Bell inequalities and density matrix for polarization entangled photons out of a two-photon cascade in a single quantum dot Matthieu Larqué, Isabelle Robert-Philip, Alexios Beveratos

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Matthieu Larqué, Isabelle Robert-Philip, Alexios Beveratos. Bell inequalities and density matrix for polarization entangled photons out of a two-photon cascade in a single quantum dot. Physical Review A, American Physical Society, 2008, 77, pp.042118. ￿10.1103/PhysRevA.77.042118￿. ￿hal-00213464v2￿

HAL Id: hal-00213464 https://hal.archives-ouvertes.fr/hal-00213464v2 Submitted on 8 Apr 2008

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. hal-00213464, version 2 - 8 Apr 2008 iebnetnldpoos[7,o oaiainentan- polarization or release may [17], decay photons This entangled 16]. time-bin [14, exciton bright relay through opposite two radiatively decays with which dot and of momentum the angular consists in which trapped biexciton, pairs excitation two-electron-hole a this each involves dots, emission quantum upon self-assembled cascade photons in system example, of four-level For pair a single cycle. as single a described a or be atom emitting system, can such single dipole In single a candidated. the good as a be (such may emission dot) dipole cascade quantum the single on a based sources from view, of From point all efficient. this more rendering much hence protocols pair, mentioned above photon the probabil- single a increased multipair containing with these of pulses ity light suppress create to to pho- possible and entangled events On it of make source 0.1). would deterministic than tons a lower hand, (usually oper- other coherence pulse the be per excitation generation to pair or photon sources producing length of rates of these low likelihood forces at ated the pairs minimize photon to multiple entanglement need multipair of The to visibility leading [13]. the pairs decreases photons which to stat- emitted emission, easy Poissonian the and the of However, from useful sitics suffer very 12]. always be they 11, may implement, [10, sources rate these generation although bandwidths maximal spectral a of narrow with sources combine non-linear 4-wave Such can on entanglement investigated. based also paramet- sources are by other mixing exper- obtained but these downconversion, were In ric photons entangle- entangled 10]. or [9, iments, 8] demonstrations [7, real swapping setups in ment teleportation implemented application quantum been advanced have world most and [3] the re- entanglement of relays Quantum using one quantum 6]. probably of are [5, lays processing realization information cryp- the quantum quantum or to from [2], ranging tography science, information tum nage htnpisaea seta olfrquan- for tool essential an are pairs photon Entangled elieulte n est arxfrplrzto enta polarization for matrix density and inequalities Bell ASnmes 25.v 86.c 10.a 03.65.Ud 81.07.Ta, 78.67.Hc, 42.50.Dv, numbers: PACS inequalities. Bell’s of increa effects violation cavity for by allowing states model excitonic our the vi of dots, for enhancement threshold quantum a self-assembled derive InAs/GaAs and standard explici processes We these of levels. dynamics relay the these between o splitting exchange diff energy population includes dephasing, deri model as such Our analytically entanglement state. destroy and entangled dots the of quantum inequalities semiconductor single in etertclyivsiaetejitpooeeto pro photodetection joint the investigate theoretically We .INTRODUCTION I. w-htncsaei igeqatmdot quantum single a in cascade two-photon NS-Lbrtied htnqee Nanostructures, et Photonique de Laboratoire - CNRS .Lrue .Rbr-hlp n .Beveratos A. and Robert-Philip, I. Larqu´e, M. ot eNzy -16 acuss FRANCE Marcoussis, F-91460 Nozay, de Route Dtd pi ,2008) 9, April (Dated: sc stastostruhtedr ttso pnflip spin entanglement or of states visibility the dark levels deteriorate the may excitonic processes) through the transitions between as exchange induc- (such population mechanisms two the a incoherent the degrade ing any of Moreover, also polarization may the photons. [26]) between vicinity correlations dipole strong the in lo charges cated fluctuating phonons with interactions with electrostatic collisions and through en- example solid-state (for due the vironment information entanglement. with recover interactions path and dephasing which structure However, the fine excitonic erase or the to to [23] pri us in magnetic allow can external [25]...) ciple optimization of growth field, (by use [24] levels electric [22], excitonic the filtering of spectral emit- linewidth splitting energy radiative the excitonic the of the within energy Reducing the [21]. via photons ted along two released the pathways were degeneracy which photons states about excitonic information provides the ex- and electron-hole lifts of such interaction 20]; anisotropy change [19, in-plane function wave by exciton anisotropic the the caused by interaction emitters, spoiled exchange single is solid-state entanglement such polarization in However, indistin- otherwise paths are decay guishable. entanglement which radiative two polarizations polarization of different existence with of the origin in resides The here single indistinguishable [18]. also successive can photons two photons from entangled obtained Time-bin be [14]. photons gled ainecag ewe h w ea ees ederive We levels. relay two the a popu- between incoherent with exchange to interacting lation subject system and four-level environment defining a solid-state by of begins paper Hamiltonian the a of relay part two following The these exci- between states. cross-dephasing the and split- between levels energy exchange tonic exciton process population as incoherent incoherent such account Several ting, in taken inequalities. been have CHSH non-optimal based the a witness on as entanglement well interesting as nevertheless matrix but density analyt- the and derive cascade biexciton ically the in probabilities tection hspprtertclyivsiae h on photode- joint the investigates theoretically paper This h ea xioi ttsadincoherent and states excitonic relay the f aiiiso h ixio-xio cascade biexciton-exciton the of babilities e h dlt fetnlmn oavalue a to entanglement of fidelity the ses rn ehnssta a pi reven or spoil may that mechanisms erent l eaetefieiyo nageetto entanglement of fidelity the relate tly lto fBl’ nqaiis ple to Applied inequalities. Bell’s of olation etedniymti n h Bell’s the and matrix density the ve niae htsotnosemission spontaneous that indicates ge htn u fa of out photons ngled n- - . 2 from such Hamiltonian a time evolution equation of the levels: (1) dephasing processes that occur simultaneously system excited on its upper state and derive the joint and attach the same information on the phase and energy photodetection probability. In section III, we quantify of these two levels with a dephasing rate denoted Γ1 and the entanglement of the photons produced by deriving an (2) dephasing processes that do not affect identically the analytical expression of the Clauser-Horne-Shimony-Holt two relay levels and whose impact depends on the polar- (CHSH) inequality as a function of the different dynam- ization of the excitonic states. These last processes will ical parameters of the four-level system, as well as the be described by polarization-dependent dephasing rates density matrix corresponding to the biexciton cascade. ΓH and ΓV . The cross-dephasing between the two relay We then stress in section IV the necessity to make use of states is therefore Γ = ΓH +ΓV . This model includes all the Purcell effect [27], in order to violate Bell’s inequali- possible dephasing processes without population modifi- ties from the cascade emission in self-assembled quantum cations that may occur. dots.

II. THEORETICAL FRAMEWORK

A. The four-level system

In the cascade emission from a four-level system, the decay paths involve two radiative transitions, one from an upper level 2 to an intermediate state 1H or 1V and the other from| i these relay states to the| groundi | statei 0 (see Fig.1). The energies of these levels 2 , 1 and 1| i | i | H i | V i are respectively denoted ~(ω1+ω2), ~(ω1+δω) and ~(ω1 − δω). We will futher assume that this 2 , 1H , 1V , 0 basis corresponds to the eigen basis of{| thei | quantumi | i dot,| i} with therefore an excitonic energy splitting 2δω but no coherent coupling between the two excitonic eigenstates FIG. 1: Schematic description of the two-photon cascade in a [21]. Radiative transitions from the biexciton in such typical four-level system with an energy splitting 2~δω of the basis release colinearly polarized photons with linear po- relay level, yielding two colinearly polarized photons (either larization denoted H and V (see Fig. 1). In the ideal H or V ). case (δω = 0), the four-level system relaxes, generating the maximally entangled two-photon state:

+ 1 Φ = ( H,ω1 H,ω2 + V,ω1 V,ω2 ) (1) | i √2 | i| i | i| i B. Dynamics of the four level system by cascade emission [1, 14]. The phase difference between the two component states H,ω1 H,ω2 and In order to account for the open nature of the four- | i| i V,ω1 V,ω2 is null, as determined by the angular mo- level system (resulting from its coupling with the phonon menta| i| of thei different involved levels and the Clebsch- and the photon reservoirs for example), we describe the Gordan coefficients [29]. Unfortunately, in realistic two- time evolution of the density operator ρ by means of the level systems (such as single quantum dots for example), following master equation in the Lindblad form [31]: the relay levels are split (δω = 0). Furthermore, relax- ation mecanisms between the6 two relay states 1 and | H i dρ 1 can occur (for example from spin flip processes). = [iH,ρ] + ( r + d + flip)ρ (2) V dt − L L L They| i will be accounted for by two phenomenological de- cay rates Γflip δΓflip, that will be latter supposed to In the previously described eigen basis ± be equals (which is a good approximation for a small 2 , 1H , 1V , 0 of the four-level system, the excitonic energy splitting). {|hamiltoniani | i | Hi |hasi} the form: In addition, the relay levels and the upper level may be subject to sudden, brief and random fluctuations of their energies without population exchange (arising, for H = (ω1 δω) 1V 1V +(ω1+δω) 1H 1H +(ω1+ω2) 2 2 example, from collisions with thermal phonons). In our − | ih | | ih | | (3)ih | model, the ground level 0 is chosen as the reference in The Lindblad operators include three contributions. The energy and phase. Dephasing| i of the upper level 2 is first one describes the interaction of the emitter with the | i described by the dephasing rate Γ2. On the two relay electromagnetic field by emitting photons, whenever it levels 1H and 1V , we distinguish two dephasing pro- undergoes a transition from its upper state to the relay cesses| withouti population| i exchange between these relay levels or from the relay levels to the ground state. This 3 radiative relaxation is accounted for by the following Li- The operator S∆ = 1H 1H 1V 1V is related to ouvillian: population difference| betweenih | − the | twoih excitonic| relay states. The two other Pauli’s matrices S = ı 1 1 Q | V ih H |− γ1 γ2 ı 1H 1V and SP = 1H 1V + 1V 1H correspond to r = ( ( 0 1p )+ ( 1p 2 )) (4) the| ih quadratures| of the| dipoleih | between| ih these| two relay L = 2 L | ih | 2 L | ih | p XH,V states. The A matrix is given by: where γ1 and γ2 are respectively the radiative decay γ1 2Γ 0 0 rates between the relay states and the ground state and − − flip between the upper level and the relay levels. We as- A = 0 µQ 2δω (9)  0 −2δω µ  sume that these decay rates do not depend on the de- − − P cay path the photons were released along. (D)ρ =   2DρD† D†Dρ ρD†D is the Lindblad operator.L The The decay constants µP/Q are equal to − − γ1 +Γflip δΓflip + Γ. second contribution d is related to dephasing processes ± and reads: L For further reference, we define the matrix transforma- tion M(U) of V , where U is an arbitrary unitary trans- = Γ2 ( 2 2 )+ Γ ( 1 1 ) formation of the excitonic levels of the source (letting the Ld L | ih | pL | pih p| = upper and fundamental states unchanged), by p XH,V +Γ1 ( 1H 1H + 1V 1V ) (5) † L | ih | | ih | T r[US∆U ρ(t)] † This Liouvillian includes phenomenologically any de- M(U)V (t)= T r[USQU ρ(t)] (10)  †  phasing effect (e.g. phonons) occuring on the levels of the T r[USP U ρ(t)] dot without population transfers as described previously.   The last contribution flip accounts for the incoherent M(U)V(t) are the mean values (8) measured at time t coupling between the twoL relay states: under the transformed basis 2 ,U 1 ,U 1 , 0 . {| i | Hi | V i | i}

= α ( 1 1 + 1 1 ) Lflip P L | H ih V | | V ih H | C. Joint photodetection probability +β (i( 1 1 + 1 1 )) P L | H ih V | | V ih H | +βQ ( 1V 1H 1H 1V ) (6) Violation of Bell inequalities as well as the reconstruc- L | ih | − | ih | tion of the density matrix is experimentally obtained by +αQ (i( 1V 1H 1H 1V )) L | ih | − | ih | measuring the joint photon detection probabilities P±,± The phenomenological rate Γflip between the two relay on the output of a binary polarization analyzer such as a states 1H and 1V appears to be twice the sum polarizing beamsplitter. Since several points of the Bloch | i | i of the different rates αi and βi (i = P,Q) involved sphere have to be measured[1], a quater-wave plate fol- in this equation. The rate δΓflip expresses likewise lowed by a half-wave are inserted in the photons path (see as : 2(αQ αP + βQ βP ). These rates simulate fig.2). The exciton and biexciton photon are spectrally − − any unspecified process inducing an incoherent inter- separated by means of optical filters and send through action between the two relay levels with population the optical path denoted i (i=1 or 2 for the exciton and exchange before radiative relaxation. δΓflip accounts biexciton respectively) for assymetry of these processes. These non-radiative processes may include spin-flip processes and transitions through dark states (assuming that the probability for the source to be in these dark-states is small compared to the probabilities related to the optically active states).

In accordance with all these assumptions, the time evo- lution of the four-level system can be decomposed by use of master equation (2) in a set of differential equations, which reduces for the purpose of this paper to: dV = AV (7) dt FIG. 2: Experimental setup for measuring CHSH or recon- structing the density matrix In this equation, V is a vector composed of the following mean values:

T r[S∆ρ] The fast axis of the quarter-(resp. half-)wave plate is V = T r[SQρ] (8) rotated by an angle χ (resp. θ ) with respect to the hori-   i i T r[SP ρ] zontal polarization direction defined by the optical table.   4

