Quantum interference of topological states of light
Jean-Luc Tambasco, Giacomo Corrielli, Robert J. Chapman, Andrea Crespi, Oded Zilberberg, Roberto Osellame, Alberto Peruzzo
QQuantum interference of topological states of light
Jean-Luc Tambasco , Giacomo Corrielli , , Robert J. Chapman , AndreaCrespi , , Oded Zilberberg , Roberto Osellame , , Alberto Peruzzo Quantum Photonics Laboratory and Centre for Quantum Computation and Communication Technology,School of Engineering, RMIT University, Melbourne, Victoria 3000, Australia Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche,Piazza Leonardo da Vinci 32, Milano I-20133, Italy Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano I-20133, Italy Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland (Dated: April 25, 2019)Topological insulators are materials that have a gapped bulk energy spectrum, but contain protected in-gap states appearing at their surface. These states exhibit remarkable properties such as unidirectionalpropagation and robustness to noise that offer an opportunity to improve the performance andscalability of quantum technologies. For quantum applications, it is essential that the topologicalstates are indistinguishable. Here we report high-visibility quantum interference of single photontopological states in an integrated photonic circuit. Two topological boundary-states, initially atopposite edges of a coupled waveguide array, are brought into proximity, where they interfere andundergo a beamsplitter operation. We observe . ± . visibility Hong-Ou-Mandel (HOM)interference, a hallmark non-classical effect that is at the heart of linear optics-based quantumcomputation. Our work shows that it is feasible to generate and control highly indistinguishablesingle photon topological states, opening pathways to enhanced photonic quantum technology withtopological properties, and to study quantum effects in topological materials. INTRODUCTION
Research into solid-state physics has led to the discov-ery of a new phase of matter, the topological insulator ; aclass of materials that insulates in the bulk, but conductson the surface [1, 2]. This has inspired the design ofnew topological systems with unique band structures andprotected boundary-states in various effective dimensions.In particular, one-dimensional topological superconduc-tors have recently received great attention due to theirtopological boundary state, namely Majorana zero-modesthat can be harnessed for topological quantum computing[3].Since the discovery of topological phases of matter, awealth of pioneering topological systems have been demon-strated using photonics [4, 5]. Topological photonics hasthe advantage of not requiring strong magnetic fields, andfeatures intrinsically high-coherence, room-temperatureoperation and easy manipulation. To-date, several topo-logical effects have been observed using integrated pho-tonics including Majorana modes [6], chiral edge modesrobust to defects [7–13], optical Weyl points [14–16], 1Dand 2D topological pumping and topological quasicrystals[17–20], as well as generation and propagation of singlephotons [21, 22].Concurrently, photonics has a long-standing goal toimplement quantum computation. Quantum interferenceof single photons at a 50:50 beamsplitter is a key phe-nomenon in quantum physics and lies at the heart of linear-optical quantum computation [23]. This phenomenon canbe observed via the well-known Hong-Ou-Mandel (HOM)experiment [24], which has been demonstrated in inte-grated photonic devices, including on-chip beamsplitters[25–27], photonic quantum walks [28, 29], circuits display- ing Anderson localization [30], and recently in plasmonicdevices [31]. High-visibility quantum interference relieson the two input photons being totally indistinguishable.To-date, quantum interference has not been observed intopological systems.In this work, we report high-visibility quantum interfer-ence of two single-photon topological boundary-states in aphotonic waveguide array. We engineered a time-varyingHamiltonian, controlling the band structure of the de-vice and the spatial isolation of the topological states toimplement a 50:50 beamsplitter. Using this ‘topologicalbeamsplitter’, we measured Hong-Ou-Mandel interferencewith . ± . visibility, demonstrating non-classicalbehavior of topological states. RESULTS
Our device implements the off-diagonal Harper model,which describes a one-dimensional lattice that exhibitstopological boundary-states [17, 32, 33]. The time-varyingHamiltonian of this model is given as ˆ H ( t ) = N − (cid:88) n =1 κ n ( t )(ˆ a n ˆ a † n +1 + ˆ a † n ˆ a n +1 ) , (1)where ˆ a n and ˆ a † n are annihilation and creation operatorsacting on lattice site n . κ n ( t ) is the coupling strength attime t between site n and site n + 1 . In the off-diagonalHarper model, the coupling strengths follow the periodicfunction κ n ( t ) = κ (cid:2) t ) cos (cid:0) π ¯ bn + φ ( t ) (cid:1)(cid:3) , (2)where κ is the nominal coupling coefficient between twoadjacent lattice sites, ¯ b controls the periodicity of the a r X i v : . [ qu a n t - ph ] A p r IG. 1.
