Quantum simulation of novel physical phenomena including topology, non-Hermitian physics, many-body interaction, etc.^[1–3], is one of the important research fields in physics nowadays. There exist numerous platforms in optics and photonics, atoms, molecules, as well as ions, that show various demonstrations of quantum simulations that can be further extended towards quantum computations^[4–7]. However, many of these platforms support the physical dynamics in low dimensions, which is constrained by the arrangement of the hopping between lattice sites and the limited lattice size of the Hamiltonian model due to the complication in the spatial geometry. This has triggered the concept of synthetic dimensions in atomic, molecular, optical, and photonic communities in the past decade.

The synthetic-dimension concept, using the additional degree of freedom in a system to build an extra dimension, is of fundamental difference from the concept of dimension in our common sense. The synthetic dimension specifically refers to the addition of a virtual axis (onto existing spatial axes) along which a physical state can propagate or evolve, so as to increase the effective dimensionality of a Hamiltonian with an extra effective dimension. In other words, the synthetic dimension plays a similar role to a spatial dimension on which the physical state can diffuse. Therefore, as people realize nowadays, there are two major categories for constructing synthetic dimensions^[8–53] (see Fig. 1).

The first idea to construct the synthetic dimension is to use the discrete physical modes of light or atoms and design the connection between these modes. The discrete modes form the necessary virtual dimension while the wave packet of the system can transport along these discrete modes through the connectivity, i.e., the energy exchange between these discrete modes follows the designed connections, similar to particle movement or wave propagation in a real spatial dimension. One can then build a Hamiltonian on the basis of the virtual dimension constructed by the connection between discrete physical states, namely, the synthetic dimension. Importantly, the connectivity in the synthetic dimension can be made to have tunable and exotic attributes that are hard to realize in a real spatial dimension. From Sec. 2 to Sec. 5, we will show recent efforts in constructing synthetic dimensions following this idea, where various degrees of freedom including frequency, orbital angular momentum, polarization, time of light, as well as intrinsic atomic states and momentum states of atomic lattices are used with the proper connectivity built for constructing numerous Hamiltonian models. Relevant physical phenomena in quantum simulations are also discussed, with possible applications in both photonics and atomic physics.

The second idea to construct the synthetic dimension is to use a system parameter in a geometric structure. This idea, relating to well-studied concepts of dimensional reduction, is thus distinct from the former approach based on introducing new discrete states that are physically connected. Here, each physical state on a specific value of the parameter refers to the eigenstate of a Hamiltonian of interest at the amplitude of a variable. Usually, the variable is chosen as the momentum reciprocal to a particular spatial dimension in the Hamiltonian, so varying the system parameter is equivalent to tuning the momentum for a Hamiltonian. Therefore, the system parameter can form the parameter synthetic space in this geometric structure without using a particular spatial dimension to form the desired Hamiltonian. In Sec. 6, we discuss how it works in detail and why it can provide an efficient way of simulating high-dimensional physics in lower-dimensional geometric structures. Representative works are also shown to understand the essence of the parameter’s synthetic dimension.

The field of synthetic dimensions has grown rapidly in recent years. Given previous reviews and perspectives^[8–19] and breakthrough works^[20–53], the time is ripe for a comprehensive review that not only summarizes the field of synthetic dimensions but also provides an introduction to people who are new to this field. This review serves this purpose by showing the development of synthetic dimensions and discussing various advantages from all examples using different synthetic dimensions, including the potential for exploring high-dimensional physics in lower-dimensional geometric structures and the designable connectivity leading to Hamiltonians supporting effective gauge fluxes, non-Hermiticity, long-range couplings, etc. Therefore, quantum simulations may be performed in high dimensions, or the simulators can be designed with simple geometric configurations but holding complicated Hamiltonians. On the other hand, physical dynamics occurring in synthetic dimensions can be used to manipulate quantum states or optical fields in novel ways, which provides unique physical or optical performance important for future applications.

