Protocol for generating multi-photon entangled states from quantum dots in the presence of nuclear spin fluctuations
Emil V. Denning, Jake Iles-Smith, Dara P. S. McCutcheon, Jesper Mork
PProtocol for generating multi-photon entangled states from quantum dots in thepresence of nuclear spin fluctuations
Emil V. Denning, Jake Iles-Smith, Dara P. S. McCutcheon, and Jesper Mork ∗ Department of Photonics Engineering, DTU Fotonik,Technical University of Denmark, Building 343, 2800 Kongens Lyngby, Denmark Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Departmentof Electrical and Electronic Engineering, University of Bristol, BS8 1FD, UK (Dated: September 27, 2018)Multi-photon entangled states are a crucial resource for many applications in quantum infor-mation science. Semiconductor quantum dots offer a promising route to generate such states bymediating photon-photon correlations via a confined electron spin, but dephasing caused by the hostnuclear spin environment typically limits coherence (and hence entanglement) between photons tothe spin T ∗ time of a few nanoseconds. We propose a protocol for the deterministic generation ofmulti-photon entangled states that is inherently robust against the dominating slow nuclear spinenvironment fluctuations, meaning that coherence and entanglement is instead limited only by themuch longer spin T time of microseconds. Unlike previous protocols, the present scheme allowsfor the generation of very low error probability polarisation encoded three-photon GHZ states andlarger entangled states, without the need for spin echo or nuclear spin calming techniques. I. INTRODUCTION
A crucial requirement for photonic measurement-basedquantum computing schemes is a resource of entangledstates [1–10]. The generation of such states is beingpursued on various platforms; among these are continu-ous variable quantum optics [11], spontaneous paramet-ric down-conversion in nonlinear crystals [12], nitrogen-vacancy centres [13], and self-assembled semiconductorquantum dots (QDs) [14]. QDs in particular are at-tractive due to the combination of their excellent opti-cal properties [15–19], and the prospect of deterministicinteractions with single photons [15, 20]. By charginga QD with a single electron, it becomes equipped withan internal spin degree of freedom that couples to thepolarisation of optical photons [21], while also benefit-ing from highly developed optical control and readouttechniques [22–30]. Using these properties, it is possibleto generate spin–photon entanglement [31, 32], and byentangling a sequence of photons with a QD, spin–multi-photon states are generated, reducing to multi-photonentangled states once the QD spin is measured [33–35].A considerable challenge for the QD platform is posedby the interaction of the QD spin with its nuclear spin en-vironment, which gives rise to a slowly fluctuating mag-netic Overhauser field [36, 37]. Due to uncertainty inthe Overhauser field, phase coherence between the QDspin states is lost on a timescale set by the spread ofavailable Overhauser states, limiting the QD spin coher-ence to typically only a few nanoseconds [38–40] (usuallytermed the T ∗ , ensemble, or inhomogeneous dephasingtime). This renders practical implementations to gener-ate states beyond spin–single photon entanglement ex-tremely challenging in their original formulations [31– ∗ [email protected]
35, 41]. Spin coherence times can in principle be extendedbeyond T ∗ by applying spin echo or dynamical decou-pling sequences which unwind fluctuating phase evolu-tion [38, 42]. However, this not only adds operationalcomplexity, but in cases which utilise photon frequencydegrees of freedom [43], will not extend photon coherencetimes, as the Overhauser field is imprinted onto the pho-tonic component of the state not affected by echo pulses.Spin coherence may also be extended by polarisation ofthe nuclear environment [39, 44–49], though a very high( > II. DEPHASING-RESILIENT PROTOCOL
As a solution to this, we propose a QD-based protocolto generate multi-photon entangled states that is natu-rally robust against slow Overhauser field fluctuations,with the coherence being instead limited only by fasterpure-dephasing (homogeneous) processes, with a typicaltimescale of microseconds (termed the T time). The cen-tral feature of our proposed protocol is that it combines1) an external field to ensure the nuclear environmentgives rise to a fluctuating magnetic field amplitude only,with 2) narrow band excitation, which means an entan-gled state is generated in which all terms have the sameenergy. This means only a global inconsequential phase isacquired over time, thus ensuring robustness against thedominating slow nuclear spin fluctuations. We bench-mark our protocol against a multi-photon extension ofthe experimental realisations in Refs. [43, 50, 51] and thetheoretical schemes in Refs. [31, 33], showing that withrealistic noise models these cannot be scaled to create en-tanglement beyond the spin–single photon regime as theylack one or both of the above properties. Using the pro-posed protocol in combination with a suitable frequencyquantum eraser, we show that three-photon GHZ states a r X i v : . [ qu a n t - ph ] J a n FIG. 1. (a)
A QD in a polarisation-degenerate, single-sidedcavity, is exposed to an external magnetic field perpendicularto the cavity axis. (b)
Electron and hole configurations for theground states and trions, which in zero field are connected viacircularly polarised transitions. (c, d)
