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Micropillar single-photon source design for simultaneous near-unity efficiency andindistinguishability

Wang, Bi-Ying; Denning, Emil Vosmar; Gür, Uur Meriç; Lu, Chao-Yang; Gregersen, Niels

Published in:Physical Review B

Link to article, DOI:10.1103/PhysRevB.102.125301

Publication date:2020

Document VersionPeer reviewed version

Link back to DTU Orbit

Citation (APA):Wang, B-Y., Denning, E. V., Gür, U. M., Lu, C-Y., & Gregersen, N. (2020). Micropillar single-photon sourcedesign for simultaneous near-unity efficiency and indistinguishability. Physical Review B, 102(12), [125301 ].https://doi.org/10.1103/PhysRevB.102.125301

https://doi.org/10.1103/PhysRevB.102.125301https://orbit.dtu.dk/en/publications/41bcdad6-1531-4e99-9309-494847455feahttps://doi.org/10.1103/PhysRevB.102.125301

Micropillar single-photon source design for simultaneous near-unity efficiency andindistinguishability

Bi-Ying Wang,1, 2 Emil V. Denning,2 Uğur Meriç Gür,3 Chao-Yang Lu,1 and Niels Gregersen2, ∗

1Hefei National Laboratory for Physical Sciences at Microscale,University of Science and Technology of China, Hefei, Anhui, 230026, China

2DTU Fotonik, Department of Photonics Engineering,Technical University of Denmark, Ørsteds Plads,

Building 343, DK-2800 Kongens Lyngby, Denmark3DTU Elektro, Department of Electrical Engineering,

Technical University of Denmark, Ørsteds Plads,Building 348, DK-2800 Kongens Lyngby, Denmark

(Dated: September 16, 2020)

We present a numerical investigation of the performance of the micropillar cavity single-photonsource in terms of collection efficiency and indistinguishability of the emitted photons in the presenceof non-Markovian phonon-induced decoherence. We analyze the physics governing the efficiency us-ing a single-mode model, and we optimize efficiency ε and the indistinguishability η on an equalfooting by computing εη as function of the micropillar design parameters. We show that εη islimited to ∼ 0.96 for the ideal geometry due to an inherent trade-off between efficiency and indis-tinguishability. Finally, we subsequently consider the influence of realistic fabrication imperfectionsand Markovian pure dephasing noise on the performance.

I. INTRODUCTION

Within the fields of optical quantum computation1 andcommunication2, a key component is the single-photonsource3,4 (SPS) capable of emitting single indistinguish-able photons on demand. The figures of merit include theefficiency ε defined as the number of detected photons inthe collection optics per trigger as well as the indistin-guishability η of consecutively emitted photons, as mea-sured in a Hong-Ou-Mandel experiment, and their prod-uct εη governs the success probability of the multi-photoninterference experiment5,6. The spontaneous parametricdown-conversion process7 has for a long time been theworkhorse for single-photon generation within the quan-tum optics community. While simple to implement ex-perimentally, its main drawback is its probabilistic naturepreventing multi-photon interference experiments withmore than a handful of photons.

As alternative, the semiconductor quantum dot (QD)embedded in a host material8 has emerged as a promis-ing platform for scalable optical quantum informationprocessing. As a two-level system, the QD allows fordeterministic emission of single photons and also entan-gled photon pairs9 through the spontaneous emission pro-cess. However, for QDs in a bulk material the high in-dex contrast at the semiconductor-air interface reducesthe collection efficiency to a few percent. The collec-tion can be improved by placing the QD in a structuredenvironment3,4 directing the light towards the collectionoptics. Design schemes allowing for vertical emissionwith high efficiency include the photonic nanowire10–13

design with ε of ∼ 0.7 and the ”bullseye”9,14–17 planarcircular Bragg grating design with ε of ∼ 0.8516. Thesedesigns feature broadband (> 50 nm) spectral operationwhich allows for easy spectral alignment of the QD andthe cavity lines. However, a drawback of the broadband

approach is the presence of the phonon sideband in thephoton emission spectrum arising due to interaction withthe solid-state environment even at zero temperature.These photons are distinguishable and constitute around10 % of the total emission, and while they can be filteredout, this occurs at the price of reduced efficiency. An-other drawback is the proximity of the QD to the fluc-tuating charge environment18 of semiconductor-air sur-faces, which again compromises the indistinguishability.

The most successful SPS design so far is the micro-cavity pillar17,19–21 featuring a QD in a vertical λ cav-ity sandwiched between two distributed Bragg reflec-tors (DBRs). Almost 20 years ago, Barnes et al. pro-posed a model22 to quantify the efficiency in terms ofthe Purcell factor, from which one finds that the effi-ciency can be improved either by enhancing the cavityQ factor or by reducing the mode volume. A practicalchallenge for the QD-based SPS is the random spatial andspectral distribution of typical Stranski-Krastanov-grownQDs, and important technological developments were thedemonstrations of in-situ lithography23 and QD imag-ing techniques15 allowing for deterministic QD positioncontrol as well as Stark tuning21 to ensure the spectralalignment. The micropillar SPS then relies on narrow-band Purcell enhancement to ensure efficient emissioninto the cavity mode, where the narrow bandwidth (∼ 0.2nm) enables a funneling of photons into the zero-phononline (ZPL) and suppression of the phonon sideband al-lowing for high indistinguishability without any filter-ing. Additionally, the QD is less subject to chargenoise due to a larger separation between the QD andthe surfaces. These assets have led to the simultaneousdemonstrations20,21 of pure (g2(0) < 0.01) single photonemission from micropillars combining near-unity (> 0.98)indistinguishability with extraction efficiency of ∼ 0.65.While, for circular micropillars half of the emitted light

2

is lost in standard resonant excitation cross-polarizationsetups20,21 where polarization filtering is used to sup-press the laser background, this loss can be avoided us-ing more advanced techniques like spatially orthogonalexcitation and collection24 or two-colour resonant25 orphonon-assisted26 excitation.

