On-chip spin-photon entanglement based on single-photon scattering

The realization of on-chip quantum gates be-8

The realization of on-chip quantum gates between photons and solid-state spins is a key building block for quantum-information processors, enabling, e.g., distributed quantum computing, where remote quantum registers are interconnected by flying photons [1-3].Self-assembled quantum dots integrated in nanostructures are one of the most promising systems for such an endeavor thanks to their near-unity photonemitter coupling [4] and fast spontaneous emission rate [5].
Here we demonstrate an onchip entangling gate between an incoming photon and a stationary quantum-dot spin qubit.The gate is based on sequential scattering of a timebin encoded photon with a waveguide-embedded quantum dot and operates on sub-microsecond timescale; two orders of magnitude faster than other platforms [6][7][8][9].Heralding on detection of a reflected photon renders the gate fidelity fully immune to spectral wandering of the emitter.These results represent a major step in realizing a quantum node capable of both photonic entanglement generation [10-13] and on-chip quantum logic, as demanded in quantum networks [14] and quantum repeaters [15].
In a future quantum network [16], remote quantum nodes could be connected by a large web of entangled photons.Traditionally these photonic states have been generated probabilistically by fusing smaller states, which typically requires an exponential overhead of ancillary photons [17].The advent of a deterministic quantum interface between light and matter promises to radically change this notion [3].For such systems, a flying photon is funneled into a waveguide or cavity and interacts efficiently with a quantum emitter that hosts a single spin [4].Coherent manipulation of the spin state entangles it with the photon, forming the basis for deterministic quantum gates and, e.g., the generation of photonic cluster states for quantum computing [18].
So far, significant progress has been made towards this goal, particularly the realization of spin-photon entanglement [7,10,11,13,[19][20][21][22], spin-spin entangle- We find that the heralded Bell-state fidelity can be near unity for our system due to the spectral selectivity of the QD that predominantly reflects photons resonant with the transition |⇑ → |↑⇓⇑ , despite residual QD spectral diffusion visible from the broadened transmission dip in Fig. 1c (see Methods; Supplementary Note II).
To calibrate the device, we subdivide the gate protocol into two separate experiments (Fig. 2).The first experiment probes the coherent nature of single-photon scattering (Fig. 2a), whereas the second experiment investigates the spin coherence with the built-in spin-echo sequence (Fig. 2c).
To demonstrate coherent scattering in the singlephoton regime, we use a coherent state with a mean photon number per pulse n ≪ 1.We prepare a time-bin qubit using an asymmetric Mach-Zehnder interferometer where a photon is superposed between early and late temporal modes |e and |l , and scatter off the QD spin initialized in |⇑ .If the input photon of spectral width σ o /2π is much narrower than the QD linewidth Γ/2π, the photon can be fully reflected due to destructive interference in transmission [30].By interfering the temporal modes of the reflected photon using the same interferom- 173 where Mi = σ(p (−58.8 ± 4.5)% and My = (57.3± 6.6)%, where resid- where ∆ h is the ground-state splitting.Eq. (3) holds  The waveguide device used in the current experiment is terminated by two high-efficiency grating outcouplers [S1] representing reflection and transmission ports.For implementing the scattering experiment, both the incident laser and reflected signal are coupled to the same grating outcoupler (Fig. S1).To distinguish the signal from the laser background, a cross-polarized scheme is used for excitation and collection.Before entering the waveguide, the polarization of the input light is carefully optimized with a set of waveplates such that it is orthogonal to the polarization of the collected light (set by another set of waveplates on the optical setup).The input is diagonally polarized (which can also be circularly polarized), thus only 50% of the light is coupled to the grating coupler which

