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Abstract
Topological materials are capable of inherently robust transport and propagation of physical fields against disorder and perturbations, holding the promise of revolutionary technologies in a wide spectrum. Higher-order topological insulators are recently predicted as topological phases beyond the standard bulk-edge correspondence principle, however, their topological invariants have been proven very challenging to observe, even not possible yet by indirect ways. Here, we demonstrate theoretically and experimentally that the topological invariants in two-dimensional systems can be directly revealed in real space by measuring single-photon bulk dynamics. By freely writing photonic lattices with femtosecond laser, we construct and identify the predicted second-order topological insulators, as well as first-order topological insulators with fractional topological winding number. Furthermore, we show that the accumulation and statistics on individual single-particle registrations can eventually lead to the same results of light waves, despite the fact that the development of topological physics was originally based on wave theories, sharing the same spirit of wave-particle nature in quantum mechanics. Our results offer a direct fashion of observing topological phases in two-dimensional systems and may inspire topologically protected artificial devices in high-order topology, high-dimension and quantum regime.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Topological states, characterized by topological invariants associated with the intrinsic energy-momentum band structure, are deepening our understanding on properties of matter and inspiring artificial devices [1–3]. Due to the scalability and flexibility of synthetic photonic systems, topological photonics offers new manners for probing topological states [4–17]. Predicted by the standard bulk-edge correspondence principle [18–20], first-order topological insulators possess backscattering-immune edge modes with fascinating capability of resisting disorder and guiding unidirectional propagation. In contrast, second-order topological insulator (SOTI), originally constructed by introduction of negative coupling and magnetic flux [21], goes beyond the standard bulk-edge correspondence with gapped 1D edge states and especially gapless 0D corner states in two-dimensional (2D) systems.
Very recently, SOTI related to the quadruple moments is experimentally demonstrated in mechanical [22], microwave [23], electronic [24] and photonic platform [25] and SOTI related to the dipolar moments is realized in sonic system [26–28] and dielectric photonic crystals [29,30], especially in breathing kagome lattice with zero dipole moments [31,32], which reveals the fascinating topologically protected lower-dimensional edge states connecting topology with the electric polarization in 2D systems. Thus, the introduction and realization of the SOTI push the boundaries of topological physics at fundamental and device levels [21–39]. Although the unconventional confined 0D topological states pinned at the corners in 2D systems linking the manifestation of second-order topology are extensively investigated [22–32], the topological invariants of SOTI phases defined on the bulk states have not yet been observed directly. The bulk topology of SOTI can be revealed by quantized bulk polarization and Wannier centers, which is defined in momentum space [21] and therefore experimentally challenging to be probed.
In this paper, we propose and experimentally demonstrate the first direct observation of SOTI phases in 2D system in real space by measuring bulk photon dynamics. We construct SOTI with all four topological insulator phases in a silica chip by using femtosecond laser writing techniques [40,41], and we distinctly identify their 2D topological winding numbers by dynamical detection of photon evolution in bulk states. Furthermore, we realize all the measurements in single-photon regime by interfacing the photonic lattices with single-photon source and imaging systems, which may facilitate explorations on the emerging field of ‘quantum topological photonics’, a crossover between topological photonics and quantum information [42–44].
