1. Introduction

Many optoelectronic applications require non-classical light sources generating single, paired or heralded photons. Quantum dots (QDs) in micropillar cavities are of high interest for fundamental research on light-matter interaction [1] and for the realization and study of bright single photon sources [2] and low-threshold microlasers [3]. Of particular interest is their application in quantum information technology, where single photon sources play a significant role in quantum communication [4], quantum metrology [5] and fundamental quantum mechanics [6]. Also quantum light sources exhibiting two-photon emission, especially entangled photon pairs, are crucial building blocks for quantum information processing protocols [7], teleportation [8], cryptography [9] or imaging [10]. A successful generation of polarization entangled photon pairs has been realized, utilizing the QD biexciton-exciton cascade [11–14]. If the two photon emission occurs during a restricted time interval (time bin), the resulting photons are time-bin entangled, e.g. realized originally by parametric down conversion [15], and more recently for coherently excited QDs [16]. Heralded single photon sources (HSPSs) generate two subsequent photons: the detection of one photon (heralding photon) announces the second photon (heralded photon), e.g. realized by parametric down conversion [17]. Compared to conventional single photon generation, HSPSs offer specific advantages. For example, the defined photon emission timing provides the basis of multiplexing [18] and often a reduction of the background photon influence [19].

In this paper, we present pronounced photon-photon correlations between light emitted from the same emitter but into non-equivalent spatial directions. Based on these results we propose a single QD-micropillar cavity device with possible applications as a source of single photons, paired photons and even as HSPS, where two photons are produced within a defined time-bin. Even if all calculations are presented for a QD-micropillar device, the results are applicable to other systems, e.g. QDs in photonic crystals [20] or metamaterials [21], featuring spatially distinct emission channels with different light-matter coupling.

2. Quantum dot micropillar model

Our investigation was inspired by a recent experiment [22] with a QD-micropillar device exhibiting photon emission into two spatially well separated directions: either through the distributed Bragg-reflector (DBR) (axial) or perpendicular to the pillar (lateral), cf. Fig. 1(a). This specific setting which allows for a simultaneous detection in axial and lateral direction was experimentally realized recently in the single-QD cavity QED regime [22]. The micropillar itself provides bound axial modes in the cavity and photon out-coupling through different two mode continua in two spatial directions (axial and lateral mode continua). This allows for an investigation of photon-photon correlations between light emitted from the same emitter but into non-equivalent spatial directions. Most important, both continua are fed by different mechanisms: direct emission from the emitter (lateral) and indirect emission from the emitter via subsequent emission process arising from the photon loss of the cavity modes (axial). [23] The collection of the laterally emitted light is experimentally challenging. However, this issue has been mastered recently in Refs. [22,24] in the desired and necessary limit of single quantum dot spectroscopy. Of course, the detection efficiency in lateral direction needs to be optimized in corresponding experimental realizations using optimization in the micropillar geometry. E.g. one could optimize the micropillar cross-section in order to realize directed lateral emission, c.f. [25]. Furthermore, in principle the underlying mechanism of our proposal can be adapted to other important cavity structures (such as non-ideal photonic wires, photonic crystal cavities or nanobeams) for the emission of light in two directions or polarizations such as in [26].

 

Fig. 1 (a) QD-micropillar cavity: a single QD is embedded between two DBRs requiring an emission in axial and lateral direction. The QD directly couples to the lateral mode continuum $\{d_{k,lat}^{(†)}\}$. The axial cavity modes $\{b_m^{(†)}\}$ couple to an axial mode continuum $\{d_{k,ax}^{(†)}\}$. (b) QD four-level system: ground state |g〉, two spin-degenerated exciton states |e ↑〉 and |e ↓〉, biexcitonic state |f〉 with the binding energy E_B. $P_{e_{\sigma }g}(P_{f{e}_{\sigma }})$ denotes incoherent pumping and ${\gamma }_{e_{\sigma }g}({\gamma }_{f{e}_{\sigma }})$ radiative decay.

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Since [22, 24] show, that cQED controlled photon emission by single quantum dots can be experimentally addressed and analyzed in lateral direction, it should be possible to perform the photon correlation measurements and to explore the realization of a heralding single photon source.

In Fig. 1(a) we introduce our notation: Two DBRs confine the cavity photons, for an ideal circular pillar cross-section typically resulting in two energy-degenerate bound axial cavity modes m = 1, 2 with ħω₁ = ħω₂ [27,28]. The axial cavity modes (boson operators $b_m^{(†)}$) couple through the DBR to an axial mode continuum (boson operators $d_{k,ax}^{(†)}$) securing the axial photon emission to free space. In contrast, the weak photon confinement in lateral direction requires a direct emission of the QD excitation into the lateral photon continuum (boson operators $d_{k,lat}^{(†)}$). The axial cavity modes m are expanded in photon number states |n_m〉 with n_m photons. The QD many-particle states relevant for the emission are the ground |g〉, bright exciton |e_σ〉 with spins σ and biexciton state |f〉 with the binding energy E_B, cf. Fig. 1(b). The system states (axial photons and QD excitations) are |s, n₁, n₂〉 with electronic QD s ∈ {g, e_↑, e_↓, f} and photon number states for both degenerate axial modes 1 and 2.