Photonic boundary-state beamsplitter. (a) Illustrative representation of a waveguide array implementingstationary topologically boundary-states (red shaded regions) that propagate at the edges of the device. This device is usedto confirm that the boundary-state is preserved during the propagation inside the array. (b) Illustrative representation of awaveguide array implementing a ‘topological beamsplitter’ that interferes two topologically boundary-states. (c) Photonicsupermodes (eigenvectors) of the arrays at the start and end of the both devices. (d, e) Band structure (eigenenergies) along thelength of the arrays a) and b). The topological bands (B and D) are highlighted in red and the bulk bands (A, C and E) areshaded in blue. lattice, Λ( t ) controls the size of the spectral gaps andcorrespondingly defines the confinement of the boundary-state at time t , and φ ( t ) is a time-varying phase. Bycarefully choosing the value of ¯ b , gaps appear in the en-ergy spectrum of the system that allow topological pump-ing by adiabatically varying φ ( t ) [17]. Carefully choos-ing φ ( t ) ensures the appearance of topological boundarymodes on the edges of the array. Both φ ( t ) [17] and Λ( t ) can be used to manipulate the boundary-states. In thiswork we vary Λ( t ) to confine, delocalize and interfere theboundary-states; this procedure is reminiscent to changingthe length of a topological superconductor and interferingits Majorana-modes [3].An array of coupled waveguides in the nearest neigh-bor approximation implements the same tight-bindingHamiltonian as Eq. (1), where the waveguide separa-tion controls the coupling strength. We experimentallycharacterized the relationship between the waveguide sep-aration and the coupling strength κ n ( t ) (see Materialsand Methods for details), which enabled us to design anarray with the desired Hamiltonian. Because we varythe κ n terms along the length of the array, we make thetransformation from a time-varying to a distance-varyingHamiltonian with the relationship z = ctn where z is theposition, c is the speed of light and n is the waveguideeffective refractive index.Initially, two photonic states are localized at the edgesof the array; they are spectrally and spatially isolatedfrom the bulk modes, start with the same energy, andare spatially isolated from one another. Interference canoccur when these states are adiabatically delocalized fromthe edges to the bulk of the lattice by reducing the bulk gap size.We designed two devices, each consisting often waveguides with symmetric coupling strengths { κ , κ , κ , κ , c, κ , κ , κ , κ } and ¯ b = 2 / . The firstdevice has fixed Λ( z ) = 0 . to demonstrate and confirmthe confinement of the topological boundary-states, as il-lustrated in Fig. 1(a). In the second device, illustrated inFig. 1(b), we vary Λ( z ) from Λ(0) = 0 . to Λ( L/
2) = 0 . ,where L is the total length of the array. This reduces thelocalization of the two boundary modes, causing themto interfere. By tuning the central coupling coefficient( c ), a 50:50 beamsplitter is realized before relocalizingthe states to the sides of the device, where λ ( L ) = 0 . .For both devices, waveguides 1–5 (and due to symmetry,10–6) have five photonic supermodes (eigenstates), whichare shown in Fig. 1(c).Exciting the boundary-states of each array requiresinjecting into the mode labeled B in Fig. 1(c). As shownin Figs. 1(a) and (b), this is achieved by extending thetwo edge waveguides to the input facet of the chip (seeSupplementary Section S1 for details). To model thebulk-band spectrum of the photonic supermodes in Fig.1(c), we calculated the eigenvalues of both devices, shownin Figs. 1(d) and (e), along the length of the array. Theapproximate bulk energy bands are shaded in blue andthe eigenvalues corresponding to the boundary-states areplotted in red (labeled B and D). As the Hamiltonianis implemented on a photonic platform, each eigenvalueis proportional to the effective refractive index of thecorresponding photonic supermode [34]. If eigenvaluesare similar in magnitude, the corresponding eigenstateswill scatter between modes; however, eigenstates between2 IG. 2.
