II.3 Phonon-mediated transitions

Polarization-entangled twin photons from two-photon quantum-dot emission

Abstract

Semiconductor quantum dots are promising sources for polarization-entangled photons. As an alternative to the usual cascaded biexciton-exciton emission, direct two-photon emission from the biexciton can be used. With a high-quality optical resonator tuned to half the biexciton energy, a large proportion of the photons can be steered into the two-photon emission channel. In this case the degree of polarization entanglement is inherently insensitive to the exciton fine-structure splitting. In the present work we analyze the biexciton emission with particular emphasis on the influence of coupling of the quantum-dot cavity system to its environment. Especially for a high-quality cavity, the coupling to the surrounding semiconductor material can open up additional phonon-assisted decay channels. Our analysis demonstrates that with the cavity tuned to half the biexciton energy, the potentially detrimental influence of the phonons on the polarization entanglement is strongly suppressed – high degrees of entanglement can still be achieved. We further discuss spectral properties and statistics of the emitted twin photons.

pacs:
Valid PACS appear here

I Introduction

Photon pairs with a high degree of polarization entanglement are a key ingredient to a number of quantum information protocols.Gisin et al. (2002); Knill et al. (2001); Bouwmeester et al. (1997); Gisin and Thew (2007) Semiconductor quantum dots have proven their capabilities for on-demand generation of individual pairs of such polarization-entangled photons.Müller et al. (2014) However, in most high-quality semiconductor quantum-dot structures, the fine-structure splitting between exciton levels limits the achievable degree of polarization entanglement.Troiani et al. (2006) Recent achievements show that this obstacle can be overcome by applying strain to the quantum dot Trotta et al. (2015); Zhang et al. (2015); Chen et al. (2016) or by selecting those quantum dots on a given sample that possess a particularly small fine-structure splitting. As an alternative approach, it was shown theoretically that a high degree of polarization-entanglement can also be obtained by using a direct two-photon emission process from the quantum-dot biexciton inside an optical cavity – independent of the fine-structure splitting.Schumacher et al. (2012)

In quantum-dot-cavity systems where strong coupling between the electronic transitions and the optical cavity mode is realized, two photon transitions gain importance.Heinze et al. (2015); Munoz et al. (2015) Starting with the proposal del Valle et al. (2011) and experimental demonstration Ota et al. (2011) of cavity enhanced two-photon emission, first potential applications have been analyzed Thoma et al. (2016); Kuhn et al. (2016). Also interaction with phonons is particularly important in cavity systems.Glässl et al. (2013); Lüker et al. (2012); Quilter et al. (2015); Ardelt et al. (2014); Bounouar et al. (2015) Phonon mediated processes have been shown to contribute to an asymmetric spectral line-broadeningRoy and Hughes (2012) and phonon-assisted cavity feedingHohenester (2010). The precise influence of phonon-mediated processes depends on the cavity detuning and phonon-bath properties.Roy and Hughes (2011a) Also the two-photon emission from the biexciton may be influenced by this mechanism. Here, the question arises if longitudinal acoustic (LA) phonons can affect the system dynamics in such a way that the degree of entanglement of twin photons emitted from the biexciton is significantly reduced.

In this paper we analyze the effect of phonon mediated cavity (bi)exciton coupling and present a detailed analysis based on a Born-Markov approximationWilson-Rae and Imamoğlu (2002); Roy and Hughes (2012) for the coupling to the phonon bath. In the emission scheme studied, the cavity is tuned near half the biexciton energy. In this case phonon-assisted cavity feeding primarily occurs through phonon absorption, which is significantly suppressed at low temperatures. Our results show that even including phonon-assisted processes, high degrees of polarization entanglement can still be achieved from a direct two-photon emission from the biexciton. We also analyze spectral properties and statistics of the emitted photons.

