## Abstract

The low-decoherence regime plays a key role in constructing multi-particle quantum systems and has therefore been constantly pursued in order to build quantum simulators and quantum computers in a scalable fashion. Quantum error correction and quantum topological computing have been proved to be able to protect quantumness but have not yet been experimentally realized. Recently, topological boundary states are found to be inherently stable and are capable of protecting physical fields from dissipation and disorder, which inspires the application of such topological protection on quantum correlation. Here we present an experimental demonstration of topological protection of two-photon quantum states against the decoherence in diffusion on a photonic chip. By analyzing the quantum correlation of photons out from the topologically nontrivial boundary state, we obtain a high cross-correlation and a strong violation of Cauchy–Schwarz inequality up to 30 standard deviations. We further prepare different quantum sources and experimentally confirm that the topological protection is robust to the wavelength difference as well as distinguishability of two photons. Our results, together with our integrated implementation, provide an alternative way of protecting quantumness and may inspire many more explorations in “quantum topological photonics”, a crossover between topological photonics and quantum information.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Single photons inherently hold the features of a single qubit in quantum computing, they have been widely used in various quantum simulation protocols, such as quantum walk [1,2], boson sampling [3–6] and quantum fast hitting [7]. Single-particle quantum walks have a precise mapping to classical wave phenomena; however, the advantage of uniquely quantum mechanical behavior is limited due to the single walker.

In contrast, multiple indistinguishable particles can provide distinctly nonclassical correlations, and this quantum behavior becomes a computational advantage. For example, a two-particle quantum walk can be an algorithmic tool for the graph isomorphism problem [8], and the universal computation can be achieved by a multi-particle quantum walk efficiently [9]. Thus, it is crucial to preserve the nonclassical features when constructing a multi-particle quantum computer.

Quantum error correction is proposed to preserve logical quantum states in a subspace and rectify errors according to the measurement outcomes of ancillary particles [10,11]. Quantum topological computing, meanwhile, strives to store and manipulate quantum information with topological protection in a nonlocal manner using non-Abelian anyons [12]. Both of them are promising candidates in theoretical predictions but are still in their initial stage for experimental implementations [10–13].

Topological photonics, derived from the discovery of topological phases in condensed-matter physics, aims to topologically protect photons from the inevitable fabrication-induced dissipation and disorder [14,15]. Many types of topological phases have been observed, for instance, the Hall effect [16,17], edge states [18–21], topological insulators [22,23], and Weyl points [24,25], implying the capability of protecting physical fields. Inspired by these, we may question whether we can extend the protection mechanism into the quantum regime to directly protect quantum-correlated states. Together with the integrability and controllability of integrated photonic chips [8,26], the on-chip topological boundary states may provide an alternative way of effectively protecting quantumness [27–29].

In this paper, we experimentally investigate the evolution dynamics and their preservation of two-photon quantum correlated states in the topological photonic lattice on a photonic chip. We observe no distinct drop of cross-correlation in the boundary state. In contrast, the two photons in bulk states are found to dissipate into many sites, and a few best measurable sites only give a cross-correlation of 6 with very large uncertainty, with which a quantum entanglement derived will no longer be able to violate Bell inequality [30,31]. We also verify the preservation of two-photon quantum correlation in the boundary state by measuring a strong violation of Cauchy–Schwarz inequality up to 30 standard deviations [32]. By preparing different quantum sources, we further experimentally confirm that the topological protection is robust to the wavelength difference as well as distinguishability of two photons.

We construct a topological boundary state under an *off-diagonal* Harper model (also known as the Aubry–André model) [33,34]. We can describe the system by the Hamiltonian

We set the modulation amplitude $\lambda =0.5$ and the site number $N=50$. The calculated Floquet band structure as a function of phase $\varphi $ is shown in Fig. 1. We can see two topologically nontrivial boundary modes in the gap connecting different bands. If the system is excited in the leftmost site ($n=1$), all the eigenmodes of the system are excited, but only the boundary state is dominated. Under the restraint of the boundary state, the photons are confined in the boundary of the lattice. As a comparison, for the bulk state, the photons are no longer confined in one or two sites and spread to other sites like a typical quantum walk in quasi-crystal.