II. SCATTERING THEORY OF SPIN-PHOTON BELL STATE GENERATION
In this section, we discuss the state evolution of the spin-photon system upon applying the entangling gate and develop an analytical expression for the entanglement fidelity.Our strategy is similar to the approach taken in Ref. [S2], which is to first evaluate the fidelity to lowest order in perturbation theory for each of the considered errors.
In the end the full fidelity is then found by multiplying the individual fidelities.
We start with modeling a right-propagating time-bin photonic qubit α |e + β |l in a two-sided waveguide where α, β ∈ C, and the QD spin is initially in the ground state |⇓ .Following from Fig. 1, the gate protocol consists of (1) applying a Ry (π/2) spin rotation to prepare a superposition spin qubit, (2) scattering of the early photon Ŝe , (3) a Ry (π)-rotation, and finally (4) the scattering of the late photon Ŝl .Each single-photon scattering process obeys the input-output relations [S3]: where the photon in each time-bin is assumed to center around the resonant frequency of the dominant transition (|⇑ ↔ |↑⇓⇑ ) with a Gaussian spectral profile, and r 1 (t 1 ) are the reflection (transmission) operators associated with the QD vertical transition |⇑ → |↑⇓⇑ with a decay rate Γ 1 (γ Y in the main text).r 2 (t 2 ) corresponds to the diagonal transition |⇓ → |↑⇓⇑ with decay rate Γ 2 (≡ γ X ).ω 2 = ω + ∆ h is the frequency of the Raman photon emitted from the diagonal transition where ∆ h is the ground-state splitting.The symbol • denotes off-resonant scattering when the spin is in |⇓ .Using Eq. (S1), the state evolution of the spin-photon system proceeds as In the ideal scenario where the early and late pulses are identical, monochromatic and resonant, and the QD optical cyclicity is infinite with no dephasing and loss, we have: (1) r 1 → −1 (resonant photons are coherently reflected with a π-phase shift), (2) t1 → 1,r 1 → 0 (off-resonant photons are being transmitted instead of reflected), (3) t 1 → 0 (complete destructive interference in the transmission); and (4) r 2 ,r 2 , t 2 , t2 → 0 (there are no Raman photons in the reflected and transmitted modes due to high cyclicity).As such, the ideal output state of the gate becomes Heralding on either the reflection or transmission of a scattered photon projects the system into a different spin-photon Bell state.By varying the phase θ p of the photonic qubit where β/α = e iθp and |α| 2 + |β| 2 = 1, all 4 maximally entangled states can be generated by the gate.
A. Scattering coefficients for a Λ-level emitter in two-sided waveguides The scattering problem of a weak coherent state on the Λ-level emitter has been solved in Ref. [S4] and its formalism can be easily extended to directly compute the scattering coefficients in Eq. (S1).Specifically, the output field bosonic operator of the waveguide can be expressed in terms of the incident field and dynamical response of the emitter from the non-Hermitian Hamiltonian Ĥnh [S2].In a two-sided waveguide configuration, we label the field operator in the reflection port by the subscript "r", and the transmitted port by "t" (Fig. S2).Assuming that a right-propagating light field â † in,t enters the waveguide, the output field operators on the transmitted (t) and reflected (r) ports are |ω ⇑ : ) Level scheme for a QD embedded in a two-sided waveguide under the Voigt magnetic field.Γ1 (Γ2) is the radiative decay rate into the waveguide from the transition |e → |⇑ (|e → |⇓ ).Γi = Γ t i + Γ r i for i ∈ {1, 2} includes both decay rates into the transmitted ('t") and reflected ('r") waveguide modes.γi is the radiative rate into the lossy modes (Not to be confused with the radiative decay rates γY ≡ Γ1 and γX ≡ Γ2 in the main text).
The output field operators have different detunings in their denominators because of different initial spin states of the QD: If the spin is initially |⇑ , the resonant frequency is ω 1 ; If it is |⇓ then the resonant frequency required to drive the diagonal spin transition is ω 2 = ω 1 + ∆ h .σij = |j i| is the atomic operator denoting a spin-flip in the atomic state when i = j.Note that when evaluating the probability of a spin-photon state, i.e., |e ⇓ r , the corresponding scattering coefficient r e 1 (ω) is first convoluted with a Gaussian lineshape Φ 1 (ω) and integrated with respect to ω [S2].
The individual resonant scattering coefficients in the frequency domain are where the off-resonant scattering coefficients are found similarly by replacing δ 1 → δ 1 + ∆ h .