The question now arises of how the micropillar SPSextraction efficiency can be improved while maintain-ing high indistinguishability. Whereas charge noisecan be effectively suppressed using metal contacts21,27,phonon-induced decoherence is not avoided even at zerotemperature28 and must be taken into account in theoptimization. The efficiency can be improved by increas-ing the mirror reflectivities leading to a larger Q factorand in turn an increased Purcell factor. However, evenif one disregards unavoidable fabrication imperfectionslimiting the maximum achievable Q, a fundamental ob-stacle is encountered arising from the interaction withthe phonon bath: While the cavity improves the indis-tinguishability in the weak coupling regime, increasingthe Purcell factor to improve the efficiency will even-tually bring the QD-cavity system into the strong cou-pling regime, which is detrimental to the indistinguisha-bility. Specifically, in the strong coupling system, the sys-tem undergoes vacuum Rabi oscillations29, which causesthe emission spectrum to split into two hybrid polari-ton states. However, in the presence of phonon coupling,the phonon-induced transitions between the upper po-lariton state and the lower polariton state strengthenthe decoherence of the system, and hence decrease theindistinguishability30. There is thus an inherent trade-off31,32 between the achievable efficiency and indistin-guishability in the presence of phonon-induced decoher-ence. Consequently, an optimization of the micropillarSPS design requires accurate modeling of the efficiencyas well as the indistinguishability subject to phonon-induced decoherence on an equal footing, and such astudy, to our knowledge, is still absent in the literature.

In this work, we perform an investigation of the achiev-able performance of the micropillar geometry in termsof efficiency and indistinguishability. Our theoreticalmethod is based on two steps: First, we use a Fouriermodal method33,34 to rigorously calculate the efficiencyallowing for direct insight into governing physics de-scribed here using a single-mode model. Subsequently,we use the calculated optical properties of the cav-ity as parameter inputs to a non-Markovian masterequation, which is used for evaluation of the photonindistinguishability31,32. This master equation accountsfor non-Markovian phonon scattering as well as dephas-ing from charge noise. The influence of the sidewallroughness is also included as an additional phenomeno-logical scattering rate. For realistic parameters of de-phasing and sidewall roughness, we provide design spec-ifications for an optimized micropillar SPS with perfor-mance significantly improved compared to state-of-the-art.

This article is organized as follows: We introduce the

FIG. 1. Illustration of the micropillar SPS design consistingof a λ cavity sandwiched between two DBRs. The QD, rep-resented by the red dot, is placed at the center of the cavity.The coupling parameters β and γ of the single-mode model,cf. Section III A, are schematically shown.

micropillar geometry and the design recipe in Section II.The details of the optical and microscopic modeling areprovided in Section III. We present our analysis of themicropillar performance and our results in Section IV,which we discuss further in Section V followed by a con-clusion. The Appendix contains additional details of themodeling.

II. MICROPILLAR GEOMETRY

In this work, we investigate the micropillar cavity il-lustrated in Fig. 1 consisting of a QD placed in the centerr0 of a λ cavity. Light is confined vertically by the DBRsand laterally by total internal reflection. We consider anasymmetric structure with larger number nbot of layerpairs in the bottom DBR than the number ntop in thetop DBR to suppress leakage of light into the substrate.In the following, we fix the number of layer pairs nbot =40 and study the SPS performance as function of pillardiameter d and of ntop. We choose GaAs for the cavitymaterial and GaAs/Al0.85Ga0.15As for the DBR pairs asthis choice avoids the oxidation of pure AlAs. For ourdesign wavelength of λ0 = 895 nm, we use a refractiveindex for GaAs of 3.5015 (T = 4 K) and an index forAl0.85Ga0.15As of 2.9982

35. For each pillar diameter con-sidered, the thickness of the layers is chosen36 such thatthe resonance matches the design wavelength λ0. Thusthe cavity thickness is given by λ0/neff , whereas the DBRlayer thicknesses are chosen as λ0/(4neff), where neff isthe diameter-dependent effective index of the fundamen-tal mode of the section. For pillar diameters above ∼ 2.5µm, the layer thicknesses are very close to those of the1D geometry and are given by 64 nm, 75 nm and 256 nm,respectively, for the GaAs and Al0.85Ga0.15As layers inthe DBR and the GaAs cavity.

3

III. NUMERICAL MODEL

Our numerical model consists of an optical simulationperformed using a Fourier modal method to determinethe efficiency combined with an open quantum systemsmaster equation approach to evaluate the indistinguisha-bility of the emitted photons in the presence of realisticdecoherence effects.

