Abstract

The photon emission into different spatial directions of a quantum dot in a micropillar cavity is theoretically analyzed. We propose two types of photon emission statistics from a single quantum light device: (i) single photon emission into the axial, strong coupling direction and a two-photon emission into the lateral, weak coupling direction, as well as (ii) the simultaneous use of both emission directions for the temporally ordered generation of two photons within a defined time-bin constituting a heralded single photon source. Our results open up exciting perspectives for solid state based quantum light sources, which can be generalized to any quantum emitter-microcavity system featuring spatially distinct emission channels between the resonator and unconfined modes.

© 2016 Optical Society of America

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 {dk,lat()}. The axial cavity modes {bm()} couple to an axial mode continuum {dk,ax()}. (b) QD four-level system: ground state |g〉, two spin-degenerated exciton states |e ↑〉 and |e ↓〉, biexcitonic state |f〉 with the binding energy EB. Peσg(Pfeσ) denotes incoherent pumping and γeσg(γfeσ) 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 bm()) couple through the DBR to an axial mode continuum (boson operators dk,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 dk,lat()). The axial cavity modes m are expanded in photon number states |nm〉 with nm photons. The QD many-particle states relevant for the emission are the ground |g〉, bright exciton |eσ〉 with spins σ and biexciton state |f〉 with the binding energy EB, 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 gxx(2)(τ) [29]:

gxx(2)(τ)=:Ix(t)Ix(t+τ):Ix(t)Ix(t).
Here, x, x′ ∈ {ax, lat} denotes the emission direction of the intensity Ix(t)=Ex()(t)Ex(+)(t) (ax: axial, lat: lateral direction). In general, the photon-photon correlation g(2)(τ) of one mode (here in a single direction) is measured in a Hanbury-Brown and Twiss experiment [30] and discriminates anti-bunched single photons (g(2)(0) < 1), bunched photons (g(2)(0) > 1) and g(2)(0) = 1 for the coherent limit. Eq. (1) defines photon-photon correlations for emitted photons into the same (x = x′) or different spatial directions (xx′). Additionally to the unidirectional axial and lateral correlations gaxax(2)(τ) and glatlat(2)(τ) with x = x′ also mixed lateral and axial correlations gaxlat(2)(τ) and glatax(2)(τ) with xx′ are studied. The spatial cross-correlation functions gaxlat(2)(τ) and glatax(2)(τ) correlate the detection of the axial and lateral emitted photon intensity after the time delay τ, normalized by the intensity of photons in axial direction and the intensity of detected photons in lateral direction.

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 Ix (t), occurring in Eq. (1), using operators Ex=kk,xdk,x, with coefficients k,x and photon annihilation (creation) operators dk,x() of the external photon continua (x = ax, lat) with mode index k. As mentioned above, the modes dk,ax(t) and dk,lat(t) interact differently with the QD-cavity system: The axial interaction Hamiltonian

HaxSR=k,mκkbmdk,ax+h.a.
describes the photon-photon coupling κk occurring via the axial cavity boundary escape of the cavity photons bm to the outside photons dk,ax.

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

HlatSR=σ,k(Mkdk,lat|geσ|+Mkdk,lat|eσf|)+h.a..
κk,m is set to the same value for both axial cavity modes and ℳk is the same for all optical transitions of the four-level system. Furthermore, the interaction between QD excitons and axial cavity modes bm() using the dipole and rotating wave approximation reads:
HelphS=σ,m(Mgeσmbm|geσ|+Meσfmbm|eσf|)+h.a.,
with the exciton-photon coupling Mgeσm and Meσfm. We assume axial cavity modes m = 1, 2 resonant to the interband ground-exciton transition ħω1 = ħω2 = ħωeħωg.

The uncoupled electron and photon Hamiltonian H0=H0S+H0R with

H0S=sωs|ss|+mωmbmbmandH0R=kxωk,xdk,xdk,x
includes electronic QD energies ħωs, photon energies ħωm of the axial cavity eigenmodes m and ħωk,x of continuum modes k for the directions x = ax, lat.

Using the total Hamiltonian H=H0S+H0R+HelphS+HaxSR+HlatSR, the observed far-field Ex()(t) in Eq. (1) is replaced by solving the Heisenberg equation of motion for dk,x(t) and applying a Markov-approximation [32]. The expectation values A=trS(AϱS) in Eq. (1) are calculated using the reduced system density matrix ϱS 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

ρx(t+τ)US(t+τ,t)(μxϱ0Sμx)
(with the initial system density matrix ϱ0S(t)=ϱ0S). The Liouville propagator US(t+τ,t) is determined by the system (QD and axial cavity modes) Liouvillian ℒS through tUS(t,t0)=LSUS(t,t0), set by the Liouville-von-Neumann equation:
tρ=i[H0S+HelphS,ρ]+LdρLSρ,
with the dissipator ℒd describing the out-coupling of photons. The initial density matrix ϱ0S for τ = 0 is obtained from tϱ0S=Lϱ0S=0 (steady state).

In Eq. (7) the system Hamiltonian H0S+HelphS between QD excitons and axial cavity modes is included for a fully dynamical description of Eq. (4), while the dissipator ℒd (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 ℒd (Lindblad super-operator) [34] reads:

Ldρ=iγi(LiρLi12(ρLiLi+LiLiρ)),
with rates γi. The outcoupling from the axial cavity eigenmodes m to the axial continuum (Lm = bm) results in photon losses γm. Additionally, radiative decay of the excitons (biexcitons) to the lateral photon mode continuum is included using Leσg=|eσg|(Lfeσ=|feσ|) with radiative lifetimes γeσg(γfeσ), cf. Fig. 1(b).

