1. Introduction

A single quantum dot (QD) in an optical cavity interacts with a radiation field through a single photon. As a result, the observed optical response of the QD is followed by photon generation according to non-Poissonian statistics, that is, single-photon generation with a number state. If the photonic environment is modified by a reduction in the effective mode volume (V_m) and an increase in the quality factor (Q), then one can realize single-photon emission with extremely high efficiency [1–4] and photon indistinguishability [5]. These light-matter interactions are fundamentally interesting and useful in the context of single-photon sources for quantum information processing.

Recently, the optical response of single excitons embedded in semiconductor nanostructures with large Q/V_m cavities, such as micropillars [1, 2, 6–8] and defect cavities in photonic crystal (PhC) membranes [9–11], has attracted much attention. Enhanced generation of single photons and strong coupling between a QD exciton and the cavity mode have already been observed. One curious feature of the optical response is that a bright cavity emissions appears under a large detuning, which is not expected from the usual spectral formulas well known for radiation decay, namely emission into radiation modes. Several experiments have now reported bright cavity mode emissions even the exciton and the cavity mode are well separated in energy. A modified theoretical analysis was recently presented by Hughes and Yao [12], who demonstrated that the the leaky cavity mode completely dominates the photon emission decay processes for planar PhC cavities, and the radiation-mode decay is basically negligible; for the measured spectrum, this manifests in a strong and robust cavity mode emission, even for large detunings, which is caused by the efficient photon feeding from the exciton to the off-resonant cavity mode. Naturally, such exciton-cavity feeding mechanisms also affect the statistics of higher-order quantum correlation effects such as antibunching phenomena. In this regard, contradicting experimental observations have been reported for the photon statistics of the exciton and the cavity emissions. For example, Press et al. [8] reported that the quantum autocorrelation [g ⁽²⁾(τ = 0)] of the exciton and the cavity emissions altogether (zero detuning, δ = 0), which appear as doublet in the spectrum due to the strong coupling, exhibit a clear anti-bunching under resonant excitation, but no anti-bunching under excitation above the barrier bandgap. They argued that when excited by above-barrier pumping, many background emitters and/or free excitons are coupled to the cavity mode whereby anti-bunching is spoiled (note that they observe anti-bunching for both auto- and cross-correlations under resonant excitation at δ ≠ 0). In contrast, Hennessy et al. [10] report totally different behavior: under excitation above the barrier bandgap in a similar way as in Ref. [9], they observe–in the strong coupling regime–clear anti-bunching in the autocorrelation at zero-detuning, but the autocorrelation function of the cavity mode alone at finite detuning (δ ≠ 0) shows no anti-bunching. They assert that these results constitute a direct manifestation of the photon blockade effect [13].

In this work, we systematically investigate the origin of the cavity mode emission in weakly coupled planar PhC cavitry systems containing one or more QDs. We perform photoluminescence (PL) measurements for the cavities with a variety of mode energy with using two excitation lasers, one for an energy well above GaAs barrier and the other with an energy just above the InAs/InGaAs QD bandgaps, respectively. We found that in the former excitation, the bright cavity mode emissions were observed irrespective of the cavity mode and detuning energies. In the latter case, however, the cavity emissions abruptly lose their intensity when nearby excitons are not sufficiently excited. These results infer that the cavity mode emission under above the barrier excitation is dominated by the radiative recombination at broad deep-level in the barrier layers. The second-order photon correlation measurements revealed in turn that the autocorrelation of the cavity mode emission with nonzero detuning (δ ≠ 0) does not exhibit photon anti-bunching. It was found, in contrast, that the autocorrelation on the exciton mode shows anti-bunching, although it leaves a correlation of ~0.57 at τ=0. Interestingly, at zero detuning where the total emission is enhanced by a factor of nine, the autocorrelation of the overlapping exciton and cavity modes shows a better anti-bunching to a τ=0 value of 0.35. As the exciton-cavity coupling becomes more pronounced, the relative weight of such deep states recombination contribution decreases, thereby the anti-bunching behavior is recovered to a better g ⁽²⁾(0), indicating that the photon statistics becomes more non-classical. These higher-order quantum correlation effects are in qualitative agreement with a master equation model that includes two excitons coupled to a leaky cavity mode.