A. Optical modeling

We model the QD as a classical point dipole pδ(r−r0)with in-plane orientation and harmonic time dependencewith frequency ω in resonance with the cavity mode. Todetermine the efficiency, we compute the total emittedpower P = ω2 Im [p

∗ ·E(r0)] as well as the power coupledto the lens PLens, where we take into account an overlapwith a Gaussian mode to model the coupling to a single-mode fiber, as discussed in Appendix B. The collectionefficiency ε is then determined as

ε = ICPLensP

, (1)

where IC is a correction factor (cf. Section III C) takinginto account the presence of fabrication imperfections.

While the collection efficiency is rigorously describedby Eq. (1), this expression does not provide detailedunderstanding of the governing physics. For this reason,we also introduce a single-mode model4,22 to determinethe single-mode efficiency εs. The model describes theefficiency in terms of the spontaneous emission β factorgiven by the power PC coupled to the cavity mode overthe total emitted power P = PC + PR, where PR is thepower coupled to all other modes. In the weak couplingregime, the β factor can be written in terms of the Purcellfactor Fp as

β =PC

PC + PR=

FpFp + PR/P0

, (2)

where P0 is the power emission for a bulk medium. FromEq. (2) it is clear that β can be improved by increas-ing the Purcell factor, which can be achieved by enhanc-ing the cavity quality factor Q or by reducing the cavitymode volume. The second parameter of the model is thetransmission γ = PLens,C/PC, where PLens,C is the powercoupled to the lens from the cavity mode alone again tak-ing into account an overlap with a Gaussian profile. Oursingle mode model now describes the efficiency simply asthe product εs = βγ as illustrated in Fig. 1.

The calculations of β, γ and ε are performed usingan open-geometry Fourier modal method33,34. Based onan eigenmode expansion of the optical field, the methodprovides direct access to the cavity mode contribution tothe electric field strength EC(r) allowing for immediateevaluation of PC and PLens,C, see Appendix A for furtherdetails. Finally, the spontaneous emission rate into the

cavity ΓC is computed from the classical power PC us-ing the relationship37 ΓC/Γ0 = PC/P0, where Γ0 is thespontaneous emission rate in a bulk medium.

B. Microscopic modeling

We describe the QD in a solid-state environment as atwo-level system, with a ground state, |0〉, and an excitedstate, |e〉, coupled to both photon and phonon environ-ments. The cavity mode is described by a bosonic anni-hilation operator, a, coupled to the QD with a rate, g.The Hamiltonian describing the dynamics of the emitterand cavity is thus given by

H0 = ~ωe |e〉〈e|+ ~ωca†a+ ~g(a†σ + aσ†), (3)

where ωe and ωc are the QD transition frequency and thecavity resonance, respectively, and σ = |0〉〈e| is the dipoleoperator of the QD. The cavity loss due to coupling withelectromagnetic modes outside of the cavity is describedby the Hamiltonian

Hc,EM = ~∑l

(ωA,lA

†lAl + hA,la

†Al + H.c.), (4)

where Al is the bosonic annihilation operator of the lthenvironmental mode in the cavity loss channel, with fre-quency ωA,l and coupling strength hA,l to the cavitymode. Using the cavity escape rate κc = ωc/Q describingthe Markovian cavity decay into all Al modes, we com-pute the QD-cavity coupling strength as g = 12

√ΓCκc.

Similarly, the coupling of the QD to optical radiationmodes is described by

He,EM = ~∑l

(ωB,lB

†lBl + hB,lσ

†Bl + H.c.), (5)

where Bl is the bosonic annihilation operator of the lthenvironmental mode in the radiation reservoir of the QD,with frequency ωB,l and coupling strength to the emitterhB,l. The emitter–phonon coupling is described by

HP = ~∑k

[νkb†kbk + gk |e〉〈e| (bk + b

†k)], (6)

where bk is the annihilation operator for the phononmode with momentum k, frequency νk and couplingstrength to the emitter gk.

The total Hamiltonian is thus given by

H = H0 +Hc,EM +He,EM +HP. (7)

1. Indistinguishability

The indistinguishability of the photon is calculatedthrough the two-colour spectrum as31

I = P−2S

∫ ∞−∞

dω

∫ ∞−∞

dν|S(ω, ν)|2. (8)

4

This two-colour spectrum is defined as

S(ω, ν) =

∫ ∞−∞

dt

∫ ∞−∞

dt′〈E†(t′)E(t))

〉ei(ωt−νt

′), (9)

where E is the operator corresponding to the detectedelectric field. The frequency-integrated spectral power,PS, is defined as

PS =

∫ ∞−∞

dωS(ω, ω). (10)

To calculate the two-colour spectrum, we use a polaronmaster equation, which is described in the following.