Ii Theory

In this section an introduction to the theoretical description of the quantum-dot cavity system is given. This includes the general formulation of the theory, the formulation of a master equation describing the system dynamics, the system’s coupling to the environment including phonon-assisted processes, and a short discussion of the two-photon density matrix needed to study the quantum properties of the emitted photons.

ii.1 The Quantum-Dot Cavity System

The present work is focussed on the photon emission from the biexciton in a semiconductor quantum dot inside an optical resonator. To analyze this process theoretically, we model the quantum-dot cavity system as a system of four electronic configurations and two orthogonal photon cavity modes. The states of the system are schematically depicted in Fig. 1. The electronic configurations included are the biexciton (B), the two excitons (, ), and the ground state (G). The two orthogonal cavity modes are at energies and . The biexciton binding energy is and the exciton levels are split by a fine-structure splitting . The Hamiltonian of this system is given by:Carmele et al. (2010)

 HS=EG|G⟩⟨G|+EH|XH⟩⟨XH|+EV|XV⟩⟨XV|+EB|B⟩⟨B|+∑i=H,Vℏωib†ibi+[X+h.c.]. (1)

It contains the free energies of the electronic states of the quantum dot and the photons inside the cavity. The photon creation and annihilation operators are denoted by and , with , respectively. The interaction part with coupling constant is given by

 X=−g[(|XV⟩⟨G|bV−|B⟩⟨XV|bV)]−g[(|XH⟩⟨G|bH+|B⟩⟨XH|bH)]. (2)

and represents the coupling of the cavity-modes and the electronic system and induces transitions between electronic state while emitting or absorbing photons.

ii.2 The Master Equation

To analyze the decay from the biexciton configuration including coupling to the system environment we use a density matrix theory. In this approach the system dynamic is given by the following Master equation:

 ∂∂tρs=−iℏ[H,ρs]+Lcavity(ρs)+Lpure(ρs)+Lphonon(ρs). (3)

The coupling of the system to its environment and corresponding dissipative losses are included via the Lindblad terms .Lindblad (1976) These include emission of photons from the cavity on a time scale ,

 Lcavity(ρs)=κ2∑i=H,V(2biρsb†i−b†ibiρs−ρsb†ibi). (4)

Through interaction with the environment, electronic coherences experience a loss of phase information which is called pure dephasing Schumacher et al. (2012); Troiani et al. (2006)

 Lpure(ρs)=−12∑χ,χ′;χ≠χ′γχ,χ′pure|χ⟩⟨χ|ρs|χ′⟩⟨χ′|. (5)

At low temperatures, pure dephasing can be as low as a few Laucht et al. (2009). Here we assume for all electronic coherences. In high-quality cavities emission into off-resonant photon modes is strongly reduced but can not be prevented completely. These radiative losses can be included through a term giving rise to changes in electronic populations but not generating photons in the system cavity modes:

 Lrad(ρs)=−γrad⟨B⟩22∑i=XH,XV(L|G⟩⟨i|+L|i⟩⟨B|)(ρs), (6)

with . The renormalization factor accounts for the reduced radiative emission due to phonon interaction.Roy and Hughes (2011a) The decay constant can be on the order of a few del Valle et al. (2010); Laucht et al. (2009). For the results shown below we have checked that radiative losses only slightly affects the quantum efficiency of the photon emission into the cavity.del Valle et al. (2011)

In the numerical evaluation, we solve the master equation, Eq. (3), (for the phonon-assisted part see Eq. (12) below). The initial condition is that the system is in the biexciton state and the cavity is empty. This state can be experimentally prepared by non-degenerate two-color two-photon Rabi floppingStufler et al. (2006); Boyle et al. (2010) and it has recently been demonstrated that the influence of phonons can be beneficial to deterministically prepare the biexciton state.Bounouar et al. (2015) The Fock space is spanned by the joint spaces of the electronic states and the space of the photons . As expected from energy conservation, in the numerical evaluation full convergence is obtained by inclusion of photon states with up to photons per cavity mode. The expectation value of any operator is calculated by taking the trace with the density operator; for example the biexciton population is calculated as .Florian et al. (2013a) If not otherwise noted, below we use the following system parameters: , , , and . Here, we define the cavity detuning with being the degenerate two-photon resonance. In Fig. 2 results are shown for different frequencies of the cavity modes. Depending on the cavity frequency different decay channels from the biexciton are enhanced or suppressed, respectively. When the cavity is tuned to the biexciton-to-exciton transitions, pronounced Rabi oscillations are visible as the population relaxes to the exciton configurations. Further decay to the ground state is suppressed however (not shown in Fig. 2); with the finite biexciton binding energy of the exciton-to-ground state transitions are off-resonant to the cavity modes. When the cavity is resonant with the exciton-to-ground state transitions, no resonant emission from the biexciton can occur such that the decay of biexciton population is very slow. Tuning the cavity to half the biexciton energy, resonant with the higher-order degenerate two-photon biexciton-to-ground state transition, efficient relaxation to the ground state is observed while emitting two photons at once (and without creating exciton population). Below we investigate this latter process in more detail and analyze the properties of the emitted photon pair.