The Voigt-geometrymagnetic field leads to linearly polarised transitions (labelled H and V ) between hybridised levels as indicated, split by theZeeman energy b x . Shown in (c) is a spectrally narrow pho-ton resonant with the zero field transition energy ω , whichcan lead to a spin-flip Raman scattering process changing thephoton’s energy and polarisation (orange arrows in (d)), ora coherent scattering process leaving energy and polarisationunchanged (blue arrows). Occurring in superposition theseprocesses lead to spin-photon entanglement. can be generated near deterministically with near-unityfidelity, and without any active measures taken to avoidnuclear spin dephasing. Several of these microclusterscould then be efficiently transformed to a large clusterstate using only passive linear optical elements [52].Our protocol is based on a negatively charged QD ina single-sided, polarisation-degenerate cavity, operatingin the weak coupling regime. An external magnetic fieldperpendicular to the optical axis splits the QD transi-tions, and results in linearly polarised transitions to theexcited trion states. We now consider a H -polarised pho-ton incident on the cavity, with the QD in the externalmagnetic field eigenstate | φ + (cid:105) = (1 / √ |↑(cid:105) + |↓(cid:105) ), where |↑(cid:105) and |↓(cid:105) denote the ground state electron spin pro-jection along the optical axis (defining the z -direction).If the incoming photon is resonant with the bare QDtransition energy in zero field, labelled ω , there are twooff-resonant scattering possibilities. A Raman transitioncan take place, in which the spin of the QD is flipped, andthe photon frequency and polarisation are changed (or-ange arrows in Fig. 1(d)), or the photon can coherentlyscatter, leaving it and the QD unchanged (blue arrows). As such, the composite QD–photon system will evolvein superposition, and we write a single photon scatteringevent as | H, ω (cid:105) | φ + (cid:105) → | ψ (1) (cid:105) with | ψ (1) (cid:105) ≡ √ | H, ω (cid:105) | φ + (cid:105) − i | V, ω + (cid:105) | φ − (cid:105) ) , (1)where ω ± = ω ± ( b x /
2) with b x is the Zeeman split-ting, and | α, ω (cid:105) i denotes photon i in polarisation state α with frequency ω . The superscript on | ψ ( n ) (cid:105) denotes thephoton number in the scattered state.A second photon can then be sent to the QD–cavitysystem after some time, and the total composite statewill be the three-qubit entangled state (cf. App. C fordetails) | ψ (2) (cid:105) = (cid:0) | H, ω (cid:105) {| H, ω (cid:105) | φ + (cid:105) − i | V, ω + (cid:105) | φ − (cid:105)} (2)+ | V, ω + (cid:105) {− i | H, ω (cid:105) | φ − (cid:105) + | V, ω − (cid:105) | φ + (cid:105)} (cid:1) . This state is local unitary equivalent (LUE) to a three-qubit linear cluster state [8] and a GHZ state, providedthat the frequency degree of freedom is erased. For threeor more photons, the state is no longer LUE to a GHZ orlinear cluster state, though possesses a rich entanglementstructure with maximal localisable entanglement and in-finite entanglement length. Of particular note, when theQD spin is projected out of the state | ψ (3) (cid:105) in the { φ ± } basis, the remaining state is LUE to a three-photon po-larisation encoded GHZ state [53].The most important feature of Eq. (2), however, isthat each term has the same total energy. This is be-cause the first Raman process flips the spin from | φ + (cid:105) to | φ − (cid:105) , transferring energy b x from the QD to the photon.In the second spin-flip event, the opposite happens, andthe photon transfers energy b x to the QD. Consequently,the state | ψ ( n ) (cid:105) for any n consists of a large superposi-tion of trajectories that all share the same total energy nω + b x /
2, and | ψ ( n ) (cid:105) will acquire only a global phasein time. Crucially, this means that when an ensembleof states such as | ψ ( n ) (cid:105) is prepared, phase coherence be-tween terms in the superposition is protected from anyfluctuations in b x that may occur between one realisa-tion and another. In particular, for a single QD, slowvariations in the Overhauser field over time will not de-cohere | ψ ( n ) (cid:105) , allowing, for example, the generation ofthree-photon GHZ states with near-unit fidelity.As this insensitivity to nuclear spin interactions is theessential feature of our protocol, we now consider it inmore detail. The dominant coupling between the QDelectron spin and nuclear spins is the hyperfine interac-tion [54]. If this is much weaker than the electron Zee-man energy and the number of nuclear spins is large, itseffect can be modelled as a magnetic Overhauser field, B N [37], which can be added to the external field togive B = B ext + B N . Due to the large number of nu-clear spins B N evolves on a slow microsecond timescale,as compared to the characteristic nanosecond timescalegoverning the electron spin dynamics [37]. This allows usto model the Overhauser field as being stationary duringa single experimental run, but probabilistically chosenfrom w ( B i N ; ∆ B ) = 1 / (∆ B √ π ) exp (cid:2) − ( B i N ) / (2∆ B ) (cid:3) ,describing a Gaussian distribution with zero mean foreach of the Cartesian components, B i N , and with stan-dard deviation ∆ B [37]. If the external field B ext = B ext ˆx is appreciably stronger than ∆ B , we can assume that thecomponents of B N parallel to B ext dominate [37]. In sucha case nuclear spins can be included by writing the ef-fective Zeeman splitting as b x = g e µ B ( B ext + B x N ), with g e the electron Land´e factor and µ B the Bohr magneton,and with B xN averaged over using w ( B x N ; ∆ B ).To see how ensemble dephasing can arise, considera simple superposition state in the magnetic fieldeigenstate basis (1 / √ | φ + (cid:105) + | φ − (cid:105) ). For times t less than a microsecond, this state becomes | ϕ (cid:105) =(1 / √ − ib x t/ | φ + (cid:105) + e ib x t/ | φ − (cid:105) ) in a single realisa-tion. An ensemble of such states, however, samples allOverhauser fields, giving the single-spin density oper-ator (cid:37) = (cid:82) d B xN w ( B xN ; ∆ B ) | ϕ (cid:105) (cid:104) ϕ | , and we find thatcoherences decay as (cid:104) φ + | (cid:37) | φ − (cid:105) ∝ exp (cid:2) − ( t/T ∗ ) (cid:3) with T ∗ = √ / ( g e µ B ∆ B ), which for typical InGaAs QDs cor-responds to nanoseconds. Crucially, however, in our pro-tocol, states such as | ϕ (cid:105) above are never produced. In-stead, assuming that all scattering processes take placewithin the microsecond time scale over which the Over-hauser field can be considered constant, after accumu-lating n photons in the composite state, it will havethe form | ψ ( n ) (cid:105) = ( | ψ ( n )+ (cid:105) | φ + (cid:105) + | ψ ( n ) − (cid:105) | φ − (cid:105) ) / √
