. Vučkovič, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16(2), 275–285 (1999). [CrossRef]

], in which they found that the horizontal decay κh decreases exponentially with the number of the air hole layers and the vertical decay κv stays relatively constant. For a L3 cavity with number of surrounding air hole layers less than 7, the total decay is dominated by the in-plane losses. When the number of the surrounding air hole layers increases, the horizontal decay κh is decreased well below κv, and the total decay rate κ asymptotically approaches κv.

The line width κf of the propagation functions f(rp,r0,ω) for each L3 cavity with different number of surrounding air hole layers is also displayed in Fig. 4. For L3 cavities with number of surrounding air hole layers less than 7, our simulation results indicate that κf is still in excellent agreement with κ rather than κv. These results confirm that a remarkable horizontal decay does not influence the identity between the line width of M(r0,ω,d^) and that of f(rp,r0,ω) in the PC L3 cavity structures, and the key assumption proposed in [22

22. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef] [PubMed]

] that supports the triple-peak spectrum is invalid.

Figure 5
Fig. 5 (a) The local dipole spectra P(r0,ω)and (b) vacuum fluorescence spectra S(ω) for the L3 cavities with less than 7 layers of air holes surrounding the defect. The transition wavelength of the QD is resonant with the resonant wavelength λc of each cavity.
displays the local dipole spectra and the vacuum fluorescence spectra for L3 cavities with number of surrounding air hole layers less than 7. For PC L3 cavities with 5 or 6 surrounding air hole layers, the vacuum fluorescence spectra remain as doublet with two broad peaks. Comparing corresponding curves in Figs. 5(a) and 5(b), it can be seen that the valleys in the vacuum fluorescence spectra become shallow than those in the local dipole spectra, and the splitting spaces of the peak become slightly narrower. When the number of surrounding air hole layers is decreased to 4, the Q factor of the cavity is too low to achieve the strong coupling between the quantum dot and the cavity mode, and both the local dipole and vacuum fluorescence spectra become singlet.

In order to further confirm the irrationality of the assumption in [22

22. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef] [PubMed]

], we consider a classical dipole source

Pex(r,t)=dd^δ(t)δ(rr0),
(14)

located inside the cavity. The electric field induced by this dipole can be calculated by the Green’s function as
Ed(r,t)=d2πε0c20ω2G(r,r0,ω)d^eiωtdω+c.c.=d2πε00K(r,r0,ω)d^eiωtdω+c.c.Pex(r,t)ε0εr(r0)
(15)
According to the assumption in [22

22. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef] [PubMed]

], for r inside the cavity the Green’s function should take the form of Klocal, for r outside the cavity we should use Kprop. The electric field inside the cavity can be obtained as

Elocal(r,t>0)=E0(r)exp(iωctκ2t)+c.c.,
(16)

where

E0(r)=iωcd2ε0Ec(r)Ec*(r0)·d^,
(17)

and the electric field outside the cavity can be written as

Eprop(r,t>0)=E0(r)exp(iωctκv2t)+c.c.
(18)

On the boundary surface of the cavity, the electromagnetic boundary condition requires that both electric fields inside and outside the surface have the same time dependence. It can be seen in Eqs. (16)-(18) that this boundary condition is broken under the assumption in [22

22. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef] [PubMed]

].

On the other hand, we can also consider the exponential decay of the energy stored in the cavity mode

U(t)=U(0)exp(κt)=U(0)exp(ωcQt),
(19)

the total power loss of the cavity is defined by energy lost per time as

P(t)=dU(t)dt=κU(t).
(20)

This power can be separated into two vertical and horizontal parts as P(t)=Pv(t)+Ph(t), the decay rate can also be written as κ=κv+κh, and the power lost in the vertical direction can thus be written as [30

30. O. Painter, J. Vučkovič, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16(2), 275–285 (1999). [CrossRef]

]

Pv(t)=κvU(t)=κvU(0)exp(κt).
(21)

Equation (21) demonstrates that the vertical decay rate κv is defined to represent the amount of vertical loss power Pv(t) . More importantly, it is shown clearly that Pv(t) decays exponentially at the total decay rate κ rather than κv. Therefore, the vertical decay rate κv should not be used to characterize the time dependence of the cavity energy and the electric field. It is thus unreasonable to assume that the line width of the propagation function is κv.