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 Peσg and Pfeσ(LPeσg=|geσ|andLPfeσ=|eσf|), 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. γpure1=2ps) these effects are not relevant. Pure dephasing γpure [36] is included using Lpure = (|e〉〈e| − |g〉〈g|) (e.g. caused by acoustical phonons).

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

gxx(2)(τ)=trS(μxρx(t+τ)μx)trS(μxρ0μx)trS(μxρ0μx),
for all combinations of both spatial emission directions x, x′ = αx, lat, in which the source μx() are either the QD dipoles directly: μlatσ(|eσg|+|feσ|) (lateral direction) or the axial cavity mode photons μaxmbm (axial direction). Note, that a strict separation of axial and lateral emission is crucial for detecting the effects described in this paper. Crosstalk between axial and lateral photons can lead to large differences in the detected correlations (see appendix 6.3 for details). Furthermore μlat and thus the g(2) functions do not depend on Mk, since it enters the dynamic only indirectly via γrad and was canceled out during the derivation in Eq. (1) in source field expansion.

Eq. (7) determines the dynamics of ρ using the matrix elements ρn1,n2,sn1,n2,sn1,n2,s|ρ|s,n1,n2 From Eqs. (78) we obtain equations of motion for the density matrix elements ρn1,n2,sn1,n2,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 EB = 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 ħωe=ħωeωe without loss of generality for the proposed applications.

Furthermore, we assume equal exciton-photon coupling elements MMgeσmMeσfm 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 γm=0.13meV(γm1=5ps). We assume an effective phonon line pure dephasing of γpure=0.33meV(γpure1=2ps) [36]. The pumping rates and the radiative decays are Peσg(feσ)P=104 [40] and γeσg(feσ)γrad=2μeV(γrad1=333ps) [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 gaxax(2)(τ) 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 gaxax(2)(τ) and glatlat(2)(τ) [Fig. 2(a)] and the spatial cross-correlations gaxlat(2)(τ) and glatax(2)(τ) [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) gaxax(2)(τ) shows anti-correlation and glatlat(2)(τ) correlation. (b) glatax(2)(τ) reaches a maximum at τmax for the emission in axial direction. The temporal width of the maximum Δτ is characterized by the time, where glatax(2)(τ) is decayed to half the difference to the uncorrelated value 1. gaxlat(2)(τ) shows anti-correlation.

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4.1. Unidirectional axial and lateral correlation

We first discuss Fig. 2(a): The axial-axial correlation gaxax(2)(τ) shows anti-correlation (gaxax(2)(0)<1) while in the lateral direction for glatlat(2)(τ) correlation occurs (glatlat(2)(0)>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 EB, the pure dephasing γpure and the excitonphoton coupling strength M (optical Stark effect) [42], resulting in the vacuum Rabi-splitting ΔER=2(M2(γmγrad)2/16) [39]. These parameters determine the resonant (bunching) and non-resonant (anti-bunching) regimes. For a biexcitonic binding energy EB = 5 meV and the pure dephasing ħγpure = 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 glatlat(2)(τ), 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: EB = 5 meV, M = 10 μeV, ħγpure = 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 gaxlat(2)(τ) and glatax(2)(τ) 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 glatax(2)(τ) reaches the maximum glatax(2)(τmax) at the time τmax, before decreasing to 1 (for t → ∞). In this case we can use the glatax(2)(τ) 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 τmax. We can here refer to emission probabilities, since we show later with glataxax(3)(τ), that almost only a single photon is present at τmax in the axial mode. The temporal width of the maximum ∆τ determines the time, where glatax(2)(τ) 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 glatax(2)(τ), gaxlat(2)(τ) always has values smaller than one, cf. Fig. 2(b) in agreement with the discussion of gaxax(2)(τ) and glatlat(2)(τ). The behavior of glatax(2)(τ) and gaxlat(2)(τ) 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 τmax 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 glatax(2)(τ) (first lateral, second axial) is suitable as light source for the preparation of time-bin entangled/heralded photon pairs. glatax(2)(0), gaxlat(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 τmax (corresponding to glatax(2)(τmax)) and a high probability within the time interval Δτ, respectively. For increasing coupling strength M the characteristic parameters glatax(2)(τmax), Δτ and τmax describing the correlation between lateral and axial emitted photon glatax(2)(τ) change, cf. Fig. 3(b). The narrower Δτ the more defined is the time for heralded photon emission. The maximal value glatax(2)(τmax) of glatax(2)(τ) 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 τmax after an lateral emission at τ = 0 increases. The maximum position τmax of glatax(2)(τmax) and the width Δτ of glatax(2)(τ) 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 glatax(2)(τmax) 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), τmax (left axis) and glatax(2)(τmax) (right axis) of glatax(2)(τ) for different exciton-photon coupling strength M. (c) Third order correlation function glataxax(3)(τ) 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 glatax(2)(τ) 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