2. Experimental procedure

A line-defect cavity with local width modulation [14] in a 2D (planar) PhC consists of a triangular lattice of air holes in a 200 nm-thick GaAs membrane containing a single InAs/InGaAs dot-in-well (DWELL) layer [15, 16]; the QD spectral density is around < 5 dots per cavity, within a detuning range of about 2 meV. The QD layer was grown by metal organic vapor phase epitaxy. The air holes of the width modulation area in the line defect (indicated by A, B and C in Fig. 1(a)), which act as a cavity, were shifted 6, 4 and 2 nm, respectively, out-wards from their original positions. The designed PhC lattice constant (a) ranged from 310 to 330 nm and the air hole radius (r) was around 80 nm. The mode volume of this cavity is about 2.0(λ/n)³ = 0.09μm³.

 

Fig. 1. (a) Schematic of the PhC system and (b) scanning electron microscope image of line-defect cavity with local width modulation. The air holes (A, B, C) were shifted 6, 4 and 2 nm outwards from their original positions, respectively. Typical PL spectra of (c) ensemble QD and (d) QDs in PhC cavity measured at 4K. The excitation used was Ar⁺ -ion laser in both cases. Labels ‘C’ and ‘X in (d) indicate the emission form the cavity mode and single exciton, respectively.

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The samples were mounted in a continuous flow helium cryostat. The cavity mode energy was tuned by using a thin gaseous film condensation technique [17], and the sample temperature was maintained at 4K. In the PL measurements, the excitation laser was focused to a spot with a ~ 1 μm diameter through an objective lens with a numerical aperture of 0.42. The PL from the ensemble of QDs without a PhC structure show the inhomogeneous broadening with center energy of 990 meV and a full width at half maximum of ≈ 40meV [Fig. 1(c)]. This peak corresponds to the QD ground-state (s-shell) emission and the large s-p shell splitting of about 90 meV were also observed [15]. The PL emission from the cavity mode and a single QD exciton were observed with linewidths Γ_c ≈ 100μeV (Q = 10 000) and Γ_x = 70 ± 30μeV, respectively. The radiation lifetime of the as-grown QD exciton was 1.3 ns (0.003 meV in half width), which corresponds to the dipole moment of 34 Debye. Here we selected the exciton-cavity coupled systems in the weak coupling regime with no vacuum Rabi splitting, as the QDs are not positioned at an anti-node position of the electromagnetic field.

To measure the second-order autocorrelation function of the emissions, a photon stream from the sample excited by a mode-locked Ti:sapphire laser (750 nm = 1.65 eV) was fed into a fiber-based Hanbury Brown-Twiss interferometer after being passed through an in-line band-pass filter with a 1 nm linewidth. The photon signal was detected by InGaAs avalanche photodiodes (APDs) operating under photon counting mode with a 5 ns gate width and a 2 MHz repetition rate, and then coincidences were obtained by using a time-to-amplitude converter equipped with a multi-channel pulse amplitude analyzer.

3. Results and theoretical discussion

3.1. Origin of cavity mode emission

To have insights into the origin of the cavity mode emissions, we first performed PL measurements for QD cavities with various r/a ratios, i.e., the individual cavities had various mode energies, as shown in Figs. 2(a) and (b). The r/a ratio is varied from 0.258 (upper) to 0.250 (lower). We used the excitation energy of 2.54 eV (Ar⁺-ion laser, 10 μW) that is well above the GaAs barrier (1.52 eV at 4 K), and 1.17 eV (YAG laser, 30 μW), which is slightly above the InGaAs quantum well (QW) bandgap (≈1.13eV at 4K). The observed cavity modes were indicated by blue and red-shaded areas. These cavity modes were distinguished from the exciton emission by the energy shift with thin-film condensation. Note that all sharp peaks except the shaded area correspond to emission from the individual excitons.

Comparing Figs. 2(a) and (b), we found the general tendency that the cavity emissions are significantly brighter when excited high in the barrier (left panel), than excited just above the QW (right panel). Another finding is that when excited at 2.54 eV, irrespective of the r/a ratio (thus of the cavity energies), the cavity mode emission intensity exhibits no apparent tendency to increase nor decrease (left panel), though the cavity emissions abruptly lose their intensity as the cavity energy decreases below about 1060 meV (right panel).

