Fig. 1. Schematic diagram of recent developments in synthetic dimensions. Adapted from Refs. [16,21,23,25,31,32,37–39" target="_self" style="display: inline;">–39,43,53].

Fig. 2. Construction of the synthetic frequency dimension. (a) Electrons in the average Coulomb potential of an ionic crystal. (b) The discrete frequency modes. (c) The one-dimensional frequency lattice constructed by connecting discrete frequency modes. (d) A ring resonator under dynamic modulation. (e) The frequency conversion between resonant modes inside the ring induced by dynamic modulation.

Fig. 3. Experimental platforms to realize synthetic frequency dimensions. (a) The electro-optically modulated fiber ring. Adapted from Ref. [88]. (b) The electro-optically modulated on-chip waveguide. Adapted from Ref. [102]. (c) The traveling wave modulated waveguide. Adapted from Ref. [118]. (d) Four-wave mixing in a nonlinear medium. Adapted from Ref. [133].

Fig. 4. (a) The static band structure of a 1D frequency lattice. Adapted from Ref. [94]. (b) The dynamic band structure of a 1D frequency lattice. Adapted from Ref. [88]. (c) The chiral band structure of the quantum Hall ladder. Adapted from Ref. [38]. (d) The SSH model in the synthetic frequency dimension. Adapted from Ref. [99]. (e) The topological winding of a non-Hermitian band. Adapted from Ref. [40]. (f) The topological complex-energy braiding of non-Hermitian bands. Adapted from Ref. [41]. (g) The dissipative solitons in a synthetic dimension. Adapted from Ref. [161]. (h) The quantum correlation in synthetic space. Adapted from Ref. [111].

Fig. 5. (a) A main cavity with an auxiliary cavity equipped with two spatial light modulators on the two arms. (b) The coupling among OAM modes forms a tight-binding lattice. (c) The SSH lattice constructed using OAM modes in a single degenerate optical cavity. Adapted from Ref. [53]. (d) A 2D synthetic lattice with one dimension along the OAM degree of freedom and the other dimension along the spatial degree of freedom. The band structure displays a Hofstadter-Harper butterfly pattern. Adapted from Ref. [208]. (e) The construction of a 3D synthetic lattice with one OAM dimension and two spatial dimensions. Adapted from Ref. [211].

Fig. 6. (a) Left panel: the cavity model used to construct the synthetic OAM lattice. Right panel: the topological invariant obtained from the experimental data. Adapted from Ref. [216]. (b) Left panel: a synthetic OAM lattice with loss included. Right panel: the exceptional topological band. Adapted from Ref. [217]. (c) Detection of the topological invariant in a quantum walk along the synthetic OAM dimension. Adapted from Ref. [221]. (d) Left panel: the construction of the 2D synthetic lattice with x dimension being the OAM state and y dimension being the spatial position. Right panel: the topologically protected bound state with vanishing Chern numbers. Adapted from Ref. [214].

Fig. 7. (a) The schematic eigenstate distribution of Jx lattice that is used to construct the modal dimension. (b) Upper panel: the unequally spaced arrangement of 1D waveguides that can form the equally spaced (in propagation constant) modal distribution in (a). Lower panel: the helical waveguide with its oscillation period consistent with the eigenvalue interval can create the hopping between nearby modes. (c) Upper panel: the synthetic 2D lattice with one spatial dimension and one modal dimension. Lower panel: the arrangement of arrays of waveguides in 2D space. (d) The schematic phase offset between nearby arrays of waveguide. (e) Upper panel: the experimental results of light transporting in the real spatial dimension. Lower panel: the transport of light in the synthetic modal dimension. (a)–(c) Adapted from Ref. [12]. (d), (e) Adapted from Ref. [37].

Fig. 8. (a) The 2D lattice in the spatial space, where each site is a helical waveguide. (b) The synthetic 3D lattice with two spatial dimensions and one modal dimension. (c) The schematic illustration of the screw dislocation in synthetic space including the modal dimension. (a)–(c) Adapted from Ref. [44].

Fig. 9. (a) Direct laser-written waveguides in fused silica. (b) Experimental setup for using the polarization degree of freedom to verify the Hong-Ou-Mandel effect. (a), (b) Adapted from Ref. [227].

Fig. 10. (a) Synthetic time dimension created from a main loop with short and long delay lines to couple pulses. Upper panel: the main loop for time-multiplexing, which incorporates green and purple delay lines to induce the hopping between nearest-neighbor pulses. Lower panel: the lattice network in synthetic time dimension. (b) Synthetic time dimension created from two loops of unequal lengths. Upper panel: the short loop and long loop are connected by a 50/50 coupler. A phase modulator can impose phases on pulses in the long loop. Lower panel: the lattice network in synthetic time dimension, where n constitutes the 1D synthetic lattice and m denotes the evolution time.