Our simulation results indicate that the line width κfof the propagation function f(rp,r0,ω) is independent on rp. We further confirm that κf is identical to the total decay rate κrather than the vertical decay rate κv. Moreover, Eq. (21) is valid regardless of what leads to the horizontal decay κh. Therefore, the identity κf=κis still valid when the horizontal decay κh is caused by other imperfection in realistic fabrications, such as the material losses or the disorder of the photonic crystal. We can thus conclude that, the pure VRS of the coupled system of the two-level QD and PC L3 cavity do not produce triple spectrum in strong coupling regime.

Acknowledgments

We thank Dr. Chao Li for useful discussions. This work was financially supported by the National Basic Research Program of China (2010CB923200), the National Natural Science Foundation of China (Grant U0934002), and the Ministry of Education of China (Grant V200801).

References and links

1.

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002). [CrossRef] [PubMed]

2.

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]

3.

C. Monroe, “Quantum information processing with atoms and photons,” Nature 416(6877), 238–246 (2002). [CrossRef] [PubMed]

4.

J. L. O'Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009). [CrossRef]

5.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432(7014), 197–200 (2004). [CrossRef] [PubMed]

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 432(7014), 200–203 (2004). [CrossRef] [PubMed]

7.

E. Peter, P. Senellart, D. Martrou, A. Lemaître, 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(6), 067401 (2005). [CrossRef] [PubMed]

8.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]

9.

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]

10.

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001). [CrossRef]

11.

R. Johne, N. A. Gippius, G. Pavlovic, D. D. Solnyshkov, I. A. Shelykh, and G. Malpuech, “Entangled photon pairs produced by a quantum dot strongly coupled to a microcavity,” Phys. Rev. Lett. 100(24), 240404 (2008). [CrossRef] [PubMed]

12.

R. Johne, N. A. Gippius, and G. Malpuech, “Entangled photons from a strongly coupled quantum dot-cavity system,” Phys. Rev. B 79(15), 155317 (2009). [CrossRef]

13.

E. del Valle, F. P. Laussy, and C. Tejedor, “Luminescence spectra of quantum dots in microcavities. II. Fermions,” Phys. Rev. B 79(23), 235326 (2009). [CrossRef]

14.

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]

15.

W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006). [CrossRef] [PubMed]

16.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98(11), 117402 (2007). [CrossRef] [PubMed]

17.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4(11), 859–863 (2008). [CrossRef]

18.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]

19.

M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Quantum dot spectroscopy using cavity quantum electrodynamics,” Phys. Rev. Lett. 101(22), 226808 (2008). [CrossRef] [PubMed]

20.

M. Yamaguchi, T. Asano, K. Kojima, and S. Noda, “Quantum electrodynamics of a nanocavity coupled with exciton complexes in a quantum dot,” Phys. Rev. B 80(15), 155326 (2009). [CrossRef]

21.

M. Yamaguchi, T. Asano, and S. Noda, “Third emission mechanism in solid-state nanocavity quantum electrodynamics,” Rep. Prog. Phys. 75(9), 096401 (2012). [CrossRef] [PubMed]

22.

S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef] [PubMed]

23.

T. Ochiai, J.-i. Inoue, and K. Sakoda, “Spontaneous emission from a two-level atom in a bisphere microcavity,” Phys. Rev. A 74(6), 063818 (2006). [CrossRef]

24.

X.-H. Wang, B.-Y. Gu, R. Wang, and H.-Q. Xu, “Decay kinetic properties of atoms in photonic crystals with absolute gaps,” Phys. Rev. Lett. 91(11), 113904 (2003). [CrossRef] [PubMed]

25.

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70(5), 053823 (2004). [CrossRef]

26.

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1993).

27.

H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A 40(10), 5516–5519 (1989). [CrossRef] [PubMed]

28.

L. C. Andreani, G. Panzarini, and J.-M. Gérard, “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys. Rev. B 60(19), 13276–13279 (1999). [CrossRef]

29.

G. Chen, Y.-C. Yu, X.-L. Zhuo, Y.-G. Huang, H. Jiang, J.-F. Liu, C.-J. Jin, and X.-H. Wang, “Ab initio determination of local coupling interaction in arbitrary nanostructures: Application to photonic crystal slabs and cavities,” Phys. Rev. B 87(19), 195138 (2013). [CrossRef]

30.