glataxax(3)(τ)=:Ilat(t)Iax(t+τmax)Iax(t+τmax+τ)::Ilat(t)Iax(t+τmax):Iax(t)
provides information about the correlation of the single photon emission in lateral direction at time t and the two-photon emission in axial direction at time t + τmax and t + τmax + τ′. Here τ′ denotes the time between the two emitted axial photons, cf. inset of Fig. 3(c). For every delay τ′ the third order correlation function Eq. (10) shows anti-bunching ( glataxax(3)(τ)1, cf. Fig. 3(c)), indicating single photon character for the heralded emission. [44] We define a correlation time τc with the condition glataxax(3)(τc)=0.5 characterizing the range with unlikely secondary axial photon emission. The limit 0.5 is typically used as a limit to characterize single photon sources [18]. Especially, up to the correlation time τc=27 ps (for M = 30µeV) the probability for a second axial photon emission is very low and the heralded axial emissions are single photons. The time range τc depending on the exciton-photon coupling strength M. Thus, we conclude that for realistic experimental parameters, the investigated QD-micropillar system can be utilized as a HSPS. Theoretically the heralding efficiency can be investigated assuming coupling and loss constants as presented in [45]. However, these constants strongly depend on the specific experimental setup and are therefore not discussed and specified here. E.g., implementing the influence of a spectral filter in lateral direction in the numerical simulations of glatax(2)(τ) shows, that the photon, which is first emitted in lateral direction, mainly results from the biexciton-exciton decay (assuming the exciton in resonance with the cavity mode). Photons resulting from the spectrally off-resonant biexciton can be detected with much higher intensity in lateral direction compared to the axial direction, where it is blocked by the stop-band of the cavity structure.

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 Mgem=Mefm=30μeV and Mgem=Mefm=10μeV. [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 Mgem=Mefm=30μeV and Mgem=Mefm=10μeV. All signatures (bunching for glatlat(2)(τ), anti-bunching for gaxax(2)(τ) and the existence of a emission maximum for glatax(2)(τ)), 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 γrad, in particular for different quality of the used micro-resonators. In Fig. 5 the second order correlation functions gaxax(2)(τ), glatlat(2)(τ), glatax(2)(τ) and gaxlat(2)(τ) 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 glatlat(2)(0) and glatax(2)(τmax) 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 γrad1=333ps (value used in the manuscript), γrad1=1ns and γrad1=2ns: 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 gaxax(2)(τ) and glatlat(2)(τ) 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 µlat and µax a superposition of axial and lateral direction µmix = αµlat + βµax must be considered in these cases. Using this ansatz for µ in Eq. (9) the second order correlation function gmixmix(2)(τ) is investigated as:

trS(μmixρμmix)=|α|2trS(μlatρμlat)+|β|2trS(μaxρμax)+αβ*trS(μaxρμlat)+α*βtrS(μlatρμax)
In addition to pure contributions trS(μaxρμax) and trS(μlatρμlat) as before also mixed contributions of the form trS(μaxρμlat) and trS(μlatρμax) enter in trS(μmixρμmix). Note, that in contrast to the calculation of gxx(2)(τ) for gmixmix(2)(τ) 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) gmixmix(2)(τ) 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 gmixmix(2)(τ) is depicted for different coefficients α and β. For increasing α and decreasing β the correlation function gmixmix(2)(τ) interpolates from gaxax(2)(τ) to glatlat(2)(τ). Similar to gaxlat(2)(τ) in Fig. 3(a), the mixed correlation function gmixmix(2)(τ) 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 gaxax(2)(τ). 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 ρn1,n2,gn1,n2,g is given by:

tρn1,n2,gn1,n2,g=σγeσgρn1,n2,eσn1,n2,eσσPeσgρn1,n2,gn1,n2,gm=1,2[(iωm(nmnm)+γm2(nm+nm))ρn1,n2,gn1,n2,g+γmnm+1nm+1ρgg(m,1,1)+iσ(Meσgmnmρeσg)(m,0,1)Mgeσmnmρgeσ(m,1,0))],
where we introduced the compact notation ρss(1,k,j)ρn1+j,n2,sn1+k,n2,s and ρss(2,k,j)ρn1,n2+j,sn1,n2+k,s. Note, that ρss(m,k,j) still depends on the photon numbers n1, n2, n1 and n2 occurring on the left of Eq. (12). ρn1,n2,gn1,n2,g has no decay term but out-scattering due to pumping Peσg of the exciton states. The cavity loss γm causes a decay of the photon probability. Eq. (12) is driven by the radiative decay γeσg of excitonic probabilities ρn1,n2,eσn1,n2,eσ. The interaction between QD excitons and axial cavity photon modes results in the excitation of an exciton under absorption Mgeσm of an axial cavity photon and vice versa.

The equation of motion of ρn1,n2,eσn1,n2,eσ describing the exciton state (|eσ〉) occupation probability (and coherences between eσ and eσ) reads:

tρn1,n2,eσn1,n2,eσ=(γeσg+γeσg2Pfeσ+Pfeσ2)ρn1,n2,eσn1,n2,eσ+γfeσρn1,n2,fn1,n2,fδσ,σ+Peσgρn1,n2,gn1,n2,gδσ,σ+m=1,2[(iωm(nmnm)γm2(nm+nm))ρn1,n2,eσn1,n2,eσ+γmnm+1nm+1ρeσeσ(m,1,1)+iMgeσmnm+1ρgeσ(m,0,1)iMeσfmnmρeσf(m,1,0)iMeσgmnm+1ρeσg(m,0,1)+iMfeσmnmρfeσ(m,0,1)].
ρn1,n2,eσn1,n2,eσ is damped by exciton radiative decay γeσg and driven by biexciton radiative decay γfeσ. The pumping of the excitonic (biexcitonic) states leads to a increase (reduction) of the excitonic densities through the pump rates Pfeσ. The cavity loss γm 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 Mgeσm and vice versa.