By applying the projection theorem, measuring +1 in the applying the quantum-measurement projection postu- optical setup i corresponds to the detection of a photon i late. First the photon at energy ~ω2 and polarized along † † emitted by the source with the polarization Λ(θi,χi) H Λ(θ2,χ2) H is detected at time t2, which projects the | i | i ∗† where Λ(θ,χ) describes the transformation of the polar- emitter on the superposition Λ(θ2,χ2) 1 of the ex- | H i ization basis H, V when a photon successively propa- citon states 1H and 1V . Secondly, the superposition gates through{ a quarter-} and a half-wave plates rotated state evolves| ini time| untili the detection of the second by the angles θ and χ. photon at energy ~ω1 at time t1 + t2. Consequently, this conditional probability will be related to the population † Λ(θ,χ)= R(θ)T (π)R(χ θ)T (π/2)R( χ) (11) in the superposition Λ(θ1,χ1) 1 at time t2 +t1, know- − − H ing that the intermediate levels| werei in the superposition where R(x) is the rotation matrix and T (r) is the Jones ∗† Λ(θ2,χ2) 1H at time t2. All these probabilities are in- matrix of a retarder plate. tegrated over| thei photodetection time window. cos(x) sin(x) 1 0 The population at time t1 + t2 in the superposition −γ1t1 R(x)= ,T (r)= −ır (12) 1 (θ1) can be expressed as [e + S∆ (t2 + t1 t2)]/2 sin(x) cos(x) 0 e | H i h i |  −    where S∆ (t2 + t1 t2) is the first value of the vector V h i | In the following, for the sake of clarity we will denote of Eq. 8 measured under the transformation of Eq. 10 † † with U = Λ(θ1,χ1) , after a free evolution during the Λ(θ,χ) 1H the superposition of the source’s states 1H and 1 | whichi analytically corresponds to the same| i time t1 (Eq. 7) with the assumption of the inital state V init ∗† | i † V corresponding to the excitonic state Λ(θ2,χ2) H transformation Λ(θ,χ) H of the photonic state H . | i | i | i at time t2. Thus by defining the vector V0 = 1, 0, 0 Experimentally one measures the joint photodetection { } det which corresponds to the values of V measured in the probabilities P±,±(θ2,χ2,θ1,χ1) of the first photon and second photon in channels of their respective optical eigenbasis with the source in the state 1H , it follows: ± | i setups with each retarder plate rotated by θi and χi. init ∗† −1 The source is pumped at time t = 0 from its ground V = M(Λ(θ2,χ2) ) V0 state to the excited state 2 with a laser pulse shorter measured † At init | i V = M(Λ(θ1,χ1) )e V than the lifetime 1/(2γ2) of the upper state. We will futher postselect joint photodetection events correspond- S∆ (t1 0) = S∆ (t2 + t1 t2) (13) h i | h i |† At ∗† −1 ing to a sequential detection of the biexcitonic photon = [M(Λ(θ1,χ1) )e M(Λ(θ2,χ2) ) ]11 and then of the excitonic photon during one excitation cycle. In this context, the probability of joint photodetec- det where [. . .]ij denotes the matrix element on row i and col- tion P+,+(θ2,χ2,θ1,χ1) is proportional to the emission umn j. The probability P+,+(θ2,χ2,θ1,χ1) can therefore probability P+ +(θ2,χ2,θ1,χ1) of a pair of photons with , be written as follows: respective polarization orientation Λ(θ ,χ )† H , at re- i i | i spective energies ~ω2 and ~ω1, assuming that the source +∞ is in state 2 at time t = 0. This radiative transition −2γ2t2 | i P+,+(θ2,χ2,θ1,χ1)= γ2e dt2 probability can be regarded as the product of two prob- 0 +∞ Z abilities: the probability of emission of the first photon γ1 † −γ1t1 with polarization Λ(θ2,χ2) H , multiplied by the con- (e + S∆ (t1 0)dt1 (14) × 2 h i | ditional probability of radiative| i transition from the re- Z0 lay levels to the ground state with emission of a pho- † ton polarized along Λ(θ1,χ1) H . This amounts in con- Upon integration, this probability reads in the partic- | i sidering the photon cascade as a two-step process and ular case χ1 = χ2 = 0:

1 γ1 γ1(γ1 +Γflip +Γ δΓflip) P+,+(θ2,θ1)= [1 + cos(4θ1)cos(4θ2)+ 2 − 2 2 sin(4θ1)sin(4θ2)] (15) 4 γ1 + 2Γ (2δω) + (γ1 +Γ + Γ) (δΓ ) flip flip − flip

For a perfect quantum dot, P+,+(0, 0) tends toward III. QUANTIFYING TWO-PHOTON 1/2 as expected. ENTANGLEMENT AND DENSITY MATRIX