O. Painter, J. Vučkovič, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16(2), 275–285 (1999). [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: March 5, 2013
Revised Manuscript: June 21, 2013
Manuscript Accepted: September 3, 2013
Published: September 26, 2013

Citation
Yi-Cong Yu, Jing-Feng Liu, Xiao-Lu Zhuo, Gengyan Chen, Chong-Jun Jin, and Xue-Hua Wang, "Vacuum Rabi splitting in a coupled system of single quantum dot and photonic crystal cavity: effect of local and propagation Green’s functions," Opt. Express 21, 23486-23497 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23486


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science298(5597), 1372–1377 (2002). [CrossRef] [PubMed]
  2. K. J. Vahala, “Optical microcavities,” Nature424(6950), 839–846 (2003). [CrossRef] [PubMed]
  3. C. Monroe, “Quantum information processing with atoms and photons,” Nature416(6877), 238–246 (2002). [CrossRef] [PubMed]
  4. J. L. O'Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics3(12), 687–695 (2009). [CrossRef]
  5. J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature432(7014), 197–200 (2004). [CrossRef] [PubMed]
  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,” Nature432(7014), 200–203 (2004). [CrossRef] [PubMed]
  7. E. Peter, P. Senellart, D. Martrou, A. Lemaître, 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(6), 067401 (2005). [CrossRef] [PubMed]
  8. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature445(7130), 896–899 (2007). [CrossRef] [PubMed]
  9. D. Englund, A. Faraon, I. Fushman, N. Stoltz, P. Petroff, and J. Vucković, “Controlling cavity reflectivity with a single quantum dot,” Nature450(7171), 857–861 (2007). [CrossRef] [PubMed]
  10. J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys.73(3), 565–582 (2001). [CrossRef]
  11. R. Johne, N. A. Gippius, G. Pavlovic, D. D. Solnyshkov, I. A. Shelykh, and G. Malpuech, “Entangled photon pairs produced by a quantum dot strongly coupled to a microcavity,” Phys. Rev. Lett.100(24), 240404 (2008). [CrossRef] [PubMed]
  12. R. Johne, N. A. Gippius, and G. Malpuech, “Entangled photons from a strongly coupled quantum dot-cavity system,” Phys. Rev. B79(15), 155317 (2009). [CrossRef]
  13. E. del Valle, F. P. Laussy, and C. Tejedor, “Luminescence spectra of quantum dots in microcavities. II. Fermions,” Phys. Rev. B79(23), 235326 (2009). [CrossRef]
  14. 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]
  15. W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett.96(11), 117401 (2006). [CrossRef] [PubMed]
  16. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett.98(11), 117402 (2007). [CrossRef] [PubMed]
  17. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys.4(11), 859–863 (2008). [CrossRef]
  18. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics1(8), 449–458 (2007). [CrossRef]
  19. M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Quantum dot spectroscopy using cavity quantum electrodynamics,” Phys. Rev. Lett.101(22), 226808 (2008). [CrossRef] [PubMed]
  20. M. Yamaguchi, T. Asano, K. Kojima, and S. Noda, “Quantum electrodynamics of a nanocavity coupled with exciton complexes in a quantum dot,” Phys. Rev. B80(15), 155326 (2009). [CrossRef]
  21. M. Yamaguchi, T. Asano, and S. Noda, “Third emission mechanism in solid-state nanocavity quantum electrodynamics,” Rep. Prog. Phys.75(9), 096401 (2012). [CrossRef] [PubMed]
  22. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express17(5), 3322–3330 (2009). [CrossRef] [PubMed]
  23. T. Ochiai, J.-i. Inoue, and K. Sakoda, “Spontaneous emission from a two-level atom in a bisphere microcavity,” Phys. Rev. A74(6), 063818 (2006). [CrossRef]
  24. X.-H. Wang, B.-Y. Gu, R. Wang, and H.-Q. Xu, “Decay kinetic properties of atoms in photonic crystals with absolute gaps,” Phys. Rev. Lett.91(11), 113904 (2003). [CrossRef] [PubMed]
  25. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A70(5), 053823 (2004). [CrossRef]
  26. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1993).
  27. H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A40(10), 5516–5519 (1989). [CrossRef] [PubMed]
  28. L. C. Andreani, G. Panzarini, and J.-M. Gérard, “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys. Rev. B60(19), 13276–13279 (1999). [CrossRef]
  29. G. Chen, Y.-C. Yu, X.-L. Zhuo, Y.-G. Huang, H. Jiang, J.-F. Liu, C.-J. Jin, and X.-H. Wang, “Ab initio determination of local coupling interaction in arbitrary nanostructures: Application to photonic crystal slabs and cavities,” Phys. Rev. B87(19), 195138 (2013). [CrossRef]
  30. O. Painter, J. Vučkovič, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B16(2), 275–285 (1999). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited