Cavity quantum electrodynamics (QED) [1,2] 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 [3,4]. By tailoring the light-matter interactions between the quantum dot (QD) and microcavity, various striking quantum phenomena have been revealed including vacuum Rabi splitting [5–9], quantum entanglement [10–12], laser oscillation [13,14], single photon source with photon antibunching [15,16], blockade [17] and spontaneous emission control [18].

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], 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], 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 [19–21]. In contrast, another theoretical model was proposed by Hughes et al. [22] in which the vacuum fluorescence spectrum is expressed as the product of a propagation function and the local dipole spectrum [23]. 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.

Motivated by the mentioned-above controversy about the triple spectrum in the coupled system of single two-level QD and a PC cavity, we investigate the vacuum fluorescence spectra for this coupled system based upon accurate numerical simulations of the propagation and local Green’s functions. Our results reveal that the propagation function determined by the related Green’s function with the position of the detector has the same line shape as that of the multiplication factor of the PLDOS related with local Green function, i.e., they have the identical central frequency and line width. This identity is independent on the position of the detector and not influenced by the horizontal decay of the cavity. As a consequence, the pure vacuum Rabi splitting effect only results in double fluorescence spectrum. The understanding of the experimental spectral-triplet in the coupled systems need to consider 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]

Here

is the propagation function which describes the electric field intensity of at the probe point $r_p$ emitted by a classical dipole located at $r_0$ with the transition frequency $\omega$. The generalized-transverse Green function is introduced as [22,25]

where the dyadic Green’s function [26] satisfies the equation

${\epsilon }_r\left(r,\omega \right)$ is the relative dielectric constant and $c$ is the speed of light in vacuum. $P\left(r_0,\omega \right)=〈{\left[{\sigma }_{-}\left(\omega \right)\right]}^{†}{\sigma }_{-}\left(\omega \right)〉$ with ${\sigma }_{-}\left(\omega \right)={\displaystyle {\int }_0^{\infty }{\sigma }_{-}\left(t\right)e^{i\omega t}dt}$ is the local dipole or polarization spectrum. The conventional definition of the emission spectrum from a two-level exciton only consider $P\left(r_0,\omega \right)$ [27,28]. Assuming that the exciton of the QD is initially at the excited state and the field is in vacuum state, $P\left(r_0,\omega \right)$ can be obtained as

where

is the LCS and

is the level shift [24]. Here the dielectrics is lossless and ${\epsilon }_r\left(r,\omega \right)$ is real, according to Eq. (5) the imaginary part of $K\left(r_0,r_0,\omega \right)$is identical to that of $-{\left(\omega /c\right)}^{2}G\left(r_0,r_0,\omega \right)$, and then we have

where ${\Gamma }_0$ is the spontaneous emission rate of the QD in vacuum, $M\left(r_0,\omega ,\widehat{d}\right)$ is the local multiplication factor of the PLDOS and can be obtained by the ab-initio mapping [29]. The level shift can be obtained by the principle value integral of the LCS in Eq. (10).

Finally, we have

where $f\left(r_p,r_0,\omega \right)={({\omega }^2/c^2{\epsilon }_0)}^2{\left|G\left(r_p,r_0,\omega \right)\cdot d\right|}^2$.

We investigate single QD in Photonic Crystal L3 cavity following the design of [6], as depicted in Fig. 1(a). 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.

 

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 $\widehat{d}$ is along y direction. (b) The 3D schematic diagram with the locations of 15 probe points.

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The local multiplication factor of the PLDOS $M\left(r_0,\omega ,\widehat{d}\right)$ is displayed in Fig. 2(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 $M\left(r_0,\omega ,\widehat{d}\right)$ can be very well fitted by Lorentz function [29]

 

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.

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where ${\omega }_c$ is the frequency of the cavity mode, $g_c\left(r_0\right)$ is the g factor that characterizes the coupling strength between the QD and the cavity mode, $\kappa ={\omega }_c/Q$ 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 $\kappa =1.74321\times {10}^{-6}$, the Q factor is thus 139,583, and the g factor is g = 22.1GHz.