Entanglement can be quantified by several means like measurement of the concurrence, tangle of the density matrix or entanglement witness operators. A non opti- mal entanglement witness, but nevertheless experimen- 5 tally simple to measure is the Bell inequality under the which, for classically correlated states, is bounded by CHSH form which discriminate between states that can S 2. In the case of an ideal entangled source, the be explained by a Local Hidden Variable Model (LHVM) |maximum| ≤ value of the CHSH coefficient S is obtained or not. The possible violation of Bell inequalities is ex- for every set of polarization directions of each analyzer ′ perimentally easy to verify by measuring the fringes vis- verifying θ2 = x + π/16 and θ2 = x +3π/16; θ1 = x and ′ ibility [36] of two-photon coincidences as a function of θ1 = x + π/8, where x is an arbitrary rotation of both (θ1 θ2) whereas other measurements need the experi- half-wave plates. In this context and under the assump- − mental knowledge of the density matrix. Hence we shall tion of δΓflip = 0, the CHSH parameter S is given by first derive the analytical form of the CHSH inequality, the formula: then generalize the result to the derivation of the density matrix and one possible entanglement witness [37] by use of the Peres criterion [28]. γ1 γ1(γ1 +Γ + Γ) S = √2 + flip 2 2 γ1 + 2Γflip (γ1 +Γflip + Γ) + (2δω) ! A. Violation of Bell’s inequalities (18) which is, as expected, independent of the arbitrary rota- The CSHS inequality is calculated by measuring the tion x. Violation of Bell’s inequalities implies S > 2. correlation coefficient for four sets of properly chosen an- gles of a half-wave plate, and therefor the angles χi refer- ring to the quarter wave plate are set to zero and omit- ted in the rest of this subsection. From the expression of Eq. 15 one deduces all the probabilities P±,±(θ2,θ1) and compute analytically in a straightforward manner the correlation coefficient of the form: B. Density Matrix E(θ~2, θ~1) = P+,+(θ~2, θ~1)+ P−,−(θ~2, θ~1)

P− +(θ~2, θ~1) P+ −(θ~2, θ~1) (16) − , − , In the above section, the Bell inequalities have been The generalized Bell’s inequality in the Clauser-Horne- derived form the coincidence probabilities when the ex- Shimony-Holt (CHSH) formulation [33] is expressed as a citon and biexciton are detected in a linear basis. We combination of such correlations functions as: will now exploit the more general expression of the joint ′ ′ ′ photodetection probability P+,+(θ2,χ2,θ1,χ1) obtained S(θ2,θ ,θ1,θ ) = E(θ~2, θ~1) E(θ~ , θ~1) 2 1 − 2 upon integration of Eq. 14. By definition of the density ~ ~′ ~′ ~′ +E(θ2, θ1)+ E(θ2, θ1) (17) matrix, this probability can also be expressed as

ρ † P (θ2,χ2,θ1,χ1)= H H (Λ(θ2,χ2) Λ(θ1,χ1)) ρ (Λ(θ2,χ2) Λ(θ1,χ1)) H H +,+ h XX X | ⊗ · · ⊗ | XX X i where ρ is the density matrix of the pair of photon in the where basis HXX HX , HXX VX , VXX HX , VXX VX . By iden- { ρ } 1 γ1 +Γflip tifying P+,+(θ2,χ2,θ1,χ1) = P+,+(θ2,χ2,θ1,χ1) for 16 α = 2 γ1 + 2Γ well chosen set of four angles (θ2,χ2,θ1,χ1) we construct flip 1 Γ a linear system of 16 independent equation whose un- β = flip known variables are the 16 real values of ρ. In this way, 2 γ1 + 2Γflip we simply reconstruct the density matrix from the joint 1 γ1(γ1 +2Γ+Γflip) photodetection probabilities[15] and obtain a theoretical d = 2 2 2 (20) 2 (2δω) + (γ1 +Γ + Γ) (δΓ ) value of ρ. Same holds for an experimental approach. flip − flip The calculated density matrix is hence : 1 γ1δω c1 = 2 2 2 2 (2δω) + (γ1 +Γ + Γ) (δΓ ) flip − flip 1 γ1δΓflip c2 = 2 2 2 α 0 0 d ıc1 2 (2δω) + (γ1 +Γ + Γ) (δΓ ) − flip flip 0 β c2 0 − ρ = (19)  0 c2 β 0  Note that for a perfect quantum dot (δω 0 Γflip 0, Γ 0), ρ tends, as expected, toward the→ Φ+ Bell→  d + ıc1 0 0 α    → | i 6 state: ρ φ+ φ+ . and with a simple analytical form: → | ih | 0 0 0 1/2 0 1/20− 0 W = (24)  0 01/2 0  C. Quantum dot spectroscopy from quantum 1/20 0 0 tomography  −  T r[Wρ] = β d  − 1 Γflip An interesting feature of the analytical form of the den- = (25) 2 γ1 + 2Γ sity matrix arises from the fact that once computed ex- flip perimentally, one can deduce all the quantum dot param- 1 γ1(γ1 +2Γ+Γflip) 2 2 2 eters provided γ1 is measured independently. They are −2 (2δω) + (γ1 +Γflip + Γ) (δΓflip) − expressed as a function of the density matrix elements :