Fig. 11. (a) A single-loop system with three delay lines to design the Ising model. (b) The output pulse train, where the pulses are labeled by OPO1 to OPO4. (c) The couplings between pulse slots provided by three delay lines. (d) The diagram illustrating the search for the ground state, where the OPO gain reaching the minimum energy of the Ising problem triggers a collapse of the OPOs to the ground state distribution of the spins. (a)–(d) Adapted from Ref. [258]. (e) A main cavity with delay lines is used to construct the synthetic time lattice with dissipative coupling. The optical field distribution in the synthetic lattice of the 1D SSH model (f), (g) and 2D HH model (h). (e)–(h) Adapted from Ref. [244]. (i) The CW and CCW modes in the storage ring constitute the photonic qubits. The optical switches can guide the photonic qubits to scatter with an atom that is controlled by a laser. (j) The atom has a Λ-shaped three-level energy structure. (k) The transformation of the photonic qubit depicted by the Bloch sphere. (i)–(k) Adapted from Ref. [241].

Fig. 12. (a) The non-Hermitian skin effect, Anderson localization, and disorder-induced topological phase transitions in the synthetic time lattice. Adapted from Ref. [246]. (b) The correspondence between self-acceleration and the spectral geometry encircled by the complex eigenenergy. Adapted from Ref. [251]. (c) The non-Hermitian skin effect, the magnetic suppression phenomena, and the Floquet topological edge modes in the time-multiplexed network including the non-Hermicity and the magnetic flux. Adapted from Ref. [249].

Fig. 13. (a) Topological funneling of light in the synthetic time lattice. Adapt from Ref. [39]. (b) The Floquet Hofstadter butterfly pattern of the AAH model in the synthetic time lattice. (c) The topological triple phase transition by controlling one single parameter, which is the coupling coefficient β. (b), (c) Adapted from [45].

Fig. 14. (a) The shape-preserving beam transmission and non-Hermitian-induced transparency phenomena in the synthetic time lattice. Adapted from Ref. [247]. (b) The time reflection and refraction phenomena at the interface formed by the mass-flipping temporal boundary. Adapted from Ref. [51]. (c) The exponential localization at time step m=110 and the mean value of the second moment in the non-Hermitian Anderson model. Adapted from Ref. [242].

Fig. 15. (a) The optical platform including two fiber rings and various components. (b) The time-synthetic mesh lattice constructed by the two-loop setup. (c) Entropy-energy diagram based on the framework of thermodynamics. The eigen-spectrum under the positive temperature (d) and negative temperature (e). (f) The isentropic compression and expansion under the negative temperature condition. (g) The optical Joule expansion in the negative temperature regime. (a)–(g) Adapted from Ref. [50].

Fig. 16. (a) The classical random walk and the quantum random walk in the 1D lattice with annihilation traps. The survival probability for the classical random walk (b) and the quantum random walk (c). (a)–(c) Adapted from Ref. [316].

Fig. 17. (a) The proposed experimental layout to construct the synthetic dimension based on the intrinsic states of cold atoms. (b) The three magnetic sublevels of the F=1 ground state. (c) The 2D lattice with a combination of the 1D optical lattice and the synthetic spin lattice. (a)–(c) Adapted from Ref. [83]. (d) The excitation of the Rydberg states through two-photon transitions. Adapted from Ref. [360]. (e) The connection between Rydberg levels through the employment of millimeter waves on the atomic system. Adapted from Ref. [364]. (f) A schematic diagram depicting how to form a momentum state lattice for ultracold atoms by driving with a pair of counter-propagating laser beams. (g) The coupling between momentum lattice sites is achieved by two-photon Bragg transitions. (f), (g) Adapted from Ref. [368].

Fig. 18. Synthetic dimensions with low-lying atomic internal states. (a) In few-leg Hofstadter lattices based on Raman-coupled hyperfine states, researchers have observed the skipping orbits associated with topological boundary states. Adapted from Ref. [24] (cf. also Ref. [23]). (b) In larger-spin magnetic atoms such as dysprosium, one can similarly explore both bulk and boundary dynamics in spin-orbit-coupled gases. Adapted from Ref. [342]. (c) Control of atomic interactions in few-state systems has recently enabled the exploration of many-body effects, namely, the observation of universal Hall response in two-leg flux ladders. Adapted from Ref. [410]. (d) The moiré pattern and rich phase diagram resulting from a synthetic twisted bilayer structure. Adapted from Ref. [48].

Fig. 19. Synthetic dimensions in Rydberg atoms. (a) The excitation spectra and the state decomposition weights in an SSH lattice formed by microwave-coupled Rydberg levels. Adapted from Ref. [364]. (b) The topological winding number extracted from the dynamics of Rydberg-level populations in an SSH lattice. Adapted from Ref. [360]. (c) The influence of strong dipolar interactions leading to inhibited dynamics of Rydberg electrons in a synthetic flux plaquette. Adapted from Ref. [363]. (d) The observation of Aharonov-Bohm (AB) caging and its breakdown due to strong dipolar interactions in a twisted diamond lattice. Adapted from Ref. [367].