2. Model of 2D topological insulator
We integrate the topological insulators into the silica chip with varieties of modulation parameters as shown in Fig.1a-b. Specifically, using the femtosecond laser direct writing techniques which allows freely writing photonic lattices [10,45–47], we fabricate 84 lattices and each contains $10 \times 10$ waveguides in a photonic chip. The cross-section of a single lattice is shown in Fig.1b. The coordinate of each unit cell in the 2D photonic lattice is labelled as $(x,y)$. Each unit cell has four sites $(a,b,c,d)$. The Hamiltonian of this model can be written as
The Hamiltonian of the system in $x$ or $y$ direction are both protected by the chiral symmetry [29,30], i.e.,
where the chiral symmetry operators are $\Gamma _x=\sigma _3\otimes I$ and $\Gamma _y=I\otimes \sigma _3$. Such chiral symmetry feature implies that we can separately introduce two topological winding numbers where $n_{s}=(h_{s1},h_{s2})/E_{s}$ ($s=x,y$), $E_{s}=\sqrt {h_{s1}^{2}+h_{s2}^{2}}$. Based on two one-dimensional topological winding numbers, we can define a generalized 2D topological winding number $\nu$ to characterize the SOTI phases of the 2D lattice, i.e., $\nu =(\nu _x+\nu _y)/{2}$.As shown in Fig. 1(c), depending on signs of $\delta t_x$ and $\delta t_y$, the system supports four different 2D topological insulator phases. For convenience, the parametric phase spaces for $\delta t_x<0$, $\delta t_y>0$ and $\delta t_x>0$, $\delta t_y>0$ and $\delta t_x<0$, $\delta t_y<0$ and $\delta t_x>0$, $\delta t_y<0$ are denoted by Region I, Region II, Region III and Region IV, respectively. In Region II, the system is in SOTI phase characterized by an integer 2D winding number of 1 (see Supplement 1, part $1$ for the identification of SOTI phase). In contrast, in Region III, the system is in the topological trivial insulator phase with a trivial 2D winding number of 0. We find that our system also supports first-order topological insulator phase with fractional 2D winding number of $\frac {1}{2}$ in Region I and Region IV. The combined eigenmode local density of states (LDOS) of edge modes (see Supplement 1, part $2$) for four phases implies the existence of the symmetry-protected states on the boundaries of 2D lattices which are the usual hallmark of topological insulator phases with non-zero winding number [2,3].
Fig. 1. Construction and modeling of 2D topological insulators (a) Schematic diagram of integrated 2D waveguide lattices on silica chip. Single photons are injected into the entry waveguide in the central unit cell along the z(time) axis shown by black arrow. (b) Microscopic image of cross section at the output facet of lattice with designed pitches. The unit cell labels x, y represent two dimensions of the topological insulator. Each unit cell marked by black dash box consists of four sites $a_{x,y}$, $b_{x,y}$, $c_{x,y}$, $d_{x,y}$ with intra-cell coupling $J_{1x}$, $J_{1y}$ and inter-cell coupling $J_{2x}$, $J_{2y}$. (c) Phase diagram classifying topological insulators. SOTI phase, first-order topological insulator phase and topological trivial insulator phase are marked by orange, blue and green, respectively. Inset(i)-(iv) LDOS of edge modes. Combined LDOS of topologically protected edge modes for $\nu$=0.5 (i), for $\nu$=1 (ii), for $\nu$=0 (iii), for $\nu$=0.5 (iv).
3. Detection of 2D topological invariant
Importantly, the 2D topological winding number, being in charge of characterizing the phases of topological insulators, can be directly detected via single-photon dynamics in the 2D photonic lattice. We can inject heralded single-photon into the middle waveguide to excite the bulk state of the waveguide lattice, as the initial state is $|\psi (z=0)\rangle$. With the evolution of light over a distance $z$ in the waveguide lattice, we measure the photon population difference center (PPDC) associated with the final output probability distribution of photons. The PPDC operator is introduced as $\hat {P}_d=\hat {x}\Gamma _x+\hat {y}\Gamma _y$ and the PPDC value can be measured through $\bar {P}_d(z)=\langle \psi (z=0)|e^{iHz}\hat {P}_d e^{-iHz}|\psi (z=0)\rangle$. If the evolution distance is long enough, we can find that the generalized 2D topological winding number is equal to the evolution-distance-averaged PPDC,
It means that the oscillation center of $\bar {P}_{d}(z)$ in the 2D photonic lattice can directly give the value of the generalized 2D topological winding number (see Supplement 1, part $3$).We simulate the single-photon dynamical evolution in the lattice with different topological insulator phases and calculate the corresponding evolution-distance-averaged PPDC $\bar {P}_{d}(z)$, as shown in Fig. 2 (seeSupplement 1, parts $4-5$ for the influence of average PPDC at finite distances with different sizes or disorder). For topological trivial insulator phase, first-order topological insulator phase and SOTI phase, the distance-averaged of $\bar {P}_{d}(z)$ is $-0.0334$, $0.4973$ and $1.0282$ respectively, which precisely reproduces the 2D topological winding number.