3. Second order correlation function

To study the emission statistics of the QD-micropillar structure we introduce the two-time normal ordered second order correlation function $g_{x-x^{\prime }}^{\left(2\right)}\left(\tau \right)$ [29]:

In the limit of single photons in lateral and axial modes, this can be interpreted similar to a probability, which is verified later. We express the intensity I_x (t), occurring in Eq. (1), using operators $E_x^{†}={\displaystyle {\sum }_k{ℰ}_{k,x}{d}_{k,x}^{†}}$, with coefficients ℰ_k,x and photon annihilation (creation) operators $d_{k,x}^{(†)}$ of the external photon continua (x = ax, lat) with mode index k. As mentioned above, the modes $d_{k,\mathit{ax}}^{†}\left(t\right)$ and $d_{k,\mathit{lat}}^{†}\left(t\right)$ interact differently with the QD-cavity system: The axial interaction Hamiltonian

The weak lateral optical confinement leads to a direct coupling ℳ_k between QD excitons and continuum photons: [31]

The uncoupled electron and photon Hamiltonian $H_0=H_0^S+H_0^R$ with

Using the total Hamiltonian $H=H_0^S+H_0^R+H_{el-ph}^S+H_{ax}^{SR}+H_{lat}^{SR}$, the observed far-field $E_x^{(-)}(t)$ in Eq. (1) is replaced by solving the Heisenberg equation of motion for $d_{k,x}^{†}(t)$ and applying a Markov-approximation [32]. The expectation values $〈A〉=t{r}_S(A{\mathit{\varrho }}^S)$ in Eq. (1) are calculated using the reduced system density matrix ϱ^S with traced out photon reservoirs (both external mode continua). Finally, the quantum regression theorem is applied [33] for the two-time correlation function Eq. (1). It uses a τ-dependent operator

In Eq. (7) the system Hamiltonian $H_0^S+H_{el-ph}^S$ between QD excitons and axial cavity modes is included for a fully dynamical description of Eq. (4), while the dissipator ℒ_d (Lindblad description) includes the mode continua emission: A Markovian treatment of the coupling between system (QD, cavity) and the reservoirs (external axial and lateral photon continua) is sufficient, since the emission time scale is in the range of pico- to nanoseconds. This argument applies to spontaneous emission from the QD emitter, cavity loss, dephasing and pumping. The total dissipator ℒ_d (Lindblad super-operator) [34] reads:

To excite the QD micropillar device, different pump scenarios are possible. Our proposal is based on simple non-resonant optical or electrical excitation, e.g. into the wetting layer of the quantum dots, which we describe by the Lindblad operators. We do not describe a direct optical pump, so that our scheme will be applicable for electrically driven devices.

The QD is incoherently pumped by pump rates $P_{e_{\sigma }g}$ and $P_{f{e}_{\sigma }}(L_{P_{e_{\sigma }g}}^{†}=\left|g〉〈e_{\sigma }\right|\mathrm{and}{L}_{P_{f{e}_{\sigma }}}^{†}=\left|e_{\sigma }〉〈f\right|)$, cf. Fig. 1(b). Note, there are effects changing the coherence dynamics like self-quenching [35], which can not be adequately described by a two-level scheme with a direct incoherent pumping Lindblad term used here. However, for the low pumping rates P = 10⁻⁴/ps (compared to other dephasing processes e.g. ${\gamma }_{pure}^{-1}=2\mathrm{ps}$) these effects are not relevant. Pure dephasing γ_pure [36] is included using L_pure = (|e〉〈e| − |g〉〈g|) (e.g. caused by acoustical phonons).

Using the reduced system operator ρ_x(t + τ), Eq. (1) reads:

Eq. (7) determines the dynamics of ρ using the matrix elements ${\rho }_{n_1^{\prime },n_2^{\prime },s\prime }^{n_1,n_2,s}\equiv 〈n_1,n_2,s\left|\rho \right|s\prime ,n_1^{\prime },n_2^{\prime }〉$ From Eqs. (7–8) we obtain equations of motion for the density matrix elements ${\rho }_{n_1^{\prime },n_2^{\prime },s\prime }^{n_1,n_2,s}$.

4. Numerical results

As a model system we study a self-organized InAs/GaAs QD with a single exciton energy of 1.3 eV and a biexciton binding energy of E_B = 5 meV [37].

Additional numerical results (presented in the Appendix 6.1) of the second order correlation g⁽²⁾ show that a fine-structure splitting in the order of the exciton-photon coupling strength (typically μeV-range [38]) or a different coupling of the fine-structure eigenstates to the cavity modes functions is not affecting the proposed schemes, only an additional beating caused by the fine-structure splitting and different exciton-photon coupling strengths for the two QD excitons is expected. Here, in the main text, the discussion is restricted to the case without fine-structure splitting $ħ{\omega }_{e_{\uparrow }}=ħ{\omega }_{e_{\downarrow }}\equiv {\omega }_e$ without loss of generality for the proposed applications.

Furthermore, we assume equal exciton-photon coupling elements $M\equiv M_{g{e}_{\sigma }}^m\equiv M_{e_{\sigma }f}^m$ for both cavity modes m = 1, 2 and exciton spins (polarizations) [34, 39]. A photon in the axial cavity mode m has a lifetime of $\hslash {\gamma }_m=0.13\mathrm{meV}({\gamma }_m^{-1}=5\mathrm{ps})$. We assume an effective phonon line pure dephasing of $\hslash {\gamma }_{pure}=0.33\mathrm{meV}({\gamma }_{pure}^{-1}=2\mathrm{ps})$ [36]. The pumping rates and the radiative decays are $P_{e_{\sigma }g\left(f{e}_{\sigma }\right)}\equiv P={10}^{-4}$ [40] and $\hslash {\gamma }_{e_{\sigma }g(f{e}_{\sigma })}\equiv \hslash {\gamma }_{rad}=2\mu \mathrm{eV}({\gamma }_{rad}^{-1}=333\mathrm{ps})$ [36]. Simulations presented in Appendix 6.2 show, that a variation of the radiative life time (from 0.3 ns to 2 ns) does not qualitatively change the behavior of the second order correlation functions.

Before discussing the spatial dependence of g⁽²⁾(τ), we briefly note that we reproduce results for $g_{ax-ax}^{\left(2\right)}\left(\tau \right)$ well known from literature [28,40]: A pronounced anti-bunching at τ = 0 occurs for low pumping rates P populating only a single QD exciton. For increasing pumping rates P the biexcitonic state becomes occupied and can generate two photons and enters bunching [28, 40] as long exciton and biexciton transitions are detected.

