On-demand source of dual-rail photon pairs based on chiral interaction in a nanophotonicwaveguide

Freja T. Østfeldt, Eva M. Gonzalez-Ruiz, Nils Hauff, Ying Wang,Leonardo Midolo, Anders S. Sørensen, Ravitej Uppu, and Peter Lodahl∗

Center for Hybrid Quantum Networks (Hy-Q), Niels Bohr Institute,University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark.

Andreas D. Wieck, Arne Ludwig, and Rudiger SchottLehrstuhl fur Angewandte Festkorperphysik, Ruhr-Universitat Bochum, Universitatsstrasse 150, D-44780 Bochum, Germany.

(Dated: January 19, 2022)

Entanglement is the fuel of advanced quantum technology as it enables measurement-based quantum com-puting and allows loss-tolerant encoding of quantum information. In photonics, entanglement has traditionallybeen generated probabilistically, requiring massive multiplexing for scaling up to many photons. An alternativeapproach utilizing quantum emitters in nanophotonic devices can realize deterministic generation of entangledphotons. However, such sources generate polarization-entanglement that is incompatible with spatial dual-railqubit encoding employed in scalable photonic quantum computing platforms utilizing integrated circuits. Herewe propose and experimentally realize an on-demand source of dual-rail photon pairs using a quantum dotin a planar nanophotonic waveguide. The source exploits the cascaded decay of a biexciton state and chi-ral light-matter coupling to achieve deterministic generation of spatial dual-rail Bell pairs with the amount ofentanglement determined by the chirality. The operational principle can readily be extended to multi-photonentanglement generation required for efficient preparation of resource states for photonic quantum computing.

INTRODUCTION

Quantum photonics has developed significantly in thelast decade and many of the required tools for scaling-upto advanced applications in quantum-information process-ing are already available [1, 2]. Indeed, the advent ofproduction-mature advanced planar photonic-integrated cir-cuits has been trumpeted as a major argument for photonicquantum computing[3]. Efficient photon sources, however,have remained challenging, and typically photon entangle-ment is heralded by multiplexing probabilistic sources[4].An alternative approach exploits quantum emitters in pho-tonic nanostructures that allow harvesting single-photon emis-sion with near-unity efficiency[5]. This is the basis of deter-ministic photon sources that combined with optical switch-ing can be demultiplexed to produce many simultaneousphotons[6, 7]. Quantum dot (QD) sources can generate two-photon polarization-entangled Bell states on-demand by ex-ploiting the radiative decay of a biexciton state[8, 9] to real-ize high entanglement fidelity and efficiency[10, 11]. Thesesources exploit polarization correlations induced from thebiexciton radiative decay, which is not compatible with pathencoding of photonic integrated circuits[12]. Furthermore,rotationally-symmetric (in the plane perpendicular to the QDgrowth axis) photonic nanostructures are required to ensureequal coupling of orthogonal dipoles. This geometric con-straint has restricted QD entanglement sources to the po-larization degree-of-freedom, and most notably implied thatthe wide-bandwidth and highly-efficient planar nanophotonicwaveguide platform [5] is incompatible with biexciton entan-glement generation. We show that a chiral light-matter inter-face allows overcoming all of these limitations by realizing aspatial dual-rail encoded entanglement source.

The planar nanophotonic QD platform offers several salientfeatures: it provides high-efficiency and broadband sourcesof highly indistinguishable photons[13] and is directly com-patible with planar integrated photonic circuits for complexphotonic processing[14]. Integrated on-demand entangle-ment sources have however, been unattainable, since the highsource efficiency relies on the radiative coupling of the QDto a single spatial and polarization mode. This limitationcan be overcome by exploiting polarization-dependent di-rectional coupling [15] where the photons from the QD areemitted preferentially into the left or the right propagatingmode, labeled as A and B, determined by the polarization(σ±) of the optical transition, cf. Fig. 1(a). While glide-plane-symmetric photonic-crystal waveguides (GPWs)[16]and nanobeam waveguides [17] have realized unidirectionalcoupling, residual backscattering of photons and multimodeoperation may adversely affect coherent quantum phenomenasuch as entanglement. In this article, we exploit specially-designed, single-mode GPWs with terminated mode adaptersfor minimized back scattering [18, 19] to transform the in-trinsic polarization entanglement of the QD biexciton cascadeinto photon path entanglement. Hereby dual-rail path encodedentanglement sources can be realized, which may be readilyinterfaced with photonic integrated circuits for realizing ad-vanced quantum processors, cf. Fig. 1(e).

PATH ENTANGLEMENT FROM CHIRAL INTERACTIONS

Figure 1(a) illustrates the operational principle of an on-demand source of path-entangled photons. The QD biexcitonstate is excited through two-photon resonant excitation usingoptical laser pulses and decays through a correlated σ± - σ∓

arX

iv:2

109.

0351

9v2

[qu

ant-

ph]

17

Jan

2022

2

XX

X

Spectral filters

B

(a) (b)

(c)(e)

(d)

A

x

y

&

Excitation

σ+XX

σ--X

σ--XX

σ+X

XX

X+X-- S

FIG. 1. Operational principle of spatial dual-rail entanglement source in a planar photonic circuit. (a) Illustration of the devicecontaining a QD in a glide-plane waveguide (GPW). The QD is optically excited to the biexciton state (XX) that recombines through a cascadeddecay (level structure in inset) via the exciton states (X±) emitting a circularly-polarized entangled photon pair. In the GPW, circular-polarizedphotons with opposite helicity couples preferentially to only one of the directions, i.e. A or B, which enables path entanglement generation. Anon-zero fine-structure splitting S induces temporal spin flip oscillations between the exciton states, as illustrated on the inset. (b) Calculatedlocal phase Φ between the Ex and Ey components of the electric field at different spatial locations within the unit-cell of the GPW. Φ =±π/2correspond to the locations of ideal chiral coupling. Here, a is the lattice parameter of the photonic crystal. (c) Experimental setup forprojective exciton-biexciton cross-correlation measurements. Two sets of cross-correlation measurements along the same path (AXX AX ) anddifferent paths (AXX BX ) are performed. If the photons are collected from the same path, we split the signal using a 50:50 fiber beam splitteras illustrated with the orange dashed line. (d) Predicted two-photon cross-correlation dynamics (coincidence detection probability) for theexperimental parameters of S = 2π×13GHz and exciton radiative decay rate γX = 8.35ns−1 and assuming ideal chiral coupling Φ =±π/2.For details on the model see Supplementary material 5 [20]. The experimental ”tell tale” of path correlation is the phase offset of 2Φ betweenthe (AXX AX ) and (AXX BX ) correlation measurements. (e) Proposed integration of a waveguide spatial dual-rail entanglement source into anadvanced photonic integrated circuit with phase shifters (θ ) and beam splitters for quantum-information processing applications.

cascade. The GPWs are designed to feature simultaneouslyhigh directionality and coupling efficiency (i.e. β -factor) [16],as required for an on-demand entanglement source. For anideally positioned QD (Φ = ±π/2), σ+- and σ−-polarizedtransitions emit photons along opposite directions (denotedby A and B) leading to optimal path entanglement. The keygoverning parameter for the entanglement fidelity is the localphase Φ of the electric field at the position of the QD withinthe GPW with ideal performance at Φ = ±π/2 (see Supple-mentary material 5). Asymmetries in the QD, however, in-troduce finestructure in the exciton level, which modifies thedynamics and leads to a photon state of the form[21]

|ψ(τ)〉= 1√N

(ψAXX AX (τ) |AXX AX 〉+ψAXX BX (τ) |AXX BX 〉

+ψBXX AX (τ) |BXX AX 〉+ψBXX BX (τ) |BXX BX 〉),

(1)

where N is the normalisation factor and the amplitudes are

ψAXX AX (τ) =−√

2γX e−γxτ/2(

e−i(Sτ/2+2Φ)+ eiSτ/2)

ψBXX BX (τ) =−√

2γX e−γxτ/2(

e−i(Sτ/2−2Φ)+ eiSτ/2)

ψAXX BX (τ) = ψBXX AX (τ) =−2√

2γX cos(Sτ/2)e−γX τ/2 .