 

Fig. 2. (a) and (b): Spectra of bright cavity emissions from QD PCs with various r/a ratios, accompanied by the off-resonant exciton emissions, and measured at 4 K. The r/a ratio is varied from 0.258 (upper) to 0.250 (lower). The excitation energies are (a) 2.54 eV and (b) 1.17 eV, and each spectrum with the same r/a between these is from the same cavity. Blue- and red-shaded areas correspond to the cavity mode emissions. The red arrows in (b) indicate the cavity mode position, which is assumed by (a). The dotted lines show the offset line of zero intensity and the absolute value of the vertical scale is the same for all data. Insets magnify the top and bottom spectra in each figure. (c) PL spectrum of GaAs thin film without QD layer excited by 2.54 eV pump at 4 K. (d) Schematic relation between the excitation energy and optical response in our sample; the blue and red areas indicate the band tail of GaAs and InGaAs QW, respectively.

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3.1.1. Exciton “photon feeding” mechanism

Such characteristics of the cavity mode emission infer two possibilities for the underlying physics; one is photon feeding to the cavity mode from the excitons [12], and the other is the direct excitation (coupling) with some source of background light emitters. The essence of the photon feeding model for the PhC slab cavity is that the leaky cavity mode emission dominates the detected spectra, and it contains both the bare exciton resonance and the bare cavity resonance; thus the exciton “feeds” the leaky cavity mode, which then dominates the entire light emission characteristics. Considering an initially excited QD exciton and a detected field at R above the PC slab, the observed cavity-mode spectrum in the weak coupling regime can be written as [12]:

where g is the exciton-cavity coupling constant, Γ_c is the decay rate of the cavity mode, F(R) is a geometrical factor that depends on the detector location, and Γ^eff _x is an effective exciton decay rate which includes radiative coupling to the cavity and pure dephasing [18, 19]. Analyzing the cavity emission spectrum, the exciton-cavity feeding mechanism strongly depends on exciton broadening processes such as pure dephasing and radiative broadening. For example, the relative cavity emission increases as a function of temperature because of an increase of pure dephasing [20, 21].

In the presence of a background of coupled (n) excitons, then the spectrum

For the excitonic, spontaneous-emission spectra, which are emitted via background radiation modes (above the PhC slab light line), one has a much simpler spectral form

for each exciton i, where Γ_b is the background rate of the radiation-mode decay. In a PhC, this rate is typically orders of magnitude below the cavity emission rate, e.g., 0.05 μeV (Γ_b) compared to 0.1 meV (Γ_c) for typical PhC cavities. Although one can have different exciton emissions from different quantum dots, we will assume that the coupling rates from each exciton and QD is the same. We also note that F(R) may be different in general for the radiation mode and cavity mode collection, depending on the geometry of the post sample optics.

All of these spectral forms assume incoherent loading from a higher lying level state, with no optical source in the frequency of interest (measured spectrum). The expected lineshapes of the spectrum is thus as follows: a leaky cavity mode emission that is coupled to the initially-excited excitons, and a radiation-mode spectrum that contains the excitonic resonances. The cavity mode spectrum will have two resonances per exciton coupling (the product of two Lorentzian lineshapes, cf. Eq. (2)), that are the dressed cavity mode and the dressed exciton mode; the relative strength of this spectrum pair increases for small detunings, but in general the cavity mode can certainly have finite oscillator strength from a number of nearby excitons. The exciton absorption peaks, on the other hand, appear, in general, as only singlets – a single dressed exciton resonance. Therefore, in the absence of any excitons, within the spectral vicinity of the measured cavity spectrum, then we do not expect a significant excitation of the cavity mode, at least within the domain of an exciton feeding mechanism. Although it may seem unusual to talk about cavity-mode emissions and exciton-emissions, we note that in the strong coupling regime, it is now known that the cavity mode emission is easily the dominant contribution to the spectrum above a PhC slab [12]; thus in planar PC cavities, extra care is needed to account for the subtle and typically dominant role of the leaky cavity mode.

Turning now towards experiments, in Fig. 2(b), we observe at most 1~2 or no exciton lines around the expected energy of the cavity modes in smaller r/a, while the cavities with larger r/a have many exciton emission. The number of the exciton lines observed around 1040 meV seems to be smaller than observed around 1060 meV, because the higher energy edge of the s-shell energy distribution is nearly equal to 1040 meV in our sample. This low exciton density causes inefficient photon feeding. Moreover, in spite of few or no exciton lines, under above the barrier excitation the observed cavity mode emission with smaller r/a is much brighter than the calculated intensity (not shown here) relative to the photon feeding of the exciton with the estimated Γ′ of 70 μeV at 4 K. Thus we need to consider additional mechanisms for exciting the cavity mode, on top of the exciton cavity feeding mechanism [8, 12].