2, as weshow in App. C. Here | ψ ( n ) ± (cid:105) is an entangled n -photonstate, in which all terms have energy Ω + = nω orΩ − = nω + b x . This form eliminates the inhomoge-neous ensemble dephasing as described above, as thephase can be factored out of the complete state. De-phasing only occurs on a much longer timescale of the T time set by pure-dephasing processes, and typicallycorresponding to microseconds [37]. If the spin is mea-sured in the basis { φ + , φ − } while the Overhauser fieldis unchanged, the photonic state is projected to one ofthe states | ψ ( n ) ± (cid:105) , which are also robust against ensem-ble dephasing. Though we have emphasised resilienceto Overhauser field fluctuations, by the same argumentsour scheme is also robust against any other slow processesleading to energy level fluctuations, most notably thosecaused by charge noise [40, 55].Having shown that our protocol is robust against en-semble dephasing processes, we now turn our attentionto another potential imperfection, that arising from pho-ton scattering process itself, which we term the scatteringfidelity . We are interested here in a quantitative analysisof how well the entangled states in Eqs. (1) and (2) areproduced given a realistic QD–cavity model. To assessthis, we write the total Hamiltonian as ˆ H = H ( t ) + H B ,where H ( t ) is the QD–cavity Hamiltonian includinglight–matter interactions, and H B contains the magneticfield. In a frame rotating at ω we have (we set (cid:126) = 1) H ( t ) = η ( t ) e † in A + g Σ † A + H . c, with A = ( a + , a − ) T the polarisation-resolved vectorial cavity mode operator − b ext / Γ cav . . . . . S c a tt e r i n g fi d e li t y (a) . . . (b) . . . − ∆ B / ∆ max B . . . (c) FIG. 2. (a)
One photon scattering fidelity F (1) with respectto the ideal Bell state, (1 / √ | x (cid:105) | φ + (cid:105) − i | y (cid:105) | φ − (cid:105) ) as func-tion of external magnetic field in the absence of Overhauserfield fluctuations. Blue solid lines correspond to a low Q -factor cavity ( κ = 10 ns − , Q (cid:39) Q -factor cavity ( κ = 150 ns − , Q (cid:39) t = 8 / Γ cav , g = 15 ns − , g h /g e =0 . η /κ = 10 − (10 − ) for the low (high) Q cavity and g e µ B ∆ max B = 0 . − . (b) Fidelity F (1) including nuclearspin noise as a function of the degree of nuclear spin polarisa-tion. The external field has been tuned to the optimal valuefound numerically in (a). Line styles represent parametersas in (a). Circles and error bars indicate ensemble averagesand (25%, 75%) quantiles of the fidelity. (c) Fidelity F (2) with respect to the ideal spin–two-photon state, obtained byscattering two photons on the QD with a time delay of 3 t . in the circular polarisation basis, Σ = ( |↑(cid:105)(cid:104)⇑| , |↓(cid:105)(cid:104)⇓| ) T ,and g is the QD–cavity coupling strength. The incominglight is modelled as a weak coherent pulse, described bya time-dependent driving of the cavity field, taken to beGaussian, η ( t ) = η exp (cid:2) − ( t/t ) (cid:3) , and e in is the inputpolarisation Jones vector in the circular basis. The mag-netic field Hamiltonian is H B = µ B B · ( g e S e − g h S h ),with S e ( S h ) the vectorial spin operator for the electron(hole) subspace and g h the hole Land´e factor [56]. Witha numerical solution of the dynamics generated by theHamiltonian [57], the scattering fidelity for an n pho-ton state is simply F ( n ) = Tr[ ρ | ˜ ψ ( n ) (cid:105)(cid:104) ˜ ψ ( n ) | ], where ρ isthe numerically calculated QD-photon density operator,and | ˜ ψ ( n ) (cid:105)(cid:104) ˜ ψ ( n ) | the ideal maximally entangled state [58].Additional details about the dynamical model and calcu-lation of fidelities can be found in Appendices A and B.By first artificially setting the Overhauser field to zero,in Fig. 2(a) we show how the spin–one photon Bell statefidelity F (1) can be optimised by tuning the externalmagnetic field. We see that near-unity scattering fidelityis reached when the external field is approximately thecavity-enhanced QD linewdith, b ext (cid:39) Γ cav = 4 g /κ , asit is depicted in Fig. 1(c). This ensures that an incom-ing photon has a high probability of scattering off oneof the two possible transitions while also ensuring thatthey are adequately separated. In this regime, the fidelityis limited by the finite bandwidth of the input photon,since any off-centre frequency components lead to an un-evenly weighted superposition in the scattered state. InFig. 2(b) and (c), we show the fidelities F (1) and F (2) in-cluding the nuclear environment, shown as a function of . . . − ∆ B / ∆ max B . . . S c a tt e r i n g fi d e li t y (a) t/T ∗ . . . O v e r a ll fi d e li t y (b) FIG. 3. Comparison of the present scheme with existing Pro-tocols A, B and C described in the main text. (a)
