Maximal entanglement generation in spectrally distinct solid state qubits
MMaximal entanglement generation in spectrally distinct solid state qubits
Elena Callus ∗ and Pieter Kok † Department of Physics and Astronomy, The University of Sheffield, Sheffield, S3 7RH, UK (Dated: February 25, 2021)We demonstrate how to create maximal entanglement between two qubits that are encoded in twospectrally distinct solid-state quantum emitters embedded in a waveguide interferometer. The opti-cal probe is provided by readily accessible squeezed light, generated by parametric down-conversion.By continuously probing the emitters, the photon scattering builds up entanglement with a concur-rence that reaches its maximum after O (10 ) photo-detection events. Our method does not requireperfectly identical emitters, and accommodates spectral variations due to the fabrication process.It is also robust enough to create entanglement with a concurrence above 99% for 10% scatteringphoton loss, and can form the basis for practical entangled networks. A key resource for quantum computing and quantum in-formation processing is entanglement [1]. For quantumtechnologies that are implemented on a photonic plat-form, entanglement can be generated between spatiallyseparated solid-state emitters, or artificial atoms, by em-bedding them in waveguides and allowing photons to in-teract with them. The generation of entanglement be-tween artificial atoms has been well-addressed theoret-ically, and various schemes to entangle such qubits al-ready exist [2–5]. These processes require only single-or few-photon interactions and a relatively simple setupwith few optical components, and utilise “which-path”information erasure. However, these schemes require theuse of qubits that are spectrally identical, necessary forcoherent erasure of path information, and therefore anyfrequency variations between the emitters would result indegraded entanglement. Considering that current fabri-cation processes of solid-state emitters result in spectrallyinhomogeneous samples [6], the matching of sufficientlysimilar qubits adds a large overhead cost and the entan-glement process cannot occur on a large-scale. Althoughmethods such as diameter tuning [7] and strain tuning [8]can be used to tune the frequencies of the emitters, theyrequire additional technical complexity in the experimen-tal setup. Furthermore, these techniques are applicableonly for sufficiently similar emitters and generally cannotbe used for arbitrary pairings.In order to try and overcome this practical limitation,Hurst et al. [9] considered how spectral variation in emit-ters affects the entanglement outcome and demonstratedthat this inhomogeneity is not as hindering as previouslythought. They propose a simple setup involving linearoptics and show that it is possible to attain entanglementdeterministically for certain combinations of central en-ergies and line-widths by adjusting the frequency of theprobing photons. One drawback of the setup is that itrequires the use of two-mode Fock states, | n, m (cid:105) , whichare typically not easily accessible given current technolo-gies. Also, for a given Fock state, near-perfect entangle-ment is attained only for certain ranges of central energyand line-width combinations. This necessitates the useof increasingly hard to source higher-order Fock states in order to entangle certain regimes of emitter-pairings.In this paper, we show that two spectrally differ-ent emitters can be entangled with extremely high con-currence in a nearly deterministic manner by repeat-edly probing the emitters with squeezed light, generatedby means of spontaneous parametric down-conversion(SPDC) [10]. This brings us one step closer to a physicalimplementation, given the accessibility of squeezed light[11], and further eases restrictions when it comes to thematching of inhomogeneous solid-state emitters. In ad-dition, we take into consideration photon loss during thescattering process and show that very high concurrenceis still possible in non-ideal situations.The setup consists of a waveguide Mach-Zehnder in-terferometer (MZI) with a 50:50 beam splitter at eitherend, and solid-state emitters, acting as our logical qubits,embedded in each arm (see Fig. 1). The emitters havetwo long-lived low-lying spin states, |↑(cid:105) and |↓(cid:105) , and anexcited state | e (cid:105) , and are of the L -configuration, with theexcited state coupled to only one of the spin states (say,the |↑(cid:105) state). The transition between the other spin stateand the excited state is forbidden by polarisation selec-tion rules. Each qubit is prepared in the superpositionstate ( |↑(cid:105) + |↓(cid:105) ) / √
2. To ensure that photons are scat-tered only in the forward direction, the two emitters areplaced at so-called c-points in the waveguide. These are χ (2) cw laser BS 1 BS 2 photondetectors | e i|↑i |↓i E ˆ a ˆ bE , Γ E , Γ FIG. 1. Schematic of the setup: the squeezed light generatedby a χ (2) nonlinear crystal driven by a continuous-wave laserpump enters the Mach-Zehnder interferometer, where it scat-ters off the two solid-state emitters characterised by energiesand line-widths E , Γ and E , Γ . The photon measurementis made at the interferometer output, with the mode opera-tors ˆ a and ˆ b representing the output arms. Inset shows the L -configuration of the emitters, with the |↑(cid:105) state coupled tothe excited state, | e (cid:105) , and transition energy E = E , E . a r X i v : . [ qu a n t - ph ] F e b the locations where emitters exhibit highly directionalscattering of circularly polarized light due to the spatialconfinement of the electromagnetic field [12]. High levelsof directional scattering have been observed experimen-tally [12–14]. Finally, we have photon detectors placedat both output arms of the MZI, where the state of theemitters post-scattering is heralded by the photon de-tection signature. We assume that the photon detectorshave near-perfect efficiency [15, 16].For emitters at c-points, the transmission coefficientfor a photon of frequency ω scattering off a two-levelemitter is obtained from the single photon S -matrix, andis