(2)

Here, γX is the exciton radiative decay rate, S is the QD fine-structure splitting (in cyclic frequency units), and we assumethat the radiative decay rate of the biexciton is 2γX . We furtherassume that both the exciton and biexciton transitions cou-ple to the waveguide with the same phase (ΦX = ΦXX = Φ),which is a consequence of the wide operation bandwidth ofGPWs compared to the energy difference between the exci-ton and biexciton states. For ease of notation, we omit theX and XX subscripts and adapt the convention that biexci-ton precedes the exciton, i.e., AB represents AXX BX . In thecase of a QD with S 6= 0 positioned at a location with idealchiral coupling (Φ = ±π/2), a maximally-entangled path-encoded state is realized for all τ , but oscillating betweenthe two Bell states |Ψ〉(τ = 0) = 1/

√2(|AB〉+ |BA〉) and

|Ψ〉(τ = 2π/S) = 1/√

2(|AA〉+ |BB〉). The oscillation canbe suppressed either by eliminating the fine-structure splittingS[22] or by implementing time gating[23]. In path-resolvedexciton(X)-biexciton(XX) photon correlations, cf. Fig. 1(c),the two-photon coincidence probabilities | 〈AA|Ψ(τ)〉 |2 and| 〈AB|Ψ(τ)〉 |2 are predicted to oscillate with a phase differ-ence of 2Φ, cf. Fig. 1(d). In the absence of chiral cou-pling (Φ = 0) a separable state is generated, e.g., |Ψ〉(τ =0) = 1/2(|AXX 〉+ |BXX 〉)(|AX 〉+ |BX 〉) and the phase shift inthe correlation measurements is absent. The observation of aphase shift consequently constitutes the ”tell tale” for the gen-eration of path-encoded entanglement mediated by the chiral

3

coupling.

IMPLEMENTATION

The experimental demonstration is realized with singleQDs located in electrically-contacted waveguide samples forlow-noise performance[24] and Stark tuning of the chargestates. We employ self-assembled indium arsenide (InAs)QDs epitaxially grown at the center of a suspended 170nmgallium arsenide (GaAs) membrane comprising a p-i-n diodefor charge control of the QD, c.f., Supplementary material 1for details about the heterostructure. The QDs are located ran-domly across the sample with an average density of≈ 1µm−2.Figure 2(a) shows a scanning electron microscope image ofthe nanofabricated device in the GaAs membrane. The devicehas a total footprint of 50× 25µm2. The central part of thedevice contains the GPW where the studied QD is located.The GPW is designed with a lattice constant of a = 260nmand hole radius of r = 69nm. The glide plane geometry ismade by shifting the holes by half a lattice constant (a/2)compared to a regular photonic-crystal waveguide along thepropagation direction (x). Additionally, the geometry is op-timized for single-mode operation, following the proposal ofRef. [18], by shifting row 2−4 of holes outwards (along y) by0.25a

√3/2, 0.2a

√3/2, and 0.1a

√3/2, respectively, while

the radii of holes in rows 1−3 are also modified according tor1,2 = 1.17r and r3 = 0.8r. A detailed discussion of the ge-ometry is provided in Supplementary Information Fig. S2(a).The mode adapters [18] (see inset of Fig. 2(a)) are designedto minimize reflections at the nanobeam/GPW interface thatoccurs due to a large group-index mismatch The nanobeamwaveguides are terminated with two shallow-etched gratingoutcouplers [25] that are oriented by 90° with respect to eachother for orthogonally polarized free-space collection.

The sample is cooled to 1.6K in a helium closed-cyclecryostat, with electrical and optical access. The excitationlaser is focused to the sample using a high numerical aperture(NA= 0.81) microscope objective. A single QD is excitedby a pulsed Ti:Sapphire laser (20ps optical duration) whosefrequency is tuned to satisfy the two-photon resonance condi-tion of the QD biexciton. The emitted photons coupled to theGPW are directed out-of-plane at two high-efficiency gratingoutcouplers (A and B in Fig. 2(a)), and are collected by thesame objective lens, and separated from the input excitationpath using a 5 : 95 beam splitter. As the light diffracted bythe outcouplers is linearly-polarized, the 90° relative orienta-tion of the outcouplers ensures cross-polarization in collec-tion. We utilize this property to collect path-dependent emis-sions at the outcouplers A and B without compromising thecollection efficiency. Using a set of quarter and half wave-plates together with a polarizing beam splitter, we separatethe emission from the two orthogonally polarized grating out-couplers into two separate spatial modes and couple each intoa single-mode optical fiber. The emission at each of the out-couplers is spectrally filtered (bandwidth = 22GHz) and de-

tected using a superconducting nanowire single-photon detec-tor (SNSPD) with a low timing jitter < 15ps. A time tag-ging device is used to register the timestamps of the detectionevents in both paths with a resolution of 4ps. The timestampsaccumulated over an hour are processed using MATLAB toconstruct the photon correlation histograms shown in Fig. 3.The low timing jitter of the SNSPDs is crucial in resolving therapid oscillations (period ≈ 80ps) in the photon correlationsinduced by the fine-structure splitting of the QD excitons.

RESULTS

QD characterization

The GPW device is characterized by spectrally-resolvedtransmission measurements where a tunable diode laser beamis injected into the waveguide at port A and the transmittedlight is measured at port B, cf. Fig. 2(b). From the opticalexcitation of the QD we identify the spectral distance of the Xand XX emission from the high group index (ng) region. TheGPW experiences slow light with slow-down factors of 16 and14 for the XX and X transitions, respectively. This gives riseto slow-light induced Purcell enhancement and a near-unitycoupling efficiency quantified through the β -factor. We mea-sure a radiative decay rate of γX = 8.35ns−1 correspondingto a Purcell factor of about 2 leading to β > 95%, i.e. near-deterministic operation of the source[16, 18], see Supplemen-tary material 3 for the experimental data. Spectrally-resolvedemission is collected at the out-coupling grating B as a func-tion of applied gate voltage, see Fig. 2(c), whereby X andXX transitions can be identified from their identical chargetuning slopes. Distinct anti-bunching is observed in pulsedsecond-order correlation measurements (g(2)(τ)) of both theX and XX , cf. data in Fig. 2(d). This confirms the high-puritysingle-photon emission from each transition, which is a pre-requisite for entangled photon pair generation.

Path-correlation measurements

We quantify the generation of path-entangled photon pairsfrom path- and spectrally-resolved correlation measurements,with the resulting data displayed in Fig. 3(a). The experimen-tal data are very well explained by the theoretical model, cf.Supplementary material 5. The correlation function decayswith the independently measured exciton decay rate γX (seeSupplementary material 3), while the fine-structure splittingS and phase shift Φ are extracted from the analysis. A keyaspect to extract the phase, is high-precision time-matchingof photon-correlation datasets, cf. Supplementary material4. A pronounced phase shift of Φ = (0.12± 0.01)π is ex-tracted, which is the experimental evidence of chiral couplingresulting in path entanglement. For comparison, Fig. 3(b)displays the separable case of a QD in a regular photonic-crystal waveguide, where we observe no pronounced phase

4

(a)

(b)

(c) (d)

FIG. 2. Experimental characterization of QD coupled to a GPW. (a) Scanning electron microscope image of the device used for themeasurements. The central section consists of a GPW (insets show close ups of the optimized waveguide design). The GPWs are connectedto grating outcouplers A and B via nanobeam waveguides. The GPW-nanobeam interface features a mode adapter, as shown in the inset. (b)Spectrally-resolved laser transmission shown in orange from gratings A to B for identifying the slow-light region. The curve is normalized tothe transmission through a nanobeam waveguide (cf. Supplementary material 1). The green curves are the simulated group index ng for theGPW (solid line) and a nanobeam waveguide (dotted line). The wavelengths of the X and XX transitions of the QD are marked as well. (c)Gate-voltage-dependent resonance fluorescence of the QD under two-photon pulsed excitation displaying the X and XX transitions. (d) Pulsedsecond-order correlation measurements of X and XX transitions after spectral filtering of each transition (22GHz bandwidth). We extractg(2)X/XX (0)≤ 0.009 for both transitions indicating excellent single-photon purity.

shift. In this case the local electric field polarization is pref-erentially linear, meaning that pronounced chiral coupling isnot expected.