Ensembleaveraged spin–one-photon Bell state scattering fidelity F (1) as in Fig. 2(b) for protocol A (crosses) and for the presentscheme (circles, already shown in Fig. 2(b)). Red and bluelines correspond to high and low Q -factor parameters as inFig.2(b). (b) Ensemble averaged photonic state fidelity afterspin projection as a function of time, assuming unit scatteringfidelity, shown for Protocols B (stars) and C (trianges). Bothshow a rapid decay due to large spread of possible Overhauserfields, while the present scheme (circles) is unaffected due tothe form of Eq. (1) which acquires only a global phase in time. the nuclear environment polarisation, ranging from max-imally unpolarised (∆ B = ∆ max B ) to the fully polarised(∆ B = 0) regime, and for high (red, dashed curve) andlow (blue, solid) cavity Q -factors, corresponding to QDswith broad and narrow Purcell-enhanced transition lines.We see that even for an unpolarised nuclear environ-ment, fidelities of the two-photon state are above 90%for Q = 13000. Higher Q -factors are advantageous sincethey correspond to larger QD linewidths and hence largeroptimal external field strengths, which in turn mean thestrength of the external field relative to the Overhauserfield is greater. This results in increased stability of theQD eigenstructure and purity of the QD–photon scatter-ing process, while also ensuring that the Overhauser fieldleads only to fluctuations in the magnitude of the field.We emphasise that the internal photon–QD interac-tion in the protocol is in principle deterministic, withthe quantum efficiency being limited only by scattering oflight into non-cavity modes, which is heavily suppressedin moderate to high Q -cavities [17, 19]. To obtain apurely polarisation-entangled state, however, it is neces-sary to erase the frequency degree of freedom in (cid:12)(cid:12) ψ ( n ) (cid:11) .This is an unavoidable consequence of the state’s insen-sitivity to ensemble dephasing, and could be achieved,for example, using fast single-photon detectors [59, 60]or ultra fast non-linear frequency converters [51]. III. COMPARISON TO ALTERNATIVEPROTOCOLS
To benchmark our protocol, we compare it to threealternative existing schemes. The first scheme (ProtocolA) is based on coherent scattering of single linearly po-larised photons on a charged QD in the absence of an external field [31, 32]. Using our noise model, we calcu-late the scattering fidelity of this protocol in the presenceof the same realistic nuclear spin environment. The fi-delity of generating the spin–one-photon Bell state F (1) is shown in Fig. 3(a), where crosses indicate values usingProtocol A, and the circles correspond to values usingthe present dephasing-resilient scheme (already shown inFig. 2(b)). We see that the scattering fidelity of ProtocolA is generally low, reaching values close to unity only forvery high degrees of nuclear spin polarisation. The reasonfor this protocol’s sensitivity to dephasing processes canbe attributed to its lack of an external field, which leavesthe QD eigenstructure highly exposed to Overhauser fieldfluctuations.The second alternative scheme we consider (ProtocolB) is a multi-photon extension of the schemes used inRefs. [43, 50, 51], that use emission of a QD in an exter-nal in-plane field. This protocol resembles the schemewe propose here, but with the crucial difference thatsingle photon scattering in our scheme is replaced byfull π -pulse excitations followed by spontaneous emission.While the magnetic field does ensure stability of the spineigenstructure and high scattering fidelity (unlike proto-col A), the spectrally broad π -pulses mean energy is notconserved in all paths of the evolution. As we show inApp. D, the result is that the n -photon state containsterms which acquire phases that depend on the fluctuat-ing total effective Zeeman energy b x = g e µ B ( B ext + B x N )in different ways, and the state therefore loses its phasecoherence on a short T ∗ = √ / ( g e µ B ∆ B ) ∼ ns timescale,in much the same way as a single electron spin. Fideli-ties of a two-photon state obtained after excitation withtwo π -pulses followed by spin projection are shown inFig. 3(b) with stars, where unit scattering fidelity is as-sumed, and t represents time after spin projection. Alsoshown with triangles is the corresponding two-photonstate fidelity for the linear cluster state generation pro-posal of Ref. [33], Protocol C, again assuming unit scat-tering fidelity. As in the case of Protocol B, this schemeis also sensitive to ensemble dephasing of the electronspin. The form of Eq. (1), however, ensures the presentscheme does not dephase by this mechanism, leading tocoherence times well beyond nanoseconds, as shown bythe open circles.In summary, we have presented a spin-mediated multi-photon entanglement protocol which is robust againstslow Overhauser field fluctuations, meaning that coher-ence is limited to the pure spin dephasing time T ofmicroseconds, rather than the inhomogeneous dephas-ing time T ∗ of nanoseconds. With a suitable frequencyeraser, the protocol can be used as a source of high-fidelity three-photon GHZ states, which through linearoptical operations can be transformed to a universalquantum resource for measurement-based quantum com-puting [52]. We emphasise that no spin echo or nuclearpolarisation techniques are necessary, and that opticalexcitation could be achieved with readily obtainable weakcoherent laser pulses, or instead with narrowband singlephotons for deterministic operation. ACKNOWLEDGMENTS
We thank Anders S. Sørensen, Thomas Nutz, PetrosAndrovitsaneas, Dale Scerri, and Erik Gauger for helpfuldiscussions. E.V.D, J.I.-S., and J.M. acknowledge fund-ing from the Danish Council for Independent Research(DFF-4181-00416). This project has received fundingfrom the European Union’s Horizon 2020 research and in-novation programme under the Marie Sk(cid:32)lodowska-Curiegrant agreement No. 703193.