given by [17] t ( ω ) = (cid:126) ω − E − i (cid:126) (Γ − γ ) / (cid:126) ω − E + i (cid:126) (Γ + γ ) / , (1)where E is the transition frequency of the emitter, and Γand γ are the coupling rates of the emitter to the waveg-uide and the non-guided modes, respectively.In the case of zero photon loss, γ = 0 and the scatteredphoton acquires a pure phase shift. In the ideal case, thetwo solid-state emitters are identical in their energies andline-widths and their spin states are entangled by pass-ing a resonant probe photon through the interferometer.After interfering with the first beamsplitter, the photonscatters off the emitters and acquires a π phase shift fromits interaction with the |↑(cid:105) state [18, 19], whilst the state |↓(cid:105) leaves the photon unchanged. The resulting state ofthe qubits and the probe photon after the second beamsplitter is given by ( | Φ − (cid:105) ⊗ | , (cid:105) − | Ψ − (cid:105) ⊗ | , (cid:105) ) / √ | Φ − (cid:105) = ( |↑↑(cid:105) − |↓↓(cid:105) ) / √ | Ψ − (cid:105) = ( |↑↓(cid:105) − |↓↑(cid:105) ) / √ | , (cid:105) and | , (cid:105) represent the two possible photon-detection outcomes. Therefore, either photon measure-ment outcome would result in a maximally entangledbipartite state. However, if we deviate from identicalsolid-state emitters and allow for spectral variations be-tween the two, we do not obtain a maximally entangledstate. Here, the amount of generated entanglement canbe tuned by adjusting the frequency of the photon probe.We consider the two-mode squeezed vacuum as our in-put state, routinely generated by spontaneous paramet-ric down-conversion (SPDC) in a nonlinear χ (2) crystaldriven by a continuous-wave (CW) pump laser. Duringthe SPDC process, a pump photon of frequency ω p isannihilated and a signal and idler photon, with frequen-cies ω s and ω i , respectively, are created. When the pumplaser is on resonance with the crystal, the squeezing oper-ator is given by S ( ξ ) = exp[ ( ξ ∗ ˆ a ˆ b − ξ ˆ a † ˆ b † )] [20], whereˆ a , ˆ b are the mode operators for the two input arms of theMZI. When acting on vacuum, the squeezing operatorgenerates the state [21] | ψ (cid:105) = 1cosh r ∞ (cid:88) n =0 ( − e i φ tanh r ) n (ˆ a † ˆ b † ) n n ! | (cid:105) , (2)where ξ = r e i φ is determined by the material propertiesand the laser pump. We can ignore vacuum contributions since they do not affect the state of the multi-partite sys-tem in any way. We also neglect higher order photon pairproduction in the SPDC process as this occurs rarely;for typical experimental parameters and utilising a CWpump, the generation of multi-pair states as a fraction ofsingle biphotons is of the order of 10 − per Watt of pumppower [22]. This places an upper limit to how strong thecrystal can be pumped before multiple pair productionchanges the dynamics of the protocol. The low conver-sion efficiency in SPDC is due to the relative weakness ofthe signal and idler fields relative to the pump field [23].For a thin SPDC crystal, the single biphoton state canbe expressed as [24, 25] | ψ (cid:105) = (cid:90) d ω s d ω i E p ( ω s + ω i )Ψ( k s + k i − k p ) × ˆ a † ( ω s , k s )ˆ b † ( ω i , k i ) | (cid:105) , (3)where the envelope of the pump laser E p and wave num-ber correlation function Ψ are a direct consequence ofphase-matching conditions relating to the conservationof energy and momentum, respectively, in the crystal.We require that the photons are quasi-monochromatic,which can be achieved by either using a monochromaticpump beam, or by frequency filtering post-SPDC. Forquasi-monochromatic photons, the non-linear term in the S -matrix from two-photon scattering becomes negligibleand the process can be described linearly, where the to-tal phase shift accumulated during the interaction is thesum of the phase shifts imparted by the individual pho-tons. Additionally, photons with a broader bandwidthare more likely to excite the emitter due to their shortertemporal length, which may result in undesirable spon-taneous emission [9].In order to successfully reach maximal entanglement,the down-conversion process needs to be degenerate, i.e.,producing signal and idler photons with the same fre-quency. A frequency difference between the signal andidler photons would impart which-path information dur-ing the scattering process, as the total acquired phase-shift is stronger for one emitter than the other (assumingnon-identical emitters). The frequency of the generatedphotons needs to be optimised for the central energiesand line-widths of the two quantum emitters. The phaseshift imparted by the photons affects the interferenceat the second beam splitter and, consequently, the finalstate of the light-matter system. In order to successfullybuild up entanglement, we require t ( ω ) = t ( ω ), where t i ( ω ) is the transmission coefficient for the scattering inarm i . The frequency of the probe photons, therefore,needs to satisfy either of the following: (cid:126) ω = 12 (cid:20) E + E ± (cid:113) ( E − E ) − (cid:126) Γ Γ (cid:21) , (4a)or (cid:126) ω = E Γ − E Γ Γ − Γ , (4b)where E and E are the energies of the emitters, andΓ and Γ are their line-widths, respectively [see Supple-mental Material (SM) for more details [26]].Next, we consider the state of the system after N de-tection events. Let m be the number of events where both detectors register a photon and n be the number ofevents where the two photons reach the same detector,with N = m + n . Then the state of the system after the( N + 1) th probe and right before photon-detection canbe expressed as | ψ m,n (cid:105) = 14 c m,n (cid:20) (1 + t ( ω )) m ( t ( ω ) − n +1 |↑↓(cid:105) + (1 + t ( ω )) m (1 − t ( ω )) n +1 |↓↑(cid:105) (cid:21) ⊗ (cid:20)(cid:0) ˆ a † (cid:1) + (cid:16) ˆ b † (cid:17) (cid:21) | (cid:105) + 12 c m,n (cid:20) ( t ( ω ) + t ( ω )) m +1 n |↑↑(cid:105) + (1 + t ( ω )) m +1 ( t ( ω ) − n |↑↓(cid:105) + (1 + t ( ω )) m +1 (1 − t ( ω )) n |↓↑(cid:105) + 2 m +1 n |↓↓(cid:105) (cid:21) ⊗ ˆ a † ˆ b † | (cid:105) , (5)where c m,n is the respective normalization constant [seeSM for more details [26]]. Post-selecting on the photondetection, we obtain the heralded state of the