The propagation function in Eq. (4) is related to the electric dipole field determined by the Green’s function. Hughes et al. [22] proposed a simple analytical model of the generalized-transverse Green’s function $K$for planar PC cavities. They divided the decay rate of the cavity $\kappa$ into two contributions $\kappa ={\kappa }_v+{\kappa }_h$, where ${\kappa }_v$ and ${\kappa }_h$ account for vertical and horizontal decay loss, respectively. They claimed that the local Green’s function has the form $K_{\text{local}}\left(r_0,r_0,\omega \right)={\omega }_c^2{\left|E_c\left(r_0\right)\right|}^2/\left({\omega }^2-{\omega }_c^2+i\omega \kappa \right)$, while the propagation Green’s function has the form $K_{\text{prop}}\left(r_p,r_0,\omega \right)={\omega }_c^2{E}_c\left(r_p\right)E_c^{\ast }\left(r_0\right)/\left({\omega }^2-{\omega }_c^2+i\omega {\kappa }_v\right)$. For a planar PC cavity with high Q factor, $K_{\text{local}}\left(r_0,r_0,\omega \right)$ and $K_{\text{prop}}\left(r_p,r_0,\omega \right)$ tend to zero very fast with increasing $\left|\omega -{\omega }_c\right|$, then $\widehat{d}\cdot \mathrm{Im}\left[K_{\text{local}}\left(r_0,r_0,\omega \right)\right]\cdot \widehat{d}$ and ${\left|K_{\text{prop}}\left(r_p,r_0,\omega \right)\cdot d\right|}^2$ have excellent Lorentz function. Comparing with our expression in Eq. (13), this equivalently assumes that the local multiplication factor $M\left(r_0,\omega ,\widehat{d}\right)$ of the PLDOS has line width $\kappa$, while the propagation function $f\left(r_p,r_0,\omega \right)$ is of the line width ${\kappa }_v$. Moreover, it is assumed that there is considerable difference between $\kappa$ and ${\kappa }_v$. Indeed, according to these assumptions we can also obtain the triple-peak spectrum from Eq. (12) if $\kappa /{\kappa }_v>1.25$as like in [22]. However, the above-mentioned assumption has not been justified by accurate numerical simulation.

We now simulate the propagation function $f\left(r_p,r_0,\omega \right)$ 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 $f\left(r_p,r_0,\omega \right)$ are obtained by the direct numerical simulations of the Green’s function. The propagation functions $f\left(r_p,r_0,\omega \right)$ 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 ${\kappa }_f$ of the propagation function $f\left(r_p,r_0,\omega \right)$ for these points are identical to the total decay rate $\kappa$ of the cavity found from the $M\left(r_0,\omega ,\widehat{d}\right)$, 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 lists the line width ${\kappa }_f$ obtained by the fitting for all 15 probe points, and they are all identical to the total decay rate $\kappa$ 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., ${\kappa }_f=\kappa$, and that this identity is independent on the spatial location, or, in realistic experiments, the position of the detector.

Table 1. ${\kappa }_f$ factors obtained by fitting $f\left(r_p,r_0,\omega \right)$ in Eq. (13) on every probe point.

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With the simulation data of the PLDOS and the propagation function, we are able to calculate the vacuum fluorescence spectrum $S\left(\omega \right)$ 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) 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 $P\left(r_0,\omega \right)$, 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 $S\left(\omega \right)$ 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] 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.

 

Fig. 3 (a) The local dipole spectra $P\left(r_0,\omega \right)$ of the Sample#1. (b) The vacuum fluorescence spectra $S\left(\omega \right)$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).