B. Projection operators for measuring time-bin encoded photons
At the end of the entangling gate, measurements to read out the state of the photonic qubit are performed by registering detector clicks in three different detection time windows.The detection of a time-bin photon is formulated by projection operators on different photonic readout bases: where the bosonic creation operator â † e (t) represents the emission of a photon at time t in the early time-bin, and τ is the interferometric delay.The projections |e e| (|l l|) correspond to detecting photons in the side peak windows (green) (Fig. 3a), whereas |e l| refers to projection onto the middle detection window (blue central peak) where the early and late photons interfere.Since we only resolve the time-bin, the creation operator can be expressed in either the time or frequency domain.Using where the success probability P s is the trace of the output density matrix over the four basis states |i = {|e ⇑ r , |e ⇓ r , |l ⇑ r , |l ⇓ r } in the Hilbert space of the spin-photon system.It is given by Combining Eqs.(S8) and (S9) results in the formula for the gate fidelity conditioned on reflected photons

D. Perturbative form of the entanglement fidelity
The two integrals in Eq. ( S10) are the probabilities of detecting a photon of frequency ω 1 originated from the resonant scattering of the spin state |⇑ and the off-resonant reflection from |⇓ respectively.In particular, using Eq. ( S4) we find where we assume that the scattered photon is equally coupled to the reflected and transmitted modes, i.e., Γ r i = Γ t i = Γ i /2.σ o is the standard deviation of the spectral width of the incident Gaussian pulse.In evaluating Eq. ( S11) perturbatively we assume the frequency detuning δ 1 to be small compared to the QD total decay rate Γ and the ground-state splitting ∆ h for efficient light-matter interaction.

Spectral mode mismatch
By heralding on the detection of a reflected photon of frequency ω 1 within the time-bin window, the entanglement fidelity becomes immune to the spectral error due to the nonzero bandwidth σ o of the incident pulse to lowest order in perturbation theory.Using Eqs.(S10) and (S11), the resultant fidelity is Simply stated, photons which are not resonant with the QD transition will be transmitted instead of reflected.Since the gate is conditioned on the reflection of either an early or a late photon, the transmission of the photon only reduces the success probability.The gate will thus have unity fidelity as long as the dynamics of the early and late scattering events are identical.The same argument can be made for the broadening of the QD optical transition due to slow spectral diffusion compared to the QD lifetime.The spectral jittering on the QD resonance is modelled by taking δ 1 → δ 1 + δ e where δ e follows a Gaussian spectral diffusion profile N (0, σ e ) [S2].
If the entangling gate is heralded on the presence of transmitted photon; however, the fidelity becomes susceptible to the spectral mismatch error.A similar fidelity analysis shows as the spectral infidelity arises from incomplete destructive interference between the incident field and the resonantly scattered photon (t 1 = 0).Any spectral effects reducing this interference would stain the quality of the entangled state.It is important to note that despite the QD spectral reflectivity, there is still a small probability of detecting undesired Raman photons of frequency ω 2 = ω 1 +∆ h in the reflection due to the finite optical cyclicity.These photons result from the imperfect QD two-level system and are filtered out.
On the reflection port, photons could either originate from (i) resonant reflection on the spin-preserving transition (indicated by r 1 ), (ii) resonant Raman spin-flip process to |⇓ (r 2 ), or (iii) off-resonant reflection from |⇓ (r 1 ).A high cyclicity reduces the probability of resonant spin-flip process but strengthens off-resonant reflection.The undesired events (ii) and (iii) can be reduced by having a larger ground-state splitting ∆ h ≫ Γ. Coupling to lossy modes of the waveguide implies that the reflected photons are lost without being detected; as a result these events do not affect the fidelity.Effectively we find Here γ 1 is the radiative rate from the main transition |⇑ → |↑⇓⇑ which couples to lossy modes.In deriving Eq. ( S14) we define the optical cyclicity C ≡ (Γ 1 + γ 1 )/(Γ 2 + γ 2 ) [S5] and the total decay rate Γ = Γ 1 + Γ 2 + γ 1 + γ 2 where Γ 2 (γ 2 ) is the radiative decay rate into (outside) the waveguide.