emitters.Given that there is no information about which emitterhas gained a phase shift, some of the which-path informa-tion of each photon is erased, resulting in a cumulativeentanglement gain.We characterize the amount of entanglement for thetwo-qubit state post-photon detection, ρ , using the con-currence, C ( ρ ) [27]. Fig. 2 shows how the concurrenceof two non-identical emitters can reach unity by repeat-edly sending in photon pairs at the optimal frequency[Eq. (4)] and keeping track of the photon detection sig-nature. Perfect entanglement is achieved regardless ofthe photon detection signature, and there is no need toreject samples on the basis of certain measurement out-comes. Furthermore, the process does not destroy anygenerated entanglement. Instead, in the case where therehas been at least one coincident photon detection, furtherprobing will toggle the state of the qubits between | Ψ + (cid:105) and | Ψ − (cid:105) , where | Ψ ± (cid:105) = ( |↑↓(cid:105) ± |↓↑(cid:105) ) / √
2. Otherwise,if photons are registered by just one of the detectors atevery iteration, the qubits keep approaching the maxi-mally entangled state ( |↑↑(cid:105) + exp(i φ ) |↓↓(cid:105) ) / √
2, where φ is a cumulative phase determined by t ( ω ) and t ( ω ).The number of iterations needed to ensure that per-fect entanglement has been reached is closely related tothe average concurrence after the first iteration, weightedby the photon outcome probabilities, which in turn de-pends on the ratio of the line-widths of the two emittersas well as the detuning of their central energies. The tra-jectories in Fig. 2 show the random process of how theconcurrence is updated after each photo-detection event.The measurement process is characterised by a projec-tion operator, assuming perfect photon counters. Thescattering process is then repeated, where the state ofthe qubits is now the reduced density matrix after theprevious measurement projection. Given a quasi-mono- chromatic probe photon and zero photon loss, the (re-duced) density matrix of the two emitters remains pure.In some situations, such as those shown in Fig. 2, itmay be desirable to pair emitters with larger spectralvariations in order to maximally entangle in fewer detec-tion events without needing to consider the detector sig-natures. In the case of emitter pairings where the initialaverage concurrence is low, it is still possible to obtainmaximal entanglement within fewer detection events, al-beit with a lower probability, by considering the detec-tor signatures; the concurrence reaches unity when bothphotons reach the same detector. This can be seen inthe first line of Eq. (5), where such an outcome would re-sult in the Bell state | Ψ ± (cid:105) = √ ( |↑↓(cid:105) ± |↓↑(cid:105) (up to someoverall phase). When the emitters are identical, bothscattering events in the two arms impart similar phases,which leads to a very high probability of detection coin-cidences in the output. In this situation, the amount ofentanglement generated is negligble. When two photonsare found in the same detector, the concurrence jumps tounity. Therefore, we find the remarkable property thatmore dissimilar emitters can produce entanglement at afaster rate. There are regimes of δ/ Γ and Γ / Γ wheremere dissimilarity is not sufficient, but the range of emit-ters that can be entangled efficiently is vastly larger thanwhen we require that all emitters are identical.We now consider photon loss during the scattering in-teraction at either emitter, which negatively impacts theentanglement process (we still assume near-perfect pho-ton detection efficiency). The β -factor is defined as thecoupling to the non-guided modes (i.e., modes resultingin loss to the environment) as a fraction of the total fieldcoupling, β = Γ / (Γ + γ ). Once we introduce photon lossto our system, describing the scattering process becomesmore involved as the photon in the guided mode will notonly obtain a phase shift, but will also undergo a changein the probability amplitude (since γ is no longer zero).The transmission coefficient of the photon that is lost to (a) (b) (c) (d) (e) (f) 𝛿/Γ ! = 5 𝛿/Γ ! = 3 FIG. 2. Typical concurrence trajectories given a series ofphoton detection events (horizontal axis) for the lossless case( γ = 0). Γ and Γ are the emitter line-widths, and the centralenergy detuning between the two emitters is δ = | E − E | .Here, δ/ Γ = 3 (left column) and 5 (right column), and Γ / Γ is set to 1 [(a) and (b)], 3 [(c) and (d)] and 5 [(e) and (f)]. Weobserve that within some spectral parameters, more dissimilaremitters can produce entanglement at a faster rate. the environment is given by [28] t e ( ω ) = − i (cid:126) √ Γ γ (cid:126) ω − E + i (cid:126) (Γ + γ ) / . (6)One way to overcome the consequences of scatter-ing losses is to consider photon number resolving de-tectors and discard samples where photon loss has oc-cured. However, given that such detectors types are stillin the experimental stage, we consider non-number re-solving detectors in our calculations. The methodologyfor generating the trajectories is the same as for β = 1,but we will find that the qubits are in a mixed state post-photon detection. For emitter matches that reach perfectentanglement rapidly in the lossless case, it is possible toobtain over 99% concurrence within a few iterations for β ∼ . |↑↑(cid:105) and |↓↓(cid:105) , and |↑↓(cid:105) and |↓↑(cid:105) due to γ no longer being zero [see SM for more details [26]]. Per-forming a bit flip balances out the probability amplitudesof these spin states and improves the amount of possible (a) β = 0.9 (b) (c) (d) β = 0.95 β = 0.9 (a) (b) (c) (d) FIG. 3. Typical concurrence trajectories given a series ofphoton detection events for β = 0 . β =0 .
95 (right column). Γ and Γ are the emitter line-widths,and δ = | E − E | is the central energy detuning between thetwo emitters. Here, δ/ Γ = 3, Γ / Γ = 3 [(a) and (b)], andΓ / Γ = 5 [(c) and (d)]. The horizontal axis includes bothsuccessful photon-detection events and zero-photon detectionsdue to scattering losses. entanglement generation. Fig. 3 shows random concur-rence trajectories for the β < C > .