The equation of motion of ground-exciton state coherence ρn1,n2,gn1,n2,eσ is:

tρn1,n2,gn1,n2,eσ=(i(ωeσωg)γeσg2σPeσg2Pfeσ2γpure)ρn1,n2,gn1,n2,eσ+m=1,2[(iωm(nmnm)γm2(nm+nm))ρn1,n2,gn1,n2,eσ+γmnm+1nm+1ρgeσ(m,1,1)iMeσfmn1ρgf(m,1,0)iMeσgmn1+1ρgg(m,1,0)+iσMeσgmn1ρeσeσ(m,0,1)].
The ground-exciton state coherence ρn1,n2,gn1,n2,eσ is damped by radiative decay, pure dephasing and pumping. Again, the exciton-photon interaction allows photon absorption and emission.
tρn1,n2,eσn1,n2,eσ=(γeσg+γeσg2Pfeσ+Pfeσ2)ρn1,n2,eσn1,n2,eσ+γfeσρn1,n2,fn1,n2,fδσ,σ+Peσgρn1,n2,gn1,n2,gδσ,σ+m=1,2[(iωm(nmnm)γm2(nm+nm))ρn1,n2,eσn1,n2,eσ+γmnm+1nm+1ρeσeσ(m,1,1)+imdgeσnm+1ρgeσ(m,0,1)imdeσfnmρeσf(m,1,0)imdeσgnm+1ρeσg(m,1,0)+imdfeσnmρfeσ(m,0,1)]
The term ρn1,n2,eσn1,n2,eσ 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 ρn1,n2,fn1,n2,eσ leads to:
tρn1,n2,fn1,n2,eσ=(γpurei(ωeωf)γeσg2σγfeσ2Pfeσ2)ρn1,n2,fn1,n2,eσ+m=1,2[(iωm(nmnm)γm2(nm+nm))ρn1,n2,fn1,n2,eσ+γmnm+1nm+1ρfeσ(m,1,1)iMeσfmnmρff(m,1,0)+iσMeσfmnm+1ρeσeσ(m,0,1)iMeσgmnm+1ρfg(m,1,0)].
The transition amplitude-like term ρn1,n2,fn1,n2,eσ describing exciton-biexciton coherences is dephased with a pure dephasing constant γpure, 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 ρn1,n2,gn1,n2,f is given by:

tρn1,n2,gn1,n2,f=(γpurei(ωfωg)σPeσg2σγfeσ2)ρn1,n2,gn1,n2,f+m=1,2[(iωm(nmnm)γm2(nm+nm))ρn1,n2,gn1,n2,f+γmnm+1nm+1ρgf(m,1,1)+iσMeσgmnmρeσf(m,0,1)iσMfeσmnm+1ρgeσ(m,1,0)]
Similar to ρn1,n2,fn1,n2,eσ describing biexciton-exciton state coherences, the contribution ρn1,n2,gn1,n2,f describes coherences between the ground and the biexcitonic state.

The equation of motion for the biexciton state (|f〉) occupation probability ρn1,n2,fn1,n2,f is given by:

tρn1,n2,fn1,n2,f=σγfeσρn1,n2,fn1,n2,f+σPeσfρn1,n2,eσn1,n2,eσ+m=1,2[(iωm(nmnm)γm2(nm+nm))ρn1,n2,fn1,n2,f+γmnm+1nm+1ρff(m,1,1)iσMfeσmnm+1ρfeσ(m,1,0)+iσMeσfmnm+1ρeσf(m,0,1)].
ρn1,n2,fn1,n2,f is damped by a radiative decay γfeσ of the biexciton and driven by a pumping Peσf of the biexciton from excitonic states. The cavity loss γm 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.

References and links

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23. We neglect lateral mode emissions caused by photon scattering processes at defects in the micropillar.

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31. Later in the derivation of the source field expression of the g(2), the continuum modes (and thus k) in lateral direction inside a solid angle towards the detector are included.

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43. τ = 0 can be clearly identified by a discontinuity for glatax(2)(τ)=gaxlat(2)(τ). Therefore, differences in the time of flight for the two modes can be compensated by identifying this discontinuity.

44. Similar to gaxax(2)(0) the value of glataxax(3)(τ) at τ′ = 0 is not exactly zero, since there are small contributions from the biexciton cascade decay despite the off-resonance of the biexciton-exciton transition to the axial cavity modes.

45. Z. Vernon, M. Liscidini, and J. E. Sipe, “No free lunch: the trade-off between heralding rate and efficiency in microresonator-based heralded single photon sources,” Opt. Lett. 41, 788–791 (2016). [CrossRef]   [PubMed]  

46. Assuming for the exciton-photon coupling elements of the excitons a difference |MgemMgem|=20μeV 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.

