Fig. 1. Forming artificial lattices. (a) Physical states labeled by consecutive integers; (b) introducing nearest-neighbor coupling between physical states to create a one-dimensional system; (c) introducing a special long-range coupling between physical states to create a two-dimensional system^[56-57]

Fig. 2. Using different orbital angular momenta to create a synthetic dimension. (a) A degenerate cavity can be described by a synthetic lattice along the direction of the angular momenta^[58]; (b) a 2D degenerate cavity array^[61]; (c) physical implementation of degenerate cavity^[61]

Fig. 3. Using orbital angular momenta to realize quantum walks. (a) Scheme of one-dimensional quantum walk^[65]; (b) experimental scheme of 2D quantum walks ^[66]

Fig. 4. Using ring resonators to create the synthetic frequency dimension. (a) A ring resonator dynamically modulated by an electro-optic modulator (EOM) can be described by a tight-binding model of a photon along a one-dimensional lattice in the synthetic frequency dimension^[67]; (b) a one-dimensional array of ring resonators can be mapped into a tight-binding model in two-dimension, with the extra dimension being the synthetic frequency dimension^[

Fig. 5. Using ring resonators to create higher synthetic space. (a) A synthetic 3D lattice realized using 2D honeycomb array of ring resonators, each ring is subjected to an index modulation which generates a synthetic frequency dimension^[70]; (b) a sketch of an effective 4D lattice composed of a 3D resonator lattice and one synthetic frequency dimension^[77]

Fig. 6. Forming synthetic lattices with finite elements. (a) Two-ring resonators A and B, each ring has a phase modulator, in between, there is an auxiliary ring C with two phase modulators^[57]; (b) a honeycomb lattice on a cylindrical surface with a twisted boundary condition is created when considering long-range coupling in the system in Fig. (a)^[57]; (c) a single optical cavity in

Fig. 7. Using waveguides to create synthetic frequency dimension. (a) Schematic of LiNbO₃ phase modulator^[88]; (b) dispersion curve of optical modes and photonic intraband transitions in the vicinity of ω₀ achieve a one-dimensional lattice formed from waveguide modes at different frequencies^[88]; (c) dynamically adjustable 2D brick-wall lattice waveguide array and an equivalen

Fig. 8. Using nonlinear optical effects to create synthetic frequency dimension. (a) Four-wave mixing Bragg scattering in third-order nonlinear waveguide^[98]; (b) with multiple pumps present (up), evolution of the signal (down) is governed by multiple hopping coefficients across synthetic frequency lattice^[98]; (c) nonlinear Brillouin scattering in a microcavity[101

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Fig. 9. Synthetic modal dimension. (a) Coupling diagram of x-θ plane (left) and corresponding x-w (OAM) plane (right)^[103]; (b) one-dimensional lattice with a spectrum of eigenmodes with equally spaced propagation constants^[104]; (c) oscillating the lattice in the longitudinal direction causes each eigenmode to couple to its nearest neighbours, forming a lattice of coupled mo

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Fig. 10. Mapping multi-dimensional networks to 1D lattices. (a) 1D transformation structure of high-dimensional network based on similarity of Hamiltonian^[106]; (b) mapping high-dimensional networks to 1D lattices^[107]

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Fig. 11. Using multiple pulses to create synthetic photonic lattice. (a) Two fiber loops with slightly different lengths connected by a 50/50 coupler; (b) an equivalent lattice network that describes a one-dimensional synthetic lattice (n) evolves along the time axis (m)

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Fig. 12. Using multiple pulses to realize 2D quantum walks. (a) Experimental setup of a 2D quantum walk^[118]; (b) experimental implementation of 2D quantum walks with time-bin encoding and the mapping between the time-bin sequence and the 2D spatial lattice^[119]

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Fig. 13. Adiabatic pumping. (a) Sketch of setup for observation of adiabatic pumping via topologically protected boundary states^[136] ; (b) illustration of the 2D array of waveguides with z-dependent spacing ^[137]

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Fig. 14. Systems with parameter dependency. (a) 1D photonic crystal with each unit cell including four layers where the thickness of each layer depends on parameters p and q^[140]; (b) the design of photonic crystal with a PT-symmetric seven-layer unit cell, n and κ represent real and imaginary part of refractive index of each layer, respectively^[142]; (c) schematic of the unit

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