Fig. 2. Theoretical results of PPDC in 2D systems. Theoretical results of evolution-distance-averaged PPDC in topological trivial phase (a), in SOTI phase (b), in first-order topological insulator phase (c,d).
In our experiment, we map the theoretical parameters into a silica chip. The z-axis of the lattices takes the role of a temporal coordinate. We prototype the 2D lattices with different propagation distances varying from 10 to 40 mm with step of 1.5 mm, and each lattice consists of $10 \times 10$ waveguides. We prepare heralded single photons from a down-conversion source (see Supplement 1, part $8$), and excite the bulk state of the 2D topological lattices via the injection of heralded single photons into the entry waveguide in the central unit cell (Fig. 1(a)). The resulting probability distribution of single photons is imaged by an Intensified Charge-Coupled Device (ICCD) camera at the output facet.
For the phases in Region III and Region II, the measured values of $\bar {P}_{d}(z)$ fluctuate but center at $0.055 \pm 0.007$ and $1.009 \pm 0.005$ respectively, corresponding to the 2D topological winding number of 0 and 1, as shown in Fig. 3(a-b). For the phases in Region I and Region IV, the measured values of $\bar {P}_{d}(z)$ keep oscillating around $0.532 \pm 0.006$ and $0.494 \pm 0.008$ respectively, which indicates that the system is in first-order topological insulator phases with $\nu = 0.5$, as shown in Fig. 3(c-d). Although there are irregular fluctuations (see Supplement 1, part $6$), we successfully realize the direct detection of the 2D topological winding number via the single-photon bulk dynamics.
Fig. 3. Experimental results of PPDC in 2D systems. The measured evolution-distance-averaged PPDCs are $0.055 \pm 0.007$ and $1.009 \pm 0.005$ in trivial phase and SOTI phase, respectively. The measured evolution-distance-averaged PPDCs are $0.532 \pm 0.006$ and $0.494 \pm 0.008$ with respect to first-order topological insulator phase. The measured evolution-distance-averaged PPDC ($\bar {P}_{d}$) is represented by dark red line. (a) $J_{1x(1y)}=0.3,J_{2x(2y)}=0.1$ (b) $J_{1x(1y)}=0.1,J_{2x(2y)}=0.3$ (c) $J_{1x}=0.3,J_{2x}=0.1,J_{1y}=0.1,J_{2y}=0.3$ (d) $J_{1x}=0.1,J_{2x}=0.3,J_{1y}=0.3,J_{2y}=0.1$. The unit of the coupling constants is $[1/mm]$.
4. Dynamical observation of the edge modes
To further investigate the topological phases of the lattices, we focus on the edge modes according to the bulk-edge correspondence [18–20]. The combined LDOS shown in Fig. 1(c) confirms the existence of edge modes, and especially shows that four phases support four types of edge modes with different spatial distribution. The topological insulator phase with second-order topology in Region II supports a pair of edge modes in both left (right) and top (bottom) edge of the system along both x and y direction, which is distinctly different from only the bulk modes existing in trivial phase in Region III. The edge modes only emerge on the boundaries along $x (y)$ direction in first-order topological insulator phase in Region IV (I). The topological insulator also satisfies mirror-inversion symmetries in which a pair of topologically protected chiral edge modes along the same direction are preserved in the same fashion.