ii.3 Phonon-Mediated Transitions

Phonon effects are suspected to have an important influence on the photons generated in a quantum-dot cavity system with sufficiently strong coupling. Here, we investigate near resonant excitation only such that in the self-assembled InAs/GaAs quantum dots the main contribution is expected to stem from interactions with LA phonons (LO phonons are sufficiently well separated in energy).Roy and Hughes (2011a) In the following we also include this coupling to take into account phonon-assisted cavity feeding.Hohenester (2010) To include these interactions we follow the analysis by Roy and Hughes for a two-level system coupled to a quantized optical modeRoy and Hughes (2012, 2011a) and extend it to a four-level systemHargart et al. (2016); Hohenester (2010) with two cavity modes as described in Sec. II.1 above.

The Hamiltonian including the LA phonons and electron-phonon interaction is then given by

 H = HS+Hphonon+HQD−phonon (7) = HS+∑qℏωqa†qaq+∑i,q|χi⟩⟨χi|λiq(a†q+aq),

with from Eq. (1), and electronic configurations and phonon creation (annihilation) operators (). The energy of the phonons in mode is and the coupling to the quantum dot state is given by . By transformation into the polaron frame the explicit appearance of the phonons can be removed from the Hamiltonian.Florian et al. (2013b); Roy and Hughes (2011a); Hohenester (2010) The transformation into the polaron frame is done following Ref. Hargart et al., 2016 with and

 A=∑i,qλiqωq|χi⟩⟨χi|(a†q−aq). (8)

The transformed quantum-dot cavity Hamiltonian then reads

 H′ = ∑i~Ei|χi⟩⟨χi|+∑jℏωjb†jbj+⟨B⟩Xg (9) + ∑qℏωqa†qaq+ζgXg−ζuXu.

The cavity and quantum - dot part of this Hamiltonian has the same structure as Eq. (1), but with a renormalized electron-photon coupling constant .Wilson-Rae and Imamoğlu (2002) The polaron shift is assumed to be included in the electronic energies given in the Hamiltonian with , and is the thermal average of the phonon-bath displacement Roy and Hughes (2011a)

 ⟨B⟩=exp(−12∫∞0dωJ(ω)ω2coth(βℏω2)). (10)

The temperature dependence is included by . The spectral function that describes the interaction between the electrons in the quantum-dot and the acoustic phonons coupling via a deformation potential is given by

 J(ω)=∑qλ2qδ(ω−ωq)=αpω3e−ω22ω2b (11)

with and for InAs/GaAs quantum-dots.Roy and Hughes (2011a, 2012) The biexciton is assumed to couple twice with the phonon bath compared to the excitons with .Hargart et al. (2016) Additionally, is the transformed quantum dot - phonon bath Hamiltonian. Here , with and , as well as , .Roy and Hughes (2011a)

By treating the phonons in a second order Born-Markov approximation in the master equation and tracing out the phonon bath degrees of freedom, the LA-phonon interaction is now included via the extra Lindblad term in Eq. (3) withUlhaq et al. (2013); Roy and Hughes (2012, 2011a, 2011b)

 Lphonon(ρs)=−1ℏ2∑i=g,u ∫t0dτ (Xi(t)Xi(τ)ρs(t)Gm(t−τ)−Xi(τ)ρs(t)Xi(t)Gm(t−τ)+h.c.), (12)

with and .Roy and Hughes (2012) The evaluation of all dynamical quantities is done in the interaction picture following Ref. Breddermann et al., 2016

In the case of an optical transition from the exciton to its ground state the rates for phonon mediated transitions derived in Ref. Roy and Hughes, 2011a are obtained as shown in the inset of Fig. 3(b):

 Γph=2⟨B⟩2g2R[∫∞0dτe±iΔτ(eϕ(τ)−1)]. (13)

Here is the energy difference of the corresponding cavity-mode and electronic transition energies between exciton and ground state. The phonon correlation functions are given byRoy and Hughes (2011a)