Appendix A: Description of model
We consider a singly negatively charged quantum dot(QD) in a one-sided cavity, which is driven by a po-larised weak optical pulse. The cavity field is resolvedin two orthogonal circular polarisations with mode oper-ators a + and a − , satisfying [ a λ , a † λ (cid:48) ] = δ λλ (cid:48) , [ a λ , a λ (cid:48) ] =[ a † λ , a † λ (cid:48) ] = 0. We assume that the cavity is resonant withthe QD transition at a frequency of ω . Further, thecavity is coupled to the optical electromagnetic environ-ment, resolved in two polarisations with mode operators b λq , where λ = ± denotes the polarisation and q denotesthe mode index. As a basis for the QD, we use the spineigenstates projected along the z -direction, taken as theoptical axis. For the charged ground states, these are |↑(cid:105) and |↓(cid:105) , while for the corresponding trion states theyare |⇑(cid:105) and |⇓(cid:105) , denoting the heavy-hole spin states withspin projection eigenvalues J z = ± /
2. Due to isotropicstrain, the light holes with J z = ± / LH , much larger than thelinewidth of the transition, and we can ignore them inthe light-matter interaction [61]. The QD is subject to amagnetic field, B in an arbitrary direction described bythe polar (azimuthal) angle, θ ( φ ), and with a magnitudeof B . Moving to a frame rotating with the resonancefrequency, ω , the total Hamiltonian can be written asˆ H ( t ) = H ( t ) + H B + H + H I EM , with ( (cid:126) = 1) H ( t ) = η ( t ) e † in A + g Σ † A + H . c ,H B = µ B B · ( g e S e − g h S h ) ,H = (cid:88) λq ( ω q − ω ) b † λq b λq , H I EM = (cid:88) λq g q b λq a † λ + H . c ., (A1)where A is the polarisation-resolved vectorial mode oper-ator ( a + , a − ) T , Σ = ( |↑(cid:105)(cid:104)⇑| , |↓(cid:105)(cid:104)⇓| ) T is the spin-resolvedQD transition operator, g is the QD–cavity couplingrate, µ B is the Bohr magneton, g e ( g h ) is the electron(hole) Land´e factor, S e ( S h ) is the vectorial spin op-erator for the electron (hole) subspace, ω q is the fre-quency of the q ’th environmental mode and g q is thecoupling rate between the cavity and the q ’th environ-mental mode. The pulse envelope, η ( t ) is taken to be Gaussian, η ( t ) = η exp (cid:2) − ( t/t ) (cid:3) and e in the input po-larisation Jones vector in the circular basis.To write down a practical form of H B , we use thespin eigenstates as a basis. For the electron spin in theground state manifold, we use the Zeeman eigenstates de-termined by the direction of the magnetic field, | φ + (cid:105) =cos θ/ |↑(cid:105) + e iφ sin θ/ |↓(cid:105) , | φ − (cid:105) = e − iφ cos θ/ |↑(cid:105) − sin θ/ |↓(cid:105) . As for the trion spin, treating the magneticfield interaction perturbatively to first order in the pa-rameter µ B g h B/ ∆ LH , the light and heavy hole manifoldsremain uncoupled. The heavy hole eigenstates are then |⇓(cid:105) and |⇑(cid:105) with associated energies ± / µ B g h B cos θ .In this basis, H B takes the form H B = −
32 ˜ g h b cos θ ( |⇑(cid:105)(cid:104)⇑| − |⇓(cid:105)(cid:104)⇓| )+ b | φ + (cid:105)(cid:104) φ + | − | φ − (cid:105)(cid:104) φ − | ) , (A2)with b = µ B g e B and ˜ g h = g h /g e .The interaction with the electromagnetic environmentcan be simplified by applying a standard Born-Markovapproximation, corresponding to assuming a flat spec-tral density over the relevant frequency range [62]. Withthis approximation and neglecting the environmentallyinduced Lamb shift, the perturbative master equationtreating H I EM to second order is ˙ ρ ( t ) = L ( t ) ρ ( t ) with L ( t ) the time-dependent Liouvillian, L ( t ) = − i [ H ( t ) + H B , · ] + κ (cid:88) λ = ± (cid:18) a λ · a λ − { a † λ a λ , ·} (cid:19) , (A3)where κ is the cavity dissipation rate. This master equa-tion can be solved numerically to obtain the time evolu-tion of the density operator [63].The polarisation resolved reflected output modes, ξ λ ,can be calculated from the cavity mode using input-output theory [64] as ξ λ ( t ) = i e † λ e in η ( t ) κ + e † λ A with e λ the Jones polarisation vector describing the polarisationmode λ . The H and V polarisations are described by theJones vectors e H = √ (1 , T , e V = √ (1 , − T . Appendix B: Fidelity measures
With the full time evolution of the cavity-QD densityoperator, ρ ( t ), at hand, we can calculate any propertiesof the system. In particular, we can calculate the fidelityof the composite state consisting of the polarisation ofscattered light and the internal spin state of the QD.However, this fidelity cannot be evaluated directly fromthe time-resolved density operator. Care must be taken,because the light polarisation must be defined in termsof the reflected light from the cavity, described by theoutput field operators, ξ H ( t ) and ξ V ( t ).First, we consider a single photon scattered onthe QD. In general, the two-qubit space spannedby the QD spin and the polarisation of the scat-tered photon can be described by the basis B (1) = {| Hφ + (cid:105) , | V φ − (cid:105) , | Hφ − (cid:105) , | V φ + (cid:105)} , with the superscript(1) signifying that the space spans the polarisation ofone photon and the QD spin. We denote the true densitymatrix of the