OBSERVATION OF PATH ENTANGLEMENT

As an interesting note, the experiment directly probes thephase of the local electric field inside a complex photonicnanostructure. The measurements were repeated on two addi-tional QDs (cf. Supplementary material 6) and Φ≈ 0.1π wasrecorded in all cases. In a GPW, QDs are with hight probabil-ity positioned near points with Φ ≈ π/2, cf. Fig. 1(b). Theexperimental values are likely influenced by the presence ofweak residual back-reflections from the outcoupling gratings,which may be overcome by further optimization of the gratingdesign[25]. Such weak scattering reduces the directionality,since a fraction of the emitted photons is back-reflected[15].See Supplementary material 6 for discussion and estimationof this effect.

The local phase Φ is a governing parameter determiningthe quality of the generated path entanglement. The degree ofpath entanglement is quantified by the concurrence C, whereC > 0 is an entanglement criterion. Furthermore, the interplaybetween the emission-time uncertainty, fine-structure split-ting induced oscillations, and timing jitter ς of the detectorsdetermines the observable concurrence, as described by thetheoretical model of the biexciton cascaded decay in a chi-ral waveguide [21]. Figure 3(c) plots the time evolution ofthe concurrence for real experimental parameters investigat-

ing the role of chirality Φ and fine-structure splitting S. Theinitial decay is determined by the interplay between the emis-sion time and detector jitter, through the ratio γX/ς , where asmaller ratio leads to a lower entanglement degradation. Coscillates with a period of 2π/S reaching up to 0.11 for ourcurrent experiment, a testimony of the successful generationof path entanglement by the source. C can be dramatically in-creased by lowering the ratio of S/ς . With the fine-structuresplitting of S = 2π×5GHz recently reported on a similar de-vice platform[13], we estimate C≈ 0.57 without any other ex-perimental improvements (Φ and ς ). Further, droplet-etchedGaAs QDs are known to possess S ≈ 0, which could be idealcandidates for on dual-rail entanglement[10]. Increasing Φ

improves the entanglement quality significantly and C > 0.9is readily attainable for Φ = π/2, cf. Fig. 3(c). The QDposition-dependence of the maximum concurrence realizablein a GPW is displayed in Fig. 3(d), i.e. the GPW supportspath entanglement generation with near-unity fidelity within awide area of the waveguide.

In summary, we have proposed and experimentally imple-mented an on-demand planar nanophotonic waveguide sourceof dual-rail path-entangled photon pairs. The operation of thesource is based on chiral coupling of a QD to the waveguidemode leading to directional emission. The operational prin-ciple is general, and could readily be extended also to multi-photon entanglement sources, e.g., for the generation of path-encoded photonic cluster states[26]. Such deterministic en-tanglement sources are much needed for advanced quantum-information processing applications [27, 28]. For instance, inphotonic quantum computing, the most demanding task is the

5

(a) (b)

(d)(c)

FIG. 3. Observation of path-dependent correlations of the entangled state. (a) Path-dependent photon correlation measurements of QDsin GPW in two configurations, AXX BX and AXX AX . Solid lines are fits to the theoretical model. The error in Φ is the 1σ confidence intervalfitting error. (b) Similar measurements as (a) but from a QD in a regular photonic crystal waveguide with no significant chiral interaction. (c)Time-dependent concurrence C calculated for a QD in the GPW under current experimental conditions (Φ1 = 0.12π , S1 = 2π × 12.78GHz,ς = 15ps), blue solid curve. The calculation for Φ2 = π/2 and otherwise same parameters is shown as the orange solid curve. The dashedcurves correspond to the case of a QD with lower fine-structure splitting of S2 = 2π×5GHz. (d) Predicted QD position-dependent C of thepath-entangled state at time τ = 0 within the GPW exhibiting wide areas of near-perfect entanglement. The concurrence is calculated usingthe theoretical model[21], and the simulated phase from Fig. 1(b).

generation of high-quality multi-photon entangled resourcestates[29] with photonic integrated circuits providing a realis-tic route for realizing quantum processors[30]. Consequently,interfacing QD path-entanglement sources with mature inte-grated photonic circuits, as sketched in Fig. 1(e), appears aresource-efficient approach. To this end, heterogeneous chiptransfer processes may be applied in order to combine the ac-tive QD devices and the passive photonic circuits[31].

DATA AVAILABILITY

The data presented in the figures of this study are availablefrom the authors on request.

ACKNOWLEDGEMENTS

We thank Single Quantum for giving us access to the ultra-low timing-jitter superconducting nanowire single-photon de-tectors required for extracting the correlations. We gratefullyacknowledge financial support from Danmarks Grundforskn-ingsfond (DNRF 139, Hy-Q Center for Hybrid Quantum Net-works). This project has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programmesunder grant agreement No. 824140 (TOCHA, H2020-FETPROACT-01-2018) and under the Marie-Skłodowska-Curie grant agreement No. 801199. A.D.W. and A.L.acknowledge gratefully support of DFG-TRR160, BMBF -

Q.Link.X 16KIS0867, and the DFH/UFA CDFA-05-06. Weacknowledge Adam Knorr for initial sample characterization.We thank Matthew Broome for discussions in the early phaseof the project.

6

(a) (b)

p-GaAs

Al0.3Ga0.7As

i-GaAs

QDs

V

n-GaAs

34nm

1nm45nm8 nm

55nm

14nm

2 nm7 nm7 nm

(c)

FIG. 4. Electrical sample properties. (a) Layout of the membrane containing p-i-n doped layers for electrical control, to which externalelectrical contacts are bonded. The QDs are located in the center of the membrane. (b) Current-voltage (IV) curve across the diode, showingsmall amounts of leakage current. The current density is found by dividing the current by the mesa size which is 2×2mm2. (c) Transmissionof a narrow band tunable laser through a nanobeam waveguide. The oscillations comes from back reflections from the grating couplers. Thetransmission scan is used for nomalisation of the GPW transmission scan in Fig. 2(b).

SUPPLEMENTARY MATERIAL

Sample heterostructure layout and fabrication

The sample membrane is grown using molecular beam epitaxy on a (100) GaAs substrate. First a sacrificial layer withAl0.75Ga0.25As of 1371nm thickness is grown to suspend the GaAs membrane after wet etching. A layout of the membraneheterostructure, is displayed in Fig. 4(a), with the QDs in the center. P-type Carbon and n-type Silicon doped layers in themembrane form an ultra thin p-i-n diode around the QDs for charge state control and Stark tuning of the QD, which simultane-ously suppresses charge noise. We use Carbon as p-type doping due to its high doping efficiency and very low diffusivity. TheAl0.30Ga0.7As layers above and below the QD allow charging the QD near 0V and prevent high currents to flow when the diodeis used in forward bias voltage. Electrical contacts are deposited on the n and p layers and trenched to create a single mesa of2 mm × 2 mm to which the external voltage is applied, as also illustrated in Fig. 4(a). Using reactive ion etching (RIE) viasare opened to the buried n-layer and Ni/Ge/Au/Ni/Au contacts are deposited by electron-beam physical vapor deposition andannealed at 430°C. Cr/Au pads are deposited on the p-layer to form Ohmic contacts. The nanophotonic structures are fabricatedwith electron-beam lithography at 100keV (Elionix ELS 7000) and etched in the GaAs membrane using inductively-coupledplasma RIE in a BCl3/Cl2/Ar chemistry. The undercut and sample cleaning was performed using the procedure discussed in Ref.32.