We fabricate the lattice in a borosilicate glass using a femtosecond laser direct writing technique [6,7,35–38] [see Supplement 1] with phase $\varphi =0.2\pi $, where there exist two topologically nontrivial boundary modes as shown in Fig. 1. The quantum-correlated photon pairs are injected into the leftmost $(n=1)$ and the middle $(n=26)$ sites, respectively. The outgoing probability distribution and the cross-correlation quality are measured after the photons have evolved for 35 mm.

The experimental setup in Fig. 2 shows that horizontally and vertically polarized photons with a wavelength of 810.4 nm are simultaneously generated from periodically poled potassium titanyl phosphate (PPKTP) crystals via spontaneous parametric downconversion. Both of them are transformed to horizontal polarization with 25% probability after a designed polarization rotation and projection. Then they are injected into the lattice simultaneously. We measure the photon outgoing distribution with a single-photon sensitive intensified CCD (ICCD) camera. We detect the cross-correlation of the photon pairs by avalanche photodiodes (APDs) after a fiber beam splitter. More details about the quantum light source and experimental measurement can be found in Supplement 1.

The outgoing photon probability distribution of the boundary state is shown in Figs. 3(a) and 3(b), where the photon can almost only be found in the leftmost site. The coincidence of correlated photon pairs detected over 300 s at the output of the left seven sites is shown by the black dots. As we can see, the two photons simultaneously occupy the leftmost site with a high probability up to 94.6%. The photons also occupy the several sites near the leftmost site with very low probability due to the excited bulk state. Meanwhile, when we inject the quantum-correlated photon pair in the middle site, a trivial quantum walk type scenario for the bulk state is observed. As is shown in Figs. 3(c) and 3(d), the two photons spread across the lattice so that even the coincidences in a few best measurable sites are still much smaller than that in the topological boundary state.

To quantify the performance of preserving the two-photon quantum correlation, we measure the cross-correlation function ${g}_{s-i}^{(2)}={p}_{s-i}/{p}_{s}{p}_{i}$ [39] before and after our photonic chip. The ${p}_{s}$, ${p}_{i}$, and ${p}_{s-i}$ represent the detection probabilities for the signal, idler, and their coincidence, respectively. The value of ${g}_{s-i}^{(2)}$ for the leftmost site protected by the boundary state is up to $34.12\pm 1.44$, which is close to the value of $44.31\pm 0.16$ obtained before the photonic chip. For the bulk state, the highest value of ${g}_{s-i}^{(2)}$ we can get is $15.12\pm 2.14$, and the values for the other sites are very difficult to obtain due to the low coincidence as aforementioned. We increase the pump power from 4 to 17 mW to generate the photon pairs with a higher flux, and the number of coincidences increases from $4.0\times {10}^{3}$ to $19.4\times {10}^{3}$ per second, which makes it possible to measure the cross-correlation (and also the auto-correlation, which will be presented later) in the bulk state in reasonable time [see Fig. S2 in Supplement 1].

As is shown in Fig. 4, we measure more details under different pump conditions. The cross-correlation of the protected state is $10.70\pm 0.25$, apparently very close to the value of $11.47\pm 0.02$ before the photonic chip. That means the boundary state preserves not only the photon probability but also their quantum correlation. As a comparison, we measure the cross-correlation for several sites near the leftmost site in the boundary of the lattice and for eight sites with the highest photon probability in the bulk state. All the measured values are found to be as low as 6 with very large uncertainty, with which a quantum entanglement derived will no longer be able to violate Bell inequality [30,31]. This implies that the quantum features of the two-photon states tend to degenerate without the protection of a topological boundary state.

As a way of distinguishing from classical behaviors, the non-classicallity can be further revealed clearly by a violation of the Cauchy–Schwarz inequality [32]

where ${g}_{s-s}^{(2)}$ (${g}_{i-i}^{(2)}$) is the auto-correlation of the signal (idler) photon. As is shown in Table 1, we measure the inequality for the input state, the protected state, and the states outgoing from the four best measurable sites in the bulk state. Again, we can observe a clear violation of Cauchy–Schwarz inequality for the protected state but not well for the unprotected state.The two photons involved in the above measurement are distinguishable though their wavelengths being partially overlapped with each other, which is the result of the narrow linewidth and strongly correlated frequency of the source [see Supplement 1]. To demonstrate the scenario of distinguishable photons more clearly, we tune the temperature of PPKTP crystal to obtain two correlated photons with distinctly different central wavelengths and observe the topological protection of quantum correlation of the two distinguishable photons.