2. Master equation

Following previous work31,32,38,39, we make use of theBorn-Markov master equation in the polaron frame. As-suming that the emitter and cavity are resonant (ωe =ωc), the polaron master equation for the reduced densityoperator of the cavity–emitter system, ρ, is (in a refer-ence frame rotating with the resonance frequency, ωe)

∂tρ = −i[grX̂, ρ] + κcD(a) + ΓD(σ)+ 2γdD(σ

†σ) +Kph(ρ).(11)

Here, gr = gB is a renormalised emitter–cavity cou-pling rate. The renormalisation factor is given by

B = 〈B±〉0 and B± = exp[±∑k g

2k/νk(b

†k + bk)

], where

〈x〉0 = Z−1 Tr[xe−

∑k ~νk/(kBT )b

†kbk]

denotes the expec-

tation value with respect to the thermal phonon state,where T is the temperature, kB is the Boltzmann con-

stant and Z = Tr[e−

∑k ~νk/(kBT )b

†kbk]

is the partition

function. The symbol D[x] = xρx†−{x†x, ρ}/2 denotes theLindblad dissipator; Γ represents the Markovian emissionrate into the radiation modes, Bl, and γd is the pure de-phasing rate. The phonon dissipator is written as31

Kph =−g2(

[X̂, X̂ρ]χX+[Ŷ , Ŷ ρ]χY+[Ŷ , Ẑρ]χZ+H.c.), (12)

with X̂ = σ†a+ σa†, Ŷ = i(σ†a− σa†) and Ẑ = σ†σ − a†a.The coefficients are

χX =

∫ ∞0

dτΛXX(τ),

χY =

∫ ∞0

dτcos(2grτ)ΛY Y (τ),

χZ = −∫ ∞

0

dτsin(2grτ)ΛY Y (τ),

(13)

with the phonon correlation functions ΛXX(τ) =〈BX(τ)BX〉 = B2[sinhϕ(τ)−1] and ΛY Y (τ) = 〈BY (τ)BY 〉 =B2 coshϕ(τ) with BX = (B+ + B− − 2B)/2 and BY =i(B+ −B−). Here, ϕ(τ) is given by38

ϕ(τ) =∫∞

0dω J(ω)ω2

[coth

(~ω

2kBT

)cos(ωτ)− i sin(ωτ)

], (14)

where the phonon spectral density is J(ν) =∑k |gk|2δ(ν − νk) = αν3 exp

(−ν2/ν2c

)with α being

exciton-phonon coupling strength and νc =√

2vc/L de-noting the cutoff frequency40, where vc and L are thespeed of sound and the size of the QD, respectively. Here,the exciton-phonon coupling strength α depends on theQD material and on the cutoff frequency νc, which itselfis related to the size of the QD41. The Franck-Condonfactor, B2, can be written compactly in terms of thisfunction as B2 = e−ϕ(0).

The master equation, Eq. (11), allows us to calculatethe two-colour spectrum, S(ω, ν). This is done by firstcalculating the dipole spectrum,

S0(ω, ν) =

∫ ∞0

dt

∫ ∞0

dτei(ω−ν)te−iωτ

×GP(τ)〈σ†(t+ τ)σ(t)

〉,

(15)

where GP(τ) = B2eϕ(τ) is the free phonon Green’s

function and the two-time dipole correlation function,〈σ†(t+ τ)σ(t)

〉, is evaluated using the quantum regres-

sion theorem with Eq. (11). From this dipole spectrum,the detected field spectrum is calculated as31

S(ω, ν) = GF(ω, ν)S0(ω, ν) +G∗F(ν, ω)S

∗0 (ν, ω), (16)

where

GF(ω, ν) =(κc/2)

2

(iω − κc/2)(−iν − κc/2)(17)

is an optical two-colour Green’s function connecting thedipole with the field emitted from the cavity. In thePurcell-regime, where g � κc, the indistingusihabilitycan be approximated by the expression

I ' ΓtΓt + 2γt

[B2

B2 + F (1−B2)

]2, (18)

where Γt = 4g2/κc + Γ is the total Purcell-enhanced

spontaneous emission rate of the QD, γt = γd +2π(gr/κc)

2J(2gr) coth(~gr/kBT ) is a phonon-enhancedpure dephasing rate and F is the fraction of photons inthe phonon sideband not eliminated by the cavity31.

Through narrowband Purcell enhancement, the mi-cropillar SPS ensures an efficient emission into the cavitymode, where the narrow bandwidth (∼ 0.2 nm) enablesa funneling of photons into the ZPL and a suppressionof the phonon sideband allowing for high indistinguisha-bility without any filtering. To illustrate this influence,we show the spectra S(ω, ω) for a low Q and a highQ micropillars in Fig. 2 both exhibiting a narrow peakand a broadband base corresponding to the ZPL and thephonon sideband, respectively. We observe that the highQ micropillar leads to a broadened ZPL as a result ofthe large Purcell factor and a suppression of the phononsideband.

5

10-7

10-4

10-1

102

105

-5 -2.5 0 2.5 5

S(ω

,ω)

ω-ωe (meV)

Q=16000

Q=400

FIG. 2. Emission spectrum showing the zero-phonon line andthe phonon sidebands at T = 4 K. Dashed curve: Q = 400and Fp = 2; solid curve: Fp = 20 and Q = 16000. Parameters:α = 0.03 ps−2 and ~νc = 1.4 meV following Ref. 31.