3.1.2. Effect of deep states in the GaAs barrier

Quite generally, one can assume deep lying radiative defect states as background emitters in each layer of the PhC slab. Under the condition of above GaAs barrier excitation, the cut-off energy of the deep states’ density-of-states (DOS) in the barrier is extended to smaller energies, since the cavity emissions are always observed irrespective of the lattice constants. Indeed, under the 2.54 eV pump, the PL spectrum from GaAs thin film without QDs shows broad deep-states emission at 1000–1300 meV as shown in Fig. 2(c). By contrast, the sharper cut-off energy of the deep state DOS in the QW can be assumed [22], and this drops the cavity mode emission whose energy is below the DOS. With regard to the cut-off energies differing for GaAs and QW as suggested, we expect the DOS or absorption spectrum distribution as depicted in Fig. 2(d). Since, as mentioned earlier, the deep state band tail in GaAs lies deep in the GaAs gap, likely forming a quasi-continuous spectrum, the cavity emission is observed in this measurement range with GaAs barrier excitation. Similarly, the deep state spectrum extends into the InGaAs QW gap, but the cut-off energy may be higher than in the GaAs case.

3.2. Influence of cavity emission caused by deep states on the photon statistics

3.2.1. Exciton-cavity systems with various coupling strengths

Next we clarify the influence of the background photons fed by the deep states in the barrier on the photon statistics. For this purpose, the comparison of the exciton-cavity systems with various coupling strengths–that depend on the QD position in the cavity field–is essential to understand the generation mechanism of the single photon emissions with a background emission. We selected two different exciton-cavity systems, with the cavity mode energy of below 1060 meV, while measuring the detuning characteristics under GaAs barrier excitation (2.54 eV pump, 2 μW). In Figs. 3 and 4, we display cavity-tuning from off-resonant to resonant evolutions and the exciton-cavity emissions for the systems labeled as CavI and CavII. The left panels are images of PL intensity mapping with cavity mode detuning at 4 K. The vertical axis in these images represents the number of thin-film condensation cycles, that effectively corresponds to stepwise change of the detuning. We find that the both systems show the bright cavity mode emission even under fairly-large detunings, and the distinctively different coupling characteristics to the QD exciton are exhibited between these in the spectral resonances.

 

Fig. 3. PL intensity mapping for CavI with cavity mode detuning at 4 K. The r/a of this cavity corresponds to 0.248. PL spectra at A and B (dashed lines on the intensity mapping) are also shown.

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Fig. 4. PL intensity mapping for CavII with cavity mode detuning at 4 K. The r/a of this cavity is 0.250. Note that this exciton-cavity system is different device from that shown in Figs. 2(a) and (b). PL spectra at A and B (dashed lines on the intensity mapping) are also shown.

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The integrated PL intensity for CavI, when the cavity mode and the exciton are resonant (zero detuning, dotted line B in Fig. 3), increases by a factor of < 2 compared with the off resonant case (dotted line A). This is quite reasonable when one considers that it is simply the overlap of the exciton and the cavity peaks without showing any interaction between the QD exciton and the cavity mode. The position of this QD is assumed to be near the node of the electromagnetic field inside the cavity. The integrated intensity for CavII also increases as shown in Fig. 4, when both emissions at resonance (zero detuning), but now the enhancement factor is about 9, which is then in agreement with the exciton-cavity feeding mechanism discussed earlier, and a clear signature of the cavity mode emission; this indicates that the QD is located near the anti-node of the cavity field and the exciton spontaneous emission is effectively enhanced by the Purcell effect [23], as expected from a coupled QD-cavity system.

3.2.2. Photon statistics

For the two exciton-cavity systems discussed above, under the excitation above GaAs gap, we have also measured the second-order quantum autocorrelation function of the photon intensity to investigate the influence of the background emission on the photon statistics. The autocorrelation function is given by g_m ⁽²⁾(τ) = 〈I(t)I(t + τ)〉 / 〈I(t)〉² , where m stands for cavity (c), exciton (x) or resonant coupling (c+x) according to the measurement situation. On resonance (δ = 0meV) of CavI, the autocorrelation function g _c+x ⁽²⁾(0)~0.9-1 was obtained (Fig. 5(a)), which basically yields negligible antibunching (within the noise of the detection system). The absence of the non-classical correlation is attributed to independently emitted photons from the deep states through the cavity mode, since there is no clear exciton-cavity coupling for this cavity, even at zero detuning.