The contacts are connected to wire bonding pads, to which the external electrical connections are bonded. The quality of thediode structure and the electrical contacts is checked by recording a current-voltage (IV) curve of the sample as seen in Fig. 4(b).A low leakage current density of < 0.09µAmm−2 under reverse bias is observed, suggesting near-ideal diode behavior. Theemission from the neutral exciton is observed when the diode is operated at around 0.24V in forward bias, where the currentdensity is lower than 0.16µAmm−2. Such current densities are sufficiently low to achieve a stable and low-noise biasing of thequantum dots, that does not add significant spectral diffusion.

The sacrificial layer thickness (1370nm) of the sample deviates from the optimal thickness of 1150nm[25], which reduces thegrating efficiency to ≈ 10%. The impedance mismatch leads to back reflections from the grating couplers into the waveguide asdiscussed in the main text. We can readily observe the effect of this impedance mismatch from the laser transmission spectrum(c.f. Fig. 4(c)) of a rectangular cross-section nanobeam waveguide (width = 300nm; length = 55µm) terminated on eachend with a grating coupler. The back reflections from the grating coupler results in the formation of a weak cavity across thewaveguide that result in the fringes with a free spectral range of 1.43nm. We estimate the reflectivity of the gratings from thefringe amplitude A to be r2 = (1−A)/(1+A)≈ 0.3 for both exciton and biexciton emission.

Glide-plane waveguide characterization

The glide-plane symmetry of the GPW features strong in-plane circular polarization in regions of high Bloch mode amplitude.As a consequence, efficient (β → 1) and highly directional coupling is abundant in a GPW, i.e. the waveguides are engineered tohave chiral light-matter coupling of high Purcell enhancements in large regions within every photonic crystal lattice unit cell [16].

The TE-like forward-propagating Bloch mode’s electric field ek,n(r) of mode index n, at position r, and wavevector k is

7

given by ek,n(r) = ek,n,x(r)ex + ek,n,y(r)ey, where ex and ey are the cartesian unit vectors. The modes satisfy the Bloch modenormalization

∫d3r ‖ek,n (r)‖2

ε (r) = 1, where ε (r) is the dielectric permeability. The Bloch mode can also be expressed byek,n,x(r) = |ek,n,x(r)|eiφk,n,x and ek,n,y(r) = |ek,n,y(r)|eiφk,n,y , where φk,n,x and φk,n,y are the phases of the field component along xand y axes, as defined in Fig. 1(a) in the main text. Assuming time-reversal symmetry the backward-propagating mode is givenby e−k,n(r) = e∗k,n(r).

We calculate the band diagrams ωk,n and the corresponding Bloch modes ek,n(r) of the GPW using commercially availablefinite element software (COMSOL Multiphysics). The model consists of a single supercell with periodic boundary conditions(PBC) along the waveguide direction (ex) and domains of perfectly matched impedance layers (PML) to confine the model. Alayer of a perfectly magnetic conductor (PMC) in the vertical symmetry plane of the waveguide reduces the model’s numericalcomplexity, while obeying the TE-like Bloch mode profile. The parameters used for the simulation are summarized as a tablein Fig. 5(a). The membrane thickness t is determined during the epitaxial growth while the photonic crystal lattice parametersare defined during the electron beam lithography step in the nanofabrication routine. In the numerical computations, we usethe dielectric constant of GaAs at cryogenic temperatures based on the values reported in Ref. 33. The resulting band diagramis shown in Fig. 5(b), where the two fundamental waveguide modes are plotted as solid orange lines. The blue shaded areain the band diagram represents the bulk mode and the orange shaded area corresponds to the light cone The bands plotted asdotted lines are the higher order waveguide modes that we do not investigate. Further, we determine the group index for of thewaveguide modes as ng(ωk,n) = c/|∇kωk,n| using the Hellmann-Feynmann theorem [34]:

ng(ωk,n) =2c(Uek,n +Uhk,n)∣∣∣∫Vs

d3r Re(e∗k,n(r)×hk,n(r))∣∣∣ , (3)

where Vs is the total super cell volume, hk,n(r) is the Bloch mode’s magnetic field, and Uek,n (Uhk,n ) is the time averaged electric(magnetic) field energy of the Bloch mode.

We characterize the fabricated device through a spectrally-resolved transmission measurement, where a narrow bandwidthcontinuously tunable laser is coupled into the waveguide at grating coupler A and the light transmitted through the waveguideis collected, coupled off-chip by grating B, and measured using a photodiode. The measured laser transmission spectrum of aGPW normalized to that of a nanobeam waveguide is shown in Fig. 5(c) along with the calculated group index ng (a portion ofthis scan is also shown in Fig. 2(b) in the main text).

We identify the spectral center of the transmission gap corresponding to the frequency with the highest group index ng. Anincrease in ng leads to a loss in transmission due to in-plane optical scattering that scales quadratically with the group index,i.e. T ∝ n2

g [35, 36]. We neglect the dependency on the Bloch mode profile and its dispersion as their contribution is negligiblewithin the spectral width of the transmission gap[19]. We identify the wavevectors kX and kXX of the exciton and biexciton bytheir spectral location relative to the frequency with highest ng (identified in Fig. 5(c)) and determine the group indices for Xand XX as ng,X = 14.3 and ng,XX = 16.1. The corresponding electric field norm for ng,X is shown in Fig. 5(d). For the designparameters of the GPW, we observed a 58nm blue shift of the frequency with maximum ng relative to the experimental data,which is consistent with previous results[37].

Chiral coupling quantifiers such as the directionality[18] Dk,n(r) and the local phase Φk,n(r) can be extracted from the Blochmodes. The phase Φk,n(r) = φk,n,x(r)−φk,n,y(r) is the relative phase between the electric field components, while the direction-ality is defined as

Dk,n (r) =(|ek,n (r) · eσ+ |2−|e−k,n (r) · eσ+ |2)/|ek,n(r)|2

=2sinΦk,n|ek,n,x(r)||ek,n,y(r)|/(|ek,n,x(r)|2 + |ek,n,y(r)|2) (4)

where eσ+ = (ex + iey)/√

2 and eσ− = (ex− iey)/√

2 are the circularly-polarised unit vectors. The calculated directionality forthe exciton is shown in the lower panel of Fig. 5(d). Fig. 1(c) of the main text shows the calculated Φ.

8

xy

(b) (c)(a) (d)

FIG. 5. GPW characterization. (a) Table of simulation parameters. Here, t is the membrane thickness. Geometry of a unit cell is displayed onthe side. The waveguide is formed in the center by repetition along x axis as in Fig. 1(a). (b) Band diagram of GPW from numerical simulationusing COMSOL and the parameters in (a). The geometry is optimized according to Ref. 18 in order to obtain single mode waveguide bands.The waveguide modes are displayed with solid lines. The light cone is shaded in orange, while bulk modes are shaded blue. Higher ordermodes are shown with dotted lines. (c) Identification of the two pass-band regions of the device obtained by scanning a narrow-band laserin a transmission measurement. The emission wavelengths of the X and XX are marked. Green curves are the simulated group index ngfor the GPW (solid line) and a nanobeam waveguide (dotted line). (d) Top: Local electric field amplitude calculated using COMSOL, theparameters from (a) and ng,X = 14.3 estimated from the band diagram and the spectral location from the band edge in (b) and (c). Bottom:Local directionality Dk,n (r) as defined in the main text.

Quantum dot characterization

The QD was identified with above band exciation, using a pulsed diode laser at 780nm. The spectrally-resolved emissionfrom the neutral exciton is seen in Fig. 6(a) as a function of applied gate voltage where the emission wavelength at the centerof the charge plateau is at ∼ 917.45nm. Two parallel lines of emission are visible in the map, which correspond to the finestructure split exciton lines. Fig. 6(b) shows a cut-through of the emission spectrum along 0.284V. The spectrum is fitted to asum of two Voigt profiles, which are individually shown as solid light blue lines. The center of the Voigt profiles are highlightedwith a dashed line, which allows estimating the fine structure splitting S = 2π × 13.5(5)GHz. Due to the limited spectralresolution of the spectrometer, we observe a small deviation from the value extracted using photon correlation measurements of2π×12.78GHz.