99 within the first 10iterations is 10 −
50% for the shown configurations. Thisoccurs after several simultaneous photon measurementsat both detectors.Next, we address challenges to the physical implemen-tation of the proposed scheme. We have seen how pho-ton loss degrades the entanglement generation processand decreases the rate at which samples can be suc-cessfully entangled. However, the results can be im-proved by implementing a bit-flip after every photon de-tection event. Alternatively, one can make use of photon-number-resolving detectors and discard samples where aphoton is lost.Another physical issue that needs to be taken into ac-count is the finite coherence time of the solid-state emit-ters, which is affected by mechanisms such as the spin-orbit and nuclear-spin interactions [29]. For semiconduc-tor quantum dots, this coherence time is relatively short,ranging between >
100 ns and several microseconds [30–33]. For nitrogen-vacancy centres in diamond, the co-herence time can be in the millisecond range [34–36],and exceeding half a second when enhanced by meansof decoupling pulsing to suppress the spin decoherence (a) (b)
FIG. 4. Typical concurrence trajectories given a series ofphoton detection events (horizontal axis) for the lossless case( γ = 0) when probing with both | , (cid:105) and | , (cid:105) states, where | , (cid:105) states occur ∼
15% of the time. Here, δ/ Γ = 3 (a) and 5(b), Γ / Γ is set to 1, and the photon detectors are assumed tobe number-resolving. The shaded region represents the areaunder the curve in Fig. 2. This shows that higher-order pairproduction in the down-conversion process is not detrimentalto entanglement generation. [37]. The emitters must survive long enough for the en-tanglement generation to take place. This means thatSPDC in the weak photon generation limit may be tooslow. Strong pump amplitudes will create multiple pairs,however, and we must take into account the effect offour-photon scattering. Fig. 4 compares the concurrenceof the emitters when probing with just single biphotonstates (i.e., | , (cid:105) ) with possible higher-order pair produc-tion, using photon-number resolving detectors. The gen-eration of multi-biphoton states does not disrupt the en-tanglement generation process, but rather may enhanceit.In conclusion, we have presented a way to maximallyentangle two solid-state quantum emitters via a cumula-tive entanglement generation protocol, taking into con-sideration the inhomogeneity arising from the fabrica-tion process. Our protocol does not require that wediscard entanglement due to undesired photon detectionoutcomes. We also have accounted for scattering photonlosses and show that the results are still promising: forcertain emitter pairings, it is still possible to create en-tanglement with a concurrence of > .
99 and <
10% scat-tering losses within a small number of photon-detectionevents. Additionally, the setup is relatively simple andcan be implemented using current technology. We findthat in trying to generate entanglement, we have moreflexibility than previously thought; in fact, larger energydetuning or line-width variations might result in fasterentanglement generation with higher concurrence, bring-ing us closer to solid state entanglement as a viable tech-nology for quantum information processing.The authors thank J. Iles-Smith and D.L. Hurst forvaluable discussions. E.C. is supported by an EPSRCstudentship. P.K. is supported by the EPSRC QuantumCommunications Hub, Grant No. EP/M013472/1. ∗ ecallus1@sheffield.ac.uk † p.kok@sheffield.ac.uk[1] R. Jozsa and N. Linden, Proceedings of the Royal Societyof London. Series A: Mathematical, Physical and Engi-neering Sciences , 2011 (2003), quant-ph/0201143.[2] S. D. Barrett and P. Kok, Physical Review A , 060310(2005), quant-ph/0408040.[3] C. Cabrillo, J. I. Cirac, P. Garc´ıa-Fern´andez, andP. Zoller, Physical Review A , 1025 (1999), quant-ph/9810013.[4] S. Bose, P. L. Knight, M. B. Plenio, and V. Vedral, Phys-ical Review Letters , 5158 (1999), quant-ph/9908004.[5] L.-M. Duan and H. J. Kimble, Physical Review Letters , 253601 (2003), quant-ph/0301164.[6] Y. Arakawa and M. J. Holmes, Applied Physics Reviews , 021309 (2020).[7] T. Heuser, J. Große, A. Kaganskiy, D. Brunner, andS. Reitzenstein, APL Photonics , 116103 (2018).[8] L. Zhai, M. C. L¨obl, J.-P. Jahn, Y. Huo, P. Treutlein,O. G. Schmidt, A. Rastelli, and R. J. Warburton, Ap-plied Physics Letters , 083106 (2020), 2008.11735.[9] D. L. Hurst, K. B. Joanesarson, J. Iles-Smith, J. Mork,and P. Kok, Physical Review Letters , 023603 (2019),1901.03631.[10] P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A.Walmsley, New Journal of Physics , 093011 (2008),0807.1409.[11] U. L. Andersen, T. Gehring, C. Marquardt, andG. Leuchs, Physica Scripta , 053001 (2016),1511.03250.[12] R. J. Coles, D. M. Price, J. E. Dixon, B. Royall,E. Clarke, P. Kok, M. S. Skolnick, A. M. Fox, and M. N.Makhonin, Nature Communications , 11183 (2016),1506.02266.[13] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeu-tel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller,Nature , 473 (2017), 1608.00446.[14] B. Lang, D. M. Beggs, and R. Oulton, Philosophi-cal Transactions of the Royal Society A: Mathematical,Physical and Engineering Sciences , 20150263 (2016),1601.04591.[15] A. E. Lita, A. J. Miller, and S. W. Nam, Optics Express , 3032 (2008).[16] D. Fukuda, G. Fujii, T. Numata, K. Amemiya,A. Yoshizawa, H. Tsuchida, H. Fujino, H. Ishii,T. Itatani, S. Inoue, and T. Zama, Optics Express ,870 (2011).[17] J.-T. Shen and S. Fan, Physical Review A , 062709(2007).[18] A. Nysteen, D. P. S. McCutcheon, M. Heuck, J. Mørk,and D. R. Englund, Physical Review A , 062304(2017), 1612.04803.[19] S. Fan, S¸. E. Kocaba¸s, and J.-T. Shen, Physical ReviewA , 063821 (2010), 1011.3296.[20] H. Seifoory, S. Doutre, M. M. Dignam, and J. E. Sipe,Journal of the Optical Society of America B , 1587(2017).[21] B. L. Pieter Kok, Introduction to Optical Quantum In-formation (Cambridge University Press, 2010).[22] J. Schneeloch, S. H. Knarr, D. F. Bogorin, M. L.Levangie, C. C. Tison, R. Frank, G. A. Howland, M. L.
