Topological photonics promises unique and robust designs for novel photonic functionalities by rendering device performance immune to degradation induced by fabrication imperfections or environmental changes [1,2]. Topological notions in photonics are inspired by condensed matter physics, and in particular the foundational concept of topological insulators [3–6]. This novel phase of matter is insulating in the bulk, yet conveys surface currents without any dissipation or backscattering, even in the presence of defects and disorder. In 2008, the key features of the electronic quantum Hall effect were adapted to light waves in the proposal of a photonic analogue of the anomalous quantum Hall effect [7,8]. These groundbreaking ideas were first implemented in photonic crystals for the microwave regime [9,10], where topological propagation of microwaves along the edge of the system was demonstrated to form a scatter-free channel.

This demonstration of robust light transport along the edge of a photonic system proved the concept of unidirectional waveguiding based on topology. The first implementation of a photonic topological insulator employed helical waveguides in a honeycomb geometry [11] and was soon followed by a realization based on coupled resonator optical waveguides [12]. Subsequently, numerous peculiar topological phenomena such as anomalous topological insulators [13,14], 4D topological Hall physics [15], Weyl points [16], topological Anderson insulators [17], topological insulators in synthetic dimensions [18], as well as non-Hermitian topological physics [19,20] and topological quantum physics [21,22], were observed and reported.

The majority of one-dimensional systems realized with static waveguide arrays are based on the Su–Schrieffer–Heeger (SSH) model [23], since it is the only topologically nontrivial system in 1D where the time reversal operator squares to one [24]. As expressed in the bulk-edge correspondence, the topology of the lattice interior determines whether topological edge states can exist at the system’s boundary. In photonics, such edge states were experimentally realized for the first time in the form of Shockley surface states [25]. SSH lattices were explored in the context of high harmonic generation [26], periodic driving in Floquet systems [27,28], and soliton states [29]. Yet another concept utilizes the framework of optical supersymmetry transformations [30] to connect different SSH-type lattices [31], which triggers topological phase transitions. Also, the SSH model is often used to study the impact of non-Hermitian concepts on topology [19,20].

More recently, the evolution of quantum states in photonic topological systems began to attract considerable interest [21], with a particular emphasis on preserving the quantum coherence due to topological protection [12,22,32–36]. Coupled waveguide lattices have proven to be an exceptionally versatile platform for exploring the evolution of photonic quantum states in complex settings within the framework of photonic quantum walks [37,38]. Beyond the average photon number in the individual channels, i.e., the expectation value of the photon number distribution, an important measure for the quantum behavior of multiphoton states in waveguide lattices is higher-order correlations, which relate to the joint probabilities of finding the photons at particular locations [39–45]. These correlations generally become highly nontrivial for indistinguishable photons as soon as several paths lead to the same average photon number in the channels, as quantum path interference occurs.

In our work, we experimentally demonstrate two-photon quantum correlations in an SSH waveguide lattice. We explore the specific features of such correlations for the topological as well as the trivial edge and compare them to the homogeneous case. Moreover, we evaluate the inter-particle distance for both edges and find that the topological edge gives rise to larger average distances compared to the trivial edge.