The QD exciton lifetime is measured in a time-resolved fluorescence measurement. The emission from both the fine-structure-split lines are collected at grating B and detected on an SNSPD, which leads to the histogram displayed in Fig. 6(c). The his-togram is fitted to a single exponential, which yields a decay rate of 8.35ns−1. Given the average decay rate of 3.9(7)ns−1 for theQDs in the unstructured bulk samples [38], the measured exciton lifetime is consistent with the estimated Purcell enhancementof ≈ 2. The single-exponential character of the decay curve (cf. Fig. 6(c)) indicates that the two orthogonal linear dipoles havea similar decay rate, and therefore a single exciton decay rate (γX ) suffices in the analysis.

Time matching of correlation data

The correlation measurements were performed with low-timing jitter (< 15ps) SNSPDs (Single Quantum) in order to resolvethe fast oscillations arising from the fine-structure splitting. A crucial component in extracting the chirality induced phase shiftin correlations is precise time matching of different datasets. Even a small variation in the optical path length between the twograting outcouplers results in a timing mismatch between different measured correlation datasets, as seen in inset of Fig. 7(a),which impedes the extraction of the phase shift. We collect timestamps of photodetections on the two detectors and processthe timestamps to construct coincidence histograms with 4ps bin size over a large time window of 100µs. The step-by-stepprocedure implemented for time-matching different correlation histograms is described below with the help of two datasets,AXX BX and AXX AX . The former correlates exciton emission collected at grating A with that of the biexciton emission collectedat grating B while the latter correlates exciton and biexciton emission collected at the same grating (A and A).

Step 1 - Find laser repetition rate: In order to align the peak at zero time across datasets, we utilize the full timespan,i.e., 100µs, to estimate the laser repetition rate τrep of each dataset using fast fourier transform (FFT). Given the timespanof the dataset, we estimate a FFT resolution of 7kHz. With a laser repetition rate of 76.148MHz, this FFT resolutioncorresponds to an imprecision of ≈ 1.3ps in the laser repetition period, i.e., τrep = 13.1323±0.0013 ns.

9

(a) (b) (c)

FIG. 6. Quantum dot aboveband spectroscopy (a) Frequency voltage map of the QD neutral exciton fluorescence under pulsed above bandexciation. The two fine structure split levels are visible as two parallel lines, where the spectrometer resolution of ∼ 25pm limits the visibility.(b) Cut through the frequency voltage map at 0.248V. The sum of two Voigt profiles is fitted to the spectrum, and plotted with light blue line.The centers of the fitted Voigt profiles are indicated with dotted lines and are split by 13.5(5)GHz. (c) Time resolved fluorescence measurementof the exciton under pulsed non-resonant excitation showing the decay rate of the QD. The decay is fitted to an exponential function convolvedwith the instrument response function, which is shown in blue. The resulting fit is shown with a solid orange line with the extracted decay ratedisplayed on the plot.

Step 2 - Refine laser repetition rate and find time zero: From the correlation histogram of the dataset, say AXX AX , weselect two time-windows of τrep width (estimated in Step 1), each containing a side peak. These time-windows contain thecoincidence peaks at the beginning and the end of the dataset, i.e., at τ =±50µs. By cross-correlating these time-windowsseparated by 100µs, we can improve the precision in estimating τrep. If the cross-correlation function is maximized at zerolag, then the error in τrep is minimized and τ = 0 is precisely estimated. If not, then the lag at which the cross-correlationfunction maximizes is the amount of adjustment to be added to τrep.

Step 2.1 - Laser repetition rate and time-zero estimation of the second dataset: The procedure described in Steps 1and 2 is repeated independently for AXX BX dataset. While one could imagine reusing the same repetition rate across alldatasets and estimate only time zero point, this naive approach fails across most datasets. As the datasets are collectedacross several days, the repetition rate of the pulsed (Ti:Sapphire) laser showed small variations on the order of a kHz (i.e.,τrep varies by few ps). This small variation of τrep results in errors in time zero estimation of the datasets.

Step 3 - Align two separate datasets: We follow the similar procedure as Step 2, but cross correlate the peak at τ =−50µsfrom the dataset AXX BX with that of AXX AX . As discussed in Step 2.1, in order to account for the different laser repetitionperiods, we rebin the two datasets onto a common time axis. Cross-correlating the datasets on this common time axis givesthe relative lag in τ = 0 position between them. The lag that maximizes the cross-correlation is used to shift the time-axisof the AXX BX dataset. The resulting aligned datasets AXX BX and AXX AX are shown in Fig. 7(b,c), which shows excellentoverlap of the correlation datasets over the complete time span of ±50µs. The performance of the time alignment andτrep estimation procedure of the two datasets is further verified by dividing the peaks at time delay = ±50µs in the datasetAXX BX with their counterpart in the dataset AXX AX as shown in Fig. 7(d), that results in a time-independent ratio thathighlights the overlaps of the datasets.

Fig. 7(b) shows a pixel-perfect overlap of the individual peaks in datasets AXX AX and BXX AX upon time matching at alltime delays −50µs ≤ τ ≤ 50µs. As we do not observe any shift between the datasets AXX AX and AXX BX over the entire timespan of 100µs, we estimate that the misalignment is smaller than the resolution of the time tagger. This corresponds to atemporal alignment precision better than 40 ppb, thereby enabling us to capture the phase shifts of oscillations in the datasetsvery accurately. Crucially, we remove not only the linear time shift between the data but also a nonlinear time shift that arisesfrom imprecision in repetition rate variations. While the former is typically easy to handle and requires only a cross-correlationcalculation of the entire time span, the latter necessitated the multistep time matching procedure described above. Given theoscillation frequency of 12.78(3)GHz for the QD presented in the main text, it corresponds to a period of 78ps. As the timematching accuracy is < 1.3ps (worst-case error estimated in Step 1, but improved in Step 2 to < 1ps), we conclude that thesystematic error from time misalignment in phase shift estimation is minor.

10

-100 -50 0 50 100Time delay [ns]

0

500

1000

1500

2000R

aw X

-XX

Coi

nc. c

ount

s AXXBXAXXAX

(a) (b) (c)

-0.5 0.0 0.50

2000

0

100

AB: 0AA: 0

-0.50 -0.25 0.00 0.25 0.50Time delay [ns]

0

100

Nor

m. X

X-X

Coi

nc. c

ount

s

AB: -50 sAB: +50 sAA: -50 sAA: +50 s

0.5

1.0

1.5

2.0AB(-50 s)/AB(+50 s)

AA(-50 s)/AA(+50 s)

-0.50 -0.25 0.00 0.25 0.50Time delay [ns]

0.5

1.0

1.5

2.0AB(-50 s)/AA(-50 s)

AB(-50 s)/AA(+50 s)

FIG. 7. Time matching of different correlation data sets. (a) Raw data from Fig. 3(a) of the main text, which are offset in time due todifferent optical path lengths and spectral filtering in the two configurations AXX BX and AXX AX . Side peaks where the cascaded bunching isnot present are visible at the repetition rate of the laser of 76.1MHz. (b) Side peaks at τ ∼±50µs (bottom panel) and≈ 12ns (top panel) on thetwo configurations after time matching where the high degree of overlap of all measurement configurations is demonstrated with an estimatedaccuracy better than 40 ppb. (c) Ratio of peaks in each dataset at 50µs and −50µs showing accurate estimation of the laser repetition rate andhence the time-zero of the dataset. The increase in the fluctuations about 1 of the ratio at longer time delays is due to the smaller number ofdetected coincidences.