IV. RESTORATION OF ENTANGLEMENT THROUGH CAVITY EFFECTS c1 δω = γ1 4(d2 + c2 c2) 1 − 2 c2 In the above section we have analytically derived the δΓflip = γ1 2 2 (21) CHSH inequality as well as the density matrix for the 2(d2 + c c ) 1 − 2 biexciton-exciton cascade emission from a single semi- 2β Γflip = γ1 conductor quantum dot. Although we could discuss on 4β 1 the density matrix as a function of the internal parame- − 2 2 d(1 2d 4β +4dβ)+2(c1 c2)(2β 1) ters of the QD, we choose to discus the CHSH inequal- Γ = γ1 − − 2 2 2 − − ity since it is an intuitive entanglement witness with a 4(d + c1 c2)(4β 1) − − simple experimental realization. Equation 18 indicates that polarization entanglement in the cascade emission from a biexciton in a self-assembled quantum dot may be affected by the relative contribution of three processes D. Entanglement witness with respect to the exciton radiative lifetime 1/γ1: the mutual coherence between the two non-degenerate exci- Apart of the CHSH inequality other entanglement wit- tonic levels described by a cross-dephasing time 1/Γ, the nesses can be constructed and following [37] we define an excitonic energy splitting giving rise to quantum beats entanglement witness as T r[Wρ] where W is an opera- with a time period 2π/2δω and the incoherent popula- tor. In the case where W is an optimal witness, the above tion exchange between the two bright excitons with a mentioned quantity is negative if ρ is an entangled state. decay time 1/Γflip. Entanglement does not depend on As proposed in [28] we define the partial transpose ρT2 the biexciton radiative rate (γ2) and among all the de- 0 phasing processes taken into account in our model, only of an arbitrary density matrix ρ0 as follows : the cross-dephasing between the excitonic levels affects the visibility of entanglement. The analytical expression ρ0 = ρ ij kl of S given by (18) also confirms that polarization en- 0 i,j,k,l| ih | i,j,k,l=H,V tanglement from the biexciton cascade in self-assembled X quantum dots is exclusively affected by the dynamics and ρT2 = ρ ij kl (22) 0 0 k,j,i,l mutual coherence of the excitonic states. = | ih | i,j,k,lXH,V For quantum dots with no excitonic energy splitting (δω = 0) and in absence of cross-dephasing (Γ = 0) and incoherent population exchange (Γ =0), the S quan- As demonstrated in [28], if ρT2 has a negative eigen- flip tity reaches its maximum value of 2√2 and the photons value λ associated to the eigenvector ν then the density emerging from the biexcitonic cascade are maximally en- matrix ρ represents an entangled state.| i Thus defining tangled [14]. Conversely, for quantum dots whose exci- W = ν ν T2 we have | ih | tonic states are splitted and which are affected by spin- dependent dephasing mechanisms and incoherent popu- T2 lation exchange between the exciton bright states, the T r[Wρ0]= T r[ ν ν ρ0 ]= λ< 0 (23) | ih | S parameter rapidly decreases so that the two photons emitted are only partially entangled or even only corre- In our case we choose for ρ0 the density matrix lated in one preferred polarization basis corresponding to φ+ φ+ toward which the biexciton density matrix ρ the polarization eigenbasis of the dot [21]. |of ourih model| tends to. This gives a non optimal witness As an example, the characteristic excitonic lifetimes but already less demanding than the Bell’s inequalities 1/γ1 of InAs quantum dots embedded in GaAs are typ- 7 ically on the order of 1 ns [34] and the excitonic energy tum beats period (Fγ1 2δω), the cross-dephasing time ≫ splitting 2~δω is of the order of few µeV [35] correspond- (Fγ1 Γ) and the decay time of incoherent excitonic ≫ ing to quantum beat periods lower than few hundreds population exchange (Fγ1 Γflip), it should be possible ps. Numerous observations also indicate that the exciton to preserve the quantum correlations≫ between the two re- spin relaxation is quite negligible on the timescale of the combination paths. We consider here that both excitonic exciton lifetime and may reach values of about 10 ns or transitions releasing either H or V -polarized photons are even higher [38, 39]. The mutual coherence time 1/Γ is accelerated by cavity effects with the same spontaneous likely to be longer than few ns [40], since it shall involve emission enhancement factor F . For dots subject to a hypothetical spin-dependent dephasing processes. These spontaneous emission enhancement of its excitonic tran- typical values indicate that the main ingredient affect- sition by a factor F = 10 (see solid line on Fig. 3), S val- ing entanglement is the excitonic energy splitting; they ues higher than 2.6 should be achievable for null exciton imply that in an experimental setup involving bare InAs energy splitting (2~δω = 0). In such microcavity source quantum dots, the S quantity is lower than 2 and tests of however, violation of Bell’s inequalities (S > 2) requires the Bell’s inequalities on the two photons emerging from the use of quantum dots with an excitonic energy split- the biexciton cascade will not lead to any violation of ting smaller than 7 µeV . With a Purcell effect of F = 30, the CHSH inequality (see dashed line on Fig. 3). Even S reaches the value of 2.76 close to its maximum value of for relatively small exciton energy splitting (2~δω higher 2√2 for dots with no exciton energy splitting, and Bell’s than few µeV ), the S value tends to 1.2, a value signifi- inequalities are violated for quantum dots displaying an cantly lower than the S = √2 limit of perfectly correlated energy splitting up to 20 µeV (see Fig. 4). photons without any hidden variables. The incoherent population exchange between the excitonic relay levels destroys entanglement and the emitted photons are a statistical mixture of HH , V V , HV , V H states. Even for bare quantum{| dotsi with| noi | excitoni | energyi} split- ting, entanglement is spoiled by cross-dephasing and in- coherent population exchange between the two bright ex- citonic states: the maximum value of S on Fig. 3 for such quantum dots reaches only 2.06, a value very close to the classical limit of 2.