 ϕ(t)=∫∞0dωJ(ω)ω2(coth(βℏω2)cos(ωt)−isin(ωt)). (14)

The effect of the phonon-assisted processes on the decay of an exciton is shown in Figs. 3(a) and (b) for temperatures of K and K. Figure 3 shows that even for an off-resonant cavity mode, faster decay of exciton populations is found with increasing temperature by the increased interaction with the phonon bath. Also, at relatively low temperatures, a cavity detuned to the red leads to a faster decay of the exciton densities compared to the blue-detuned case. This is because the phonon absorption from the phonon bath (needed to assist the exciton decay for blue-detuned cavity) is strongly reduced at low temperatures. This asymmetry is also visible in the inset. The lower the temperature the more pronounced this asymmetry is. Already at , phonon assisted processes involving phonon absorption and emission are almost balanced such that the decay of exciton population at in Fig. 3 is almost the same for postive and negative detuning. At in Fig. 3 , the role of phonon-assisted processes is only weak when phonon absorption is required for emission into a cavity mode that is tuned to the blue.

ii.4 The Two-Photon Density Matrix

To know the state of the emitted photons the decay path from the biexciton to the ground state must be known. In general, it is possible that the biexciton decays purely via a single cavity mode by emitting two photons of the same polarization such that the photons are in the state or or the biexciton can decay by emitting two photons of different polarization, resulting in a or state. The two-photon density matrix contains full information about the quantum state of the two emitted photons and is obtained experimentally by quantum-state tomography based on photon correlation measurements.Edamatsu (2007) The two-photon density matrix is calculated as the double time integral

 ρ2phij,kl=∬G(2)ij,kl(t,τ) dtdτ, (15)

of the second order photon autocorrelation function

 G(2)ij,kl(t,τ) = ⟨b†i(t)b†j(t+τ)bk(t+τ)bl(t)⟩ (16) = tr(ρsb†i(t)b†j(t+τ)bk(t+τ)bl(t)).

We use the quantum regression theorem to calculate these two-time expectation values.Carmichael (2002) The diagonal elements of (16) contain information about the photon statistics where the off-diagonal elements contain information about the polarization entanglement of the two photons. The two-photon density matrix fulfils . In the system studied here, the two-photon density matrix only contains up to four non-zero matrix elements (cf. Fig. 4) and can be simplified accordingly to , with . In this case the degree of polarization entanglement can be measured by the concurrenceHorodecki et al. (2009)

 C=2|ρH,V|. (17)

Figure 4 shows the two-photon density matrix for a maximally entangled state and a state resulting from the emission of a quantum-dot cavity system with finite fine structure splitting of the two exciton states. In the latter case the decay is favored through the exciton over the exciton, which results in a slight increase of the contribution. Also as a result of the exciton splitting, the ”which-path” information for the biexciton decay is revealed such that the off-diagonal elements and with them the concurrence (as a measure of polarization entanglement) is reduced.

To gain insight into the statistics of emitted photons, one has to consider the time integrated function

 g2i,i(τ):=∫G(2)ii,ii(t,τ) dt. (18)

This photon correlation function gives the probability to detect another photon with delay after the first photon was detected and is a useful measure to evaluate the statistical properties of the emitted photons.

Iii Results & Discussion

In this section we discuss the main results obtained for the photons emitted from a quantum dot biexciton when embedded inside a cavity with its resonance tuned close to half the biexciton energy. In Sec. III.1 polarization entanglement is discussed for the ideal and several non-ideal cases, and in Secs. III.2 and III.3 photon statistics and spectral properties, respectively.

iii.1 Polarization Entanglement

The polarization-entanglement of the emitted photons depends on various system parameters. In order to give a detailed picture, the dependence on fine-structure splitting, temperature, biexciton binding energy, different loss mechanisms, cavity quality, and cavity detuning are discussed in the following. The concurrence, Eq. (17), is used to quantify the degree of entanglement. The calculated concurrence as a function of the excitonic fine-structure splitting is shown in Fig. 5 for different values of cavity-quality and temperature. The cavity quality determines the cavity enhancement of the direct two-photon emission process over the cascaded decay. For a sufficiently high cavity quality, the two-photon process dominates the emission from the biexciton such that no ”which-path” information is revealed and a consequently a high degree of entanglement is achieved that is insensitive to exciton fine structure splitting. For a lower quality cavity the biexciton mostly decays through the biexciton-exciton cascade such that the usual sensitivity of the polarization entanglement on the fine structure splitting is recovered.Hafenbrak et al. (2007) Importantly, at low temperatures and high quality cavity, we obtain high degrees of polarization entanglement very similar to the case where phonon-assisted cavity feeding is neglected.Schumacher et al. (2012) With the cavity tuned near half the biexciton energy the biexciton to exciton transition is red-shifted from the cavity mode. Thus at low temperatures phonon-assisted cavity feeding leading to a decay through the cascade is suppressed as the probability for phonon absorption from the phonon bath is very low. With increasing temperature the phonon bath population increases and polarization entanglement is reduced for both high and low quality cavity.