Phonon-induced pure dephasing
The interaction of the QD with a phononic environment results in the broadening of the zero-phonon line and a broad phonon sideband [S6-S9].The latter can be filtered out while the former contributes to the reflection of incoherent photons which scramble the phase coherence of the spin-photon Bell state.The incoherent photons are only slightly broadened and thus cannot easily be removed by filters.
We follow the approach in Ref. [S2] and model this incoherent process as Markovian decoherence given by a dephasing rate γ d with the Lindblad operator √ 2γ d σee where |e s ≡ |↑⇓⇑ is the atomic excited state.The dephasing leads to a quantum jump to the excited state (with a dephasing probability P ωi γ d ) followed by the decay to either of the two hole ground states with probabilities set by the transition rates Γ i /Γ.The emitted photon into the waveguide is represented by a normalized photon density matrix ρ ωi γ d .This is described by the density matrix where ρ is the density matrix without a dephasing quantum jump.Initially there are also incoherent photons of frequency ω 2 due to finite optical cyclicity but these are subsequently filtered out together with phonon sidebands.
ρ ω1 γ d ⊗ |⇑ ⇑| is the photon density matrix resulting from the incoherent dephasing with a probability given by Here we take /2 since the dephasing effectively becomes an additional decay channel; however, the probability for the incoherent excited state to decay is governed only by the branching ratio thus To evaluate the effect of pure dephasing in the gate protocol, it is instructive to consider the propagation of the error as there are two separate scattering events which will both lead to incoherent decay.Since Eq. (S15) depends on whether there is a quantum jump to the excited state, we can assume that pure dephasing occurs primarily when the incident photon is resonant with the QD state since the excited state is unlikely to be populated via off-resonant scattering.As such, using Eq.(S15) there are two additional incoherent density matrices in the normalized output reduced density matrix Using Eq. ( S16) with |α| = |β| = 1/ √ 2, the entanglement fidelity under pure dephasing is