As shown in Fig. 5(a), the wavelength of the horizontal polarization photon generated from crystal is 811.2 nm, while that of vertical polarization photon is 809.6 nm. The two photons are prepared to horizontal polarization with the setup shown in Fig. 2. With the same measurement method, we obtain the result of the correlation function of photons from the outgoing boundary state and the bulk state. As is shown in Fig. 5(b), the measured ${g}_{s-i}^{(2)}$ of photons in the leftmost site of the boundary state is $24.08\pm 1.90$, which approaches that of the source with ${g}_{s-i}^{(2)}=30.00\pm 0.30$, while the ${g}_{s-i}^{(2)}$ of photons in the bulk state and the sites near the leftmost site have low values. The results suggest that the performance of the boundary state in protecting the quantum correlation is robust to the wavelength difference of the two photons.

As we know, if the two photons are indistinguishable, the quantum effects, such as bunching, will affect the evolution of photons in waveguide lattice, and the process shown in inset $\mathbf{i}$ in Fig. 2 can also be considered the Hong–Ou–Mandel interference. To investigate the quantum correlation behavior of two indistinguishable photons in the boundary state and bulk state, we prepare a frequency-uncorrelated photon pair source by pumping a $\beta -{\mathrm{BaB}}_{2}{\mathrm{O}}_{4}$ (BBO) crystal and applying bandpass coherent filters. We inject the obtained two-photon state into the lattice with the similar experimental setup to the one shown in Fig. 2. As shown in Fig. 6(a), the joint spectrum of the source implies that we can obtain two indistinguishable photons at a central wavelength of 810 nm. We further verify the indistinguishability by performing Hong–Ou–Mandel interference. The two photons are prepared with horizontal polarization and then are interfered in a fiber beam splitter. The coincidence count of the photons behind the beam splitter changing with the delay line is shown in Fig. 6(b). The visibility quantifies the contrast of interference and is defined as $V={P}_{0}/{P}_{T}$ for the Hong–Ou–Mandel dip, where the ${P}_{0}$ and ${P}_{T}$ are the coincidence probability at zero and large delays, respectively. The high visibility up to 86.9 implies that a good indistinguishability is obtained.

We tune the delay lines of one of the photons and measure the coincidence count of the two photons outgoing from the boundary state and bulk state, respectively. As shown in Fig. 6(c), the coincidence count at zero delay is twice as large as that at the large delay for the protected site in a boundary state, and the same result is presented but with one-ninth count for the unprotected site in the bulk state [see Fig. 6(d)]. Considering that the counts of the signal photon and the idler photon are unchanged with the delay, the value of ${g}_{s-i}^{(2)}$ of indistinguishable photons is twice as large as that of distinguishable photons. The physics behind the difference between indistinguishable and distinguishable photons is the quantum bunching effect, which enhances the probability of obtaining a two-photon state 2 times from 25% (shown in inset $\mathbf{i}$ in Fig. 2) to 50%. Such enhancement manifests on preparing a two-photon state, and the ability of the boundary state to protect quantum correlation against the decoherence in diffusion in coupled waveguides remains the same for the indistinguishable and distinguishable photons.

In summary, we experimentally demonstrate topological protection of two-photon quantum correlation on a photonic chip. The measurement on the cross-correlation and the violation of Cauchy–Schwarz inequality indicate that the two-photon quantum-correlated states can be well preserved in the topologically nontrivial boundary state and substantially thermalize in the bulk state. We also confirm that the topological protection is robust to the wavelength difference as well as distinguishability of the two photons.

Our results, together with works from the same period [40–42], extend the protection mechanism of topological phases into thequantum regime to directly protect quantum-correlated states, representing an emerging and alternative way of protecting quantumness. Further work includes the protection of quantum entanglement or the implementation in higher dimensional structures.

## Funding

National Natural Science Foundation of China (NSFC) (11690033, 11761141014, 61734005); National Key R&D Program of China (2017YFA0303700); Science and Technology Commission of Shanghai Municipality (STCSM) (15QA1402200, 16JC1400405, 17JC1400403); Shanghai Municipal Education Commission (SMEC) (16SG09, 2017-01-07-00-02-E00049).

## Acknowledgment

The authors thank Roberto Osellame and Jian-Wei Pan for helpful discussions. X.-M. J. acknowledges support from the National Young 1000 Talents Plan.

See Supplement 1 for supporting content.

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