Fig. 2 shows numerical results for the unidirectional correlations $g_{ax-ax}^{\left(2\right)}\left(\tau \right)$ and $g_{lat-lat}^{\left(2\right)}\left(\tau \right)$ [Fig. 2(a)] and the spatial cross-correlations $g_{ax-lat}^{\left(2\right)}\left(\tau \right)$ and $g_{lat-ax}^{\left(2\right)}\left(\tau \right)$ [Fig. 2(b)] for a pumping strength of P = 10⁻⁴/ps and a coupling strength of M = 10 μeV [22, 39], where the system operates in the weak coupling regime.

 

Fig. 2 Second order correlation functions for a exciton-photon coupling strength M = 10 μeV: (a) $g_{\mathit{ax}-\mathit{ax}}^{(2)}(\tau )$ shows anti-correlation and $g_{\mathit{lat}-\mathit{lat}}^{(2)}(\tau )$ correlation. (b) $g_{\mathit{lat}-\mathit{ax}}^{(2)}(\tau )$ reaches a maximum at τ_max for the emission in axial direction. The temporal width of the maximum Δτ is characterized by the time, where $g_{\mathit{lat}-\mathit{ax}}^{(2)}(\tau )$ is decayed to half the difference to the uncorrelated value 1. $g_{\mathit{ax}-\mathit{lat}}^{(2)}(\tau )$ shows anti-correlation.

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4.1. Unidirectional axial and lateral correlation

We first discuss Fig. 2(a): The axial-axial correlation $g_{\mathit{ax}-\mathit{ax}}^{\left(2\right)}\left(\tau \right)$ shows anti-correlation $(g_{\mathit{ax}-\mathit{ax}}^{\left(2\right)}\left(0\right)<1)$ while in the lateral direction for $g_{\mathit{lat}-\mathit{lat}}^{\left(2\right)}\left(\tau \right)$ correlation occurs $(g_{\mathit{lat}-\mathit{lat}}^{\left(2\right)}\left(0\right)>1)$, cf. Fig. 2(a). The two regimes are explained by considering the importance of the ratio of two and one photon processes: In axial direction, the source of the emission intensity are photons coupled from the resonant exciton into the cavity mode [41]. Here, to obtain bunching, the resonance conditions between the biexciton-exciton, the exciton-ground state transition and the discrete axial cavity mode energy is not fulfilled without additional requirements: It is determined by the interplay of the biexcitonic binding energy E_B, the pure dephasing γ_pure and the excitonphoton coupling strength M (optical Stark effect) [42], resulting in the vacuum Rabi-splitting $\mathrm{\Delta }{E}_R=2\sqrt{\left(M^2-\hslash {\left({\gamma }_m-{\gamma }_{\mathit{rad}}\right)}^2/16\right)}$ [39]. These parameters determine the resonant (bunching) and non-resonant (anti-bunching) regimes. For a biexcitonic binding energy E_B = 5 meV and the pure dephasing ħγ_pure = 0.33meV, the biexciton-exciton state transition is off-resonant to the axial cavity modes. Thus, the spectral overlap is reduced and less photons are created in the axial cavity modes. In comparison to the two photon processes starting from the biexciton state, the resonant single photon process is more likely, which for axial emission results in anti-bunching.

In lateral direction, correlated photon emission occurs for $g_{\mathit{lat}-\mathit{lat}}^{\left(2\right)}\left(\tau \right)$, cf. Fig. 2(a), because of the broad lateral emission continuum. Hence, the emission process is not sensitive to the transition energy and both, excitonic and biexcitonic transitions contribute and correlated photon emission dominates.

We conclude the following: In axial direction the photon out-coupling occurs through the axial cavity modes, where resonance conditions for two photon processes are not fulfilled. In lateral direction the photons are emitted directly to the lateral mode continuum, i.e. the resonance condition is always fulfilled. For systems with a non-vanishing biexcitonic binding energy, a moderate exciton-photon coupling and pure dephasing for instance controllable by temperature (here: E_B = 5 meV, M = 10 μeV, ħγ_pure = 0.33 meV) the photon statistics can be chosen between dominant two- or one-photon processes by exploiting the emission in different directions. This way, one single device acts as a single photon source in one direction and a two-photon source in the other emission direction.

Key ingredients for all applications in this paper are: the detuning of axial photons from the single exciton to biexciton transition and the different time scales for axial emission (fast) and lateral emission (slow).

4.2. Axial and lateral cross correlation

Unlike the unidirectional emission, the spatial cross-correlations $g_{\mathit{ax}-\mathit{lat}}^{\left(2\right)}\left(\tau \right)$ and $g_{\mathit{lat}-\mathit{ax}}^{\left(2\right)}\left(\tau \right)$ characterize the emission of photons in axial and photons in lateral direction with the time delay τ. [43] As illustrated in Fig. 2(b) the correlation function $g_{\mathit{lat}-\mathit{ax}}^{\left(2\right)}\left(\tau \right)$ reaches the maximum $g_{\mathit{lat}-ax}^{\left(2\right)}\left({\tau }_{max}\right)$ at the time τ_max, before decreasing to 1 (for t → ∞). In this case we can use the $g_{lat-ax}^{\left(2\right)}\left(\tau \right)$ for judging about the conditional probability of detecting first an lateral photon and then an axial photon: since the denominator of Eq. (9) describes a constant intensity over τ, this means, that the probability of detecting a second axial photon (after a lateral photon at τ = 0) increases with τ and reaches its maximal value at the time τ_max. We can here refer to emission probabilities, since we show later with $g_{lat-ax-ax}^{\left(3\right)}\left(\tau \prime \right)$, that almost only a single photon is present at τ_max in the axial mode. The temporal width of the maximum ∆τ determines the time, where $g_{lat-ax}^{\left(2\right)}\left(\tau \right)$ is decayed to half the difference to the uncorrelated value 1. Thus, ∆τ is a measure of the time interval (time bin), in which the axial photon is emitted after the lateral photon.