Spin dephasing
In this section, we investigate how the decoherence of the spin states affects the entanglement fidelity.Specifically we consider the dephasing of the QD spin ground states, due to the presence of an external Overhauser field effectively formed by a neighboring nuclear ensemble.This effect causes a superposition spin qubit to precess on the equatorial plane at a random frequency δ g slower than the QD decay rate, which is modelled by applying a time evolution operator T (∆t) = exp −iδ g Ŝz ∆t on the superposition spin state, where Ŝz = σz /2 [S2].In the course of the entangling gate, a π-pulse is applied between two scattering events to ensure the precession of the spin is reversed and thus the spin is eventually refocused.In theory, the superposition qubit starts to precess at t 0 and the π-rotation pulse is applied at t π .The spin is then refocused and read out at t r where t r − t π = t π − t 0 = ∆t must be satisfied for the perfect echo condition.In the experiment, a rotation pulse Ri = Ry,φr (π/2) is applied at t r to project the spin state onto one of its poles thus preventing further precession.
To understand how spin echo works for the gate, we introduce the spin-echo operator Ûecho ≡ T (t r −t π ) Ry (π) T (t π − t 0 ) which transforms the spin states into With Eq. (S19), the output state in Eq. (S2) becomes Eq. (S20) implies that the phase coherence between |e ⇓ r and |l ⇑ r depends on (i) the accumulated phase from spin precession, and (ii) the phase acquired from the early and late single-photon scattering events which is determined by the exact time of scattering occurred within the optical pulse.The former is effectively removed by the echo sequence as 2t π − t r − t 0 = 0, whereas the latter is made equal by interfering the time-bins with a matching time delay τ = 11.8 ns on the detection path.Since the time-bin qubit is created and measured using the same interferometer setup, by having an equal time delay τ e = τ d = τ for the excitation and detection paths, the interferometer temporally picks out events in which the exact time of scattering is in the same position of the pulse, i.e., r e 1 (t ′ ) = r l 1 (t ′ ) for some time t ′ ∈ Φ 1 (t) within the optical pulse.Therefore, the coherence of the spin-photon Bell state is well-preserved.
where ρ ⇓ is the initial spin density matrix and ρ − ≡ |− s −| s .E π/2 is the output density matrix.In addition to the incoherent spin flip with a probability p π/2 we here include known imperfections of the rotation pulse Ri y (π/2), which has a fidelity of F π 2 to coherently rotate the spin to the superposition state |+ s and a probability of 1 − F π 2 to project onto |− s .The fidelity of coherent spin rotation is determined by the limitations of the two-photon Raman scheme, which is dominated by the spin coherence time [S12]: for a pulse duration of T r,π/2 .
The probability of introducing a depolarizing error p π/2 during a Ry (π/2) rotation is estimated by integrating the exponential distribution over the pulse duration for a given incoherent spin-flip rate κ: The exponential distribution describes the probability of a random spin-flip occurring in a certain time period, where the spin-flip event is assumed not to depend on how much time has passed in the protocol (i.e. it is memory-less).
Similarly, for a Ry (π) pulse applied on an arbitrary spin state ρ s , where the initial spin density matrix is and p π is the probability of introducing the depolarizing error during a Ry (π) rotation found similarly as in Eq. (S23).
Using Eq. (S24) and ρ 1 = ρ 2 = ρ 3 = 0, ρ 4 = 1, the total π-rotation pulse fidelity which includes the contribution from both coherent and incoherent spin-flip processes can be estimated to be Using experimental values for the incoherent spin-flip rate κ = 0.021 ns −1 and the spin coherence time T * 2 = 23.2ns which are extracted in separate experiments [S12], we then estimate F π,total ≈ 91.6% for T r,π = 7 ns.For T * 2 = 21.4 ns estimated in Ref. [S5], the corresponding π-rotation fidelity is 91.2% which agrees well with the experiment (91%).Now we consider the evolution of the spin-photon system during the entangling gate.The protocol begins by preparing a time-bin photonic qubit ρ p and a spin state in ρ s : where the basis states that span the spin-photon density matrix are {|e ⇑ r , |e ⇓ r , |l ⇑ r , |l ⇓ r } conditioned on the detection of a reflected photon, as the events in which photons are transmitted do not contribute to the fidelity.The terms which are of interest are highlighted in red.As an example we evaluate one of the matrix elements |e ⇑ r e ⇑| r : For an ideal state of |ψ ideal = (α |e ⇓ r − β |l ⇑ r ) where |α| = |β| = 1/ √ 2, the entanglement fidelity is given by for κ ≪ Ω where Ω is the spin-rotation Rabi frequency and ΩT π = π for a π-pulse.Using the relevant parameters: T r,π = 7 ns, T r,π/2 = 3.5 ns, κ = 0.021 ns −1 and T * 2 = 23.2ns, we find F theory κ = 82.94%from the analytical form in Eq. (S29) taking r 1 = 1 and r1 = 0.