C. Modeling of fabrication imperfections

While the figures of merit of the micropillar SPS canbe computed accurately using the formalisms presentedabove, the influence of unavoidable fabrication imperfec-tions leading to discrepancy42,43 between the predictionof the modeling and the experimentally measured perfor-mance should be taken into account. The imperfectionscan be described phenomenologically by introducing ad-ditional loss channels due to sidewall roughness and ma-terial absorption44. The quality Q factor is then writtenas19,45

1

Q=

1

Qint+

1

Qedge+

1

Qabs, (19)

where the intrinsic Qint is computed for the perfectstructure43, and Q−1edge and Q

−1abs represent the contribu-

tions to loss from sidewall scattering and material ab-sorption, respectively. Since Fp is linearly proportionalto Q, the introduction of imperfections leads to a cor-rected Purcell factor given by F cp = Fp(Q/Qint) and inturn a corrected β factor βc = F cp/(F

cp+PR/P0). We then

model the correction IC to the efficiency in the presenceof imperfections as IC = β

c/β. Moreover, the correctedindistinguishability is obtained by evaluating the decayrate κc = ω/Q in Eq. (11) using the Q given by Eq. (19).

We model the sidewall scattering using a phenomeno-logical model44 discussed in Appendix D, where κ is aphenomenological constant proportional to the sidewallscattering rate Q−1edge. The effect of material absorption

loss can be modeled either using Q−1abs in Eq. (19) orby introducing an imaginary part43,46 to the refractiveindex. However, the effect of material loss is typicallynegligible43,46 in micropillars with Q factors below ∼ 105and is thus ignored in this work.

IV. RESULTS

In this section, we first present calculations of thesingle-mode model parameters to elucidate the physicsgoverning the collection efficiency, after which we presentresults from full simulations of the efficiency and indis-tinguishability taking into account fabrication imperfec-tions. All calculations are performed at a numerical aper-ture (NA) of 0.82 for the first lens. Following Ref. 31,we use the parameters ~νc = 1.4 meV and α = 0.03 ps2corresponding to a typical QD of size 25 nm41.

A. Analysis using the single-mode model

We first present the cavity Q factor as function of thepillar diameter d and of the number of layer pairs ntopin the top DBR in Fig. 3(a) for the ideal structure. Anincrease in ntop leads to a higher reflectivity resultingin an increased cavity Q factor. For large diameters,the Q factor approaches the value of a planar cavitywhereas for short diameters it drops due to poor modaloverlap of the HE11 mode in the cavity section and thecorresponding Bloch mode in the DBR sections. Oscil-lations are observed for the largest values of ntop arisingfrom coupling36,43,47 with higher-order Bloch modes ofthe DBRs. This behavior is also observed for the Purcellfactor Fp illustrated in Fig. 3(b), where the reductionof the pillar diameter d and in turn the mode volumeadditionally leads to an overall increase in Fp until thedrop of the Q at around d ∼ 1.2 µm sets in. As ex-pected, the increase in the Purcell factor Fp is directlyreflected in the calculated spontaneous emission β factorpresented in Fig. 3(c). However, the observed oscilla-

103

104

105

1 2 3 4

(a)

Q

d (µm)

ntop=172125

0

180

360

540

1 2 3 4

(b)

Fp

d (µm)

ntop=172125

0.76

0.84

0.92

1

1 2 3 4

(c)

β

d (µm)

ntop=172125

0

0.5

1

1 2 3 4

(d)

γ

d (µm)

ntop=172125

FIG. 3. Computed cavity Q factor (a), Purcell factor Fp (b)and spontaneous emission β factor (c) as well as the trans-mission γ (d) to the lens as a function of pillar diameter dfor different numbers of layer pairs ntop in the top DBR. Theideal structure with γd = κ = 0 is considered.

6

tions in β are not originating from variations in Q butrather from diameter-dependent variations in the emis-sion PR to background modes, and they are reduced inmagnitude with increasing Fp in agreement with Eq. (2).While the best coupling β to the cavity is obtained forthe diameter of ∼ 1.1 µm corresponding to the maximumPurcell factor in Fig. 3(b), the transmission γ presentedin Fig. 3(d) for this d is modest due to its highly diver-gent output beam profile48. As the diameter increases,the divergence is reduced and γ is improved. Also, whereβ is maximum for ntop = 25, an increasing top mirrorreflectivity leads to an increasing leakage of light intothe substrate and thus a reduction of γ for ntop = 25 asshown in Appendix E.

We thus observe that there exists a trade-off betweenthe coupling β to the cavity and the transmission γ to thelens. Since the efficiency is the product of the two, onemay ask if the overall performance can be optimized bychoosing a large diameter and by increasing the reflectiv-ity of both DBRs while keeping nbot comfortably largerthan ntop to avoid leakage of light into the substrate. Inthe following, we will see that this is not the case.

B. Analysis using the full model

Having established our understanding of the variationsof β and γ, we now present the collection efficiency com-puted using the full model Eq. (1). As shown in Ap-pendix C, excellent agreement between the single-modemodel and the full model is observed, and the variationsin the efficiency can thus be fully understood from thebehavior of β and γ. The collection efficiency ε = βγis presented in Fig. 4(a) for the ideal structure withoutfabrication imperfections. The reduction in ε for low di-ameters due to the drop in γ is clearly observed, and weobserve that an efficiency of ∼ 0.95 is obtained in a broadregime for diameters above ∼ 2 µm and ntop ∼ 21.