The biexciton binding energy is one of the quantum dot’s intrinsic properties, which can be modified by, e.g., material composition, growth conditions, or post growth by applying strain and electrical fieldsTrotta et al. (2012); Juska et al. (2013). The dependence of the polarization entanglement on the biexciton binding energy is shown in Fig. 6. At low temperature and for the parameters studied here we find that the concurrence slightly decreases with increasing biexciton binding energy. The increase in biexciton binding energy also leads to an increased detuning of the degenerate two-photon transition from the single-photon resonances, which comes at the loss of the resonance enhancement of the two-photon process from the near-by single-photon transitions. This leads to an overall slower decay of the biexciton. Consequently, with increasing temperature when phonon-assisted transitions more efficiently feed the cascaded decay, an increased biexciton binding energy is actually found to reduce the polarization entanglement of the emitted photons even more than at low temperatures.

In addition to the electronic properties of the quantum-dot discussed above, for the emission scheme discussed here, tuning the optical cavity near the two-photon resonance is very important for generating highly entangled photon pairs. Figure 7 shows the concurrence for different detunings of the cavity mode from the two-photon resonance condition. The highest concurrence is obtained for , on resonance with the two-photon emission process. If the cavity energy is detuned from this ideal condition, the cascaded decay takes over the emission dynamics and the concurrence is reduced accordingly. We note that the asymmetry in Fig. 7 for positive and negative detuning is caused by the near-by single-photon resonances of the cascade.

In the remainder of this section we would like to further discuss the role of the different loss mechanisms included in the calculations discussed above. Figure 8 shows the concurrence when the different loss mechanisms are selectively switched off. The most fundamental loss is caused by the loss of photons from the cavity. Only including this mechanism, a maximally entangled state is obtained at zero fine-structure splitting. Loss of electronic coherence (pure dephasing) limits the maximally achievable entanglement also at zero fine-structure splitting. A further overall reduction of the concurrence is caused by the coupling to the phonon bath. At low temperatures, this effect causes a nearly constant offset on the energy range of considered here. The peak in the concurrence at zero fine-structure splitting is caused by the overlap of the excitons with a line with given by the pure dephasing. With increasing temperature this peak is broadened by the phonon contributions. A further loss mechanism not included in the results discussed above is caused by the emission of photons into optical modes other than those cavity modes explicitly considered part of the system. Emission into these leaky modes mostly reduces the total brightness of the quantum-dot cavity system as a source of entangled photon pairs. If loss into leaky modes through Eq. (5) is explicitly included in our calculations it generally slightly increases the concurrence of those photons emitted from the system cavity modes. Other than that the concurrence shows the same dependence on system parameters as in the scenario with no radiative decay present. Finally, we would like to note, that in all scenarios studied, polarization entanglement can be further increased by spectral filtering such that only those photons from the direct two-photon transition are detected.