Characterization of the stationary boundary-state device and the topological beamsplitter.
The outputof the chip is characterized using laser light and a CCD camera. (a) The normalized output intensity distribution of thestationary boundary-state. (b, c) The normalized output intensity distribution of the topological beamsplitter with injectioninto the left and right inputs respectively. the energy bands are resilient to scattering.Our devices were fabricated using the direct-write tech-nique [35, 36] as it enables high-precision control of thewaveguide coupling coefficients. The direct-write tech-nique is implemented by translating a borosilicate chipwhile focusing a femtosecond laser into the bulk (seeMethods and Materials for more details on the chip fabri-cation).We characterized each device using laser light at
808 nm ,to match the wavelength of our single photon source, andmeasured the output with a CCD camera. We calcu-lated the fidelity between the measured output distribu-tion across the whole array and the simulated result as F = (cid:80) i (cid:112) P S i P M i , where P S i ( P M i ) is the simulated (mea-sured) intensity of light at the output of waveguide i afternormalization. The intensity distribution is equivalent tothe output single-photon probability distribution. Herethe simulation is based on the physical parameters ofour device. We note that depicting the boundary-statesupermodes (B and D in Fig 1(c)) as being confined totwo-waveguide is an approximation and, in a real device,they exponentially decay beyond the edge waveguides—this phenomenon is inherent to any bound mode in aspectral gap.Figure 2(a) shows the measured output intensity andsimulation results for the stationary topological boundary-state when injecting into the left even-mode eigenstate,and the fidelity is F = 97 . . Figure 2(b) shows theresults for the topological beamsplitter. We measuredfidelity for the left and right input of F = 96 . and F = 97 . respectively. These fidelities are very highand are mainly limited by fabrication imperfections.In the quantum interference experiment, we employed a coupling setup to collect photons from the outer waveg-uides (1,2 and 9,10). When selecting only these waveg-uides, we calculated the reflectivity of the topologicalbeamsplitter to be 49.9% (50.7%) for the left (right) in-put, which is very close to 50%—a requirement for highvisibility quantum interference.Figure 3(a) shows our setup for measuring HOM inter-ference. We generated pairs of photons using a free-spacespontaneous parametric down conversion (SPDC) sourcebefore coupling into two polarization maintaining opticalfibers (PMF). Narrow-band filters are inserted to ensurethe photon wavelengths are matched and one fiber is posi-tioned on a translation stage to enable a tunable delay (seeSupplementary Section S2 for full details on the photonpair source). We measure the visibility of the interferenceby controlling the distinguishability of the photons withthe tunable delay.We first injected single photon pairs into a commerciallyavailable fiber coupled 50:50 beamsplitter (FBS) withPMF for the input, ensuring the photons have the samepolarization when they interfere. The output fibers arecoupled to single photon avalanche photodiodes (APDs)that emit an electrical pulse when a photon is detected.Coincidence measurements of the APD signals are per-formed with a timing card. We measured the HOM dipshown in Fig 4(a) with visibility V FBS = 94 . ± . .Error bars on the plot are calculated using Poissonianstatistics (see Supplementary Section S3 for HOM diperror calculation). Accidental coincidences due to strayambient light and dark counts were detected and sub-tracted from the signal by applying an electronic timedelay to one detector. As the beamsplitter reflectivity isclose to ideal ( r = 49 . ± . ), the visibility is limited3 IG. 3.