The computed indistinguishability η in the presenceof phonon-induced decoherence for the otherwise idealstructure is shown in Fig. 4(b). The computed indistin-guishability initially increases from 17 to 21 layer pairsas the influence of dephasing is overcome by the in-crease in Fp in agreement with Eq. (18). However, forntop ≥ 21 the high Q factor results in significant dropsin η for decreasing pillar diameter d due to the onsetof the strong coupling regime (4g & κc) detrimental tothe indistinguishability31. This drop becomes more pro-nounced as ntop increases and the decay rate κc decreases.

To answer our previous question, we realize that theincrease of bottom mirror nbot layer pairs to increase thetransmission γ for ntop ≥ 25 may improve the efficiencybut will simultaneously bring us even further into theregime of strong coupling detrimental to η. On the otherhand, a good performance with η > 0.995 is obtained ina regime at the boundary of strong coupling, in the caseof ntop = 21 for d larger than ∼ 2 µm. Overall, η is in-creased towards unity by decreasing κc and by increasing

0.1

0.4

0.7

1(a)

ε ntop=17

21

25

0.93

0.96

0.99

1 2 3 4

(b)

ηd (µm)

ntop=17

21

23

25

FIG. 4. Computed efficiency ε (a) and indistinguishability η(b) versus pillar diameter d for different layer pairs ntop inthe top DBR for the ideal geometry with γd = κ = 0.

the diameter, which should be chosen sufficiently large toensure weak coupling. However, as shown in Figs. 3(b,c)the increase in d ultimately leads to a reduction in effi-ciency. An optimization of the micropillar performancein terms of ε and η is thus necessary to identify the op-timal performance.

C. Optimization

To optimize the efficiency and indistinguishability onan equal footing, we consider as figure of merit the prod-uct εη presented for the ideal case without charge noiseand sidewall roughness in Fig. 5(a). Here, by directlyreading the maximum of εη in the numerical simula-tion, the best performance of εη ∼ 0.95 is obtained forntop = 21 at various peak d positions in the interval [2,4]µm, where the observed variations are direct reflectionsof those for the β factor in 3(b). In this regime, we haveε ∼ 0.95 and η ∼ 0.997, in good agreement with theperformance optimum identified in Ref. 31.

We now introduce imperfections initially in the formof a finite dephasing rate ~γd = 0.086 µeV49 due tonon-phononic Markovian decoherence mechanisms, e.g.charge noise due to an unstable charge environment18.Compared to the ideal case, the indistinguishability isreduced by the additional charge noise decoherence mech-

7

0.6

0.8

1(a)

ideal

εη

ntop=17

21

25

0.6

0.8

1(b)

CN

εη

0.4

0.6

0.8

1

1.6 2.4 3.2 4

(c)

SR+CN

εη

d (µm)

0.9

0.95

1

1 2 3 4

ntop=21

η

d (µm)

CNSR+CN

0.96

0.98

1

1 2 3 4

ntop=21

η

d (µm)

idealCN

FIG. 5. The computed figure of merit εη as function of pillardiameter d for different layer pairs ntop in the top DBR for(a) the ideal case, (b) in the presence of charge noise (CN)only and (c) with both charge noise and sidewall roughness(SR) present. The insets compare the indistinguishability inthe three cases for ntop = 21. Parameters: ~γd = 0.086 µeV;κ = 0.2 µm.

anism. As shown in the inset of Fig. 5(b), the reductionin η increases with d due to the corresponding reductionin the Purcell factor in agreement with Eq. (18). Theresulting figure of merit εη is shown in Fig. 5(b), whereεη is reduced to ∼ 0.94 for ntop = 21 and d ∈ [2,3.25]µm, where ε ∼ 0.95 and η ∼ 0.99.

As a second element of imperfection, we now also takesidewall roughness into account by introducing a rela-tively large phenomenological constant κ = 0.2 µm19,and the results are plotted in Fig. 5(c). We observe thata maximum εη of∼ 0.93 is obtained for ntop = 21 and d ∈

[3.05,3.9] µm, where ε ∼ 0.945 and η ∼ 0.98. Whereasthe charge noise only influenced the indistinguishability,the sidewall imperfections are detrimental to both ε andη through the reductions in IC and Fp due to the de-crease in Q. The reduction of η is shown in the insetof Fig. 5(c), where the impact of the increase in sidewallscattering Q−1edge with decreasing diameter is directly ob-served.

While charge noise can be overcome by increasing Fp,this ultimately will lower η due to the onset of strongcoupling. Similarly, while the effect of sidewall imperfec-tions is reduced for increasing pillar diameter, this alsoleads to a decrease in Fp and in turn the efficiency. Thedetrimental effects of charge noise and sidewall imperfec-tions thus cannot be completely overcome in the designprocess alone, and they should be addressed in the fab-rication, e.g. by introducing metal contacts21 for chargestabilization and using a reduced Al content in the DBRsto avoid oxidation.

