Dimensionality is one of the fundamental properties of our physical world; systems of high spatial dimensions can exhibit novel and exotic phenomena not found in their low-dimensional counterparts. One of the central themes in contemporary research in physics and engineering is to understand and exploit such phenomena unique to high spatial dimensions. A direct observation of these phenomena, however, would require the explicit construction of the underlying high-dimensional system, and therefore, in many cases, is constrained by the complication of the system’s spatial geometry, particularly for systems beyond three spatial dimensions.

Within this context, the concept of the synthetic dimension has been introduced and developed over the past decade. In a recent review led by Luqi Yuan, Avik Dutt, and Bryce Gadway published in Photonics Insights^[1], the authors, including some of the leading scientists in this emerging field, provided an authoritative and comprehensive overview on the theoretical and experimental progress in the field of synthetic dimensions, bringing to the audience a unique perspective that bridges interdisciplinary engagement.

The concept of the synthetic dimension concerns the utilization of a non-spatial degree of freedom of a physical system as an extra dimension. With this perspective, additional dimensions are effectively “synthesized” and appended to existing spatial dimensions in the system, making it possible to study high-dimensional physics on low-dimensional platforms. Moreover, the connectivity in the synthetic dimensions can be designed with unparalleled tunability, flexibility, and reconfigurability that would otherwise be challenging to realize in real spatial dimensions. Using the synthetic dimension, researchers have demonstrated simulations of classical and quantum physics in high dimensions and complicated models using relatively simple experimental setups, as well as technological applications in the manipulation of wave propagation and quantum information processing.

In their comprehensive review^[1], the authors summarize two categories of implementations of the synthetic dimension (Fig. 1): using discrete physical modes of photons and atoms, where each mode is considered as a spatial lattice site, and using continuous system parameters, where each parameter is considered as a spatial lattice momentum.

In an optical system, the discrete-state degrees of freedom that can be used to create synthetic dimensions include the frequency^[2–4], orbital angular momentum (OAM)^[5], transverse mode^[6], and polarization^[7] of light. Along the frequency axis, different longitudinal modes inside a ring resonator are viewed as spatial lattice sites and coupled via electro-optic modulations. Different OAM states inside a resonator can be viewed as spatial lattice sites and coupled via spatial light modulators. Different transverse modes within a group of evanescently coupled waveguides can be viewed as spatial lattice sites and coupled via geometrical deformations of the waveguides. Finally, orthogonal polarizations of light can be viewed as two lattice sites and coupled via the engineering of the material birefringence. Using these degrees of freedom, there have been successful studies of high-dimensional topological physics, non-Hermitian physics, quantum walks, quantum correlations of photons, and nonlinear effects such as the generation of soliton states and frequency combs.

In addition, the temporal degree of freedom of light can be used to create synthetic dimensions as well^[8]. A group of coupled fiber loops can be formed within which a sequence of light pulses propagates. Different light pulses in the sequence correspond to different spatial lattice sites, and couplings between neighboring sites are created when these pulses interfere at the couplers between the fiber loops, and can be further adjusted by varying the length differences of the loops. Besides the capabilities of demonstrating high-dimensional topological physics and non-Hermitian physics, the synthetic temporal dimension also enables the simulation of thermodynamic processes involving thermalization and negative temperatures^[9].

In atomic systems, the discrete-state degrees of freedom, such as low-energy internal states^[10], Rydberg states^[11], and momentum states^[12], are important ingredients for the synthetic dimension. In these approaches, a spatial array of atoms is typically prepared at first, and synthetic dimensions are created beyond this spatial dimension. One example of the low-energy internal states is the hyperfine states of the atoms induced by Zeeman splitting, which can be coupled via a coherent Raman process induced by external lasers. The hyperfine states of atoms in individual spatial lattice sites can be precisely resolved. Rydberg states of atoms can be coupled via microwaves to form a synthetic dimension. The Rydberg synthetic dimensions have the advantages that the coupling between different Rydberg states can be separately controlled, and that along this synthetic dimension, the hopping only involves the excitation of the atoms rather than the atoms themselves. Alternatively, on the dispersion relationship E(k) of the atoms, different momentum states can be coupled to form a synthetic dimension as well via two-photon Bragg transitions induced by two counter-propagating lasers. Compared with optical synthetic dimensions, importantly, the strong and stable interactions between atoms facilitate the exploration of many-body phenomena in high-dimensional systems. Phenomenal examples include the studies related to the fractional quantum Hall effect in atomic synthetic dimensions^[13].

In our discussions above, the essence of constructing synthetic dimensions from discrete physical modes lies in the design of the coupling connectivity between the modes. An alternative approach is to leverage the degrees of freedom inherently embedded in continuous parameters of the system. Such parameters, including parameters of structural geometries, material properties, or external conditions, can be interpreted as additional “quantum momenta” that complement the spatial quantum momenta (kx, ky, kz), creating a high-dimensional Brillouin zone. This idea is closely connected to the well-studied concept of dimension reduction and dimension extension. In this spirit, there have been demonstrations of adiabatic evolutions and topological charge pumping^[14] by slowly varying the parameters in a photonic waveguide system. On the other hand, if the parameters are kept static, the system represents a projection of a high-dimensional model under this choice of parameters. The high-dimensional model is thus studied one projection at a time. Along this line, researchers have observed three-dimensional Weyl points^[15], five-dimensional Yang monopoles^[16], and non-Hermitian exceptional topologies^[17] in nanophotonic structures.

The synthetic dimension, which interprets non-spatial degrees of freedom as extra dimensions in addition to spatial ones, has proven a powerful toolkit in the exploration of high-dimensional physics on low-dimensional platforms. It makes accessible simulations of nontrivial high-dimensional classical and quantum physics, and also sheds light on potential applications to control particles and waves in unconventional ways using principles from such fundamental physics. For example, miniaturized on-chip devices may benefit from the simplification of the spatial geometries by synthetic dimensions^[18]. In the classical regime, studies of topological effects and non-Hermitian physics in optical synthetic dimensions have found applications in optical communication, sensing, and lasing^[19]. The progress in quantum mechanics and atomic and optical synthesis dimensions has contributed to the studies of quantum computations and quantum networks^[7].

The field of synthetic dimensions is currently at a time of rapid growth and transformative advances. We believe that this comprehensive review will inspire new directions and emerging opportunities across the physical science and engineering communities.