σ+XX

σ--X

σ--XX

σ+X

XX

X+X-- S

HXX

VX

XX

X2

X1

Sħ

VXX

HX

ωXX

ωX

0

0 0

FIG. 8. Energy level structure of the biexciton |XX〉 including a fine structure splitting S, in different basis representations. The left diagramshows the circular polarization basis of the emitted photons (σ±), in which the two exciton states |X±〉 are degenerate in energy. Here thefine-structure splitting induces oscillations between the two exciton states at a rate ∝ 2π/S, as indicated in the figure. Right diagram is in arotating frame with non-degenerate linearly polarized basis states

∣∣X1,2⟩. The degeneracy is lifted by the fine-structure splitting energy.

Theoretical model for the state of the biexciton decay in a waveguide

A path-entangled state can be generated by coupling the emitted photons from a biexciton cascade to a GPW waveguideusing chiral light-matter coupling. The fine-structure splitting (S), as well as imperfect chiral coupling determine the quality ofentanglement. We have modelled the final output state and quantified the quality of the entanglement generated. The details ofthe calculations will be published elsewhere[21].

Fig. 8 shows the biexciton level structure represented in two different basis descriptions of the exciton states: either two degen-erate states |X±〉 coupled with circularly-polarized transitions (σ±) or non-degenerate states |X1,2〉 coupled by linearly-polarizedtransitions (H,V ). In the presence of QD asymmetry, the states |X1,2〉 are the eigenstates, while in the former representation thetwo degenerate eigenstates will oscillate with a characteristic timescale 2π/S. This basis is therefore not stationary in time. Todescribe the time dependence, we thus use the rotating frame with non-degenerate linear exciton basis |X1,2〉 for the completemodelling of the system.

A wavefunction approach is applied in order to model the experiment in the non-degenerate linear exciton basis by followingthe theoretical framework put forward in Ref. 39. The obtained wavefunction components associated with the different decaypaths are expressed by Eqs. (2), where we have assumed that the decay rates γXX = 2γX are the same in horizontal and verticalpolarizations, and that they couple equally well to paths A and B. This assumption of similar decay rate for the horizontal andvertical is motivated by the fact that experimentally we only observe a single exponential decay. We have further assumed thatthe chiral phase ΦX = ΦXX = Φ is identical for biexciton and exciton photons, which is motivated by the comparable ng at the Xand XX emission wavelengths. Finally, we have solved the system in the limit in which S� γX , i.e. the interference that occursbetween the decay of the exciton levels (|X〉1 and |X〉2) is neglected compared to the effect of the fine-structure splitting when

11

calculating the evolution of their amplitudes. This approximation is well applicable to our system since S = 2π×12.78(3)GHzis an order of magnitude larger than the decay rate γX = 8.35(6)ns−1, i.e. Γ≡ γX/(2π) = 1.33GHz. From Eqs. (2) in the maintext we calculate the correlation probabilities

PAXX AX (τ) = |ψAXX AX (τ)|2 = 4γ

2X e−γX τ (1+ cos(Sτ +2Φ))

PBXX BX (τ) = |ψBXX BX (τ)|2 = 4γ

2X e−γX τ (1+ cos(Sτ−2Φ))

PBXX AX (τ) = PAXX BXX (τ) = |ψBXX AX (τ)|2 = |ψAXX BX (τ)|

2 = 4γ2X e−γX τ (1+ cos(Sτ)) ,

(5)

which are corresponding to the experimentally recorded correlation data. Here, τ > 0 is the time delay between the biexcitonand exciton emissions. Eqs. (5) have been used for fitting the time-matched correlation data discussed in to extract the chiralphase Φ as shown in Fig. 3 of the main text. Note that for perfect chiral coupling Φ = ±π/2, which leads to PAXX AX (τ) =PBXX BX (τ) and out-of-phase oscillations compared to PBXX AX (τ) = PAXX BX (τ). On the contrary, in the non-chiral case (Φ =0,±π) all correlations are oscillating in phase. As exemplary cases we consider these two extreme situations. Ideal chiralcoupling leads to the state

|ψ(τ)〉Φ=π/2 =

12

(cos(

Sτ

2

)(|AB〉+ |BA〉)+ isin

(Sτ

2

)(|AA〉+ |BB〉)

), (6)

which is a maximally path-entangled state. This state oscillates between the Bell states |ψ+〉 = 1/√

2 |AB〉+ |BA〉 and |φ+〉 =1/√

2 |AA〉+ |BB〉 with a frequency determined by the fine structure splitting S of the QD. The non-chiral case corresponds to:

|ψ(τ)〉Φ=0,π =

12(|AB〉+ |BA〉+ |AA〉+ |BB〉) = 1

2(|A〉+ |B〉)X (|A〉+ |B〉)XX , (7)

which is a separable state. Non-perfect chiral coupling will lead to a partially-entangled state that can be quantified by theconcurrence C.

Entanglement measures

We quantitatively evaluate the degree of the path-entanglement generated in our experiment. For this purpose, we evaluatedependence various entanglement measures of the state as a function of the chiral phase Φ, as well as the emission and detectiontime jitters. In particular, the Von Neumann entropy of a two-dimensional bipartite state ρ can be obtained as

H(ρ) =−(P1 log2 (P1)+P2 log2 (P2)) , (8)

where P1 and P2 are the eigenvalues of the reduced density matrix of the system. For compact notation, we omit the X and XXsubscripts and follow the convention AB = AXX BX . We calculate the density matrix ρ = |ψ〉〈ψ| from the pure state describedin Eq. (1) and trace out the biexciton subsystem to get the reduced density matrix

ρX = TrXX (ρ) =1N

(|ψAA|2 + |ψAB|2 ψAAψ∗BA +ψABψ∗BB

ψ∗AAψBA +ψ∗ABψBB |ψBA|2 + |ψBB|2). (9)

On the other hand, conservation of the trace under the diagonalization of the reduced density matrix implies that

Tr(ρ

2x)= P2

1 +P22 = P2

1 +(1−P1)2 , (10)

which allows us to get the eigenvalues directly as a function of the reduced density matrix ρX

P1,2 =12

(1+√

2Tr(ρ2

X

)−1)=

12

(1±√

1− 4N2 |ψAAψBB−ψABψBA|2

). (11)

The concurrence C of the state ρ is defined as

C(ρ) = max{0,λ1−λ2−λ3−λ4} , (12)

where λi are the square root of the eigenvalues of ρ ˜ρ in descending order and ˜ρ = (σy⊗ σy)ρ∗(σy⊗ σy) with ⊗ representing

the direct product and σi being a Pauli matrix, where i ∈ x,y,z. Substituting ρ = |ψ〉〈ψ| from Eq. (1) and the results from Eqs.(2), the concurrence is

C(ρ) =2N|ψAAψBB−ψABψBA|=

sin2 (Φ)

1+ cos(Sτ)cos2 (Φ). (13)

12

0 /4 /2 3 /4Phase

0.0

0.2

0.4

0.6

0.8

1.0

Eige

nval

ues

P1P2

0 /4 /2 3 /4Phase

0.0

0.2

0.4

0.6

0.8

1.0

Deg

ree

of E

ntan

glem

ent

ConcurenceV. N. Entropy

FIG. 9. (a) Von Neumann entropy and concurrence of the path-entangled state as a function of the chiral phase Φ at τ = 0. (b) Correspondingeigenvalues P1 and P2 at τ = 0.

Eq. (13) shows how the fine-structure splitting S and partial chirality Φ influence the concurrence of the entangled state. A QDlocated at a perfectly chiral position within the waveguide, i.e. Φ = ±π/2, leads to C = 1, which agrees with the fact that theresultant biphoton state oscillates between the two maximally entangled states as discussed above. Similarly, QD coupled to anon-chiral waveguide (Φ = 0,±π) leads to C = 0. Partial chiral coupling results in imperfect entanglement, where 0 < C < 1with a sinusoidally varying concurrence arising from fine-structure-splitting induced oscillations. Both entanglement measures,concurrence C and von Neumann entropy H(ρ), are plotted in Fig. 9(a) as a function of the chiral phase Φ.