 

Fig. 5. Measured second-order autocorrelation function of (a) g _c+x ⁽²⁾(τ) for CavI, and (b) g_c ⁽²⁾(τ), (c) g_x ⁽²⁾(τ), and (d) g _c+x ⁽²⁾(τ) for CavII, respectively. The detuning of (b) and (c) are both δ = 0.6 meV. Each dashed line indicates g ⁽²⁾(τ) = 1. All cases correspond to above barrier excitation (2.54 eV pump).

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For CavII, when off-resonant (δ = 0.6 meV), then anti-bunching was also not observed where g_c ⁽²⁾(0) ~1 as shown in Fig. 5(b). The absence of the non-classical correlation in this case is attributed to cavity coupling to multiple exciton feeders, as well as coupling to deep states; the former mechanism is supported by model calculations and will be discussed in more detail below. As can been seen, the off-resonant exciton with a detuning energy of δ = 0.6 meV, for CavII, exhibited partial anti-bunching with g _x ⁽²⁾(0) = 0.57 [Fig. 5(c)], indicating that quantum mechanical coupling prevails. This may, however, be similar to uncoupled QD exciton case, and thus there may be little effect from the cavity. Interestingly, at zero detuning, the autocorrelation of the overlapping exciton and cavity modes for CavII shows a better anti-bunching with g _c+x ⁽²⁾(0) = 0.35, as shown in Fig. 5(d). This is due to stronger exciton-cavity coupling in CavII compared to that of CavI. As the exciton-cavity coupling is stronger, the relative weight of the deep states recombination contribution decreases, thereby the anti-bunching behavior is recovered to a smaller g ⁽²⁾(0) value.

To help explain the basic coupling mechanisms for the various autocorrelation scenarios, we have computed example autocorrelation functions g ⁽²⁾(τ) of the cavity mode and several excitons [24], using the master equation solution for the coupled cavity (â, â^†) and exciton operators ($\widehat{\sigma }$ +,$\widehat{\sigma }$ ^-). The cavity-mode autocorrelation function g_c ⁽²⁾(τ) is obtained from 〈â^†(t)â^†(t + τ)â(t + τ)â(t)〉, which in general has resonance behavior at the bare cavity mode frequency and the bare exciton mode frequency. The exciton autocorrelation function g_x ⁽²⁾(τ) can also be calculated from 〈$\widehat{\sigma }$ +(t)$\widehat{\sigma }$ +(t + τ)$\widehat{\sigma }$ ^-(t + τ)$\widehat{\sigma }$ ^-(τ)〉, essentially using a similar model to that highlighted earlier for deriving the analytical spectra. In the model, we include one or two excitons and a leaky cavity mode, using representative decay rates and g that closely match those from experiment. The model parameters are chosen as representative examples, but the general findings are found to be fairly general.

 

Fig. 6. Calculated second-order autocorrelation function g ⁽²⁾(τ) for exciton (blue dashed curve) and cavity mode (red solid curve). The positive delay times are only shown here because of symmetry. The parameters used are exciton-cavity coupling g = 8 μeV, pure dephasing γ′ = 40 μeV, exciton radiative broadening Γ_x = 2 μeV, and repetition time of 2.4 ns. Each system includes (a) one exciton with δ = 0.6 meV, (b) two excitons with δ = 0.6 and 1.0 meV, and (c) with δ = 0 and 1.0 meV. (d) g ⁽²⁾(τ) of (b) highlight the early time dynamics, showing rapid oscillations; note there are also fast oscillations of the cavity mode autocorrelation in (a), but these are hard to see on the timescale shown, and they have little qualitative influence.