Fanto, and P. M. Alsing, Journal of Optics , 043501(2019), 1807.10885.[23] C. Couteau, Contemporary Physics , 291 (2018),1809.00127.[24] S.-Y. Baek and Y.-H. Kim, Physical Review A , 043807(2008).[25] O. Kwon, Y.-S. Ra, and Y.-H. Kim, Optics Express ,13059 (2009).[26] .[27] S. Hill and W. K. Wootters, Physical Review Letters ,5022 (1997), quant-ph/9703041.[28] E. Rephaeli and S. Fan, Photonics Research , 110(2013).[29] J. Fischer, M. Trif, W. Coish, and D. Loss, Solid StateCommunications , 1443 (2009), 0903.0527.[30] J. Yoneda, K. Takeda, T. Otsuka, T. Nakajima, M. R.Delbecq, G. Allison, T. Honda, T. Kodera, S. Oda,Y. Hoshi, N. Usami, K. M. Itoh, and S. Tarucha, NatureNanotechnology , 102 (2018), 1708.01454.[31] J. H. Prechtel, A. V. Kuhlmann, J. Houel, A. Ludwig,S. R. Valentin, A. D. Wieck, and R. J. Warburton, Na-ture Materials , 981 (2016). [32] J. Houel, J. H. Prechtel, A. V. Kuhlmann, D. Brunner,C. E. Kuklewicz, B. D. Gerardot, N. G. Stoltz, P. M.Petroff, and R. J. Warburton, Physical Review Letters , 107401 (2014), 1307.2000.[33] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,and A. C. Gossard, Science , 2180 (2005).[34] E. D. Herbschleb, H. Kato, Y. Maruyama, T. Danjo,T. Makino, S. Yamasaki, I. Ohki, K. Hayashi, H. Mor-ishita, M. Fujiwara, and N. Mizuochi, Nature Commu-nications , 3766 (2019).[35] J. M. Zadrozny, J. Niklas, O. G. Poluektov, and D. E.Freedman, ACS Central Science , 488 (2015).[36] K. D. Jahnke, B. Naydenov, T. Teraji, S. Koizumi,T. Umeda, J. Isoya, and F. Jelezko, Applied PhysicsLetters , 012405 (2012), 1206.4260.[37] N. Bar-Gill, L. Pham, A. Jarmola, D. Budker, andR. Walsworth, Nature Communications , 1743 (2013),1211.7094.[38] W. K. Wootters, Physical Review Letters , 2245(1998), quant-ph/9709029. SUPPLEMENTAL MATERIAL: MAXIMAL ENTANGLEMENT GENERATION IN SPECTRALLYDISTINCT SOLID STATE QUBITSScattering amplitudes
The scattering coefficient gained by a photon (upon scattering to a guided mode) is [19] t ( ω ) = (cid:126) ω − E − i (cid:126) (Γ − γ ) / (cid:126) ω − E + i (cid:126) (Γ + γ ) / , (SM.1)where E is the transition energy of the emitter, and Γ and γ are the coupling rates to the guided and non-guidedmodes, respectively. When a photon is lost to the environment upon scattering, it gains a probability amplitude givenby [28] t e ( ω ) = − i (cid:126) √ Γ γ (cid:126) ω − E + i (cid:126) (Γ + γ ) / . (SM.2)In the case of two-photon scattering, the scattering is characterised by a linear term and a bound state term. For thezero-, one- and two-photon loss cases, the post-scattering wavefunctions become [28]˜ β ( ω , ω ) = 12 t ( ω ) t ( ω ) [ β ( ω , ω ) + β ( ω , ω )]+ i √ Γ2 π s ( ω ) s ( ω ) (cid:90) d k [ s ( k ) + s ( ω + ω − k )] β ( k, ω + ω − k ) , (SM.3)˜ β e ( ω , ω ) = t ( ω ) t e ( ω ) β ( ω , ω ) + t e ( ω ) t ( ω ) β ( ω , ω )+ √ Γ π s ( ω ) s e ( ω ) (cid:90) d k [ s ( k ) + s ( ω + ω − k )] β ( k, ω + ω − k ) , (SM.4)and ˜ β ee ( ω , ω ) = 12 t e ( ω ) t e ( ω ) [ β ( ω , ω ) + β ( ω , ω )]+ i √ Γ2 π s e ( ω ) s e ( ω ) (cid:90) d k [ s ( k ) + s ( ω + ω − k )] β ( k, ω + ω − k ) , (SM.5)respectively, where β ( ω , ω ) is the original wavepacket, such that (cid:82) d ω d ω | β ( ω , ω ) | = 1, and s ( ω ) = (cid:126) √ Γ (cid:126) ω − E + i (cid:126) (Γ + γ ) / , (SM.6) s e ( ω ) = (cid:126) √ γ (cid:126) ω − E + i (cid:126) (Γ + γ ) / . (SM.7)Since we consider identical monochromatic photons where β ( ω , ω ) = δ ( ω − ω ) δ ( ω − ω ), the integral of the boundstate term vanishes. This is because monochromatic photons have an infinite temporal spread and therefore do notexcite the emitter, resulting in no spontaneous emission since the emitter remains in the ground state (indeed, theterm s ( ω ) is related to the spontaneous emission of the emitter into the waveguide [19]).Integrating the post-scattering wavefunctions for the monochromatic case, the scattering amplitudes simplify to (cid:90) d ω d ω ˜ β ( ω , ω ) = t ( ω ) , (SM.8) (cid:90) d ω d ω ˜ β e ( ω , ω ) = 2 t ( ω ) t e ( ω ) , (SM.9) (cid:90) d ω d ω ˜ β ee ( ω , ω ) = t e ( ω ) . (SM.10) Emitter density matrix