References

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  1. J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004).
    [Crossref] [PubMed]
  2. O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
    [Crossref] [PubMed]
  3. J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
    [Crossref] [PubMed]
  4. J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
    [Crossref]
  5. J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
    [Crossref]
  6. G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, V. Schettini, N. Gisin, S. V. Polyakov, and A. Migdall, “Improved implementation of the alicki–van ryn nonclassicality test for a single particle using Si detectors,” Phys. Rev. A 79, 044102 (2009).
    [Crossref]
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    [Crossref] [PubMed]
  8. D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
    [Crossref]
  9. T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
    [Crossref] [PubMed]
  10. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
    [Crossref] [PubMed]
  11. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
    [Crossref] [PubMed]
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    [Crossref]
  15. I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
    [Crossref] [PubMed]
  16. H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
    [Crossref] [PubMed]
  17. A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010).
    [Crossref]
  18. M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
    [Crossref]
  19. G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
    [Crossref]
  20. L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
    [Crossref]
  21. W. D. Newman, C. L. Cortes, and Z. Jacob, “Enhanced and directional single-photon emission in hyperbolic metamaterials,” J. Opt. Soc. Am. B 30, 766–775 (2013).
    [Crossref]
  22. A. Musiał, C. Hopfmann, T. Heindel, C. Gies, M. Florian, H. A. M. Leymann, A. Foerster, C. Schneider, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, “Correlations between axial and lateral emission of coupled quantum dot-micropillar cavities,” Phys. Rev. B 91, 205310 (2015).
    [Crossref]
  23. We neglect lateral mode emissions caused by photon scattering processes at defects in the micropillar.
  24. F. Hargart, M. Müller, K. Roy-Choudhury, S. L. Portalupi, C. Schneider, S. Höfling, M. Kamp, S. Hughes, and P. Michler, “Cavity-enhanced simultaneous dressing of quantum dot exciton and biexciton states,” Phys. Rev. B 93, 115308 (2016).
    [Crossref]
  25. F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
    [Crossref]
  26. Y. Ota, R. Ohta, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Vacuum rabi spectra of a single quantum emitter,” Phys. Rev. Lett. 114, 143603 (2015).
    [Crossref] [PubMed]
  27. M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
    [Crossref]
  28. M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
    [Crossref]
  29. A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
    [Crossref]
  30. R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
    [Crossref]
  31. Later in the derivation of the source field expression of the g(2), the continuum modes (and thus k) in lateral direction inside a solid angle towards the detector are included.
  32. M. Kira and S. Koch, Semiconductor Quantum Optics (Cambridge University, 2011).
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  33. F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
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  34. C. Gies, F. Jahnke, and W. W. Chow, “Photon antibunching from few quantum dots in a cavity,” Phys. Rev. A 91, 061804 (2015).
    [Crossref]
  35. Y. Mu and C. M. Savage, “One-atom lasers,” Phys. Rev. A 46, 5944–5954 (1992).
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  36. P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
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  37. S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
    [Crossref]
  38. R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
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  39. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007).
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  40. S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statisticsof a single quantum dot laser,” Opt. Express 18, 9909–9921 (2010).
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  41. S. Schumacher, J. Förstner, A. Zrenner, M. Florian, C. Gies, P. Gartner, and F. Jahnke, “Cavity-assisted emission of polarization-entangled photons from biexcitons in quantum dots with fine-structure splitting,” Opt. Express 20, 5335–5342 (2012).
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  42. T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
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  43. τ = 0 can be clearly identified by a discontinuity for glat−ax(2)(−τ)=gax−lat(2)(τ). Therefore, differences in the time of flight for the two modes can be compensated by identifying this discontinuity.
  44. Similar to gax−ax(2)(0) the value of glat−ax−ax(3)(τ′) at τ′ = 0 is not exactly zero, since there are small contributions from the biexciton cascade decay despite the off-resonance of the biexciton-exciton transition to the axial cavity modes.
  45. Z. Vernon, M. Liscidini, and J. E. Sipe, “No free lunch: the trade-off between heralding rate and efficiency in microresonator-based heralded single photon sources,” Opt. Lett. 41, 788–791 (2016).
    [Crossref] [PubMed]
  46. Assuming for the exciton-photon coupling elements of the excitons a difference |Mge↑m−Mge↓m|=20μeV 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.

2016 (2)

F. Hargart, M. Müller, K. Roy-Choudhury, S. L. Portalupi, C. Schneider, S. Höfling, M. Kamp, S. Hughes, and P. Michler, “Cavity-enhanced simultaneous dressing of quantum dot exciton and biexciton states,” Phys. Rev. B 93, 115308 (2016).
[Crossref]

Z. Vernon, M. Liscidini, and J. E. Sipe, “No free lunch: the trade-off between heralding rate and efficiency in microresonator-based heralded single photon sources,” Opt. Lett. 41, 788–791 (2016).
[Crossref] [PubMed]

2015 (4)

Y. Ota, R. Ohta, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Vacuum rabi spectra of a single quantum emitter,” Phys. Rev. Lett. 114, 143603 (2015).
[Crossref] [PubMed]

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

A. Musiał, C. Hopfmann, T. Heindel, C. Gies, M. Florian, H. A. M. Leymann, A. Foerster, C. Schneider, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, “Correlations between axial and lateral emission of coupled quantum dot-micropillar cavities,” Phys. Rev. B 91, 205310 (2015).
[Crossref]

C. Gies, F. Jahnke, and W. W. Chow, “Photon antibunching from few quantum dots in a cavity,” Phys. Rev. A 91, 061804 (2015).
[Crossref]

2014 (1)

H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
[Crossref] [PubMed]

2013 (3)

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
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W. D. Newman, C. L. Cortes, and Z. Jacob, “Enhanced and directional single-photon emission in hyperbolic metamaterials,” J. Opt. Soc. Am. B 30, 766–775 (2013).
[Crossref]