post-scattering state of the single photon–spin system in this basis by ρ (1) . We denote by χ (1) theideal density operator corresponding to the pure state | ψ (1)pure (cid:105) = α | Hφ + (cid:105) + β | V φ − (cid:105) . In the basis B (1) it takesthe form χ (1) = | α | αβ ∗ α ∗ β | β | . (B1)The fidelity between two density operators, ρ and ρ takes the form F = tr( ρ ρ ), if at least one of the den-sity operators is pure. In our case, χ (1) is pure and wemay write the fidelity as F (1) = | α | ρ (1)11 + | β | ρ (1)22 + 2 Re (cid:110) αβ ∗ ρ (1)21 (cid:111) , (B2)which shows that we only need to calculatefour matrix elements of ρ (1) to evaluate the fi-delity. To evaluate these matrix elements, wedefine the joint spin-polarisation expectation val-ues (cid:104) S P λP (cid:48) λ (cid:48) (cid:105) = (cid:82) d t (cid:104) ξ † P ( t ) ξ P (cid:48) ( t ) σ λλ (cid:48) ( t ) (cid:105) / N (1) with M λλ (cid:48) = | φ λ (cid:105)(cid:104) φ λ (cid:48) | the input intensity normalisation N (1) = (cid:82) ∞−∞ d t ξ ∗ in ( t ) ξ in ( t ) = (cid:112) π/ η t /κ , obtainedusing ξ in ( t ) = iη ( t ) /κ . This normalisation accountsfor the fact that we model the incoming photon as aweak coherent pulse. We then find that the F (1) can becalculated as F (1) = | α | (cid:104) S H + H + (cid:105) + | β | (cid:104) S H + V − (cid:105) + 2 Re[ α ∗ β (cid:104) S V − V − (cid:105) ] . Now we shall consider the scattering of a second photonon the cavity after some time, τ . In App. C, we calculatethe explicit state, but here we shall simply use the generalform for the ideal scattered state | ψ (2)pure (cid:105) = α | H H φ + (cid:105) + β | H V φ − (cid:105) + γ | V H φ − (cid:105) + δ | V V φ + (cid:105) , (B3)with corresponding density operator χ (2) = | ψ (2) (cid:105)(cid:104) ψ (2) | .Note that the coefficients α and β are not those entering χ (1) . In analogy with the single-photon scattering case,we shall denote the true post-scattering density operatorby ρ (2) . The fidelity becomes F (2) = | α | (cid:104) S HH + HH + (cid:105) + | β | (cid:104) S HV − HV − (cid:105) + | γ | (cid:104) S V H − V H − (cid:105) + | δ | (cid:104) S V V + V V + (cid:105) + 2 Re (cid:110) α ∗ β (cid:104) S HV − HH + (cid:105) + γ ∗ δ (cid:104) S V V + V H − (cid:105) + e ib x τ (cid:2) α ∗ γ (cid:104) S V H − HH + (cid:105) + α ∗ δ (cid:104) S V V + HH + (cid:105) + β ∗ γ (cid:104) S V H − HV − (cid:105) + β ∗ δ (cid:104) S V V + HV − (cid:105) (cid:3)(cid:111) , (B4) with (cid:104) S P QλP (cid:48) Q (cid:48) λ (cid:48) (cid:105) = 1 N (2) ( τ ) (cid:90) d t (cid:104) ξ † P ( t ) ξ † Q ( t + τ ) σ λλ (cid:48) ( t + τ ) ξ Q (cid:48) ( t + τ ) ξ P (cid:48) ( t ) (cid:105) and N (2) ( τ ) = (cid:90) ∞−∞ d t ξ ∗ in ( t ) ξ ∗ in ( t + τ ) ξ in ( t + τ ) ξ in ( t )= 1 κ (cid:90) ∞−∞ d t | η ( t ) | | η ( t + τ ) | . Due to the symmetries of the proposed protocol as de-scribed in the main text, the fidelity turns out to be in-dependent of the photon separation time, τ . Appendix C: Multi-photon entanglement structure1. Unitary dynamics
The interaction between a string of n photons and theQD can be entirely described by the unitary scatteringoperator, U , describing the asymptotic composite stateresulting from of a single-photon scattering event. To find U , we have numerically calculated the post-scatteringstate of four orthogonal initial conditions using the meth-ods described in Appendices A and B, | H, ω (cid:105) | φ + (cid:105) U −→ √ (cid:0) | H, ω (cid:105) | φ + (cid:105) − i (cid:12)(cid:12) V, ω + b H (cid:11) | φ − (cid:105) (cid:1) , | V, ω (cid:105) | φ + (cid:105) U −→ √ (cid:0) | V, ω (cid:105) | φ + (cid:105) − i (cid:12)(cid:12) H, ω + b H (cid:11) | φ − (cid:105) (cid:1) , | H, ω (cid:105) | φ − (cid:105) U −→ √ (cid:0) | H, ω (cid:105) | φ − (cid:105) + i (cid:12)(cid:12) V, ω − b H (cid:11) | φ + (cid:105) (cid:1) , | V, ω (cid:105) | φ − (cid:105) U −→ √ (cid:0) | V, ω (cid:105) | φ − (cid:105) + i (cid:12)(cid:12) H, ω − b H (cid:11) | φ + (cid:105) (cid:1) . (C1)To establish the full unitary operator, we would need tofind the evolution of initial conditions with photon fre-quencies ω ± b as well. However, to this end we are onlyinterested in the scattering dynamics of photons resonantwith the zero-field QD transition at ω . In particular,when restricting the discussion to H -polarised input pho-tons, we only need to know how U works on | H, ω (cid:105) | φ ± (cid:105) .We then write the total scattered state as | ψ ( n ) (cid:105) = (cid:89) j U j | H, ω (cid:105) · · · | H, ω (cid:105) n | φ + (cid:105) , (C2)where U j acts on the j ’th photon and the QD. In particu-lar, for two photons, we obtain the state in Eq. (1) of themain text. Generally speaking, the n -photon entangledstate has the form | ψ ( n ) (cid:105) = √ ( | n, + (cid:105) | φ + (cid:105) + | n, −(cid:105) | φ − (cid:105) ).Here, | n, + (cid:105) contains all superpositions of polarisationpermutations with an even number of y -polarised pho-tons, where each term in the superposition has a totalphotonic energy of nω . Similarly, | n, −(cid:105) contains allterms with an odd number of y -polarised photons andall terms have a photonic energy of nω + b x .