Timing jitter

Finally we consider the effect of detection time jitter on the entanglement measure, which arises due to the uncertainty in theemission time of the photons, and therefore the detection. We assume a Gaussian distribution of the detection times, motivatedby the Gaussian instrument response function of the SNSPDs, with a ς standard deviation centered around a time τ ′. Thepostive-time-averaged (τ ′ > 0) unnormalised matrix element of the density matrix ρ is

¯ρi jkl =∫

∞

0dτ′e−(τ ′−τ)2

2ς2ψi j(τ,0,τ ′)ψ∗kl(τ,0,τ

′), (14)

where the subscripts represent the collection gratings {i, j,k, l} ∈ {A,B}. The normalization factor of the state N is

N =∫

∞

0dτ′e−(τ ′−τ)2

2ς2 N, (15)

where N is the total number of detected photons. The concurrence C is calculated from the density matrix element by numericallycomputing ¯ρ ˜

ρ under the same approximation as earlier S� γX . Figure 3(c,d) of the main text show the expected temporal andspatial dependence of the calculated C, respectively. As expected, we observe that the ratio S/ς influences the maximum Cachievable at a given chiral phase Φ. The smaller effect of timing-jitter at shorter times on C observed in Fig. 3(c) is a result ofaccepting only positive time intervals τ ′ > 0, which effectively constrains the uncertainty at τ ′ ≈ 0.

Modelling the experimental data and additional correlation measurements

We fit the correlation data in Fig. 3 to the theoretical model presented in using Eq. (5). The parameters entering the model arethe exciton decay rate γX , the fine structure splitting S and the position of zero time delay τ0. For the data for the same gratingwe further introduce that phase Φ, whereas this does not play a role for different gratings. The exciton decay rate is extractedseparately from Fig. 6(c). The fine structure splitting estimated in Fig. 6(b) is used as a starting parameter. Furthermore Eq.(5) is convolved with a Gaussian function of width ς accounting for the finite detector timing resolution (jitter), and multipliedwith an over all amplitude. First we fit the A-B correlations, where Φ = 0 is fixed. Hereby all other parameters are extracted andused as fixed parameters in the fit of the A-A correlations, where the phase is determined. The analysis is performed with Pythonusing lmfit, and the uncertainty in all plots is the 1ς confidence interval.

13

(a) (b)

FIG. 10. Correlation measurements of QD2 (a) and QD3 (b) in similar GPWs shown as points and the model as solid lines. The field atposition of both QDs exhibit a phase shift as indicated in the figure. Note that the two QDs have different fine structure splittings leading todifferent oscillation periods.

0.0

0.5

1.0

AXXBX

0.0

0.5

1.0AXXAX

0.0

0.5

1.0

XX-X

Nor

mal

ized

Coi

ncid

ence

cou

nts

[a.u

.]

BXXBX

-100 0 100 200 300 400 500Time delay [ps]

0.0

0.5

1.0BXXAX

FIG. 11. Correlation measurements of QD1 in all four configurations of collection gratings shown as points. All data are time-matched withrespect to the data set in the A-B configuration. Solid lines show the results from the theoretical model in Eq. (5) for each configuration, basedon the fit result of Fig. 3(a).

We found and measured the XX-X correlation phase shift of two additional QDs in two other GPW devices with slightlydifferent design parameters, cf. data in Fig. 10. We denote these QD2 and QD3, which have exciton emission wavelengths at924.25nm and 921.55nm, respectively. The X emission from QD2 is 0.25nm from the high ng region on the high wavelengthside, and has a significantly smaller fine structure splitting, leading to slower oscillations. QD3 on the other hand is around 5nmfrom the high ng region on the low wavelength side and has a large S, as seen from the faster oscillations.

In addition to the correlation data presented in Fig. 3 we performed measurements in the remaining BXX BX and BXX AXconfigurations as well for that QD. The data are presented in Fig. 11. According to the theory (see Eq. (5)) in the same gratingconfigurations (AXX AX and BXX BX ) we should observe the same phase but with opposite signs. The solid line shows this casesuperimposed on the data using the fitted phase from Fig. 3(a), where it can be seen that the model describes the data well.

14

Cause of reduction in the phase

We attribute the deviation of the measured Φ = 0.12π from the likely phase values (Φ ≈ 0.5π) c.f. Fig 1(b), to the weakback reflections from the grating outcouplers. This is due to a non-optimal design as discussed in , where we also estimate thereflectively r2. As discussed in the main text these back reflections will reduce the chirality, due to partial cancellation of thelongitudinal component of the electric field. We split the effects into an ideal phase Φ0 (corresponding to Fig. 1(b) in main text)and a reduced phase Φ, which is the one we can observe in our correlations measurements Φ.

We model the back reflections from the two grating outcouplers as a cavity with reflectance r in both ends. The propagating-mode component of the electric field is thus modified by added contributions from consequent cavity reflections. The totalaccumulated component ek,x of the electric field yields

ek,x = |ek,x|(

eiφx + rei(−φx+θ)+ r2ei(φx+θ+θ ′) + r3ei(−φx+2θ+θ ′) + ...), (16)

and similarly for the y component of the field, where θ +θ ′ ≡ Θ is the total propagation phase in the cavity. Note that here wehave opposite phases ±φx of the forward and backward propagating fields since these modes are the time reverse of each other.Solving the geometric series we obtain

ek,x = |ek,x|eiφx + rei(θ−φx)

1− r2ei(θ+θ ′). (17)

We then calculate the ratio ek,x/ek,y between the electric field components in order to relate the reduced phase Φ = φx−φy andthe initial phase Φ0 = φ0,x− φ0,y with the reflectance of the cavity. Since we only observe a single exponential decay of theexciton state, c.f. Fig. 6(c), we constrain the modulus of the x and y components of the field to be the same. This implies thatthe propagation phase equals the initial phase Φ. The reflectance can then be calculated to be

r =sin( 1

2 (Φ0−Φ))

sin( 1

2 (Φ0 +Φ)) . (18)

Note that we have set φy = φy = 0 for simplicity.We measure a reflectance of r =

√0.3 and a phase of Φ = 0.12π . Applying Eq. (18) we obtain that the QD was sitting at

Φ0 = 0.37π . Locations with a phase of 0.37π are likely in the GPW used for the experiment, as seen in the simulation in Fig.1(b) of the main text. This means that the reduction in phase is can be explained by the measured reflectivity (30%) through Eq.18.

ACKNOWLEDGEMENT

We thank Single Quantum for giving us access to the ultra-low timing-jitter superconducting nanowire single-photon detectorsrequired for extracting the correlations. We gratefully acknowledge financial support from Danmarks Grundforskningsfond(DNRF 139, Hy-Q Center for Hybrid Quantum Networks). This project has received funding from the European Union’sHorizon 2020 research and innovation programmes under grant agreement No. 824140 (TOCHA, H2020-FETPROACT-01-2018) and under the Marie-Skłodowska-Curie grant agreement No. 801199. A.D.W. and A.L. acknowledge gratefully supportof DFG-TRR160, BMBF - Q.Link.X 16KIS0867, and the DFH/UFA CDFA-05-06. We acknowledge Adam Knorr for initialsample characterization. We thank Matthew Broome for discussions in the early phase of the project.

REFERENCES

∗ lodahl@nbi.ku.dk[1] J. L. O’Brien, A. Furusawa, and J. Vuckovic, Photonic quantum technologies, Nat. Photon. 3, 687 (2009).[2] R. Uppu, L. Midolo, X. Zhou, J. Carolan, and P. Lodahl, Quantum-dot-based deterministic photon–emitter interfaces for scalable photonic

quantum technology, Nature Nanotechnology 16, 1308 (2021).[3] T. Rudolph, Why I am optimistic about the silicon-photonic route to quantum computing, APL Photonics 2, 30901 (2017).