2012 (3)

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

S. Schumacher, J. Förstner, A. Zrenner, M. Florian, C. Gies, P. Gartner, and F. Jahnke, “Cavity-assisted emission of polarization-entangled photons from biexcitons in quantum dots with fine-structure splitting,” Opt. Express 20, 5335–5342 (2012).
[Crossref] [PubMed]

2011 (1)

P. K. Pathak and S. Hughes, “Coherent generation of time-bin entangled photon pairs using the biexciton cascade and cavity-assisted piecewise adiabatic passage,” Phys. Rev. B 83, 245301 (2011).
[Crossref]

2010 (3)

A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010).
[Crossref]

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
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S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statisticsof a single quantum dot laser,” Opt. Express 18, 9909–9921 (2010).
[Crossref] [PubMed]

2009 (4)

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, V. Schettini, N. Gisin, S. V. Polyakov, and A. Migdall, “Improved implementation of the alicki–van ryn nonclassicality test for a single particle using Si detectors,” Phys. Rev. A 79, 044102 (2009).
[Crossref]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
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J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
[Crossref]

2008 (2)

P. Machnikowski, “Theory of two-photon processes in quantum dots: Coherent evolution and phonon-induced dephasing,” Phys. Rev. B 78, 195320 (2008).
[Crossref]

F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
[Crossref]

2007 (1)

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

2006 (1)

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
[Crossref] [PubMed]

2005 (1)

R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
[Crossref] [PubMed]

2004 (5)

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
[Crossref]

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

I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
[Crossref] [PubMed]

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
[Crossref] [PubMed]

2003 (1)

S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
[Crossref]

2001 (1)

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

2000 (2)

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
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S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity qed,” Phys. Rev. Lett. 85, 2392–2395 (2000).
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1998 (1)

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
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1995 (2)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
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P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
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1992 (1)

Y. Mu and C. M. Savage, “One-atom lasers,” Phys. Rev. A 46, 5944–5954 (1992).
[Crossref] [PubMed]

1956 (1)

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[Crossref]

Akopian, N.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
[Crossref] [PubMed]

Albert, F.

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

Arakawa, Y.

Y. Ota, R. Ohta, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Vacuum rabi spectra of a single quantum emitter,” Phys. Rev. Lett. 114, 143603 (2015).
[Crossref] [PubMed]

Arie, I.

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

Arnold, C.

O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
[Crossref] [PubMed]

Arnold, F.

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

Aßmann, M.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Avron, J.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
[Crossref] [PubMed]

Bahgat Shehata, A.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

Bakker, M. P.

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

Barve, A. V.

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

Bayer, M.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Benyoucef, M.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
[Crossref]

Berlatzky, Y.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
[Crossref] [PubMed]

Berstermann, T.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Bimberg, D.

R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
[Crossref] [PubMed]

S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
[Crossref]

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

Borri, P.

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

Boschi, D.

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[Crossref]

Bouwmeester, D.

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

Branca, S.

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[Crossref]

Branczyk, A. M.

A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010).
[Crossref]

Brida, G.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, V. Schettini, N. Gisin, S. V. Polyakov, and A. Migdall, “Improved implementation of the alicki–van ryn nonclassicality test for a single particle using Si detectors,” Phys. Rev. A 79, 044102 (2009).
[Crossref]

Carmele, A.

A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
[Crossref]

Chow, W. W.

C. Gies, F. Jahnke, and W. W. Chow, “Photon antibunching from few quantum dots in a cavity,” Phys. Rev. A 91, 061804 (2015).
[Crossref]

Clark, A.

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Coldren, L. A.

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

Collins, M.

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Cortes, C. L.

Daniel, V.

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

De Martini, F.

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[Crossref]

de Riedmatten, H.

I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
[Crossref] [PubMed]

Degiovanni, I. P.

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Tosi, A.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

Traina, P.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

Troiani, F.

F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
[Crossref]

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[Crossref]

Ulm, G.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
[Crossref]

Ulrich, S. M.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
[Crossref]

Unold, T.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
[Crossref] [PubMed]

V., W. L.

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

van Exter, M. P.

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

Vernon, Z.

Vo, T.

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Vuckovic, J.

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

Weihs, G.

H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
[Crossref] [PubMed]

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[Crossref] [PubMed]

Weinfurter, H.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Wieck, A. D.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
[Crossref] [PubMed]

Wiersig, J.

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
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M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
[Crossref]

Woggon, U.

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

Xiong, C.

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Yamamoto, Y.

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

Zbinden, H.

I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
[Crossref] [PubMed]

Zeilinger, A.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Zheng, S.-B.

S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity qed,” Phys. Rev. Lett. 85, 2392–2395 (2000).
[Crossref] [PubMed]

Zrenner, A.