2. Entanglement structure of spin–multi-photonstate
If we assume that the photon frequency degree of free-dom is erased and can be factored out of the remainingstate, the generating scattering transformation, (C1), be-comes non-unitary and takes the form G j = √ ( QD ⊗ j − Y QD ⊗ X j ), with Y QD = i ( | φ − (cid:105)(cid:104) φ + | − | φ + (cid:105)(cid:104) φ − | ) and X j = | H (cid:105)(cid:104) V | j + | V (cid:105)(cid:104) H | j . Using this form of the scat-tering operator, we can write down the n -photon–spinentangled state. To do so, we change notation by defin-ing the computational basis for the photon polarisationas | H (cid:105) k = | (cid:105) k , | V (cid:105) k = | (cid:105) k and | φ + (cid:105) = | (cid:105) , | φ − (cid:105) = | (cid:105) for the QD spin. The ket subscripts k = 1 , · · · , n shallbe used for the photonic qubits, while k = 0 denotes thespin qubit. We shall use i k ∈ { , } to denote the valueof a qubit in the computational basis, i = ( i , · · · , i m )denotes an m -bitstring and | i (cid:105) S the corresponding statewith respect to the qubits in the ordered set S . Further,we shall neglect normalisation factors for ease of nota-tion. The n -photon scattered state can then be writtenas | ψ ( n ) (cid:105) = (cid:89) j G j | , · · · , (cid:105) { , ··· ,n } = | (cid:105) (cid:88) i ∈ S e ( n ) | i (cid:105) { , ··· ,n } − i | (cid:105) (cid:88) i ∈ S o ( n ) | i (cid:105) { , ··· ,n } , (C3)with S e ( n ) = {| i , · · · , i n (cid:105) | (cid:80) k i k = 2 m, m ∈ N } and S o ( n ) = {| i , · · · , i n (cid:105) | (cid:80) k i k = 2 m + 1 , m ∈ N } . Bysingling out the k ’th, l ’th and m ’th photonic qubits fromthe sums, we can rewrite this state as | ψ ( n ) (cid:105) = (cid:12)(cid:12) c (cid:48) + (cid:11) klm | R + (cid:105) + (cid:12)(cid:12) c (cid:48)− (cid:11) klm | R − (cid:105) , (C4)with the k, l, m -qubit states (cid:12)(cid:12) c (cid:48) + (cid:11) = | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) , (cid:12)(cid:12) c (cid:48)− (cid:11) = | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) and the residualqubit states | R + (cid:105) = | (cid:105) (cid:88) i ∈ S e ( n − | i (cid:105) { , ··· ,n }\{ k,l,m } − i | (cid:105) (cid:88) i ∈ S o ( n − | i (cid:105) { , ··· ,n }\{ k,l,m } , | R − (cid:105) = | (cid:105) (cid:88) i ∈ S o ( n − | i (cid:105) { , ··· ,n }\{ k,l,m } − i | (cid:105) (cid:88) i ∈ S e ( n − | i (cid:105) { , ··· ,n }\{ k,l,m } . (C5) The states (cid:12)(cid:12) c (cid:48)± (cid:11) klm are local unitary equivalent to three-qubit linear cluster states, which are local unitary equiv-alent to three-photon GHZ states [65]. This is seen byapplying the Hadamard transformation, H = | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | − | (cid:105)(cid:104) | to the l ’th photon, H l (cid:12)(cid:12) c (cid:48)± (cid:11) { klm } = | c ± (cid:105) { klm } where | c ± (cid:105) are the two orthogonal cluster states | (cid:105) ± | (cid:105) + | (cid:105) ∓ | (cid:105) + | (cid:105) ∓ | (cid:105) + | (cid:105) ± | (cid:105) .From this form, we can easily calculate all single-qubit re-duced density operators, which all take the form ρ k = .Furthermore, we can calculate the two-qubit reduceddensity operators, which for two photonic qubits, kl , takethe form ρ kl = |B (cid:105)(cid:104)B | + |B (cid:105)(cid:104)B | with the two Bell states |B (cid:105) = | (cid:105) + | (cid:105) , |B (cid:105) = | (cid:105) + | (cid:105) . Two-qubit reduceddensity operators involving the spin qubit, take the sim-ilar form ρ k = |B (cid:48) (cid:105)(cid:104)B (cid:48) | + |B (cid:48) (cid:105)(cid:104)B (cid:48) | with the rotated Bellstates |B (cid:48) (cid:105) = | (cid:105) − i | (cid:105) , |B (cid:48) (cid:105) = | (cid:105) − i | (cid:105) . Fromthese reduced density operators, we can show that thegenerated state is not local unitary equivalent to a linearcluster state for more than three qubits. This is due tothe necessary condition for local unitary equivalence thatall reduced density operators must also be local unitaryequivalent [66]. Since for a linear cluster state with morethan three qubits there exist indices kl such that ρ kl = ,the two states cannot be local unitary equivalent.However, from (C4), we can infer that performing lo-cal projective measurements on any n − n − Appendix D: Analysis of Protocol B