Unlike $g_{lat-ax}^{(2)}(\tau )$, $g_{ax-lat}^{(2)}(\tau )$ always has values smaller than one, cf. Fig. 2(b) in agreement with the discussion of $g_{ax-ax}^{(2)}(\tau )$ and $g_{lat-lat}^{(2)}(\tau )$. The behavior of $g_{lat-ax}^{(2)}(\tau )$ and $g_{ax-lat}^{(2)}(\tau )$ can be understood as follows: Since an axial emission requires a transformation of excitons into photons of the axial cavity mode, the axial emission is temporally delayed compared to a direct photon emission from the QD in lateral direction, cf. inset of Fig. 2(a). Therefore the time τ_max is related to the time the single exciton to ground state transition needs (after emission of a lateral photon) to feed the cavity photon for axial emission after the lateral photon was emitted.

The photon pair emission indicated by $g_{lat-ax}^{(2)}(\tau )$ (first lateral, second axial) is suitable as light source for the preparation of time-bin entangled/heralded photon pairs. $g_{lat-ax}^{(2)}(0)$, $g_{ax-lat}^{(2)}(0)$ are small, but finite. Both correlations have a small nonzero contribution from off-resonant transitions. The corresponding photons have to be generated at two different well-defined times [15]. Fig. 3(a) illustrates a proposal for an heralded single photon source (HSPS): A detection of a photon in lateral direction heralds a second (final) photon in axial direction after the average delay time τ_max (corresponding to $g_{lat-ax}^{(2)}({\tau }_{max})$) and a high probability within the time interval Δτ, respectively. For increasing coupling strength M the characteristic parameters $g_{lat-ax}^{(2)}({\tau }_{max})$, Δτ and τ_max describing the correlation between lateral and axial emitted photon $g_{lat-ax}^{(2)}(\tau )$ change, cf. Fig. 3(b). The narrower Δτ the more defined is the time for heralded photon emission. The maximal value $g_{lat-ax}^{(2)}({\tau }_{max})$ of $g_{lat-ax}^{(2)}(\tau )$ strongly increases with M, cf. dashed curve in Fig. 3(b), which means that the probability for a heralded photon emission in axial direction at τ_max after an lateral emission at τ = 0 increases. The maximum position τ_max of $g_{lat-ax}^{(2)}({\tau }_{max})$ and the width Δτ of $g_{lat-ax}^{(2)}(\tau )$ decrease for increasing coupling strength M, cf. dashed-dotted and solid curve in Fig. 3(b). The quality of the HSPS rises for increasing exciton-photon coupling, since the correlated axial photon emission probability after measuring the heralding lateral photon increases and a more accurate determination of the emission time is possible. However, very high values of $g_{lat-ax}^{(2)}({\tau }_{max})$ may also indicate, that the uncorrelated processes become less probable (here axial emission decreases due to the increased detuning between axial cavity mode and single photon transition).

 

Fig. 3 (a) HSPS-proposal based on a QD-micropillar cavity system. (b) Characteristic parameters Δτ (left axis), τ_max (left axis) and $g_{\mathit{\text{lat}}-\mathit{\text{ax}}}^{(2)}({\tau }_{max})$ (right axis) of $g_{\mathit{\text{lat}}-\mathit{\text{ax}}}^{(2)}(\tau )$ for different exciton-photon coupling strength M. (c) Third order correlation function $g_{\mathit{\text{lat}}-\mathit{\text{ax}}-\mathit{\text{ax}}}^{(3)}(\tau \prime )$ for M = 30 µeV showing anti-bunching.

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Finally, to decide if the setup of Fig. 3(a) is suitable for a HSPS, one important point needs to be clarified, since $g_{\mathit{\text{lat}}-\mathit{\text{ax}}}^{(2)}(\tau )$ does not determine, whether the heralded axial photons are emitted as bunched or as the required single photons. We already discussed the g⁽²⁾ correlation function in the context of heralding a single photon and discussed emission probabilities. A discussion as a probability is only valid, if the axial photon emitted after the lateral photon is a single photon. For the HSPS-quality, the anti-bunching strength of the photons after the emission of the first photon is experimentally measured. Accordingly, to estimate the capability of the HSPS-proposal in Fig. 3(a), the third order correlation function

5. Conclusion

In summary, we presented a theory of the photon-photon correlation function characterizing the emission process in two non-equivalent spatial directions: axial (strong) and lateral (weak light-QD coupling). Based on the results of the unidirectional second order correlation function, we propose an integrated QD-micropillar device exhibiting a single photon emission in axial direction and a two-photon emission lateral direction. More important, spatially crossed correlations indicate the generation of two photons within a defined time-bin providing a time-bin heralded single photon source based on a QD-micropillar cavity system. Beside QD-micropillar devices also systems of other research areas can match the required prerequisites such as QDs in photonic crystals [20] or emitter embedded inside metamaterials [21].