Fig. 20. Synthetic lattices based on laser-coupled atomic momentum states. (a) The realization of disorder-induced topology in momentum lattices. Adapted from Ref. [32]. (b) The observation of time reflection and refraction based on engineering a temporal boundary in momentum lattices. Adapted from Ref. [451]. (c) The transition between flat-band localization and Anderson localization in a synthetic momentum-state Tasaki lattice. Adapted from Ref. [453]. (d) The observation of chiral dynamics in a synthetic momentum lattice with non-Abelian gauge fields. Adapted from Ref. [382]. (e) The observation of nonlinear self-trapping when the mean-field interaction shift U exceeds the tunneling bandwidth 4J. Adapted from Ref. [392]. (f) The observation of flux-dependent nonlinear self-trapping in a momentum-state flux ladder. Adapted from Ref. [456].

Fig. 21. (a) The valley Hall effect and the chiral edge states of the Fock-state lattice. Adapted from Ref. [480]. (b) The valley and anomalous Hall effects in a strained honeycomb lattice. Adapted from Ref. [482]. (c) The Haldane chiral edge currents and valley Hall effect in the Fock-state lattice. Adapted from Ref. [46].

Fig. 22. (a) The averaged reflectivity and the normalized probability versus ϕ by selectively probing the flat or chiral band. Adapted from Ref. [498]. (b) The phase transition in the honeycomb superradiance lattice. Adapted from Ref. [500]. (c) The dynamic localization and delocalization in the Floquet superradiance lattice. Adapted from Ref. [503].

Fig. 23. (a) The 3D surface states for the circuit implementation of a 4D topological insulator. Adapted from Ref. [517]. (b) The surface states of the 3D Weyl circuit. Adapted from Ref. [518]. (c) The corner states of the quadrupole insulators. Adapted from Ref. [524]. (d) The enhanced third-order harmonic generation in the nonlinear SSH lattice. Adapted from Ref. [525].

Fig. 24. (a) Schematics of topological adiabatic pumping in waveguide lattices. (b) Mode evolutions as the function of the pumping parameter ϕ. The inset pictures illustrate the mode profile during the evolution process. (a), (b) Adapted from Ref. [555]. (c) Band diagram of a topological pumping in the 2D lattice manifesting the 4D integer quantum Hall effect. Adapted from Ref. [31]. (d) Nonlinear Thouless pumps with quantized soliton transport. Adapted from Ref. [42]. (e) Asymmetric topological pumps with nonparaxial conditions. Adapted from Ref. [561]. (f) Non-Abelian Thouless pumps. Adapted from Ref. [564].

Fig. 25. (a) Optical Weyl points in a 1D photonic crystals with two structural parameters as additional dimensions. Adapted from Ref. [569]. (b) Charge-2 Dirac points in a 1D optical superlattice with two structural parameters forming two synthetic dimensions. Adapted from Ref. [570]. (c) Fermi arc reconstruction in a 1D dielectric trilayer grating with the relative displacements between adjacent layers as two synthetic momenta. Adapted from Ref. [572]. (d) Type-II Weyl points and interface states in a 1D optical SWG waveguide lattice, where two structural parameters controlling the coupling and on-site energy serve as parameter dimensions. Adapted from Ref. [553]. (e) One-way fiber in a 3D Weyl photonic crystal using the angle of helical modulation to construct a 4D synthetic space. Adapted from Ref. [575]. (f) 4D second Chern crystal realized in 2D photonic crystals with two extra synthetic translation dimensions. Adapted from Ref. [576]. (g) 5D Yang monopoles in 3D metamaterials, which includes two bi-anisotropy material parameters as synthetic dimensions. Adapted from Ref. [43].

Fig. 26. (a) Exceptional points formed in parameter spaces. Adapted from Ref. [153]. (b) Realization of EP in different photonic systems. Top panel: in coupled waveguides. Adapted from Ref. [578]. Bottom panel: in micro resonators. Adapted from Ref. [301]. (c) Encircling the EP with different directions gives rise to asymmetric mode switching. Adapted from Ref. [25]. (d) Parameter synthetic dimension realized in PT-symmetric photonic crystal systems. Adapted from Ref. [590]. (e) Realization of Weyl exceptional ring and non-Hermitian Weyl interface modes in synthetic parameter waveguide lattices. Adapted from Ref. [591]. (f) Exceptional surfaces in parameter synthetic dimensions with magnon polaritons. Adapted from Ref. [592].

Fig. 27. (a) Forming synthetic dimension utilizing the translational degree of freedom of a 2D photonic crystal’s unit cell. Adapted from Ref. [593]. (b) A topological rainbow concentrator thus can be realized and demonstrated in integrated nanophotonic chips. Adapted from Ref. [595]. (c) Robust and broadband optical coupler using a topological pump. Adapted from Ref. [603]. (d) A topological splitter for high-visibility quantum interference of single-photon states. Adapted from Ref. [604]. (e) A topological lasing in a 1D coupled ring resonator array that can be mapped to a 2D non-Hermitian Chern insulator with synthetic dimension. Adapted from Ref. [605].

Fig. 28. (a) Topological pump with ultracold bosonic atoms in a 2D angled optical superlattice. Adapted from Ref. [30]. (b) Non-Abelian Thouless pumping with structural modulated acoustic waveguides. Adapted from Ref. [566]. (c) Weyl points in a 1D sonic crystal. Adapted from Ref. [612]. (d) Realizing topological pumping in elastic waves. Adapted from Ref. [618].