FIG. 4: CHSH inequality as a function of the energy splitting of the exciton line and its spontaneous emission exaltation F , for a single quantum dot with T1 = 1/γ1 = 1 ns in bulk material, 1/Γflip = 10 ns and 1/Γ = 2 ns.

Figure 4 shows the value of S as a function of 2~δω and F for values of γ1, Γ and Γflip considered above as typical of currently available quantum dots. The results confirm that the main ingredient degrading entanglement is the exciton fine structure. However, FIG. 3: CHSH inequality as a function of the energy splitting of the exciton line, for a single quentum dot in bulk material reducing the exciton energy splitting within the exciton (dashed line) and subject to a Purcell effect with F = 10 linewidth is not experimentally sufficient and hardly (continuous line). Dotted line corresponds to the classical allows for violation of Bell’s inequalities. Violation of limit of 2. For these two curves, T1 = 1/γ1 = 1 ns, 1/Γflip = the CHSH inequalities requires a combination of cavity 10 ns and 1/Γ = 2 ns. effects enhancing the excitons spontaneous emission and techniques leading to a reduction of the exciton energy splitting (such as growth optimization [25] or Nevertheless, restoration of entanglement and im- use of external magnetic [23] or electric [24] field). For provement of its visibility can be achieved by reducing the typically available quantum dots, a Purcell factor of the excitonic radiative lifetime of the quantum dot by a factor order of 10 exalting equally both excitons transitions, of F through its introduction in a resonant microcavity would be sufficient for reaching values of S higher and the exploitation of the Purcell effect [41]: by making than the classical limit of 2. Yet, the generation of the excitonic spontaneous emission faster than the quan- maximally-entangled photons (S = 2√2) with a single 8 quantum dot is precluded by all decoherence mechanisms dot: The presence of the microcavity enhances the such as cross-dephasing between the exciton states and spontaneous emission rate of the excitonic transition, incoherent population exchange between the two bright so that emission of the second photon arises before any excitons. Maximally entangled states could however still quantum beat, cross-dephasing or incoherent population be obtained out of non-maximally entangled states by transfer between the excitonic radiative states. For use of entanglement purification [42]. experimentally accessible regime, violation of Bell’s inequalities can be achieved with real quantum dots, provided a small excitonic energy splitting (lower than few µeV ) and a Purcell factor of the order of 10. Such V. SUMMARY AND CONCLUSION Purcell factors and excitonic energy splitting have already been achieved, indicating that the possibility We have shown analytically that in the two-photon of realizing polarization-entangled photons with semi- cascade from the biexciton in a single semiconductor conductor quantum dots embedded in microcavities is quantum dot, solely the dynamics and coherence of the totally accessible with available technology. excitonic dipole governs the visibility of polarization entanglement. We have derived Bell inequalities under the CHSH form, as well as the density matrix of such a Acknowledgements: Numerous helpful discussions state. In bare quantum dots, polarization entanglement with I. Abram, S. Laurent, O. Krebs, P. Voisin, V. is spoiled not only by the energy splitting of the relay Scarani and F. Grosshans are gratefully acknowledged. level but also by the incoherent population exchange This work was partly supported by the NanoSci-ERA Eu- and cross-dephasing between the two bright relay states. ropean Consortium under project “NanoEPR”. We also The use of a microcavity can restore the generation acknowledge support of the “SANDiE” Network of Ex- of polarization-entangled photons from the quantum cellence of the European Commission.