Zero photon-loss case ( β = 1) In the case of no photon losses, the post-measurement state of the qubit remains pure, regardless of the detectionoutcome. This is due to our choice of using a single frequency biphoton state and the absence of scattering losses –each detection measurement corresponds to a single projection measurement. (Note that this no longer holds whenusing higher order biphoton states.) We therefore can make use of the usual wavefunction and ket notation to obtaina general expression for the state of the qubits after any number of photon-measurement events.We start with both emitters in the state √ ( |↑(cid:105) + |↓(cid:105) ), giving an initial state | ψ (cid:105) = 12 ( |↑(cid:105) + |↓(cid:105) ) ⊗ ( |↑(cid:105) + |↓(cid:105) ) ⊗ ˆ a † ˆ b † | (cid:105) , (SM.11)where ˆ a † and ˆ b † are the creation operators for the upper and lower input arms of the interferometer, both withfrequency ω .The state after the first beam-splitter (ˆ a → √ (cid:16) ˆ a + ˆ b (cid:17) and ˆ b → √ (cid:16) ˆ a − ˆ b (cid:17) ) is | ψ (cid:105) = 14 ( |↑(cid:105) + |↓(cid:105) ) ⊗ ( |↑(cid:105) + |↓(cid:105) ) ⊗ (cid:16) ˆ a † + ˆ b † (cid:17) (cid:16) ˆ a † − ˆ b † (cid:17) | (cid:105) = 14 ( |↑(cid:105) + |↓(cid:105) ) ⊗ ( |↑(cid:105) + |↓(cid:105) ) ⊗ (cid:20)(cid:0) ˆ a † (cid:1) − (cid:16) ˆ b † (cid:17) (cid:21) | (cid:105) . (SM.12)The state post-scattering is then | ψ (cid:105) = 14 (cid:0) t ( ω ) |↑(cid:105) + |↓(cid:105) (cid:1) ⊗ ( |↑(cid:105) + |↓(cid:105) ) ⊗ (cid:0) ˆ a † (cid:1) | (cid:105)−
14 ( |↑(cid:105) + |↓(cid:105) ) ⊗ (cid:0) t ( ω ) |↑(cid:105) + |↓(cid:105) (cid:1) ⊗ (cid:16) ˆ b † (cid:17) | (cid:105) . (SM.13)The state after the second beam-splitter is given by | ψ (cid:105) = 18 (cid:20) ( t ( ω ) − t ( ω )) |↑↑(cid:105) + ( t ( ω ) − |↑↓(cid:105) + (1 − t ( ω )) |↓↑(cid:105) + 0 |↓↓(cid:105) (cid:21) ⊗ (cid:20) (cid:0) ˆ a † (cid:1) + (cid:16) ˆ b † (cid:17) (cid:21) | (cid:105) + 14 (cid:20) ( t ( ω ) + t ( ω )) |↑↑(cid:105) + ( t ( ω ) + 1) |↑↓(cid:105) + (1 + t ( ω )) |↓↑(cid:105) + 2 |↓↓(cid:105) (cid:21) ⊗ ˆ a † ˆ b † | (cid:105) . (SM.14)Making use of projection measurements once again, a measurement outcome heralds one of the following states: | ψ (1 , (cid:105) = | ψ (0 , (cid:105) = ( t ( ω ) − t ( ω )) |↑↑(cid:105) + ( t ( ω ) − |↑↓(cid:105) + (1 − t ( ω )) |↓↑(cid:105) + 0 |↓↓(cid:105) (cid:112) [ | t ( ω ) − t ( ω ) | + | t ( ω ) − | + | − t ( ω ) | ] (SM.15)or | ψ (1 , (cid:105) = ( t ( ω ) + t ( ω )) |↑↑(cid:105) + ( t ( ω ) + 1) |↑↓(cid:105) + (1 + t ( ω )) |↓↑(cid:105) + 2 |↓↓(cid:105) (cid:112) [ | t ( ω ) + t ( ω ) | + | t ( ω ) + 1 | + | t ( ω ) | + 4] . (SM.16)We then repeated the probing process and replace the initial state of the emitters with either of the heralded states.Let us consider the system after N = m + n detection events, where m is the number of events where both detectorsregister a photon and n be the number of events where the two photons reach the same detector. Then we can expressthe state after the ( N + 1) th probe and right before photon-detection as | ψ m,n (cid:105) = 14 c m,n (cid:20) ( t ( ω ) + t ( ω )) m ( t ( ω ) − t ( ω )) n +1 |↑↑(cid:105) + (1 + t ( ω )) m ( t ( ω ) − n +1 |↑↓(cid:105) + (1 + t ( ω )) m (1 − t ( ω )) n +1 |↓↑(cid:105) (cid:21) ⊗ (cid:20)(cid:0) ˆ a † (cid:1) + (cid:16) ˆ b † (cid:17) (cid:21) | (cid:105) + 12 c m,n (cid:20) ( t ( ω ) + t ( ω )) m +1 n |↑↑(cid:105) + (1 + t ( ω )) m +1 ( t ( ω ) − n |↑↓(cid:105) + (1 + t ( ω )) m +1 (1 − t ( ω )) n |↓↑(cid:105) + 2 m +1 n |↓↓(cid:105) (cid:21) ⊗ ˆ a † ˆ b † | (cid:105) (SM.17)where c m,n is the respective normalization constant given by c m,n = (cid:20) (cid:12)(cid:12) ( t ( ω ) + t ( ω )) m ( t ( ω ) − t ( ω )) n (cid:12)(cid:12) + (cid:12)(cid:12) (1 + t ( ω )) m ( t ( ω ) − n (cid:12)(cid:12) + (cid:12)(cid:12) (1 + t ( ω )) m (1 − t ( ω )) n (cid:12)(cid:12) + | m n | (cid:21) / . (SM.18)Next we justify the limitation placed on the choice of frequency: by selecting a frequency that satisfies t ( ω ) = t ( ω ),(SM.17) simplifies to | ψ m,n (cid:105) = 14 c m,n (cid:20) (1 + t ( ω )) m ( t ( ω ) − n +1 |↑↓(cid:105) + (1 + t ( ω )) m (1 − t ( ω )) n +1 |↓↑(cid:105) (cid:21) ⊗ (cid:20)(cid:0) ˆ a † (cid:1) + (cid:16) ˆ b † (cid:17) (cid:21) | (cid:105) + 12 c m,n (cid:20) ( t ( ω ) + t ( ω )) m +1 n |↑↑(cid:105) + (1 + t ( ω )) m +1 ( t ( ω ) − n |↑↓(cid:105) + (1 + t ( ω )) m +1 (1 − t ( ω )) n |↓↑(cid:105) + 2 m +1 n |↓↓(cid:105) (cid:21) ⊗ ˆ a † ˆ b † | (cid:105) . (SM.19)Therefore, the choice of frequency allows us to obtain a maximally entangled Bell state | Ψ ± (cid:105) = ( |↑↓(cid:105) ± |↓↑(cid:105) ) / √ φ ) |↑↑(cid:105) + |↓↓(cid:105) . An expressionfor the relative phase can be obtained by considering how long it takes for the emitters to reach this state: if thisstate is reached after M consecutive coincident photon-detections, then exp(i φ ) = t M ( ω ) = t M ( ω ) . The frequency ω that satisfies the condition t ( ω ) = t ( ω ) is given by either of the following: (cid:126) ω = 12 (cid:20) E + E ± (cid:113) ( E − E ) − (cid:126) Γ Γ (cid:21) , (SM.20) (cid:126) ω = E Γ − E Γ Γ − Γ . (SM.21) General case ( β ≤ We can express the state of the system using the density matrix formalism (this is due to the possibility of obtaininga mixed state post-photon detection), which allows us to obtain a general expression for how the qubits evolve asthe probing process is repeated. We choose a frequency that satisfies Eq. (SM.21), where t ( ω ) = ± t ( ω ) no longerholds since γ is no longer zero. Furthermore, it may be the case that the three frequency choices result in differentscattering amplitudes for a given emitter pairing, and therefore, in different trajectories. We suppress the scatteringamplitude notation so that t i ( ω ) → t i and t e,i ( ω ) → t e,i .We start off with an arbitrary density matrix for the qubits, ρ emitters , and a photon in each input arm of theinterferometer, ˆ a and ˆ b : ρ = ρ emitters ⊗ (cid:2) ˆ a † ˆ b † | (cid:105) (cid:104) | ˆ a ˆ b (cid:3) , (SM.22)where ρ emitters = (cid:104)↑↑| (cid:104)↑↓| (cid:104)↓↑| (cid:104)↓↓| |↑↑(cid:105) c c c c |↑↓(cid:105) c c c c |↓↑(cid:105) c c c c |↓↓(cid:105) c c c c . (SM.23)For the first iteration, c ij = for all i, j .Interacting with the first beamsplitter, where ˆ a → √ (cid:16) ˆ a + ˆ b (cid:17) and ˆ b → √ (cid:16) ˆ a − ˆ b (cid:17) , the state evolves to ρ = ρ emitters ⊗ (cid:20) (cid:0) ˆ a † (cid:1) − (cid:16) ˆ b † (cid:17) (cid:21) | (cid:105) (cid:104) | (cid:2) ˆ a − ˆ b (cid:3) . (SM.24)0The state after the scattering interactions at the emitters is ρ = 14 (cid:88) i,j =1 , , ρ ( M i , M j ) ⊗ (cid:2) M i | (cid:105) (cid:104) | M j (cid:3) + 14 (cid:88) i,j =1 , , ρ ( N i , N j ) ⊗ (cid:2) N i | (cid:105) (cid:104) | N j (cid:3) − (cid:88) i,j =1 , , (cid:8) ρ ( M i , N j ) ⊗ (cid:2) M i | (cid:105) (cid:104) | N j (cid:3) + ρ ( N i , M j ) ⊗ (cid:2) N i | (cid:105) (cid:104) | M j (cid:3)(cid:9) , (SM.25)where M = (cid:34) (cid:35) ˆ a ˆ a ˆ r ˆ r , (SM.26) N = ˆ b ˆ b ˆ r ˆ r , (SM.27)and where ˆ r i is the photon annihilation operator to a reservoir around the emitter in arm i which represents scatteringlosses, and ρ ( ˆ m, ˆ n ) is the respective density matrix of the qubits associated with the scattered optical state ˆ m † | (cid:105) (cid:104) | ˆ n .The scattering amplitudes are given by the following transformations: |↑↑(cid:105)|↑↓(cid:105)|↓↑(cid:105)|↓↓(cid:105) ⊗ (cid:0) ˆ a † (cid:1) | (cid:105) −→ t |↑↑(cid:105) t |↑↓(cid:105)|↓↑(cid:105)|↓↓(cid:105) ⊗ (cid:0) ˆ a † (cid:1) | (cid:105) + t t e, |↑↑(cid:105) t t e, |↑↓(cid:105) ⊗ ˆ a † ˆ r † | (cid:105) + t e, |↑↑(cid:105) t e, |↑↓(cid:105) ⊗ (cid:16) ˆ r † (cid:17) | (cid:105) , (SM.28) |↑↑(cid:105)|↑↓(cid:105)|↓↑(cid:105)|↓↓(cid:105) ⊗ (cid:16) ˆ b † (cid:17) | (cid:105) −→ t |↑↑(cid:105)|↑↓(cid:105) t |↓↑(cid:105)|↓↓(cid:105) ⊗ (cid:16) ˆ b † (cid:17) | (cid:105) + t t e, |↑↑(cid:105) t t e, |↓↑(cid:105) ⊗ ˆ b † ˆ r † | (cid:105) + t e, |↑↑(cid:105) t e, |↓↑(cid:105) ⊗ (cid:16) ˆ r † (cid:17) | (cid:105) . (SM.29)The state then interacts with the second beamsplitter, leaving the state of the qubits unaltered and only changingthe optical state: ρ = 14 (cid:88) i,j =1 , , ρ ( M i, , M j, ) ⊗ (cid:2) M i, | (cid:105) (cid:104) | M j, (cid:3) + 14 (cid:88) i,j =1 , , ρ ( N i, , N j, ) ⊗ (cid:2) N i, | (cid:105) (cid:104) | N j, (cid:3) − (cid:88) i,j =1 , , (cid:8) ρ ( M i, , N j, ) ⊗ (cid:2) M i, | (cid:105) (cid:104) | N