# Fileset

[entangleSuppl_20130629.pdf](https://mdr.nims.go.jp/filesets/db83834d-f9c6-4e4e-a8e8-1ef1a939589e/download)

## Creator

[Takashi Kuroda](https://orcid.org/0000-0001-6445-7673), [Takaaki Mano](https://orcid.org/0000-0002-6955-260X), Neul Ha, Hideaki Nakajima, Hidekazu Kumano, Bernhard Urbaszek, Masafumi Jo, Marco Abbarchi, [Yoshiki Sakuma](https://orcid.org/0000-0001-6804-7217), Kazuaki Sakoda, Ikuo Suemune, Xavier Marie, Thierry Amand

## Rights

©2013 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Symmetric quantum dots as efficient sources of highly entangled photons: Violation of Bell's inequality without spectral and temporal filtering](https://mdr.nims.go.jp/datasets/f2f824a1-df2d-4253-abc0-2d1f493cdc78)

## Fulltext

Supplementary information onSymmetric quantum dots as efficient sources of highly entangled photons: violation ofBell’s inequality without spectral and temporal filteringTakashi Kuroda,1, 2, ∗ Takaaki Mano,1 Neul Ha,1, 2 Hideaki Nakajima,1, 3Hidekazu Kumano,3 Bernhard Urbaszek,4 Masafumi Jo,1 Marco Abbarachi,1Yoshiki Sakuma,1 Kazuaki Sakoda,1 Ikuo Suemune,3 Xavier Marie,4 and Thierry Amand41National Institute for Materials Science, 1 Namiki, Tsukuba 305-0044, Japan2Graduate School of Engineering, Kyushu University, Japan3Research Institute for Electronic Science, Hokkaido University, Sapporo 001-0021, Japan4Université de Toulouse, INSA-CNRS-UPS, LPCNO,135 avenue de Rangueil, 31077 Toulouse, France(Dated: July 5, 2013)Supplementary Figure 1. (a) The X line PL spectrum de-tected by a charge-coupled devise detector (black diamonds)and its Lorentzian fit (red line). (b) The XX line PL spec-trum. (c) Peak energies of X (EX, blue circles) and XX (EXX,red diamonds) as a function of the angle of linearly-polarizedprojection. The shaded area shows the radiatively-limitedspectral width (1.2 µeV in FWHM). They are studied in thesame dot as that used in the correlation measurement.DH φVRAL(a) (b)Supplementary Figure 2. Photon correlations using linearpolarization bases. (a) The correlation visibility (C) of lin-ear polarizations for different polarization angles. The angleused in these plots is defined as the azimuth angle (φ) of apolarization state that moves in the equator of the Poincarésphere (φ = 0 for H and φ = 90◦ for D). Note that φ is dou-ble the angle of a polarization analyzer. There is no signifi-cant dependence of C on the polarization angle within error.The higher C value for circular bases, indicated by an arrow,is related to exciton depolarization due to nuclear fluctua-tions. See, Supplementary Discussion for detail. (b) Normal-ized coincidence counts as a function of the X polarizationangle (φX) for four different values of the XX polarization(φXX). The error bars include only Poissonian noise. Thesinusoidal fits are also shown by lines. The S parameter isestimated to be 2.03± 0.08 for these polarization sets. The Svalue is lower than that estimated in the main text becauseof the use of different measurement bases and the fact thatCR/L > CH/V ≈ CD/A.2Supplementary DiscussionSource of entanglement degradation. The degreeof quantum interference is suppressed by the presence ofincoherent light. If its spectrum coincides with the XXand X lines, unwanted photons would reach the detec-tors and mask the interference. However, we expect thephoton noise to be negligible because of the fact that thecorrelation visibility is independent of the bandpass ofthe spectral filters, which are varied from 60 to 200 µeV.Another potential source of entanglement degradation,which has recently been pointed out, is recapture [1, 2].In this process, the intermediate X state is re-excited tothe XX state before recombination, along with the cap-ture of hot carriers or the reabsorption of pump light.Recapture leads to the emission of more than two pho-tons per excitation cycle and these photons act as noise.However, we do not expect recapture to play a majorrole in the present case. The experiments were carriedout at sufficiently low excitation power, and we observean asymmetric coincidence profile, which confirms thatthe photon pairs come from a clean cascade. Here wediscuss the effect of exciton depolarization on quantuminterference, which has thus far been overlooked.Exciton depolarization is associated with asymmetricenvironmental motion, which includes the charge and nu-clear fluctuations in the vicinity of a dot. Any asymmet-ric perturbation can lift the exciton degeneracy. Notethat, if the energy shift is constant, the time-resolvedemissions show temporal oscillations in the polarizationdegree due to the superposition of the split eigenstates.If the energy shift changes randomly, the polarizationdegree becomes quenched through ensemble (time) av-eraging. The correlation visibility is thus estimated asΓ1/(Γ1 + ∆), where Γ1 is the recombination rate, and ∆is the standard deviation of the fluctuating exciton splits.The effect of a slowly varying environment is probably thelast remaining source which limits the degree of quantuminterference in the solid-state photon source.We observe that the correlation visibility is higher for acircular polarization basis than linear polarization bases,i.e., CR/L > CH/V = CD/A. The same signature hasbeen observed previously [1, 2]. We interpret this orien-tation dependent depolarization in terms of the fluctuat-ing electric fields (with localized charges) and magneticfields (with nuclei). Since a vertical electric field does notbreak the rotational symmetry, only an in-plane electricfield gives rise to the degeneracy lift of the exciton state.A frozen in-plane field along x splits the exciton state intolinearly polarized states, which lead to the observation ofCx/y > Cx+45◦/y+45◦ = CR/L. If the electric field fluc-tuates randomly in all directions, the polarization degreebecomes isotropic so that CH/V = CD/A = CR/L.In a GaAs quantum dot the carrier spins are interact-ing with the nuclear spins of the atoms that form the dot[3]. In the absence of optical pumping, the mean valueof the nuclear spin polarization (represented by an effec-ABABPBS PBSWP WPΓSupplementary Figure 3. Photon correlation setup. Pho-ton pairs are generated with a probability of Γ. Each photon isprojected to a specific measurement base, using an appropri-ate wave plate (WP) followed by a polarization beam splitter(PBS).tive magnetic field Bn called Overhauser field) is zero,but the carrier spins are subject to the field fluctuationsδBn, which is in the tens of mT range. Since the nuclearspin configuration changes randomly on a ms time-scale,we average over all possible orientations of δBn in thepresent experiments.The effect of δBn on the bright X states depends onthe orientation [4]. (i) A transverse (in-plane) compo-nent (δBTn ⊥ [111]) induces electron spin precession andcouples the bright X state |+1〉 (|−1〉) to the dark Xstate |+ 2〉 (| − 2〉). The bright states and the darkstates are separated by the isotropic electron hole ex-change splitting δ0 ≈ 350 µeV in our dot sample [5]. Thecoupling strength and the associated energy renormal-ization scale as the ratio |δn|/δ0, where |δn| = geµB |δBn|with ge = 0.5 for our dot [6] and µB is the Bohr magne-ton (µB = 57.9 µeV/T). In our system a typical value of|δBn| = 10 mT corresponds to |δn| = 0.25 µeV and hence|δn| � δ0. As a consequence, the effect of the in-planecomponent of δBn on the X energy and polarization isnegligible. (ii) A longitudinal component (δBLn ‖ [111])has two effects. First, it increases the FSS from the ini-tial value of δ1 to√δ21 + (geµBδBn)2. If δ1 is negligi-ble, the energy shift is approximated as |δn|. Second,the X eigenstates become more circular and as a resultCR/L > CH/V = CD/A. Using |δn| = 0.25 µeV one wouldexpect to find CR/L − CH/V = |δn|/Γ1 = 0.1, whichis in excellent agreement with the experimental results.Thus, the nuclear field fluctuations introduce an addi-tional anisotropy that needs to be controlled to optimizethe degree of entanglement.Supplementary MethodNormalization procedure used in the correlationanalysis. A typical setup for measuring the polarizationcorrelations is shown in Supplementary Fig. 3. Photonpairs, which are generated with a rate of Γ, are dividedinto two photons, either of which is projected on a polar-ization base of A, and the other is projected on a polar-ization base of B. Polarization beam splitters transmitthe projected photons, and reflect the orthogonal com-plements, Ā and B̄. The number of coincidence per