Spin readout error
The non-ideal spin readout by optical pumping is also considered to be one of the dominant sources of imperfections as it directly influences the spin readout basis.Due to finite optical cyclicity, optically pumping of the main transition can unfavourably result in an opposite outcome by flipping the spin state: where the readout fidelity is estimated to be F R = 96.6%[S12].Using Eqs.(S27) and (S30), the resulting entanglement fidelity under both rotation error and imperfect spin readout is F theory κ,R = 80.24%.From here it is apparent that the dominant infidelity results from incoherent spin flips (17%).To further investigate the influence of this error we plot the entanglement and π-rotation fidelities as a function of the spin-flip rate κ (Fig. S3), which indicates a linear dependence in the perturbative regime where κ ≪ Ω. Dashed black line shows the corresponding fidelities when κ = 0.021 ns −1 .An order of magnitude reduction in κ would lead to an improved entanglement fidelity of .This can be understood as the probability of n disjoint successful scattering events.To describe the effect of this error, we adopt a phase-damping model E d where Here Ŝ is the scattering matrix acting on the spin-photon density matrix and s i is the scattering amplitude obtained from Ŝ. E d introduces dephasing only to the QD spin state thus the photonic component is traced out before applying the phase-damping channel.Now we follow the same approach in Sec.II D 5 and consider propagation of the dephasing error in the protocol: Similarly, the entanglement fidelity under the driving-induced dephasing is found to be where the probability of successful scattering at zero detuning is given by Eq. (S4): To estimate the infidelity in the experiment, we first extract the average number of photons in the pulse n = Sn c T p Γ = 0.0496 × 0.2976 × 2 ns × 2.48 ns −1 ≈ 0.0732 where the relevant parameters are obtained from saturation measurements (Sec.IV).Given that optically cyclicity C = 14.7 and radiative loss rate γ 1 = γ 2 = 0.05 ns −1 , the experimental infidelity is estimated using the exact form in Eq. (S33) to be 1 − F theory n = 6.34%.

E. Overall fidelity and gate efficiency
Assuming perfect manipulation of the hole spin state, the fidelity of the entangling gate is expressed by: which is estimated to be 96.2% with Γ = 2.48 ns −1 , ∆ h = 2π × 7.3 GHz [S5] and γ d = 0.092 ns −1 (fitted in Sec.III).
This predominantly reflects the infidelity from phonon-induced pure dephasing 1−F theory (estimated), σ e = 0.3 ns −1 (estimated), the entangling gate efficiency is represented by the success probability where and is estimated to be 33.3%.

III. PHOTON VISIBILITY
Here we derive an analytical form of the visibility as a function of the QD pure dephasing rate.In the experiment, a time-bin encoded qubit (a weak coherent state) is scattered by a QD spin embedded in a two-sided photonic-crystal waveguide, and is subsequently measured by an asymmetric Mach-Zehnder interferometer with equal time delay as the qubit.The visibility is therefore a measure of the temporal overlap between the time-bins of the scattered pulses.
To model this, we consider the scattering of the time-bin photon with the QD and project the output state onto the photonic X-bases.The initial state of the system is expressed as |in = (|e + |l )/ √ 2 ⊗ |⇑ .Here we have neglected the multi-photon components from the coherent state since we are interested in the effect of pure dephasing.For a complete modelling of the photon visibility, however, one should include the effect of multi-photon scattering and inelastic contributions [S14].With Eq. (S4) the output state becomes where the superscript prime (') represents a scattered photon of frequency ω 2 = ω 1 and the subscript "r" ("t") indicates a reflected (transmitted) photon.We then seek the photonic density matrix by tracing out the spin degree of freedom, the transmitted photons as well as the wrong frequency state ω 2 .For ease of computation the scattering coefficients are replaced by C i where i refers to the time-bin, thus Now (S39) is used to evaluate the middle-bin intensity in detector D2(D1): where the output photon state is projected onto the superposition state âe (t) ± e iθp âl (t) which is equivalent to adding a phase shifter on the long path of the excitation interferometer and interfering both bins.Setting θ p = 0 implies projecting the output state into the p ± = |±X p ±X| p bases as described in the main text.The projected state is then traced out in both the early and late time bases.The photon visibility is defined as the normalized contrast of the middle-bin intensity when θ p = 0: To further simplify the above expression, we consider the scattering events of the early and late bins to be identical, i.e., with the same scattering coefficient C e = C l = r 1 .as justified in Sec.II D 4. Therefore, under this assumption the photon visibility becomes unity in the single-photon regime.