iii.2 Photon Statistics

The photon statistics reveal information about the temporal emission properties in a photon mode, here the mode. The photon statistics are calculated as the time-integrated second order correlation function given in Eq. (18). The computed results of are shown in Figs. 9 and 10 for two different temperatures and for varying detunings of the cavity from the two-photon resonance. The fine-structure splitting is set to . We find that for low temperature, Fig. 9, a clear anti-bunching of the photons is observed if the cavity is resonant with the exciton to ground state transition . In this case the photon density inside the cavity is comparatively small at all times and the biexciton to exciton transition is far off-resonant and consequenctly inefficient because the first photon is emitted into a blue-detuned cavity mode. Thus, these photon pairs show the characteristic anti-bunching of the cascaded decay at low temperatures. If the biexciton to exciton transition is resonant with the cavity mode at , the transition to the ground state is suppressed by the exciton binding energy of and consequently the second photon is emitted into a red-detuned cavity mode, which benefits from phonon assisted cavity feeding. Here, a slowly decaying is observed for long delays , with the maximum at . If the cavity is resonant with the two-photon transition at , the photon emission is bunched since the two photons are generated at the same time. At higher temperature, in Fig. 10, features are smeared out as for all detunings phonon-assisted transitions are enabled, leading to more complex decay dynamics. The asymmetry caused by different phonon absorption and emission rates is clearly visible. Especially, in the case of , the first photon from the biexciton to the exciton is emitted faster into the cavity mode than at low temperatures. As a result a more bunching-like photon statistics can be observed. However, direct two photon emission is possible only if the cavity is tuned to the two photon resonance.

iii.3 Emission Spectra

Below we discuss the spectral properties of the emitted photons. To analyze the spectral shape of the emitted photons the physical cavity emission spectrumEberly and Wódkiewicz (1977); del Valle et al. (2011); Breddermann et al. (2016) is calculated from the two-time photon correlation function as

 SC(ω)=R∫T0dt∫T−t0dτ⟨b†i(t)bi(t+τ)⟩eiωτ. (19)

The photon correlation function is calculated using the quantum regression theorem,Carmichael (2002) with sufficiently large to obtain a time integrated spectrum after the system has fully relaxed to its ground state. The calculated spectra are shown in Fig. 11. The fine-structure splitting is zero in these calculations, such that the emission in both cavity modes is identical. The cavity is tuned to the two-photon resonance and the temperature is at . For low cavity quality (upper row) the emission is only visible at the biexciton to exciton transistion () and at the ground state to exciton transition (). Including the coupling to the phonon bath, the emission lines are altered to a slightly broadened and slightly asymmetric shape. Without phonon coupling the areas under the two emission peaks from the cascade are identical. For a high-quality cavity, a third emission peak is visible at the cavity frequency. This peak mostly stems from the direct two-photon biexciton to ground state transition and gains additional smaller contributions from phonon-assisted cavity feeding when the coupling to the phonon bath is included. The two-photon emission line is slightly shifted from the ideal two-photon resonance condition because of the strong internal coupling between states.del Valle et al. (2010) Even with finite fine structure splitting, the two photons emitted into the central peak are highly entangled in their polarization state. However, for finite fine-structure splitting, phonon-assisted emission slightly lowers this entanglement as it mixes in photons emitted from the cascade, e.g., through terms in the corresponding Lindblad term, Eq. (12), which carries the path information of the decay.

Iv Conclusions

We have presented a detailed analysis of the generation of polarization-entangled photons from quantum-dot biexcitons via cavity-assisted two-photon emission. In particular we have studied the dependence of the polarization entanglement on system design and parameters, such as cavity quality and frequency and biexciton binding energy. We have further analyzed photon statistics and spectral properties of the emitted photons. A focus of our present study lies on the role that different loss mechanisms including pure dephasing, loss into leaky cavity modes, and phonon-assisted cavity feeding at finite temperatures play for the achievable polarization entanglement. Tuning the cavity to half the biexciton energy, for a bound biexciton the biexciton to exciton transition is red-shifted relative to the cavity mode. Therefore at low temperatures with low probability for phonon absorption from the bath, feeding the biexciton-exciton cascade through phonon-assisted processes is strongly suppressed. As a consequence, even in high-quality cavities where phonon-assisted processes are strongest, at low temperature the emission can efficiently be channelled into the two-photon emission process such that a high degree of polarization entanglement is achieved. Radiative loss reduces the overall quantum efficiency but only slightly alters the entanglement properties of the photons emitted from the system cavity mode. With increasing temperature, a detrimental influence of the coupling of the system to the bath of LA phonons on the achievable polarization entanglement is found.

Acknowledgements.
We acknowledge valuable discussion with Christopher Gies, Matthias Florian, Paul Gartner, and Frank Jahnke from the University of Bremen. We gratefully acknowledge financial support from the DFG through the research center TRR142 and doctoral training center GRK1464, from the BMBF through Q.com 16KIS0114, and a grant for computing time at Paderborn Center for Parallel Computing. Stefan Schumacher further acknowledges support through the Heisenberg programme of the DFG.

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