V. DISCUSSION

While our simulations identify geometrical parametersleading to predicted performance significantly beyondstate-of-the-art, the exact performance obtained exper-imentally will depend on unknown experimental param-eters. Even though the exciton-phonon coupling strengthα and the cutoff frequency νc are QD-dependent, previ-ous investigations have revealed that the optimum cavitylinewidth for maximum indistinguishability is indepen-dent of the QD size in the 10-40 nm range41. Futhermore,the magnitude of the Markovian noise term as well asthe unavoidable fabrication imperfections influence theperformance. Nevertheless, our study reveals that thediameter range from 2.5 to 4 µm for the ntop = 21 con-figuration leads to the optimal micropillar SPS perfor-mance, and in our study, this conclusion is again notaltered by the presence of realistic pure dephasing orfabrication imperfections, which instead simply lead toan overall degradation of the performance by a few per-cent. In the ideal performance regime, εη in Fig. 5 dis-play variations of ∼ 5 % as function of d. Interestingly,these variations are not caused by the well-establisheddiameter-dependent variations43,47 in Q due to couplingwith higher-order Bloch modes, and an adiabatic cavitydesign50 will thus not eliminate the variations. Rather,Figs. 3(c,d) show that the εη variations appear due tostrong variations with d of the background emission ratePR/P0, which is often simply assumed equal to unity. Abetter physical understanding of the background emis-sion rate in micropillars is thus desirable but beyond thiswork.

However, even with such understanding, the micropil-lar geometry does not provide a mechanism to completelysuppress the background emission. The consequence isthat the trade-off between efficiency and indistinguisha-bility is unavoidable, and a figure of merit εη for the mi-

8

cropillar above ∼ 0.96 is out of reach. While the micropil-lar geometry presently remains the champion device forefficient generation of single indistinguishable photons,this limitation motivates further investigation of alter-native design strategies such as the photonic nanowireSPS51,52, where the background emission can be con-trolled using dielectric screening effects53.

Whereas we predict an εη ∼ 0.96, experimentallydemonstrated figures of merit20,21 are significantly be-low this value. This discrepancy is partly due to thecross-polarization setup in resonant excitation used inthese works, reducing the efficiency by a factor of 2.This reduction is not a fundamental limitation and canbe avoided using e.g. phonon-assisted excitation26 forthe rotationally symmetric micropillars considered inthis work or by the introduction of an elliptical cross-section17. Even so, these methods will not overcome theinherent trade-off between efficiency and indistinguisha-bility of the micropillar SPS. Nevertheless, an optimiza-tion of also the elliptical micropillar design in terms ofefficiency and indistinguishable polarized photon emis-sion remains of interest but is beyond this work.

VI. CONCLUSION

In summary, by applying a Fourier modal method anda polaron master equation approach, we have numericallycalculated the efficiency and indistinguishability of themicrocavity pillar single-photon source. To elucidate themechanism governing the efficiency, we have presented asingle-mode model, which fully captures the physics interms of the initial spontaneous emission to the cavitymode and the subsequent transmission to the first lens.By varying the number of DBR layer pairs and the diam-eter, we have observed the onset of the trade-off betweenefficiency and indistinguishability predicted previously,and we have identified design parameters leading to amaximum figure of merit εη of the product of the effi-ciency ε and the indistinguishability η of ∼ 0.96 for theideal geometry. In addition, we have investigated the in-fluence of pure dephasing as well as sidewall scattering onthe performance. Using realistic values for these mech-anisms, we have shown that, while the performance isdegraded by imperfections, the optimum design parame-ters are unchanged.

ACKNOWLEDGEMENT

The authors thank Jean-Michel Gérard for fruitful dis-cussions. EVD and NG were supported by the Inde-pendent Research Fund Denmark, grant No. DFF-4181-00416 and DFF-9041-00046B. BW acknowledges supportfrom the China Scholarship Council. This work wassupported by the Danish National Research Foundationthrough NanoPhoton - Center for Nanophotonics, grantno. DNRF147.

APPENDIX

In the Fourier modal method, the geometry is dividedinto layers uniform along a propagation z axis, and theelectric field in each layer is expanded upon the forwardand backwards propagating eigenmodes e±j (r⊥)e

±iβjz

that are solutions to the wave equation for the layer as-suming uniformity along z. The fields on each side of theinterface between layers q and q+1 are related by inter-

face reflection ¯̄Rq,q+1 and transmission¯̄Tq,q+1 matrices

determined from the boundary conditions at the inter-face, and the propagation through a layer of length L isdescribed by a propagation PL matrix with elements e

iβjL

in its diagonal. The total reflection matrix ¯̄Rt (¯̄Rb) for

the top (bottom) DBR mirror is obtained from interfaceand propagation matrices in an iterative manner usingthe S matrix formalism. The total field at the positionz0 + δz just above the QD (δz → 0) is given by

E(r⊥, z0 + δz) =∑

j

[aje

+j (r⊥) + bje

−j (r⊥)

], (20)

where the field amplitude vectors ā and b̄ represent thefield expansion just above the QD. The forward propa-gating field vector ā is computed from the point dipolepδ(r − r0) using the reciprocity theorem and takinginto account multiple round trips inside the cavity34.The backwards propagating field is obtained as b̄ =¯̄PL/2

¯̄Rt¯̄PL/2ā.