unit3time (NAB) is expressed by,NAB = Pr(A ∩B)Γ ηAηB , (1)where Pr(A∩B) is the joint probability that one photonis projected along A, and the other is projected alongB. ηA (ηB) is the counting efficiency for A (B) projectedphotons. The final purpose of this measurement is theestimation of Pr(A ∩ B). Equation 1 contains an un-known factor of ΓηAηB , which can be determined sepa-rately in principle. However, for a dot-based source, thisfactor fluctuates in time (due to the small vibration ofthe micro-objective setup), and its precise determinationis practically impossible. Here we describe that adequatetwo-step normalizations ideally remove the effect of thefluctuations.The first normalization is based on the number of ac-cidental coincidence. It is counted simultaneously with“true” coincidence, and appears as side peaks in the co-incidence histogram. Hereafter, we assume that XX pho-tons are projected on A, and X photons are projected onB. The number of accidental coincidence is given by,N sideAB = pAΓXX ηA × pBΓX ηB , (2)where pA is the probability of finding A polarized XXphotons, and pB is that of finding B polarized X photons.We are able to determine pA and pB by performing astandard photoluminescence measurement. ΓXX (ΓX) isthe probability that XX (X) occupies the dot. Note thatΓ = ΓXX ≤ ΓX for the dot-based sources [7]. We definea normalized coincidence count as,nAB ≡NABN sideAB=1ΓX pApBPr(A ∩B). (3)Note that the factor ηAηB in Eq. 1 disappears in Eq. 3.Thus, nAB is not affected by the fluctuation of collectionefficiencies.Equation 3 still contains a parameter of ΓX that fluc-tuates with time. The relevant effect will be removedby the second normalization, which uses the number ofcoincidence with the orthogonal complement. Here we si-multaneously count B̄ polarized X photons and integratenAB̄ along with nAB . We define Pr(B|A) as the condi-tional probability that X photons are projected along B,given that XX photons are projected along A, so thatPr(B|A) =Pr(A ∩B)pA, (4)According to the definition of projection measurements,we havePr(B|A) + Pr(B̄|A) = 1. (5)The above relation allows us to normalize nAB and nAB̄ ,and finally obtainPr(B|A) =pB nABpBnAB + pB̄nAB̄, (6)Pr(B̄|A) =pB̄ nAB̄pBnAB + pB̄nAB̄, (7)and thusPr(A ∩B) =pApB nABpBnAB + pB̄nAB̄, (8)Pr(A ∩ B̄) =pApB̄ nAB̄pBnAB + pB̄nAB̄. (9)Note that the above forms are free from any fluctuatingfactor.To test the CHSH inequality, we measure the degreeof correlation for A projected photons and B projectedphotons, i.e.,CA/B = Pr(A∩B)−Pr(A∩ B̄)−Pr(Ā∩B) +Pr(Ā∩ B̄).(10)Then, the CHSH equality is given byS = CA/B + CA′/B − CA/B′ + CA′/B′ < 2. (11)In the present experiment, we confirmed that both XXand X emissions are unpolarized within an error of 5 %.If we assume a perfectly unpolarized source so that pA =pĀ = pB = pB̄ = 1/2, the above forms become,Pr(A ∩B) =12nABnAB + nAB̄, (12)Pr(A ∩ B̄) =12nAB̄nAB + nAB̄. (13)They are used to plot Fig. 3(a) in the main text andSupplementary Fig. 1(b).∗ kuroda.takashi@nims.go.jp[1] R. J. Young, R. M. Stevenson, A. J. Hudson, C. A. Nicoll,D. A. Ritchie, and A. J. Shields, Phys. Rev. Lett. 102,030406 (2009).[2] A. Dousse, J. Suffczyǹski, A. Beveratos, O. Krebs,A. Lemâıtre, I. Sagnes, J. Bloch, P. Voisin, and P. Senel-lart, Nature 466, 217 (2010).[3] T. Belhadj, T. Kuroda, C.-M. Simon, T. Amand, T. Mano,K. Sakoda, N. Koguchi, X. Marie, and B. Urbaszek, Phys.Rev. B 78, 205325 (2008).[4] R. M. Stevenson, C. L. Salter, A. Boyer de la Giroday,I. A. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields,“Coherent entangled light generated by quantum dots inthe presence of nuclear magnetic fields,” arXiv:1103.2969(2011).[5] G. Sallen, B. Urbaszek, M. M. Glazov, E. L. Ivchenko,T. Kuroda, T. Mano, S. Kunz, M. Abbarchi, K. Sakoda,D. Lagarde, A. Balocchi, X. Marie, and T. Amand, Phys.Rev. Lett. 107, 166604 (2011).[6] M. V. Durnev, M. M. Glazov, E. L. Ivchenko, M. Jo,T. Mano, T. Kuroda, K. Sakoda, S. Kunz, G. Sallen,L. Bouet, X. Marie, D. Lagarde, T. Amand, and B. Ur-baszek, Phys. Rev. B 87, 085315 (2013).[7] T. Kuroda, T. Belhadj, M. Abbarchi, C. Mastrandrea,M. Gurioli, T. Mano, N. Ikeda, Y. Sugimoto, K. Asakawa,N. Koguchi, K. Sakoda, B. Urbaszek, T. Amand, andX. Marie, Phys. Rev. B 79, 035330 (2009).