To describe the evolution of quantized light, the single-photon Hamiltonian for a coupled waveguide array reads [48] H=ℏcn0∑k[β0(a^k†a^k+12)+∑lCk,la^k†a^l],(5)where c is the speed of light, n0 is the refractive index of the material, β0 is the propagation constant in the individual waveguides, a^k† and a^k are the creation and annihilation operators in waveguide k, and Ck,l is the coupling between waveguides k and l, where only nearest-neighbor coupling is assumed, that is, l=k±1. The Heisenberg equations of motion in the co-moving frame, describing the evolution of a single photon in the lattice, then take the following form: iddza^k†(z)+∑lCk,la^k†=0,(6)with z as propagation distance in our optical setting, which acts as the evolution coordinate. Equation (6) can be formally integrated, such that one obtains the analytic solution for the a^k†(z): a^k†(z)=∑lUk,l(z)a^l†(0),(7)with the unitary propagation matrix Uk,l(z) given by the expression Uk,l(z)=(eizC)k,l.(8)By its definition, Uk,l(z) can be interpreted as the probability amplitude for a photon injected in guide l to be detected in channel k after propagation over the distance z. Hence, when a single photon is launched in waveguide k, the average photon number n¯m(z) in channel m computes to n¯m(z)=⟨a^m†(z)a^m(z)⟩=|Um,k(z)|2.(9)When launching two photons in waveguides k and l, the average photon number is n¯m(z)=|Um,k(z)|2+|Um,l(z)|2.(10)This is the incoherent sum of the average photon numbers for single photon inputs in channels k and l. Hence, quantum path interference is never visible in the average photon number, regardless of the nature of the input state. In fact, the photon number evolves just like a classical intensity. In contrast, the two-photon correlation function Γk,l of the photon outputs does depend on the input state and reveals this interference. This function is defined as Γm,n(z)=⟨a^m†(z)a^n†(z)a^n(z)a^m(z)⟩(11)and, hence, reads Γm,n(z)=|Um,k(z)Un,l(z)+Um,l(z)Un,k(z)|2.(12)When launching two indistinguishable photons into the lattice with one of the photons launched in channel k and the other in channel l, the function Γm,n(z) relates to the joint probability of finding one photon in guide m and one in l at the same time via Pm,n=11+δm,nΓm,n.

Along the length of the sample (150 mm), the fabricated structures comprise 41 waveguides and consist of three sections. The central section contains the actual lattice under investigation, that is, either an SSH lattice or a homogeneous lattice [see Fig. 2(B)]. The first and third sections are fanning structures that serve to match the fiber array spacing of 82 μm, with which the single photons are launched into and collected from the chip. The input fanning feeds the waveguides in which the photons are launched, whereas the output fanning collects the photons from a broader region in the lattice.

In our experiments, we employed a photon pair source at λ=815nm using a standard type I spontaneous parametric downconversion source with a visibility of 94%. A BiBO crystal is pumped by a 100 mW, 407.5 nm laser diode producing |H⟩ polarized single-photon pairs collected by polarization maintaining fibers. Commercial V-groove fiber arrays with a pitch of 82 µm were used to couple the photons into the chip as well as to collect them at the output facet from the individual waveguides. Using high-NA multimode fibers in order to feed the photons to the respective avalanche photodiodes ensures low coupling losses at the output side of the chip. From the data of the photodiodes, the photon probability distribution at the output as well as the inter-channel correlations can be computed using a correlation device (Becker–Hickl) and standard software (LabView, MATLAB). Multiphoton events can be neglected, as for instance, when sending one photon from the downconversion pair into a chip the ratio of probabilities of two count events compared to single click events is 10−4 at the output of the system. Moreover, as the ratio of single clicks to zero-photon events was measured to be 0.03, the possibility of weak coherent input states can be excluded with high confidence.

We compare this to the two-photon correlations in a homogeneous lattice, where all hoppings between the waveguides are identical (the dimerization is zero). In this case, the photons inevitably exhibit bunching and are found with high probability close to each other, irrespective of whether they were launched into the bulk of the lattice [Fig. 4(G), first measured in Ref. [40]) or close to the edge of the lattice (Fig. 4(H)], without being affected by the topology of the lattice.

In our work, we observed two-particle quantum correlations of indistinguishable photons at the edge and in the bulk of an SSH waveguide lattice and compared them to the correlations in a homogeneous lattice. We find that, whereas at the trivial edge in the SSH lattice the bunching behavior of the indistinguishable photons remains unchanged, the bunching is significantly distorted at the topological edge. Although one photon remains at the edge with high probability and the other mostly penetrates into the bulk, the joint probability of finding the photons close to each other after propagating through the sample has approximately the same magnitude as the probability that the photons leave the sample far away from each other. Therefore, the existence of the edge state at the topological edge in the SSH model has a significant influence on the quantum interference of indistinguishable photons.

Our work sheds new light on the quantum features of topologically nontrivial lattices and may help to find new applications in preparing and transmitting complex quantum states. There are several questions that can be asked. What behavior can be observed in higher-dimensional topological systems, and in particular higher-order topological insulators? What is the impact of topological edge states on states with larger number of photons? And what happens in the case of non-Hermiticity, in particular parity-time symmetry? These and related problems are now in reach to be explored experimentally.

Acknowledgment. The authors would like to thank C. Otto for preparing the high-quality fused silica samples employed in this work.