A. Visibility expression including pure dephasing
Following from the discussion in Sec.II D 3, we can now take into account the effect of phonon-induced pure dephasing.In essence, the resulting spin-photon density matrix is the sum of coherent and incoherent parts as described by Eqs.(S15) and (S16).The advantage of the formalism in (S15) is that its effect can be straightforwardly included in Eq. (S39).Accordingly, the new photonic density matrix becomes where the last two terms correspond to dephasing occuring during the single-photon scattering of either the early or late time-bin.The effect of pure dephasing on the multi-photon component is not considered due to its polynomial dependence on the mean photon number per pulse n, which is negligible as n ≪ 1.Note that the incoherent photon does not interfere with other photons since Tr âe ρ ω1 γ d ,e |l ∅| l â † l = Tr âe ρ ω1 γ d ,e × Tr(|l ∅| l ) = 0.This means only the total intensity is affected and Eq.(S41) can be simplified as This indicates that in the single-photon scattering limit where n ≈ 0, the y-intercept of the visibility curve in Fig. 2b is given by the pure dephasing rate per QD total decay rate Γ.Here Γ = 2.48 ns −1 is measured in Ref. [S5].A linear fit of the data gives a y-intercept of V p (n = 0) = 0.926 ± 0.003 implying γ d ≈ (0.092 ± 0.004) ns −1 .Apart from measuring the photon visibility, another approach to probe the single-photon nature of the scattering process is through QD saturation measurement, in which the QD response is observed by scanning the power of the input qubit laser.From fitting the scattered signal, the mean photon number per pulse n can be extracted, where n ≪ 1 indicates the scattering occurs in the single-photon regime.To mimic the entangling gate experiment, we prepare a single pulse of 2 ns duration and scatter on a QD spin initialized in either |⇑ or |⇓ .Due to the QD spin-dependent reflectivity, the input photon which is resonant with the QD transition |⇑ → |↑⇓⇑ is coherently reflected.By time-gating on the reflected signal (Fig. S4a; green shaded region) and increasing the input power, the QD prepared in |⇑ becomes saturated (Fig. S4b).The intensity in the reflected signal is fitted assuming a two-level system between |⇑ → |↑⇓⇑ : where Ω 1 is the Rabi frequency driving the transition |⇑ → |↑⇓⇑ , δ is the probe laser detuning.b 1 , b 2 and b 3 are free fit parameters.In particular, b 1 is the calibrating parameter that associates the Rabi frequency to the input power P via Ω 1 = √ b 1 P which includes optical losses on the excitation path.
Eq. (S44) holds when Γ ≫ κ g and T p ≫ Γ −1 where κ g is the effective spin-flip rate between the hole ground states and T p is the qubit pulse duration.The first condition implies that the main transition is eventually saturated as the QD decays faster than the spin can recycle, thus |⇓ effectively becomes dark.This is generally true since κ g is typically on the order of 10 −7 ns −1 at the plateau center voltage [S15], which is lower than Γ = 2.48 ns −1 .The second condition ensures that the QD decays back to |⇑ before the next scattering event within the pulse.When the driving pulse is sufficiently long, i.e., T p = 2 ns > Γ −1 = 0.4 ns with increasing power, the QD saturates similarly as when being driven by a weak continuous-wave laser.In addition, a finite optical cyclicity leads to a resonant spin-flip into the dark state |⇓ reflecting a photon of frequency ω 2 = ω 1 which is filtered out, thus only reducing the total intensity included in b 3 and not affecting the scaling of the relevant parameters in Eq. (S44).

FIG. 1 .
FIG. 1. Operational principle of the photon-scattering gate generating spin-photon entanglement.(a) A coherently controlled spin in a QD (red) inside a photonic-crystal waveguide, where a Bell state (cyan lines) is generated upon conditional detection of a reflected photon.(b) QD level diagram.The excited state |↑⇓⇑ predominantly decays into |⇑ with rate γY as γY ≫ γX .The wavelength of the main transition is 945 nm.Coherent control of the metastable hole spin ground states (magenta arrows, Rabi frequency Ω) is realized via two-photon Raman processes by a detuned laser.(c) Transmission measurement of the QD at By = 2 T when preparing the spin state in either |⇑ or |⇓ .(d) State evolution at different points in time during gate operation.At t1, the QD spin (red) is prepared in a superposition state.At t2, spin-dependent QD scattering occurs for the early time-bin |e .A π-rotation of the spin at t3 is followed by scattering of the late time-bin |l photon pulse at t4.The two distinct Bell states φ − ( ψ − ) are generated conditioned on the detection of a reflected (transmitted) photon.