6. Appendix

6.1. Influence of the fine-structure splitting

In quantum-dot micropillar devices often a fine structure splitting of the excitonic states occurs. The influence of a fine-structure splitting on the second order correlation function and the selective coupling of the fine-structure eigenstates to the cavity modes has to be investigated in order to verify the applicability of the proposals. Therefore, we performed additional calculations with a non-zero fine-structure splitting and for different exciton-photon coupling strengths for the two QD excitons to compare (Fig. 4) with the results neglecting a fine-structure splitting and with the same exciton-photon coupling strengths presented in Sec. 4. Fig. 4(b) shows the second order correlation functions for a exemplary fine-structure splitting of 20 µeV and exciton-photon coupling elements $M_{g{e}_{\uparrow }}^m=M_{e_{\downarrow }f}^m=30\mu e\mathrm{V}$ and $M_{g{e}_{\downarrow }}^m=M_{e_{\uparrow }f}^m=10\mu e\mathrm{V}$. [46] Apart from an oscillating behavior (caused by a beating of the two transition frequencies), the results are qualitatively similar to the correlation functions without a fine-structure splitting presented in Sec. 4 in Fig. 4(a). Thus, we can conclude, that the influence of a fine-structure splitting does not affect the validity of the functionality proposed in Sec. 4, but may show additional features of a beating due to a combination of the fine-structure splitting and different exciton-photon coupling strengths for the two QD excitons.

 

Fig. 4 Second order correlation functions (a) without a fine-structure splitting and for a constant exciton-photon coupling strength M = 10 µeV and (b) for a fine-structure splitting of 20 µeV and the exciton-photon coupling elements $M_{g{e}_{\uparrow }}^m=M_{e_{\downarrow }f}^m=30\mu e\mathrm{V}$ and $M_{g{e}_{\downarrow }}^m=M_{e_{\uparrow }f}^m=10\mu e\mathrm{V}$. All signatures (bunching for $g_{lat-lat}^{(2)}(\tau )$, anti-bunching for $g_{ax-ax}^{(2)}(\tau )$ and the existence of a emission maximum for $g_{\mathit{\text{lat}}-\mathit{\text{ax}}}^{(2)}(\tau ))$, important for the applications, are still visible in (b).

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6.2. Influence of the radiative life time

The influence of the radiative lifetime on the second order correlation function is discussed in this section to ensure that the proposed applications are still possible for varying radiative decay rates γ_rad, in particular for different quality of the used micro-resonators. In Fig. 5 the second order correlation functions $g_{ax-ax}^{(2)}(\tau )$, $g_{lat-lat}^{(2)}(\tau )$, $g_{lat-ax}^{(2)}(\tau )$ and $g_{ax-lat}^{(2)}(\tau )$ are depicted for different radiative life times (from 333 ps to 2 ns). The results show that the qualitative behavior does not change with increasing radiative lifetime (the values of $g_{lat-lat}^{(2)}(0)$ and $g_{lat-ax}^{(2)}({\tau }_{max})$ even increase) and thus the feasibility of the theoretical proposals presented in Sec. 4 is possible for many systems featuring different radiative lifetimes.

 

Fig. 5 Second order correlation functions for different radiative life times ${\gamma }_{rad}^{-1}=333\text{ps}$ (value used in the manuscript), ${\gamma }_{rad}^{-1}=1\text{ns}$ and ${\gamma }_{rad}^{-1}=2\text{ns}$: The qualitative behavior does not change with increasing radiative life time.

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6.3. Superposition of axial and lateral correlation function

In experiments, where correlation functions like $g_{\mathrm{ax}-\mathrm{ax}}^{(2)}(\tau )$ and $g_{\mathrm{lat}-\mathrm{lat}}^{(2)}(\tau )$ are measured, the detection of the emission only in a defined spatial direction is a demanding task since scattered light from the other directions may also arrive at the detector. In general, superpositions of photons emitted in axial and lateral direction can be the quantity, which is measured in experiments. In addition to the unintentional mixing of scattered axial and lateral emitted photons during the detection, also a controlled adjustment of the detector angle δ is possible, as depicted in Fig. 6(a). As consequence, instead of the pure lateral and axial contributions µ_lat and µ_ax a superposition of axial and lateral direction µ_mix = αµ_lat + βµ_ax must be considered in these cases. Using this ansatz for µ in Eq. (9) the second order correlation function $g_{mix-mix}^{(2)}(\tau )$ is investigated as:

(11)$${\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{mix}}^{†}\rho {\mu }_{\mathrm{mix}})={\left|\alpha \right|}^2{\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{lat}}^{†}\rho {\mu }_{\mathrm{lat}})+{\left|\beta \right|}^2{\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{ax}}^{†}\rho {\mu }_{\mathrm{ax}})+\alpha \beta *{\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{ax}}^{†}\rho {\mu }_{\mathrm{lat}})+\alpha *\beta {\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{lat}}^{†}\rho {\mu }_{\mathrm{ax}})$$

In addition to pure contributions ${\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{ax}}^{†}\rho {\mu }_{\mathrm{ax}})$ and ${\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{lat}}^{†}\rho {\mu }_{\mathrm{lat}})$ as before also mixed contributions of the form ${\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{ax}}^{†}\rho {\mu }_{\mathrm{lat}})$ and ${\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{lat}}^{†}\rho {\mu }_{\mathrm{ax}})$ enter in ${\mathrm{tr}}_{\mathrm{S}}({\mu }_{\mathrm{mix}}^{†}\rho {\mu }_{\mathrm{mix}})$. Note, that in contrast to the calculation of $g_{x-x^{\prime }}^{(2)}(\tau )$ for $g_{\mathrm{mix}-\mathrm{mix}}^{(2)}(\tau )$ the constant prefactors α and β in Eq. (11) are relevant and determine interference effects between axial and lateral direction. Therefore, α and β in Eq. (11) cannot be omitted as the prefactors like in the case with strict axial and strict lateral emission. Via the coefficients α and β all superpositions between axial and lateral directions can be studied. Choosing α = 1 and β = 0 (α = 0 and β = 1) results again in the pure lateral-lateral (axial-axial) correlation function.

 

Fig. 6 (a) Scheme of the measurement of the axial and lateral second order correlation function. (b) $g_{\mathrm{mix}-\mathrm{mix}}^{(2)}(\tau )$ for different coefficients α and β and an exciton-photon coupling strength of 10 μeV. The superposition of axial and lateral direction generates curves between the pure axial-axial and lateral-lateral correlation functions.