Experimental setup for interfering topological boundary-states. (a) Setup to characterize the indistinguishablyof the photon pairs generated from a spontaneous parametric down-conversion (SPDC) source. The photons are interfered in a50:50 beamsplitter via polarization maintaining fibers (PMF). The output of the beamsplitter is pigtailed with single mode fiber(SMF) connected to single photon avalanche photodiodes (APDs). Coincidence counts are measured between the two detectorswith a timing card. (b) To perform the indistinguishability measurements of single photon topologically protected states, thepigtailed beamsplitter in a) is replaced with the topological beamsplitter device. We used PMF, multimode fibers (MMF) andfree-space lenses to couple photons to the device.FIG. 4.
Measurements of indistinguishability. a) HOMinterference of single photons using a commercially availablefiber pigtailed 50:50 beamsplitter with a visibility of . ± . .b) HOM interference on the topological beamsplitter with avisibility of . ± . . The error bars shown are based onPoissonian statistics. predominately by the spectral distinguishability of thegenerated photon pairs. We then replaced the 50:50 fiber beamsplitter withour waveguide chip, as shown in Fig. 3(b). We injectedsingle photons simultaneously into both boundary-statesof the topological beamsplitter (TBS) and varied thedelay such that we could perfectly match the arrival times.We measured the HOM dip shown in Fig. 4(b) with avisibility of V TBS = 93 . ± . ; this gives a relativevisibility V relative = V TBS V FBS of . ± . , confirming thatthe quantum interference of topological boundary-statesin our device is close to ideal. The measurement noisefor the topological beamsplitter chip is increased due tocoupling losses leading to a significantly lower count rateand, consequently, a decreased signal-to-noise ratio. DISCUSSION
In this work, we have demonstrated that single photonslocalized to topological boundary-states can undergo high-visibility quantum interference. To this aim, we employeda laser-written photonic circuit that represents one of themost complex examples of a continuous waveguide arraywith engineered coupling coefficients varying along thepropagation direction. This technology enables futurestudies of quantum effects in topological materials thatare challenging or impossible to probe due to, for example,large magnetic field requirements or excessive noise [2].Moreover, the TBS could be extended to other topologicalmodels (such as the Su, Schrieffer and Heeger (SSH) modelof a one-dimensional dimer chain [37]). We anticipatethat the TBS presented in this work will combine withother leading works in topological photonics [8, 22] to helpsolve challenges currently faced in quantum photonics,including pump filtering for photon generation and robustphoton transport.
Acknowledgments:
G.C., A.C. and R.O. acknowledge4nancial support by the ERC-Advanced Grant CAPABLE(Composite integrated photonic platform by femtosecondlaser micromachining; grant agreement no. 742745) andby the H2020-FETPROACT-2014 Grant QUCHIP (Quan-tum Simulation on a Photonic Chip; grant agreement no.641039). O.Z. acknowledges financial support from theSwiss National Science Foundation (SNSF). A.P. acknowl-edges support from the Australian Research Council Cen-tre of Excellent for Quantum Computation and Commu-nication Technology (CQC T), project no. CE170100012,an Australian Research Council Discovery Early CareerResearcher Award, project no. DE140101700 and anRMIT University Vice-Chancellor‘s Senior Research Fel-lowship.
MATERIALS AND METHODSDevice design and simulation
The relationship between waveguide separation andcoupling coefficient is characterized with a test chip con-taining varying spaced waveguides. This relationshipfollows an exponential decay κ = ae − bd , where a = 115 cm − and b = 0 . µ m − are experimentally measuredconstants and d is the separation between the waveguides.We can invert this function to find the waveguide separa-tion necessary to achieve the desired coupling coefficientsin Eq. 2. These κ n ( z ) coupling coefficients control thetransfer of the topological boundary-state from the sides of the array to the center.We numerically optimize the coupling coefficient c inEq. 1 such that the boundary-states couple with 50%probability. This implements a 50:50 beamsplitter opera-tion.Finally, the waveguide separations are adjusted to trans-fer the boundary-states back to the sides of the array. Device fabrication
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