Photonic topological insulator in synthetic dimensions


Topological phases enable protected transport along the edges of materials, offering immunity against scattering from disorder and imperfections. These phases have been demonstrated for electronic systems, electromagnetic waves1,2,3,4,5, cold atoms6,7, acoustics8 and even mechanics9, and their potential applications include spintronics, quantum computing and highly efficient lasers10,11,12. Typically, the model describing topological insulators is a spatial lattice in two or three dimensions. However, topological edge states have also been observed in a lattice with one spatial dimension and one synthetic dimension (corresponding to the spin modes of an ultracold atom13,14,15), and atomic modes have been used as synthetic dimensions to demonstrate lattice models and physical phenomena that are not accessible to experiments in spatial lattices13,16,17. In photonics, topological lattices with synthetic dimensions have been proposed for the study of physical phenomena in high dimensions and interacting photons18,19,20,21,22, but so far photonic topological insulators in synthetic dimensions have not been observed. Here we demonstrate experimentally a photonic topological insulator in synthetic dimensions. We fabricate a photonic lattice in which photons are subjected to an effective magnetic field in a space with one spatial dimension and one synthetic modal dimension. Our scheme supports topological edge states in this spatial-modal lattice, resulting in a robust topological state that extends over the bulk of a two-dimensional real-space lattice. Our system can be used to increase the dimensionality of a photonic lattice and induce long-range coupling by design, leading to lattice models that can be used to study unexplored physical phenomena.

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Fig. 1: Forming a one-dimensional lattice in a synthetic modal dimension.
Fig. 2: Two-dimensional synthetic-space lattice.
Fig. 3: Experimentally observed evolution of a topological edge state in synthetic dimensions and in the trivial synthetic lattice.
Fig. 4: Propagation of a wavepacket along an edge state of synthetic space.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.


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This work was supported by the German–Israeli DIP Program, by an Advanced Grant from the European Research Council and by the Israel Science Foundation. E.L. is grateful for the support of the Israel Academy of Sciences through an Adams Fellowship.

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Nature thanks Ling Lu and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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All authors contributed significantly to this work.

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Correspondence to Mordechai Segev.

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Extended data figures and tables

Extended Data Fig. 1 Increasing the dimensionality of a two-dimensional lattice of waveguides.

a, One-dimensional columns arranged in a two-dimensional lattice. The columns oscillate along the black paths in the direction indicated by the triangles. Red and blue colours indicate different columns in the same unit cell. b, Representation of a in synthetic dimensions. c, Lattice with two oscillation frequencies, oscillating in the x direction according to Δx and with index of refraction changing periodically according to Δn. d, Representation of c in synthetic dimensions.

Extended Data Fig. 2 Light occupancy in the leftmost column when a Gaussian beam is launched from the middle of the lattice.

a, b, Illustrated intensity (colour bar; peak intensity is set to 1) of the initial Gaussian beam launched in real space (a) and synthetic space (b). c, Measured output intensity of the leftmost column for the input shown in a and b. d, Measured interference of the lattice output at the leftmost column with the input beam, showing that all of the seven peaks are in phase (the dashed line represents equal phase), indicating that the light at the leftmost column occupies the lowest spatial mode of the column. e, Measured normalized intensity cross-section (blue line) at the leftmost column of the lattice output, compared with the theoretically calculated (tight-binding) intensity profile of the first mode of the column (orange circles). The sum of peak intensities is normalized to 1.

Extended Data Fig. 3 Light occupancy in the leftmost column when a Gaussian beam is launched from the left side of the lattice.

a, b, Illustrated intensity pattern (colour bar) of the initial Gaussian beam launched in real space (a) and in synthetic space (b). c, Measured intensity distribution at the leftmost column. d, Measured interference pattern at the lattice output at the leftmost column, showing that, unlike in Extended Data Fig. 2d, the seven intensity peaks are not in phase and therefore occupy higher modes of the array. e, Zoom-in on the red rectangle in d, showing that the light peaks are not aligned on the equal-phase line (dashed line). f, Cross-section (blue line) of the intensity pattern at the leftmost column of the lattice output, compared with the calculated intensity profile of the first mode (orange circles) where the sum of peak intensities is normalized to 1. The mismatch shows that the light is occupying modes higher than the first mode.

Extended Data Fig. 4 Floquet spectrum of equation (1).

a, Eigenvalues obtained numerically for a 20 × 20 site, displaying two topological gaps. b, Real-space amplitude distribution of an eigenstate residing in the gap (edge state marked in a). c, The same eigenstate as in b as it appears in synthetic space (unitary transformation of b), showing clearly that the marked state is an edge state. The parameters used are R = 5 μm, Ω = 69.1448 m−1, coupling in y cy = 10.9431 m−1 and k0 = 1.9 × 107 m−1.

Extended Data Fig. 5 Disorder in synthetic space.

a, Typical disorder introduced to the Hamiltonian of a one-dimensional lattice. The Hamiltonian is in real-space matrix representation (the colour bar corresponds to the real part of the value of the matrix element). b, The disorder shown in a after transformation to synthetic space. c, Standard deviation and mean value of the Bott index averaged over 10 random disorder realizations for the regular topological insulator (blue line) and for the topological insulator in synthetic dimensions (red line). The red bracket indicates half of the bandgap. d, e, A disorder (black sites) that is confined spatially in real space is not confined in synthetic space (blue sites).

Extended Data Fig. 6 Exciting the anticlockwise edge state.

a, Initial excitation of the first mode in the rightmost column of the topological lattice. b, Output intensity (colour bar) corresponding to a. c, Direction of propagation of the light in synthetic space. df, The same as ac, but for the trivial lattice. CW, clockwise; ACW, anticlockwise.

Extended Data Fig. 7 Experimental results for a sample fabricated with different parameters.

Illustration of an incident narrow Gaussian beam directed towards the input facet of a synthetic-space topological lattice fabricated with parameters different from those used in the experiments presented in the main text (here, ϕ = π/4, Ω = 91 m−1 and R = 17.5 μm). a, b, Input beam launched at different locations in real space (a) and synthetic space (b). c, d, Experimentally imaged (c) and simulated (d) beam intensity at the output facet of the lattice. e, Synthetic-space description corresponding to the evolution of the wavepacket as it propagates in the lattice.

Extended Data Fig. 8 Stages in the preparation of the experimental model.

a, Horizontal cross-section of the intensity of a propagating mode of a single waveguide (orange line) and the cross-section of the mode in the simulations (blue line). The image on the right is the intensity distribution of the mode in the direction of the cross-section. b, Same as a, but for the vertical cross-section of the propagating mode of a single waveguide. c, Coupling versus distance in the horizontal direction. d, Coupling versus distance in the vertical direction. e, Mode propagation constants of the Jx lattice based on measured experimental parameters (blue circles). The dashed line is a guide for the eye.

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This file contains Supplementary Text regarding analytic derivation of the lattice in synthetic dimensions.

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Lustig, E., Weimann, S., Plotnik, Y. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356–360 (2019).

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