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In Fig. 6(b) the mixed correlation function $g_{\mathrm{mix}-\mathrm{mix}}^{(2)}(\tau )$ is depicted for different coefficients α and β. For increasing α and decreasing β the correlation function $g_{\mathrm{mix}-\mathrm{mix}}^{(2)}(\tau )$ interpolates from $g_{\mathrm{ax}-\mathrm{ax}}^{(2)}(\tau )$ to $g_{\mathrm{lat}-\mathrm{lat}}^{(2)}(\tau )$. Similar to $g_{\mathrm{ax}-\mathrm{lat}}^{(2)}(\tau )$ in Fig. 3(a), the mixed correlation function $g_{\mathrm{mix}-\mathrm{mix}}^{(2)}(\tau )$ with α ≠ 0 and β ≠ 0 shows two time scales (a fast increase and a shorter time scale to reach the value of one) resulting in a temporal maximum. Fig. 6(b) illustrates, that already a lateral contribution of α = 0.1 together with an axial contribution of β = 0.9 is sufficient to prevent anti-bunching occuring in $g_{\mathrm{ax}-\mathrm{ax}}^{(2)}(\tau )$. Only when the scattered light of the lateral direction is shielded, the anti-bunching effect in axial direction occurs. Therefore, to detect the emission only in one exact spatial direction, a careful experimental setup is necessary (elimination of the scattered light) with a defined angle. Otherwise almost any behavior in between the two cases can be observed.

6.4. Equations of motion

In the following we present a complete set of the density matrix equations. The equation of motion for the ground state (|g〉) occupation probability ${\rho }_{n_1^{\prime },n_2^{\prime },g}^{n_1,n_2,g}$ is given by:

(12)$$\begin{array}{l}{\partial }_t{\rho }_{n_1^{\prime },n_2^{\prime },g}^{n_1,n_2,g}={\displaystyle \sum _{\sigma }{\gamma }_{e_{\sigma }g}}{\rho }_{n_1^{\prime },n_2^{\prime },e_{\sigma }}^{n_1,n_2,e_{\sigma }}-{\displaystyle \sum _{\sigma }{P}_{e_{\sigma }g}}{\rho }_{n_1^{\prime },n_2^{\prime },g}^{n_1,n_2,g}\\ -{\displaystyle \sum _{m=1,2}[\left(i{\omega }_m(n_m-n_m^{\prime })+\frac{{\gamma }_m}{2}(n_m+n_m^{\prime })\right){\rho }_{n_1^{\prime },n_2^{\prime },g}^{n_1,n_2,g}}\\ +{\gamma }_m\sqrt{n_m+1}\sqrt{n_m^{\prime }+1}{\rho }_g^g(m,1,1)\\ +\frac{i}{\hslash }{\displaystyle \sum _{\sigma }\left(M_{e_{\sigma }g}^{m\ast }\sqrt{n_m^{\prime }}{\rho }_{e_{\sigma }}^g)(m,0,-1)-M_{g{e}_{\sigma }}^m\sqrt{n_m}{\rho }_g^{e_{\sigma }}(m,-1,0)\right)}],\end{array}$$

where we introduced the compact notation ${\rho }_{s^{\prime }}^s(1,k,j)\equiv {\rho }_{n_1^{\prime }+j,n_2^{\prime },s^{\prime }}^{n_1+k,n_2,s}$ and ${\rho }_{s^{\prime }}^s(2,k,j)\equiv {\rho }_{n_1^{\prime },n_2^{\prime }+j,s^{\prime }}^{n_1,n_2+k,s}$. Note, that ${\rho }_{s^{\prime }}^s(m,k,j)$ still depends on the photon numbers n₁, n₂, $n_1^{\prime }$ and $n_2^{\prime }$ occurring on the left of Eq. (12). ${\rho }_{n_1^{\prime },n_2^{\prime },g}^{n_1,n_2,g}$ has no decay term but out-scattering due to pumping $P_{e_{\sigma }g}$ of the exciton states. The cavity loss γ_m causes a decay of the photon probability. Eq. (12) is driven by the radiative decay ${\gamma }_{e_{\sigma }g}$ of excitonic probabilities ${\rho }_{n_1^{\prime },n_2^{\prime },e_{\sigma }}^{n_1,n_2,e_{\sigma }}$. The interaction between QD excitons and axial cavity photon modes results in the excitation of an exciton under absorption $M_{g{e}_{\sigma }}^m$ of an axial cavity photon and vice versa.

The equation of motion of ${\rho }_{n_1^{\prime },n_2^{\prime },e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}$ describing the exciton state (|e_σ〉) occupation probability (and coherences between e_σ and $e_{{\sigma }^{\prime }}$) reads:

(13)$$\begin{array}{l}{\partial }_t{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}=\left(-\frac{{\gamma }_{e_{\sigma }g}+{\gamma }_{e_{{\sigma }^{\prime }}g}}{2}-\frac{P_{f{e}_{\sigma }}+P_{f{e}_{{\sigma }^{\prime }}}}{2}\right){\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}\\ +{\gamma }_{f{e}_{\sigma }}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,f}{\delta }_{\sigma ,{\sigma }^{\prime }}+P_{e_{\sigma }g}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,g}{\delta }_{\sigma ,{\sigma }^{\prime }}\\ +{\displaystyle \sum _{m=1,2}[\left(-i{\omega }_m(n_m-{n^{\prime }}_m)-\frac{{\gamma }_m}{2}(n_m+{n^{\prime }}_m)\right)}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}\\ +{\gamma }_m\sqrt{n_m+1}\sqrt{{n^{\prime }}_m+1}{\rho }_{e_{{\sigma }^{\prime }}}^{e_{\sigma }}(m,1,1)\\ +\frac{i}{\hslash }{M}_{g{e}_{{\sigma }^{\prime }}}^m\sqrt{{n^{\prime }}_m+1}{\rho }_g^{e_{\sigma }}(m,0,1)-\frac{i}{\hslash }{M}_{e_{\sigma }f}^m\sqrt{n_m}{\rho }_{e_{{\sigma }^{\prime }}}^f(m,-1,0)\\ -\frac{i}{\hslash }{M}_{e_{\sigma }g}^{m\ast }\sqrt{n_m+1}{\rho }_{e_{{\sigma }^{\prime }}}^g(m,0,1)+\frac{i}{\hslash }{M}_{f{e}_{{\sigma }^{\prime }}}^{m\ast }\sqrt{{n^{\prime }}_m}{\rho }_f^{e_{\sigma }}(m,0,-1)].\end{array}$$