15

[4] H.-S. Zhong, Y. Li, W. Li, L.-C. Peng, Z.-E. Su, Y. Hu, Y.-M. He, X. Ding, W. Zhang, H. Li, L. Zhang, Z. Wang, L. You, X.-L. Wang,X. Jiang, L. Li, Y.-A. Chen, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, 12-photon entanglement and scalable scattershot boson sampling withoptimal entangled-photon pairs from parametric down-conversion, Phys. Rev. Lett. 121, 250505 (2018).

[5] P. Lodahl, S. Mahmoodian, and S. Stobbe, Interfacing single photons and single quantum dots with photonic nanostructures, Rev. Mod.Phys. 87, 347 (2015).

[6] F. Lenzini, B. Haylock, J. C. Loredo, R. A. Abrahao, N. A. Zakaria, S. Kasture, I. Sagnes, A. Lemaitre, H.-P. Phan, D. V. Dao, P. Senellart,M. P. Almeida, A. G. White, and M. Lobino, Active demultiplexing of single photons from a solid-state source, Laser Photon. Rev. 11,1600297 (2017).

[7] H. Wang, Y. He, Y.-H. Li, Z.-E. Su, B. Li, H.-L. Huang, X. Ding, M.-C. Chen, C. Liu, J. Qin, J.-P. Li, H. Y.-M., C. Schneider, M. Kamp,C.-Z. Peng, S. Hofling, C.-Y. Lu, and J.-W. Pan, High-efficiency multiphoton boson sampling, Nat. Photon. 11, 361 (2017).

[8] B. Oliver, C. Santori, M. Pelton, and Y. Yamamoto, Regulated and entangled photons from a single quantum dot, Phys. Rev. Lett. 84,2513 (2000).

[9] N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, Entangled photon pairs fromsemiconductor quantum dots, Phys. Rev. Lett. 96, 130501 (2006).

[10] D. Huber, M. Reindl, S. F. Covre Da Silva, C. Schimpf, J. Martın-Sanchez, H. Huang, G. Piredda, J. Edlinger, A. Rastelli, and R. Trotta,Strain-Tunable GaAs Quantum Dot: A Nearly Dephasing-Free Source of Entangled Photon Pairs on Demand, Phys. Rev. Lett. 121,033902 (2018).

[11] J. Liu, R. Su, Y. Wei, B. Yao, S. F. C. da Silva, Y. Yu, J. Iles-Smith, K. Srinivasan, A. Rastelli, J. Li, and X. Wang, A solid-state source ofstrongly entangled photon pairs with high brightness and indistinguishability, Nat. Nanotechnol. 14, 586 (2019).

[12] A. Politi, J. C. Matthews, M. G. Thompson, and J. L. O’Brien, Integrated quantum photonics, IEEE Journal of Selected Topics in QuantumElectronics 15, 1673 (2009).

[13] R. Uppu, F. T. Pedersen, Y. Wang, C. T. Olesen, C. Papon, X. Zhou, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, and P. Lodahl,Scalable integrated single-photon source, Sci. Adv. 6, eabc8268 (2020).

[14] W. Bogaerts, D. Perez, J. Capmany, D. A. Miller, J. Poon, D. Englund, F. Morichetti, and A. Melloni, Programmable photonic circuits,Nature 586, 207 (2020).

[15] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Chiral quantum optics,Nature 541, 473 (2017).

[16] I. Sollner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kirsanske, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe,and P. Lodahl, Deterministic photon-emitter coupling in chiral photonic circuits, Nat. Nanotechnol. 10, 775 (2015).

[17] 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, Chirality ofnanophotonic waveguide with embedded quantum emitter for unidirectional spin transfer, Nat. Commun. 7, 11183 (2016).

[18] S. Mahmoodian, K. Prindal-Nielsen, I. Sollner, S. Stobbe, and P. Lodahl, Engineering chiral light–matter interaction in photonic crystalwaveguides with slow light, Opt. Mater. Express 7, 43 (2017).

[19] N. V. Hauff, S. Hughes, H. Le Jeannic, P. Lodahl, and N. Rotenberg, Chiral quantum optics in broken-symmetry and topological photoniccrystal waveguides, arXiv:2111.02828 (2021).

[20] See Supplemental Material at [URL] for details.[21] E. M. Gonzalez-Ruiz and et al., (2021), In preparation.[22] Y. H. Huo, A. Rastelli, and O. G. Schmidt, Ultra-small excitonic fine structure splitting in highly symmetric quantum dots on GaAs (001)

substrate, Appl. Phys. Lett. 102, 152105 (2013).[23] T. Huber, A. Predojevic, M. Khoshnegar, D. Dalacu, P. J. Poole, H. Majedi, and G. Weihs, Polarization entangled photons from quantum

dots embedded in nanowires, Nano Lett. 14, 7107 (2014).[24] F. T. Pedersen, Y. Wang, C. T. Olesen, S. Scholz, A. D. Wieck, A. Ludwig, M. C. Lobl, R. J. Warburton, L. Midolo, R. Uppu, and

P. Lodahl, Near Transform-limited Quantum Dot Linewidths in a Broadband Photonic Crystal Waveguide, ACS Photonics , 2343–2349(2020).

[25] X. Zhou, I. Kulkova, T. Lund-Hansen, S. L. Hansen, P. Lodahl, and L. Midolo, High-efficiency shallow-etched grating on GaAs mem-branes for quantum photonic applications, Appl. Phys. Lett. 113, 251103 (2018).

[26] N. H. Lindner and T. Rudolph, Proposal for Pulsed On-Demand Sources of Photonic Cluster State Strings, Phys. Rev. Lett. 103, 113602(2009).

[27] P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, Experimental one-wayquantum computing, Nature 434, 169 (2005).

[28] K. Azuma, K. Tamaki, and H. K. Lo, All-photonic quantum repeaters, Nat. Commun. 6, 6787 (2015).[29] S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant,

F. Pastawski, T. Rudolph, and C. Sparrow, Fusion-based quantum computation, arXiv:2101.09310 (2021).[30] A. W. Elshaari, W. Pernice, K. Srinivasan, O. Benson, and V. Zwiller, Hybrid integrated quantum photonic circuits, Nat. Photon. 14, 285

(2020).[31] N. H. Wan, T.-J. Lu, K. C. Chen, M. P. Walsh, M. E. Trusheim, L. De Santis, E. A. Bersin, I. B. Harris, S. L. Mouradian, I. R. Christen,

E. S. Bielejec, and D. Englund, Large-scale integration of artificial atoms in hybrid photonic circuits, Nature 583, 226 (2020).[32] L. Midolo, T. Pregnolato, G. Kirsanske, and S. Stobbe, Soft-mask fabrication of gallium arsenide nanomembranes for integrated quantum

photonics, Nanotechnology 26, 484002 (2015).[33] S. Adachi, GaAs, AlAs, and AlxGa1−xAs: Material parameters for use in research and device applications, Journal of Applied Physics

58, R1 (1985).[34] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Molding the flow of light, Princeton Univ. Press, Princeton, NJ (2008).[35] S. Hughes, L. Ramunno, J. F. Young, and J. Sipe, Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication

disorder and photon group velocity, Phys. Rev. Lett. 94, 033903 (2005).

16

[36] M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, Disorder-induced incoherent scatteringlosses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the beer-lambert law, Phys. Rev. B80, 195305 (2009).

[37] T. Pregnolato, Deterministic quantum photonic devices based on self-assembled quantum dots, Ph.D. thesis, Niels Bohr Institute, Univer-sity of Copenhagen (2019).

[38] X.-L. Chu, T. Pregnolato, R. Schott, A. D. Wieck, A. Ludwig, N. Rotenberg, and P. Lodahl, Lifetimes and quantum efficiencies ofquantum dots deterministically positioned in photonic-crystal waveguides, Adv. Quantum Technol. 3, 2000026 (2020).

[39] S. Das, L. Zhai, M. Cepulskovskis, A. Javadi, S. Mahmoodian, P. Lodahl, and A. S. Sørensen, A wave-function ansatz method forcalculating field correlations and its application to the study of spectral filtering and quantum dynamics of multi-emitter systems,arXiv:1912.08303 (2019).