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Figure 6 shows an example of the g(²⁾(τ) simulations in the weak coupling regime with (a) one coupled exciton, and (b–c) two lower-lying coupled excitons by using the following parameters; exciton-cavity coupling g = 8 μeV, pure dephasing rate Γ′ = 40 μeV and exciton radiative broadening Γ_x = 2μeV (the effective exciton rate in a spectrum is the sum of these processes), and cavity decay Γ_c = 100 μeV. For numerical convenience, the repetition time is set as short as 2.4 ns, but this has no effect on the amount of antibunching. The autocorrelation function of the off-resonant exciton g _x1 ⁽²⁾(τ) with an example δ = 0.6 meV, as shown in Fig. 6(a), reasonably reproduces the trends of the experimental date in Fig. 5(c); namely, antibunching of around 0.5–0.6 is observed near zero delay. In this figure, since no classical background emitter is considered, g_c ⁽²⁾(τ) also shows the anti-bunching because the cavity mode emission is fed by the coupled exciton ‘x1’. In contrast, when we have another off-resonant exciton ‘x2’ with δ = 1 meV as a background emitter (which is assumed to be not coupled to x1), g_c ⁽²⁾ (τ = 0) becomes closer to 1 [Fig. 6(b)], and this in again in agreement with the measurements shown in Fig. 5(b). When one of these excitons couples resonantly to the cavity mode, g_c ⁽²⁾(㰔) (or g _x1 ⁽²⁾(τ)) recovers and reduces to about 0.4 as shown in Fig. 6(c); such behavior is also observed experimentally Fig. 5(d). This reduced antibunching is because of the significant Purcell effect, which prevails over the detrimental effect of pure dephasing and the background transitions. The absence of the non-classical correlation in CavI: g _c+x ⁽²⁾(τ = 0) [Fig. 5(a)] can be due to very small coupling factor between a single exciton and the cavity mode. Note that the leaky cavity mode becomes dressed in the presence of the exciton, and exchanges energy with the cavity mode. In Fig. 6(d), we see the oscillation in the early time dynamics of Fig. 6(b). These quantum interference effects arise from the coupling between the ‘x1-c’ and ‘x2-c’ exchange mechanisms, and the period of this oscillation corresponds to the energy difference between x1 and x2; in the frequency domain, these oscillations give rise to the two peaked spectral of a coupled exciton-cavity pair as discussed earlier. However, since the time resolution of InGaAs APD is limited to about several-hundred picoseconds, the theoretical g ⁽²⁾(τ = 0) is averaged out around τ = 0 in practical experiments. The details of this model will reported elsewhere.

The deep states are often significantly spatially localized and strongly coupled to the bulk and/or localized phonons. Due to such mechanisms, their optical spectra are characterized by phonon related bands and a broad distribution in energy. Specifically, the photon emission peak resides in a lower energy side than the absorption onset, the energy difference of which is called the Frank-Condon (Stokes) shift. The optical cross sections of the deep states are typically in a range 10^-18-10^-15 cm² [22]. Adopting a photon energy of 1.05eV, we have a corresponding range of the oscillator strength of 0.002< f <2.45 (dipole moment d of 0.45< d < 14 Debye) and the lifetime of 8.5ns< τ <8.5 μs, much longer than that of the exciton (τ_x). Assuming the defect density of 10¹⁰ cm², we roughly estimate ~ 50 defects in the cavity mode volume. As the cavity mode is resonant to the exciton, the exciton lifetime is decreased by the Purcell effect. However, the deep state emission rate may not be significantly modified, because the much smaller dipole moments give only a negligibly small coupling, and also because they do not always reside in the antinode position. With this consideration, the deep state emissions fed to the cavity mode seems to change from 9 to 90% depending on the oscillator strength. The remaining photon sources may thus be the off-resonant excitons (< 5 in the mode) and/or guided photons around the cavity within an area much larger than the cavity mode. This scenario, however, do not quantitatively explain the observed bright cavity mode emissions, which is left to future work to be clarified.

4. Summary

We have studied the origin of bright cavity mode emission with regarding the photon statistics in weakly coupled QD - PhC cavity systems. We found that when excited above the GaAs barrier, then exciton cavity feeding as expected from incoherent loading cannot solely account for the bright cavity mode emission, and thus the cavity mode emissions with nonzero detuning δ ≠ 0 are most likely dominated by radiative recombination of the deep-level defects in the barrier layers. The g ⁽²⁾(τ) measurements revealed in turn that while the autocorrelation on the cavity mode emission alone with δ≠0 does not exhibit photon anti-bunching, the exciton autocorrelation revealed a clear partial anti-bunching–although it leaves g ⁽²⁾(τ)(0) ~0.57. At zero detuning, where the total emission is enhanced by a factor of nine, the autocorrelation of the overlapping exciton and cavity modes shows a better (lower) anti-bunching of around g ⁽²⁾(τ)(0) ~0.35. As the exciton-cavity coupling is stronger, the relative weight of such background recombination contribution decreases, thereby the anti-bunching behavior is recovered to a better value, indicating that the photon statistics becomes more non-classical. These higher order quantum statistics, for coupled cavity-exciton systems, are supported by a model that includes several excitons and a leaky cavity mode.