Zwinkels, J. C.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
[Crossref]

Appl. Phys. Lett. (2)

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

J. Opt. Soc. Am. B (1)

Metrologia (1)

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
[Crossref]

Nat Commun. (1)

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Nat. Commun. (2)

O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
[Crossref] [PubMed]

H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

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

Nature (4)

J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004).
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J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[Crossref]

New J. Phys. (2)

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
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A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010).
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Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (4)

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C. Gies, F. Jahnke, and W. W. Chow, “Photon antibunching from few quantum dots in a cavity,” Phys. Rev. A 91, 061804 (2015).
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Phys. Rev. B (8)

S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
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M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
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A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
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F. Hargart, M. Müller, K. Roy-Choudhury, S. L. Portalupi, C. Schneider, S. Höfling, M. Kamp, S. Hughes, and P. Michler, “Cavity-enhanced simultaneous dressing of quantum dot exciton and biexciton states,” Phys. Rev. B 93, 115308 (2016).
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A. Musiał, C. Hopfmann, T. Heindel, C. Gies, M. Florian, H. A. M. Leymann, A. Foerster, C. Schneider, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, “Correlations between axial and lateral emission of coupled quantum dot-micropillar cavities,” Phys. Rev. B 91, 205310 (2015).
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P. Machnikowski, “Theory of two-photon processes in quantum dots: Coherent evolution and phonon-induced dephasing,” Phys. Rev. B 78, 195320 (2008).
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P. K. Pathak and S. Hughes, “Coherent generation of time-bin entangled photon pairs using the biexciton cascade and cavity-assisted piecewise adiabatic passage,” Phys. Rev. B 83, 245301 (2011).
[Crossref]

Phys. Rev. Lett. (11)

I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
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S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity qed,” Phys. Rev. Lett. 85, 2392–2395 (2000).
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D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
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T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

Y. Ota, R. Ohta, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Vacuum rabi spectra of a single quantum emitter,” Phys. Rev. Lett. 114, 143603 (2015).
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D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007).
[Crossref] [PubMed]

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
[Crossref] [PubMed]

Other (6)

τ = 0 can be clearly identified by a discontinuity for glat−ax(2)(−τ)=gax−lat(2)(τ). Therefore, differences in the time of flight for the two modes can be compensated by identifying this discontinuity.

Similar to gax−ax(2)(0) the value of glat−ax−ax(3)(τ′) at τ′ = 0 is not exactly zero, since there are small contributions from the biexciton cascade decay despite the off-resonance of the biexciton-exciton transition to the axial cavity modes.

Assuming for the exciton-photon coupling elements of the excitons a difference |Mge↑m−Mge↓m|=20μeV 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.

Later in the derivation of the source field expression of the g(2), the continuum modes (and thus k) in lateral direction inside a solid angle towards the detector are included.

M. Kira and S. Koch, Semiconductor Quantum Optics (Cambridge University, 2011).
[Crossref]

We neglect lateral mode emissions caused by photon scattering processes at defects in the micropillar.

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Figures (6)

Fig. 1
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 , l a t ( ) }. The axial cavity modes { b m ( ) } couple to an axial mode continuum { d k , a x ( ) }. (b) QD four-level system: ground state |g〉, two spin-degenerated exciton states |e ↑〉 and |e ↓〉, biexcitonic state |f〉 with the binding energy EB. P e σ g ( P f e σ ) denotes incoherent pumping and γ e σ g ( γ f e σ ) radiative decay.
Fig. 2
Fig. 2 Second order correlation functions for a exciton-photon coupling strength M = 10 μeV: (a) g ax ax ( 2 ) ( τ ) shows anti-correlation and g lat lat ( 2 ) ( τ ) correlation. (b) g lat ax ( 2 ) ( τ ) reaches a maximum at τmax for the emission in axial direction. The temporal width of the maximum Δτ is characterized by the time, where g lat ax ( 2 ) ( τ ) is decayed to half the difference to the uncorrelated value 1. g ax lat ( 2 ) ( τ ) shows anti-correlation.
Fig. 3
Fig. 3 (a) HSPS-proposal based on a QD-micropillar cavity system. (b) Characteristic parameters Δτ (left axis), τmax (left axis) and g lat ax ( 2 ) ( τ m a x ) (right axis) of g lat ax ( 2 ) ( τ ) for different exciton-photon coupling strength M. (c) Third order correlation function g lat ax ax ( 3 ) ( τ ) for M = 30 µeV showing anti-bunching.
Fig. 4
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 m = M e f m = 30 μ e V and M g e m = M e f m = 10 μ e V. All signatures (bunching for g l a t l a t ( 2 ) ( τ ), anti-bunching for g a x a x ( 2 ) ( τ ) and the existence of a emission maximum for g lat ax ( 2 ) ( τ ) ), important for the applications, are still visible in (b).
Fig. 5
Fig. 5 Second order correlation functions for different radiative life times γ r a d 1 = 333 ps (value used in the manuscript), γ r a d 1 = 1 ns and γ r a d 1 = 2 ns: The qualitative behavior does not change with increasing radiative life time.
Fig. 6
Fig. 6 (a) Scheme of the measurement of the axial and lateral second order correlation function. (b) g mix mix ( 2 ) ( τ ) 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.