j, (cid:3) + ρ ( N i, , M j, ) ⊗ (cid:2) N i, | (cid:105) (cid:104) | M j, (cid:3)(cid:9) , (SM.30)where M and N now change to M = ˆ a (ˆ a + 2ˆ a ˆ b + ˆ b )ˆ a ˆ r √ (ˆ a + ˆ b )ˆ r ˆ r ˆ r , (SM.31) N = ˆ b (ˆ a − a ˆ b + ˆ b )ˆ b ˆ r √ (ˆ a − ˆ b )ˆ r ˆ r ˆ r . (SM.32)We make use of positive operator-valued measures made up of orthogonal projectors, where Π(ˆ x ) = ˆ x † | (cid:105) (cid:104) | ˆ x , andassume non-photon-number resolving detectors.1A click by the detector at output arm ˆ a , with no detection in the other detector, is only possible for the projectionmeasurements Π(ˆ a ), Π(ˆ a ˆ r ) and Π(ˆ a ˆ r ). This yields the following reduced density matrix for the qubits: ρ (1 , = 1 P (1 ,
0) Tr field (cid:2)(cid:0)
Π(ˆ a ) + Π(ˆ a ˆ r ) + Π(ˆ a ˆ r ) (cid:1) ρ (cid:0) Π(ˆ a ) + Π(ˆ a ˆ r ) + Π(ˆ a ˆ r ) (cid:1)(cid:3) = 1 P (1 , (cid:26) (cid:104) ρ (cid:0) ˆ a , ˆ a (cid:1) + ρ (cid:16) ˆ b , ˆ b (cid:17) − ρ (cid:16) ˆ a , ˆ b (cid:17) − ρ (cid:16) ˆ b , ˆ a (cid:17)(cid:105) + 18 (cid:104) ρ (ˆ a ˆ r , ˆ a ˆ r ) + ρ (ˆ b ˆ r , ˆ b ˆ r ) (cid:105) (cid:27) = 1 P (1 , (cid:32) | ( t − t ) | ( t − t )( t − ∗ ( t − t )( t − ∗ t − t − t ) ∗ | t − | ( t − − t ) ∗ t − t − t ) ∗ (1 − t )( t − ∗ | t − |
00 0 0 0 + 18 | t t e, | + | t t e, | | t t e, | | t t e, | | t t e, | | t t e, | | t t e, | | t t e, |
00 0 0 0 (cid:33) ◦ ρ emitters , (SM.33)where ◦ denotes the Hadamard, or element-wise, product of the two matrices, and ρ emitters is the density matrix ofthe qubits at the start of the probing round [Eq. (SM.23)]. The probability of obtaining this measurement outcome, P (1 , P (1 ,
0) = Tr (cid:2)(cid:0)
Π(ˆ a ) + Π(ˆ a ˆ r ) + Π(ˆ a ˆ r ) (cid:1) ρ (cid:0) Π(ˆ a ) + Π(ˆ a ˆ r ) + Π(ˆ a ˆ r ) (cid:1)(cid:3) = 116 (cid:2) c | t − t | + c | t − | + c | t − | (cid:3) + 12 (cid:2) c (cid:0) | t t e, | + | t t e, | (cid:1) + c | t t e, | + c | t t e, | (cid:3) , (SM.34)where c ij are elements of the emitter density matrix ρ emitters .For the measurement outcome (0 , b ), Π(ˆ b ˆ r ) and Π(ˆ b ˆ r ), which gives the same result asfor the previous case, i.e., ρ (0 , is given by Eq. (SM.33) and P (0 ,
1) is given by Eq. (SM.34).In the case of a coincidence detection, the post-measurement density matrix is given by ρ (1 , = 1 P (1 ,
1) Tr field (cid:104)
Π(ˆ a ˆ b ) ρ Π(ˆ a ˆ b ) (cid:105) = 14 P (1 , (cid:104) ρ (ˆ a , ˆ a ) + ρ (ˆ b , ˆ b ) + ρ (ˆ a , ˆ b ) + ρ (ˆ b , ˆ a ) (cid:105) = 14 P (1 , | ( t + t ) | ( t + t )( t + 1) ∗ ( t + t )( t + 1) ∗ t + t )( t + 1)( t + t ) ∗ | ( t + 1) | ( t + 1)( t + 1) ∗ t + 1)( t + 1)( t + t ) ∗ ( t + 1)( t + 1) ∗ | ( t + 1) | t + 1)2( t + t ) ∗ t + 1) ∗ t + 1) ∗ ◦ ρ emitters , (SM.35)where P (1 ,
1) = Tr (cid:104)
Π(ˆ a ˆ b ) ρ Π(ˆ a ˆ b ) (cid:105) = 14 (cid:2) c | ( t + t ) | + c | ( t + 1) | + c | ( t + 1) | + 4 c (cid:3) . (SM.36)Note that this reduced matrix represents a pure state.Finally, we have the case of no photon detection due to scattering losses at the emitters, which results in the state: ρ (0 , = 1 P (0 ,
0) Tr field (cid:2)(cid:0)
Π(ˆ r ) + Π(ˆ r ) (cid:1) ρ (cid:0) Π(ˆ r ) + Π(ˆ r ) (cid:1)(cid:3) = 14 P (0 , | t e, | + | t e, | | t e, | | t e, | | t e, | | t e, | | t e, | | t e, |
00 0 0 0 ◦ ρ emitters , (SM.37)2where P (0 ,
0) = 14 (cid:2) c ( | t e, | + | t e, | ) + c | t e, | + c | t e, | (cid:3) . (SM.38)The process can now be repeated from Eq. (SM.23), where we replace ρ emitters with one of the four possible resultingqubit density matrices, ρ ( i,j ) , depending on the photon-detection outcome. Concurrence
We use concurrence to measure the amount of entanglement that is generated between the two qubits: it is anentanglement monotone that can be easily applied to mixed states. The concurrence of two qubits in the state ρ isgiven by [38] C ( ρ ) = max(0 , λ − λ − λ − λ ) , (SM.39)where λ , . . . , λ are the eigenvalues, in decreasing order, of R = (cid:113) √ ρ ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) √ ρ. (SM.40)In the case of a pure state, | ψ (cid:105) = c |↑↑(cid:105) + c |↑↓(cid:105) + c |↓↑(cid:105) + c |↓↓(cid:105) , the concurrence simplifies to C ( | ψ (cid:105) ) = | (cid:104) ψ | σ y ⊗ σ y | ψ ∗ (cid:105) | = 2 | c c − c c | ..