(
Dated: June 16, 2022) CONTENTS I. Cross-polarization scheme on waveguides 2 II.Scattering theory of spin-photon Bell state generation 2 A. Scattering coefficients for a Λ-level emitter in two-sided waveguides 3 B. Projection operators for measuring time-bin encoded photons 4 C. Formula for the operational fidelity 5 D. Perturbative form of the entanglement fidelity 6 FIG. S1.Scanning Electron Microscope (SEM) image of the waveguide and polarizations of the input and reflected light.Dark grey arrows denote the predefined polarizations of the grating couplers.

5.2
Incoherent spin-flip error and finite T * The next error concerns spin decoherence induced by the red-detuned spin rotation laser and due to finite spin coherence time T * 2 .The former effect has been observed in Refs.[S10, S11] which results in power-dependent spinflips, thereby destroying the coherence of the spin qubit during spin rotations.Despite its exact origin being not fully resolved, its effect on the spin coherence and the fidelity can be approximated by modelling the spin-flip error by a depolarizing channel E s depol , with the probability of undergoing a random spin-flip p dependent on the incoherent spin-flip rate κ and the duration of the respective rotation pulse T r .The action of the depolarizing channel on a density matrix ρ is denoted by E depol (ρ) = (1 − p)ρ + pI/2, where I is the identity matrix.As an example, after applying a Ry (π/2) pulse on a spin state initialized in |⇓ , the spin density matrix transforms according to FIG.S3.Fidelity plot against the spin-flip rate.
γ d as the off-resonant reflection error Γ 2 /∆ 2 h is comparably small.Together with the incoherent spin-flip and the readout errors discussed in Sec.II D, we estimate the overall entanglement fidelity F theory total the experimentally achieved value (74.3 ± 2.3)% including error margins.The remaining infidelity is likely a combination of imperfect spin initialization (1.4%) [S11], and non-Gaussian pulse shaping of the input photon which are not included in the theory.Using C = 14.7, optical pulse duration T p = 1/2σ o = 2 ns, γ 1 = 0.05 ns −1

FIG. S4 .
FIG.S4.Saturation measurement to calibrate the mean photon flux.(a) Time-resolved histogram of the measurement sequence.A 2 ns pulse gets reflected from a QD prepared in |⇑ via optical spin pumping followed by a π-rotation pulse.The reflected signal is time-gated (green shaded region) and recorded for each input power.Peaks at around 100 and 215 ns are laser scatter from the time-bin interferometer and the optical breadboard respectively.The spin readout at 300 ns maintains the same duty cycle as the entangling gate experiment and does not affect the gated counts.(b) Gated fluorescence in the reflection as a function of the input pulse power.Blue (red) circles are summed counts over a time window of 3 ns, when the QD spin is prepared in |⇑ (|⇓ ).Fitted (black solid line) using Eq.(S44).Around 0.075 nW is used for a single pulse in the entangling gate experiment.

Fitting
photon flux leading to an excited state population of 1/4[S16].Therefore, with γ d = 0.092 ns −1 (see Sec. III), a beta-factor of β = 0.95 (estimated), and S = b 1 × b 2 × P ≈ 0.0496 is the saturation parameter for an input power of P = 0.075 nW used for a single pulse in the entangling experiment, we estimate n F ≈ 0.0148 and the average number of photons in the scattering pulse n = n F T p Γ = 0.073 ≪ 1.A scaling factor between n and the input power S n = 10 −2 b 1 b 2 n c T p Γ/2 ≈ 0.005/nW (assuming transmission of 10 −2 with a neutral-density filter) is also obtained and subsequently adopted in Fig.2b.