[1] A.Aspect, Ph. Grangier and G. Roger, Phys. Rev. Lett. [19] D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, 49, 91-94 (1982) and D. Park, Phys. Rev. Lett. 76, 3005 (1996) [2] A. K. Ekert, Phys. Rev. Lett. 67, 661 - 663 (1991) [20] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gor- [3] D. Collins, N. Gisin and H. de Riedmatten, J. Mod. Opt bunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. 52, 735-753 (2005) L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and [4] C. H. Bennett, G. Brassard, C. Crpeau, R. Jozsa, A. F. Schfer, Phys. Rev. B 65, 195315 (2002) Peres and W. K. Wootters, Phys. Rev. Lett. 70, 1895 - [21] C. Santori, D. Fattal, M. Pelton, G. S. Solomon and Y. 1899 (1993) Yamamoto, Phys. Rev. B 66, 045308 (2002) [5] P. Shor, SIAM J. Computing 26, 14841509 (1997) [22] N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. [6] L.K. Grover, Phys. Rev. Lett. 79, 325 - 328 (1997) Avron, D. Gershoni, B. D. Gerardot and P. M. Petroff, [7] R. Ursin, T. Jennewein, M. Aspelmeyer, R. Kaltenbaek, Phys. Rev. Lett. 96, 130501 (2006) M. Lindenthal and A. Zeilinger, Nature 430, 849 (2004) [23] R.M. Stevenson, R.J. Young, P. Atkinson, K. Cooper, [8] O. Landry, J. A. W. van Houwelingen, A. Beveratos, H. D.A. Ritchie and A.J. Shields, Nature (London) 409, 179 Zbinden and N. Gisin, JOSA B 24, 398-403 (2007) (2006) [9] J.- .W Pan, D Bouwmeester and H Weinfurter Phys. Rev. [24] B. D. Gerardot, S. Seidl, P. A. Dalgarno, R. J. War- Lett. 80, 3891 (1998) burton, D. Granados, J. M. Garcia, K. Kowalik and O. [10] M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon Krebs, Appl. Phys. Lett. 90, 041101 (2007) and H. Zbinden” Nature Phys 3 3, 692 (2007) [25] R. Seguin, A. Schliwa, S. Rodt, K. Ptschke, U. W. Pohl, [11] X. Li, P. L. Voss, J. E. Sharping and P. Kumar Phys. and D. Bimberg, Phys. Rev. Lett. 95, 257402 (2005) Rev. Lett. 94, 053601 (2005) [26] A. Berthelot, I. Favero, G. Cassabois, C. Voisin, C. De- [12] J. Fulconis, O. Alibart, J. L. O’Brien, W. J. Wadsworth lalande, Ph. Roussignol, R. Ferreira and J. M. G´erard, and J. G. Rarity Phys. Rev. Lett. 99, 120501 (2007) Nature Physics 2, 759 (2006) [13] V. Scarani, H. de Riedmatten, I. Marcikic, H. Zbinden [27] J.M. G´erard, B. Sermage, B. Gayral, B. Legrand, E. and N. Gisin, Eur. Phys. J. D. 32, 129 (2005) Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 [14] O. Benson, C. Santori, M. Pelton and Y. Yamamoto, (1998) Phys. Rev. Lett. 84, 2513 (2000) [28] A. Peres, Phys. Rev. Lett. 77, 1413 (1996) [15] D. F. V. James,P. G. Kwiat, W. J. Munro and A. G. [29] E. S. Fry, Phys. Rev. A 8, 1219 (1973) White, Phys. Rev. A, 64, 052312 (2001) [30] J. Bylander, I. Robert-Philip and I. Abram, Eur. Phys. [16] E. Moreau, I. Robert, L. Manin, V. Thierry-Mieg, J. M. J. D 22, 295 (2003) Grard and I. Abram, Phys. Rev. Lett. 87, 183601 (2001) [31] G. Lindblad, Commun. Math. Phys. 48, 119 (1976) [17] C. Simon and J.P. Poizat, Phys. Rev. Lett. 94, (2005) [32] D. Bohm, Phys. Rev. 85, 166 (1951) 030502 [33] J. F. Clauser, M. H. Horne, A. Shimony, and R. A. Holt, [18] M. Larqu, A. Beveratos and I. Robert-Philip Eur. Phys. Phys. Rev. Lett. 23, 880 (1969) J. D. 47, 119 2008 [34] J.M. Grard, O. Cabrol, B. Sermage, Appl. Phys. Lett. 9

68, 3123 (1996) Voisin, arXiv:0801.0571 (2008) [35] W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. [40] A. J. Hudson, R. M. Stevenson, A. J. Bennett, R. J. Reuter and A. D. Wieck, Phys. Rev. B 69, 161301(R) Young, C. A. Nicoll, P. Atkinson, K. Cooper, D. A. (2004) Ritchie and A. J. Shields, Phys. Rev. Lett. 99, 266802 [36] I. Marcikic, H. de Riedmatten,W. Tittel, H. Zbinden, M. (2007) Legr, and N. Gisin, Phys. Rev. Lett. 93, 180502 (2004) [41] F. Troiani, J. I. Perea and C. Tejedor, Phys. Rev. B 74, [37] P. Hyllus, O. Ghne, D. Bru, and M. Lewenstein1, Phys. 235310 (2006) Rev. A. 72, 012321 (2005) [42] JW. Pan, S. Gasparoni, R. Ursin, G. Weihs and A. [38] M. Paillard, X. Marie, P. Renucci, T. Amand, A. Jbeli Zeilinger, Nature 423, 417 (2003) and J. M. Grard, Phys. Rev. Lett. 86, 1634 (2001) [39] K. Kowalik, O. Krebs,A. Lematre, J. A. Gaj and P.