A protocol very similar to the one proposed in the maintext has been used for generation of entanglement be-tween a single photon and a QD [43, 50, 51]. Here, theQD is initialised in the | φ + (cid:105) ground state and excited to √ ( |⇑(cid:105) + |⇓(cid:105) ) by an H -polarised π -pulse resonant withthe transition | φ + (cid:105) ↔ √ ( |⇑(cid:105) + |⇓(cid:105) ) at ω − b x /
2. As thisstate decays, a photon is emitted, which is entangled withthe spin of the QD, | ψ (1) (cid:105) = c H | H, ω − b x / (cid:105) | φ + (cid:105) + c V | V, ω + b x / (cid:105) | φ − (cid:105) (D1)with | c i | = 1 /
2. This state is protected against dephas-ing, because both terms in the superposition have thesame total energy of ω . To add a second photon to thestate, the QD is excited again. This time, we have touse a two-colour π -pulse, because the QD is in a super-position of the two ground states. Immediately after theexcitation, the system is in the state c H | H, − b x / (cid:105) |⇑(cid:105) + |⇓(cid:105)√ c V | V, + b x / (cid:105) |⇑(cid:105) − |⇓(cid:105)√ , (D2)where we have transformed to a frame rotating with ω .As the QD decays, the state becomes | ψ (2) (cid:105) = c HH | H, − b x / (cid:105) | H, − b x / (cid:105) | φ + (cid:105) + c HV | H, − b x / (cid:105) | V, + b x / (cid:105) | φ − (cid:105) + c V H | V, + b x / (cid:105) | H, + b x / (cid:105) | φ − (cid:105) + c V V | V, + b x / (cid:105) | V, − b x / (cid:105) | φ + (cid:105) , (D3)with | c αβ | = 1 / − b x /
2, whereas the two last terms have an energyof + b x /
2. In the time until the next excitation event, τ ,the state will evolve freely. Recalling that b x = b x ext + b x N ,the time evolution is | ψ (2) , τ (cid:105) = e + i ( b x ext + b x N ) τ/ (cid:104) c HH | H, − b x / (cid:105) | H, − b x / (cid:105) | φ + (cid:105) + c HV | H, − b x / (cid:105) | V, + b x / (cid:105) | φ − (cid:105) (cid:105) + e − i ( b x ext + b x N ) τ/ (cid:104) c V H | V, + b x / (cid:105) | H, + b x / (cid:105) | φ − (cid:105) + c V V | V, + b x / (cid:105) | V, − b x / (cid:105) | φ + (cid:105) (cid:105) . (D4)The fidelity with respect to | ψ (2) (cid:105) is (cid:12)(cid:12) (cid:104) ψ (2) | ψ (2) , τ (cid:105) (cid:12)(cid:12) = { b x ext + b x N ) τ ] } . On performing an en-semble average over the weight distributionof the Overhauser field, the fidelity becomes F = (cid:82) ∞−∞ d b x N w ( b x N ; δ b ) { b x ext + b x N ) τ ] } = (cid:110) e − ( τ/T ∗ ) cos( b x ext τ ) (cid:111) , with T ∗ = √ / ( g e µ B ∆ B ).Such dephasing processes will take place between allof the following excitation events. The time betweenexcitations is limited by the lifetime of the QD, and ifwe assume that this is much shorter than the coherence time, T ∗ , we may neglect dephasing between excitationsfor a few photons. However, after spin projection, thephotonic state will be subject to dephasing of the samenature. Measuring the spin in the basis { φ + , φ − } leavesthe two emitted photons in either of the two states | ψ (2)+ (cid:105) = √ (cid:104) φ + | ψ (2) (cid:105) = √ (cid:104) c HH | H, − b x / (cid:105) | H, − b x / (cid:105) + c V V | V, + b x / (cid:105) | V, − b x / (cid:105) (cid:105) , | ψ (2) − (cid:105) = √ (cid:104) φ − | ψ (2) (cid:105) = √ (cid:104) c HV | H, − b x / (cid:105) | V, + b x / (cid:105) + c V H | V, + b x / (cid:105) | H, + b x / (cid:105) (cid:105) . (D5)After the projective measurement, the states evolve as | ψ (2)+ , t (cid:105) = √ (cid:104) φ + | ψ (2) (cid:105) = √ (cid:104) c HH e + i ( b x ext + b x N ) t | H, − b x / (cid:105) | H, − b x / (cid:105) + c V V | V, + b x / (cid:105) | V, − b x / (cid:105) (cid:105) , | ψ (2) − , t (cid:105) = √ (cid:104) φ − | ψ (2) (cid:105) = √ (cid:104) c HV | H, − b x / (cid:105) | V, + b x / (cid:105) + c V H e − i ( b x ext + b x N ) t | V, + b x / (cid:105) | H, + b x / (cid:105) (cid:105) . 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