We experimentally excite the edges of the constructed $10 \times 10$ lattices by injecting the photons into entry waveguide and measure output pattern along the z direction to dynamically detect the edge modes for both directions (see Visualization 1,Visualization 2,Visualization 3,Visualization 4). In order to quantify the localization of the edge modes, we introduce the generalized return probability $\xi = (\sum _{i=n}^{n+\Delta }|\psi _i|^{2})/(\sum _{i=1}^{N}|\psi _i|^{2})$, where the $N$ is the total site number of the lattice, $n$ is the site along the edge of the lattice, and $\Delta$ is the index interval between the neighboring sites along the same edge [49]. As shown in Fig. 4(a), in Region III with trivial phase, the return probability of edge for two directions continuously decreases and finally approaches zero. In contrast, in Region II with SOTI phase, there are a pair of protected edge modes for both two directions with stably high return probability (see Fig. 4(b)). The steadily high value of $\xi$ for top edge and low value of $\xi$ for left edge in Fig. 4(c) can be distinctly differentiated from the high $\xi$ for left edge and low $\xi$ for top edge in Fig. 4(d), which matches the edge modes emerged only in $x$ and $y$ direction for first-order topology.
Fig. 4. Dynamical observation of the edge modes in 2D photonic lattices. Results of left and top edge in topological trivial phase (a), in SOTI phase (b), in first-order topological insulator phase (c,d). Images of measured single-photon probability distribution in both left and top edge for $z = 40$ $mm$ are indicated by the shade shown in the inset of (a-d). The white dash circles indicate the position where the single photons inject. The color bar normalizes the distribution with the maximal value of each measured distributions. The coupling constants in each region are the same as those in Fig. 3.
The topological protections are manifested better when evolution distance goes larger, where the return probabilities in edge modes keep stably much higher than the ones of unprotected modes. At the limit of evolution distances explored in our experiment, $z=40$ $mm$, we measure the photon distribution at the output facet to visualize the performance of the topological protection by heralded single-photon imaging (see the inset of Fig. 4(a-d)).
5. Conclusion and outlook
It should be noted that the single photons are generated by heralding one photon from a pair so that they are always injected into a lattice one by one in a particle-like behavior. By accumulating enough such individual single-particle events, we are able to see a fixed pattern [46,47]. The resulting statistics is consistent with the results obtained by the waves of light, despite the fact that the development of topological physics was originally based on wave theories. The consistence recalls the famous single-electron double-slit experiment and shares the same spirit of wave-particle nature in quantum mechanics, confirming an elegant compatibility between topological physics and quantum theories. Our work based on single-photon, single-particle level of light, is compatible with the coherent laser, which shows the universality of enabling the enormous potential and extension to classical light.
A prompt application of ‘quantum topological photonics’ is to protect quantum states against diffusion-induced loss and decoherence [42–44]. For example, single photons can be well trapped in one edge mode and quantum superposition states can also be well-preserved among a few edge modes of SOTI in 2D systems, being immune to disorder and perturbations. Thus, the built ability of directly identifying 2D topological invariants in real space is the key for developing topologically protected artificial materials and devices in the future [39].
It is of great interest to generalize our approach to a few crucial issues, including direct observation of topological phase transitions in 2D lattices, direct observation of topology in higher-dimensional systems via single-photon dynamics, real-space measurement of higher-order topological insulators, and revealing the complex Chern number in non-hermitian systems [50,51].
Funding
National Key Research and Development Program of China (2016YFA0301803, 2017YFA0303700, 2017YFA0304203, 2019YFA0308700); National Natural Science Foundation of China (No. 11690033, No. 11761141014, No. 61734005, No. 12074234); Science and Technology Commission of Shanghai Municipality (Grant No.2019SHZDZX01, Grants No. 17JC1400403); Shanghai Municipal Education Commission (2017-01-07-00-02-E00049); Changjiang Scholar Program of Chinese Ministry of Education (IRT17R70); Shanxi 1331KSC; 111 Project (D18001).
Acknowledgments
The authors thank Jian-Wei Pan for helpful discussion. X.-M.J. acknowledges the support from Zhiyuan Innovative Research Center of Shanghai Jiao Tong University, and the additional support from a Shanghai talent program.
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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