${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}$ is damped by exciton radiative decay ${\gamma }_{e_{\sigma }g}$ and driven by biexciton radiative decay ${\gamma }_{f{e}_{\sigma }}$. The pumping of the excitonic (biexcitonic) states leads to a increase (reduction) of the excitonic densities through the pump rates $P_{f{e}_{\sigma }}$. The cavity loss γ_m causes a photon decay. The interaction between QD excitons and axial cavity photon modes results in the excitation of an exciton under absorption of an axial cavity photon, e.g., via $M_{g{e}_{\sigma }}^m$ and vice versa.

The equation of motion of ground-exciton state coherence ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,e_{\sigma }}$ is:

(14)$$\begin{array}{l}{\partial }_t{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,e_{\sigma }}=\left(-i({\omega }_{e_{\sigma }}-{\omega }_g)-\frac{{\gamma }_{e_{\sigma }g}}{2}-{\displaystyle \sum _{{\sigma }^{\prime }}\frac{P_{e_{{\sigma }^{\prime }}g}}{2}-\frac{P_{f{e}_{\sigma }}}{2}-{\gamma }_{pure}}\right){\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,e_{\sigma }}\\ +{\displaystyle \sum _{m=1,2}[\left(-i{\omega }_m(n_m-{n^{\prime }}_m)-\frac{{\gamma }_m}{2}(n_m+{n^{\prime }}_m)\right){\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,e_{\sigma }}}\\ +{\gamma }_m\sqrt{n_m+1}\sqrt{{n^{\prime }}_m+1}{\rho }_g^{e_{\sigma }}(m,1,1)-\frac{i}{\hslash }{M}_{e_{\sigma }f}^m\sqrt{n_1}{\rho }_g^f(m,-1,0)\\ -\frac{i}{\hslash }{M}_{e_{\sigma }g}^{m\ast }\sqrt{n_1+1}{\rho }_g^g(m,1,0)+\frac{i}{\hslash }{\displaystyle \sum _{{\sigma }^{\prime }}{M}_{e_{{\sigma }^{\prime }}g}^{m\ast }}\sqrt{{n^{\prime }}_1}{\rho }_{e_{{\sigma }^{\prime }}}^{e_{\sigma }}(m,0,-1)].\end{array}$$

The ground-exciton state coherence ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,e_{\sigma }}$ is damped by radiative decay, pure dephasing and pumping. Again, the exciton-photon interaction allows photon absorption and emission.

(15)$$\begin{array}{l}{\partial }_t{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}=\left(-\frac{{\gamma }_{e_{\sigma }g}+{\gamma }_{e_{{\sigma }^{\prime }}g}}{2}-\frac{P_{f{e}_{\sigma }}+P_{f{e}_{{\sigma }^{\prime }}}}{2}\right){\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}\\ +{\gamma }_{f{e}_{\sigma }}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,f}{\delta }_{\sigma ,{\sigma }^{\prime }}+P_{e_{\sigma }g}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,g}{\delta }_{\sigma ,{\sigma }^{\prime }}\\ +{\displaystyle \sum _{m=1,2}[\left(-i{\omega }_m(n_m-{n^{\prime }}_m)-\frac{{\gamma }_m}{2}(n_m+{n^{\prime }}_m)\right)}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}\\ +{\gamma }_m\sqrt{n_m+1}\sqrt{{n^{\prime }}_m+1}{\rho }_{e_{{\sigma }^{\prime }}}^{e_{\sigma }}(m,1,1)\\ +\frac{i}{\hslash }{ℰ}_m\cdot {\mathbf{d}}_{g{e}_{{\sigma }^{\prime }}}\sqrt{{n^{\prime }}_m+1}{\rho }_g^{e_{\sigma }}(m,0,1)-\frac{i}{\hslash }{ℰ}_m\cdot {\mathbf{d}}_{e_{\sigma }f}\sqrt{n_m}{\rho }_{e_{{\sigma }^{\prime }}}^f(m,-1,0)\\ -\frac{i}{\hslash }{ℰ}_m^{\ast }\cdot {\mathbf{d}}_{e_{\sigma }g}\sqrt{n_m+1}{\rho }_{e_{{\sigma }^{\prime }}}^g(m,1,0)+\frac{i}{\hslash }{ℰ}_m^{\ast }\cdot {\mathbf{d}}_{f{e}_{{\sigma }^{\prime }}}\sqrt{{n^{\prime }}_m}{\rho }_f^{e_{\sigma }}(m,0,-1)]\end{array}$$

The term ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{{\sigma }^{\prime }}}^{n_1,n_2,e_{\sigma }}$ representing the exciton state correlation is damped by radiative decay of the exciton, driven by radiative decay of the biexciton and reduced due to the pumping into the biexcitonic state. Again, the electron-photon interaction allows photon apsorption and emission. The equation of motion for the biexciton-exciton state coherence ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,e_{\sigma }}$ leads to:

(16)$$\begin{array}{l}{\partial }_t{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,e_{\sigma }}=\left(-{\gamma }_{pure}-i({\omega }_e-{\omega }_f)-\frac{{\gamma }_{e_{\sigma }g}}{2}-{\displaystyle \sum _{{\sigma }^{\prime }}\frac{{\gamma }_{f{e}_{{\sigma }^{\prime }}}}{2}-\frac{P_{f{e}_{\sigma }}}{2}}\right){\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,e_{\sigma }}\\ +{\displaystyle \sum _{m=1,2}[\left(-i{\omega }_m(n_m-{n^{\prime }}_m)-\frac{{\gamma }_m}{2}(n_m+{n^{\prime }}_m)\right)}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,e_{\sigma }}\\ +{\gamma }_m\sqrt{n_m+1}\sqrt{{n^{\prime }}_m+1}{\rho }_f^{e_{\sigma }}(m,1,1)-\frac{i}{\hslash }{M}_{e_{\sigma }f}^m\sqrt{n_m}{\rho }_f^{f^{\prime }}(m,-1,0)\\ +\frac{i}{\hslash }{\displaystyle \sum _{{\sigma }^{\prime }}{M}_{e_{{\sigma }^{\prime }}f}^m}\sqrt{{n^{\prime }}_m+1}{\rho }_{e_{\sigma }}^{e_{{\sigma }^{\prime }}}(m,0,1)-\frac{i}{\hslash }{M}_{e_{\sigma }g}^{m\ast }\sqrt{n_m+1}{\rho }_f^g(m,1,0)].\end{array}$$

The transition amplitude-like term ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,e_{\sigma }}$ describing exciton-biexciton coherences is dephased with a pure dephasing constant γ_pure, radiative decays and pump induced dephasings. Via the coupling between cavity photons and excitons the term is driven by the exciton density and loses in the biexciton state.

The equation of motion for the ground-biexciton state coherence ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,f}$ is given by:

(17)$$\begin{array}{l}{\partial }_t{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,f}=\left(-{\gamma }_{pure}-i({\omega }_f-{\omega }_g)-{\displaystyle \sum _{\sigma }\frac{P_{e_{\sigma }g}}{2}-{\displaystyle \sum _{\sigma }\frac{{\gamma }_{f{e}_{\sigma }}}{2}}}\right){\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,f}\\ +{\displaystyle \sum _{m=1,2}[\left(-i{\omega }_m(n_m-{n^{\prime }}_m)-\frac{{\gamma }_m}{2}(n_m+{n^{\prime }}_m)\right)}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,f}\\ +{\gamma }_m\sqrt{n_m+1}\sqrt{{n^{\prime }}_m+1}{\rho }_g^f(m,1,1)\\ +\frac{i}{\hslash }{\displaystyle \sum _{\sigma }{M}_{e_{\sigma }g}^{m\ast }}\sqrt{{n^{\prime }}_m}{\rho }_{e_{\sigma }}^f(m,0,-1)-\frac{i}{\hslash }{\displaystyle \sum _{\sigma }{M}_{f{e}_{\sigma }}^{m\ast }}\sqrt{n_m+1}{\rho }_g^{e_{\sigma }}(m,1,0)]\end{array}$$

Similar to ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,e_{\sigma }}$ describing biexciton-exciton state coherences, the contribution ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,g}^{n_1,n_2,f}$ describes coherences between the ground and the biexcitonic state.

The equation of motion for the biexciton state (|f〉) occupation probability ${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,f}$ is given by:

(18)$$\begin{array}{l}{\partial }_t{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,f}=-{\displaystyle \sum _{\sigma }{\gamma }_{f{e}_{\sigma }}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,f}+}{\displaystyle \sum _{\sigma }{P}_{e_{\sigma }f}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,e_{\sigma }}^{n_1,n_2,e_{\sigma }}}\\ +{\displaystyle \sum _{m=1,2}[\left(-i{\omega }_m(n_m-{n^{\prime }}_m)-\frac{{\gamma }_m}{2}(n_m+{n^{\prime }}_m)\right)}{\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,f}\\ +{\gamma }_m\sqrt{n_m+1}\sqrt{{n^{\prime }}_m+1}{\rho }_f^f(m,1,1)\\ -\frac{i}{\hslash }{\displaystyle \sum _{\sigma }{M}_{f{e}_{\sigma }}^{m\ast }}\sqrt{n_m+1}{\rho }_f^{e_{\sigma }}(m,1,0)+\frac{i}{\hslash }{\displaystyle \sum _{\sigma }{M}_{e_{\sigma }f}^m}\sqrt{{n^{\prime }}_m+1}{\rho }_{e_{\sigma }}^f(m,0,1)].\end{array}$$

${\rho }_{{n^{\prime }}_1,{n^{\prime }}_2,f}^{n_1,n_2,f}$ is damped by a radiative decay ${\gamma }_{f{e}_{\sigma }}$ of the biexciton and driven by a pumping $P_{e_{\sigma }f}$ of the biexciton from excitonic states. The cavity loss γ_m causes a decay of the photon probability. Again, the QD excitons interact with the axial cavity photon modes via the exciton-photon coupling elements M.

Funding

Deutsche Forschungsgemeinschaft (DFG) (SFB 787 B1, SFB 787 “School of Nanophotonics”, SFB 910 B1, project Re2974/3-1).

Acknowledgments

We thank A. Musiał and C. Hopfmann for useful discussions. Financial support by Deutsche Forschungsgemeinschaft (DFG) through Sonderforschungsbereich 787 (S.K., M.R.), the School of Nanophotonics (S.K.) and Sonderforschungsbereich 910 (A.K.), the project Re2974/3-1 (S.R.) is gratefully acknowledged.

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46. Assuming for the exciton-photon coupling elements of the excitons a difference $|M_{ge\uparrow }^m-M_{ge\downarrow }^m|=20\mu \mathrm{eV}$ is probably a very extreme case and the differences are often smaller. However, we wanted to test the robustness of the effect with the extreme case.