Equations (18)

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g x x ( 2 ) ( τ ) = : I x ( t ) I x ( t + τ ) : I x ( t ) I x ( t ) .
H a x S R = k , m κ k b m d k , a x + h . a .
H l a t S R = σ , k ( M k d k , l a t | g e σ | + M k d k , l a t | e σ f | ) + h . a . .
H e l p h S = σ , m ( M g e σ m b m | g e σ | + M e σ f m b m | e σ f | ) + h . a . ,
H 0 S = s ω s | s s | + m ω m b m b m and H 0 R = k x ω k , x d k , x d k , x
ρ x ( t + τ ) U S ( t + τ , t ) ( μ x ϱ 0 S μ x )
t ρ = i [ H 0 S + H e l p h S , ρ ] + L d ρ L S ρ ,
L d ρ = i γ i ( L i ρ L i 1 2 ( ρ L i L i + L i L i ρ ) ) ,
g x x ( 2 ) ( τ ) = t r S ( μ x ρ x ( t + τ ) μ x ) t r S ( μ x ρ 0 μ x ) t r S ( μ x ρ 0 μ x ) ,
g lat ax ax ( 3 ) ( τ ) = : I l a t ( t ) I a x ( t + τ m a x ) I a x ( t + τ m a x + τ ) : : I l a t ( t ) I a x ( t + τ m a x ) : I a x ( t )
tr S ( μ mix ρ μ mix ) = | α | 2 tr S ( μ lat ρ μ lat ) + | β | 2 tr S ( μ ax ρ μ ax ) + α β * tr S ( μ ax ρ μ lat ) + α * β tr S ( μ lat ρ μ ax )
t ρ n 1 , n 2 , g n 1 , n 2 , g = σ γ e σ g ρ n 1 , n 2 , e σ n 1 , n 2 , e σ σ P e σ g ρ n 1 , n 2 , g n 1 , n 2 , g m = 1 , 2 [ ( i ω m ( n m n m ) + γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , g n 1 , n 2 , g + γ m n m + 1 n m + 1 ρ g g ( m , 1 , 1 ) + i σ ( M e σ g m n m ρ e σ g ) ( m , 0 , 1 ) M g e σ m n m ρ g e σ ( m , 1 , 0 ) ) ] ,
t ρ n 1 , n 2 , e σ n 1 , n 2 , e σ = ( γ e σ g + γ e σ g 2 P f e σ + P f e σ 2 ) ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + γ f e σ ρ n 1 , n 2 , f n 1 , n 2 , f δ σ , σ + P e σ g ρ n 1 , n 2 , g n 1 , n 2 , g δ σ , σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + γ m n m + 1 n m + 1 ρ e σ e σ ( m , 1 , 1 ) + i M g e σ m n m + 1 ρ g e σ ( m , 0 , 1 ) i M e σ f m n m ρ e σ f ( m , 1 , 0 ) i M e σ g m n m + 1 ρ e σ g ( m , 0 , 1 ) + i M f e σ m n m ρ f e σ ( m , 0 , 1 ) ] .
t ρ n 1 , n 2 , g n 1 , n 2 , e σ = ( i ( ω e σ ω g ) γ e σ g 2 σ P e σ g 2 P f e σ 2 γ p u r e ) ρ n 1 , n 2 , g n 1 , n 2 , e σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , g n 1 , n 2 , e σ + γ m n m + 1 n m + 1 ρ g e σ ( m , 1 , 1 ) i M e σ f m n 1 ρ g f ( m , 1 , 0 ) i M e σ g m n 1 + 1 ρ g g ( m , 1 , 0 ) + i σ M e σ g m n 1 ρ e σ e σ ( m , 0 , 1 ) ] .
t ρ n 1 , n 2 , e σ n 1 , n 2 , e σ = ( γ e σ g + γ e σ g 2 P f e σ + P f e σ 2 ) ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + γ f e σ ρ n 1 , n 2 , f n 1 , n 2 , f δ σ , σ + P e σ g ρ n 1 , n 2 , g n 1 , n 2 , g δ σ , σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + γ m n m + 1 n m + 1 ρ e σ e σ ( m , 1 , 1 ) + i m d g e σ n m + 1 ρ g e σ ( m , 0 , 1 ) i m d e σ f n m ρ e σ f ( m , 1 , 0 ) i m d e σ g n m + 1 ρ e σ g ( m , 1 , 0 ) + i m d f e σ n m ρ f e σ ( m , 0 , 1 ) ]
t ρ n 1 , n 2 , f n 1 , n 2 , e σ = ( γ p u r e i ( ω e ω f ) γ e σ g 2 σ γ f e σ 2 P f e σ 2 ) ρ n 1 , n 2 , f n 1 , n 2 , e σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , f n 1 , n 2 , e σ + γ m n m + 1 n m + 1 ρ f e σ ( m , 1 , 1 ) i M e σ f m n m ρ f f ( m , 1 , 0 ) + i σ M e σ f m n m + 1 ρ e σ e σ ( m , 0 , 1 ) i M e σ g m n m + 1 ρ f g ( m , 1 , 0 ) ] .
t ρ n 1 , n 2 , g n 1 , n 2 , f = ( γ p u r e i ( ω f ω g ) σ P e σ g 2 σ γ f e σ 2 ) ρ n 1 , n 2 , g n 1 , n 2 , f + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , g n 1 , n 2 , f + γ m n m + 1 n m + 1 ρ g f ( m , 1 , 1 ) + i σ M e σ g m n m ρ e σ f ( m , 0 , 1 ) i σ M f e σ m n m + 1 ρ g e σ ( m , 1 , 0 ) ]
t ρ n 1 , n 2 , f n 1 , n 2 , f = σ γ f e σ ρ n 1 , n 2 , f n 1 , n 2 , f + σ P e σ f ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , f n 1 , n 2 , f + γ m n m + 1 n m + 1 ρ f f ( m , 1 , 1 ) i σ M f e σ m n m + 1 ρ f e σ ( m , 1 , 0 ) + i σ M e σ f m n m + 1 ρ e σ f ( m , 0 , 1 ) ] .

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