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Although our results in Figs. 2 and 3 support ${\kappa }_f=\kappa$, this high Q L3 cavity may be a special structure with horizontal decay ${\kappa }_h=0$ so that it also satisfies ${\kappa }_f={\kappa }_v$. 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 displays the decay rate $\kappa$ obtained from the multiplication factor $M\left(r_0,\omega ,\widehat{d}\right)$, 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 $\kappa$ 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 $\kappa$ 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], in which they found that the horizontal decay ${\kappa }_h$ decreases exponentially with the number of the air hole layers and the vertical decay ${\kappa }_v$ 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 ${\kappa }_h$ is decreased well below ${\kappa }_v$, and the total decay rate $\kappa$ asymptotically approaches ${\kappa }_v$.

 

Fig. 4 The decay rate $\kappa$from the multiplication factor $M\left(r_0,\omega ,\widehat{d}\right)$ with the corresponding Q factor, and the line width ${\kappa }_f$ of the propagation functions $f\left(r_p,r_0,\omega \right)$, for different numbers of air hole layers in the L3 cavity.

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The line width ${\kappa }_f$ of the propagation functions $f\left(r_p,r_0,\omega \right)$ 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 ${\kappa }_f$ is still in excellent agreement with $\kappa$ rather than ${\kappa }_v$. These results confirm that a remarkable horizontal decay does not influence the identity between the line width of $M\left(r_0,\omega ,\widehat{d}\right)$ and that of $f\left(r_p,r_0,\omega \right)$ in the PC L3 cavity structures, and the key assumption proposed in [22] that supports the triple-peak spectrum is invalid.

Figure 5 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.

 

Fig. 5 (a) The local dipole spectra $P\left(r_0,\omega \right)$and (b) vacuum fluorescence spectra $S\left(\omega \right)$ 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 ${\lambda }_c$ of each cavity.

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In order to further confirm the irrationality of the assumption in [22], we consider a classical dipole source

located inside the cavity. The electric field induced by this dipole can be calculated by the Green’s function as

where

and the electric field outside the cavity can be written as

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].

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

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

This power can be separated into two vertical and horizontal parts as $P\left(t\right)=P_v\left(t\right)+P_h\left(t\right)$, the decay rate can also be written as $\kappa ={\kappa }_v+{\kappa }_h$, and the power lost in the vertical direction can thus be written as [30]

Equation (21) demonstrates that the vertical decay rate ${\kappa }_v$ is defined to represent the amount of vertical loss power $P_v\left(t\right)$ . More importantly, it is shown clearly that $P_v\left(t\right)$ decays exponentially at the total decay rate $\kappa$ rather than ${\kappa }_v$. Therefore, the vertical decay rate ${\kappa }_v$ 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 ${\kappa }_v$.

Our simulation results indicate that the line width ${\kappa }_f$of the propagation function $f\left(r_p,r_0,\omega \right)$ is independent on $r_p$. We further confirm that ${\kappa }_f$ is identical to the total decay rate $\kappa$rather than the vertical decay rate ${\kappa }_v$. Moreover, Eq. (21) is valid regardless of what leads to the horizontal decay ${\kappa }_h$. Therefore, the identity ${\kappa }_f=\kappa$is still valid when the horizontal decay ${\kappa }_h$ 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.

In summary, we have studied the vacuum fluorescence spectrum from the coupled system of a single two-level QD and PC L3 cavity based on accurate numerical simulations of the propagation and local Green’s functions. It is found that the propagation function related to the position of the detector has a Lorentz line shape, whose central frequency and line width are identical to those of the PLDOS. Moreover, this identity is independent on the position of the detector outside the slab of the PC cavity, and is not influenced by the horizontal decay ${\kappa }_h$. Based on the simulation results and brief theoretical analysis, we have confirmed that the assumption proposed in [22] about the propagation and local optical Green’s function is invalid. Therefore, the pure VRS leads to only double peaks rather than triplet in the vacuum fluorescence spectrum in strong coupling regime. In order to understand the spectral triplet observed in the experiments, it is necessary to consider other physics mechanism beyond the VRS. Our work should stimulate further research interest on this topic.