A. Cavity contribution to the field

To determine the contribution EC from the cavitymode to the total field given in Eq. (20), we define

the eigenvectors c̄+k of the roundtrip matrix¯̄R+ =

¯̄PL/2¯̄Rb

¯̄PL¯̄Rt

¯̄PL/2 as solutions to the eigenproblem

¯̄R+c̄+k = λ+k c̄

+k , (21)

where λ+k are eigenvalues with |λ+k | < 1. For normalized

eigenvectors, the cavity roundtrip eigenvector c̄+M corre-sponding to the fundamental cavity mode is easily identi-fied as the one with the first element having a maximumvalue such that |c+1M| ≥ |c

+1k|. For the forward propagat-

ing field in (20), we have∑j

aje+j (r⊥) =

∑j,k

c+jkαke+j (r⊥), (22)

where ᾱ = (¯̄c+)−1ā, whereas for the backward propagat-ing field, we have∑

j

bje−j (r⊥) =

∑j,k

γ−jkαke−j (r⊥), (23)

where ¯̄γ− = ¯̄PL/2¯̄Rt

¯̄PL/2¯̄c+. The total field can then be

written as

E(r⊥, z0) =∑j,k

c+jkαke+j (r⊥) +

∑j,k

γ−jkαke−j (r⊥), (24)

9

and the cavity mode contribution is given by

EC(r⊥, z0) =∑

j

c+jMαMe+j (r⊥) +

∑j

γ−jMαMe−j (r⊥).

(25)

B. Far field calculation

The electric field at the surface z = zS of the micropil-lar is computed from the electric field expansion Eq. (20)using the total transmission matrix of the top DBR. Wethen determine the far field EF(θ, φ) on the surface ofa sphere S of radius R using a standard near field tofar field transformation54. To take into account the lim-ited NA of the lens, we set EF(θ > θMax, φ) = 0 withθMax = sin

−1(NA). Following the procedure in Refs. 12and 13, we then consider a Gaussian beam in the paraxiallimit in a focal plane z = zF given by

Eg(x, y, zF) =

√2cµ0πw20

exp

(−x

2 + y2

w20

)x̂, (26)

Hg(x, y, zF) =1√cµ0

ẑ×Eg, (27)

where w0 is the beam waist and the integrated power inthe plane z = zF is normalized to 1. We then computethe associated far field EgF and H

gF on the surface S, and

the overlap with the Gaussian beam is then given as

O(w0, zF) = R2∫ (

E∗F,θHgF,φ − E∗F,φH

gF,θ

)sin θdθdφ, (28)

where the subindices θ and φ indicate the field compo-nents in spherical coordinates. Finally, the transmissionto the lens taking into account the coupling to the Gaus-sian profile is given by

PLens =1

2Max|O(w0, zF)|2. (29)

The power PLens,C coupled to the lens from the cavity isobtained by calculating EF(θ, φ) using the cavity contri-bution to the electric field given by Eq. (25) instead.

C. Comparison of efficiency and the one fromsimplified model

We compare the calculated efficiency ε obtained fromthe full model Eq. (1) and the efficiency εs from thesingle-mode model in Fig. 6. Excellent agreement be-tween the two models is observed, indicating that thecontribution from radiation modes to the collection oflight in a Gaussian mode at the first lens is negligible.

D. Modeling of sidewall roughness

The sidewall scattering Q−1edge can be computed using

a model44 taking into account the ratio of the mode in-

0.1

0.4

0.7

1

1 2 3 4

d (µm)

ntop=17 ε

εs

ntop=25 ε

εs

FIG. 6. The full efficiency ε and the single-mode efficiency εscomputed using ntop = 17 and 25 top DBR layer pairs. Theideal structure with κ = 0 is considered.

102

103

104

105

1 2 3 4

d (µm)

ntop=17 QintQ

ntop=21 QintQ

ntop=25 QintQ

FIG. 7. The intrinsic cavity quality factor Qint and the Qfactor taking into account sidewall scattering as a function ofpillar diameter d for κ = 0.2 µm.

tensity at the micropillar edge and its diameter d as

Q−1edge = κJ20 (ktd/2)

d/2. (30)

Here, κ is a phenomenological constant, J0 is theBessel function of the first kind, kt is defined askt =

√n2k20 − β21 with n being the refractive index

of GaAs, k0 = ω/c and β1 denoting the propagatingconstant of the fundamental HE11 mode in the cavitysection. The Q factor in the presence of sidewall rough-ness is then determined from Eq. (30) and the intrinsicquality factor Qint using Eq. (19) and is presented inFig. 7. The sidewall roughness represents an additionaldecay channel of the light in the cavity leading to areduction of Q, which becomes more significant for highintrinsic quality factors. As the pillar diameter increases,this influence decreases as the relative overlap of theHE11 mode with the boundary is reduced.

10

0.82

0.88

0.94

1

15 17 19 21 23 25 27 29

ntop

ε

β

γ

FIG. 8. The efficiency ε, the spontaneous emission β factorand the transmission γ as a function of top DBR layer pairsntop for a pillar diameter d = 3.2 µm with κ = 0.

E. Dependence on the top DBR thickness

The efficiency, the spontaneous emission β factor andthe transmission γ are presented in Fig. 8 as function ofthe number of top DBR layer pairs. As ntop increases,the cavity Q factor and in turn the Purcell factor areimproved leading to an increasing β. However, as thetop mirror reflectivity becomes comparable to that of thebottom DBR, an increasing fraction of light penetratesinto the substrate reducing the transmission γ. We thushave a trade-off between β and γ, where an optimumefficiency of ε ∼ 0.95 is obtained for ntop ∼ 21.

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