As probably the simplest way to describe many interesting physical phenomena in condensed matter systems, the tight-binding model has shown great success in describing the electronic dynamics within the one-electron approximation in the solid material^[54]. An electron in a solid experiences interactions with nearby positive charges and other electrons. These interactions can be combined and treated as one average Coulomb potential for each electron as illustratively expressed in Fig. 2(a), where the potential has periodic negative dips at positions of the atomic nuclei labeled by the index m [see Fig. 2(a)]. Classically, the electron is trapped at each potential dip (for example, the mth dip), but quantum mechanics show that the wave packet of the electron can tunnel to nearby potential dips [(m±1)th]. Such a process can be described by the mathematical expression as gmam+1†ameiϕm, where gm is the tunneling strength, am (am+1†) is the annihilation (creation) operator, and ϕm is the hopping phase. Therefore, the tight-binding lattice model is built and the corresponding second-quantized Hamiltonian for the electronic system is^[55]He=∑mVmam†am+∑m(gmam+1†ameiϕm+h.c.),(1)where Vm is the onsite potential at the mth position. This model successfully reveals the property of electronic systems in a concise way; the tight-binding model has been widely extended to the optical system with the aid of coupled-mode theory^[56–64]. The key ingredient here is to determine appropriate discrete optical modes and then find a way to connect them in order^[8,15]. In the spatial domain, optical modes hosted in distinct photonic structures provide discrete modes, such as waveguide modes in waveguide arrays^[65,66] and resonant modes in resonator arrays^[65,67]. With these elements, one can consider the connectivity between them from the spatial overlap between two nearby modes, which can be described by the coupled-mode theory for guided-wave optics^[60,61,68] and coupled optical resonators^[57,59,69], respectively. The construction of synthetic dimensions using discrete optical modes with engineered connectivity follows a similar idea, which utilizes different degrees of freedom of light, including frequency, orbital angular momentum, waveguide supermodes, etc., to build the effective tight-binding model along the synthetic dimension. Nevertheless, different from the construction of lattices using photonic structures in real-space dimensions, synthetic dimensions may provide unique aspects of advantages such as complex and/or long-range hoppings, extra reconfigurable flexibility, and the potential towards higher dimensions (>3), as we discuss below.

This section is organized as follows. We show the use of the frequency degree of freedom to construct the synthetic frequency dimension in Sec. 2.1, where we discuss theoretical approaches and experimental platforms, along with their associated physical phenomena and application perspectives. In Sec. 2.2, we review the process of taking the orbital angular momentum of light to build the synthetic dimension, with both theoretical methods and experimental progress provided. We further talk about the way to use the modal states in waveguides to design the couplings along the synthetic dimension and show the resulting experimental results in Sec. 2.3. We discuss other degrees of freedom of light that may be used to construct the synthetic dimension in photonics in Sec. 2.4.

In this subsection, we review recent progress in constructing the so-called synthetic frequency dimension^[70,71]. We will first outline the theoretical models and then showcase different experimental platforms that have been used to construct relevant synthetic lattices. Various physical phenomena and potential applications in the synthetic frequency dimension are also discussed.

In building the synthetic frequency dimension, one focuses on the construction of a tight-binding model for photons along the frequency axis of light. Here, we introduce the most attractive synthetic model in recent times, where one or more ring resonators are used to implement the frequency lattice. A ring resonator supports a set of discrete resonant modes at different frequencies, which can mimic the model of electrons on a discrete lattice of sites under the average Coulomb potential [see Fig. 2(b)]. These frequency modes are isolated and disconnected from each other if no frequency conversion happens^[72]. To induce the hopping effect or the connectivity between frequency modes, frequency conversion between such resonant frequency modes is required. The nearest-neighbor hopping or the connectivity between two nearby frequency modes can then form a one-dimensional (1D) synthetic lattice in the frequency dimension, which is the essential conceit behind the synthetic frequency dimension [see Fig. 2(c)].

There are several ways to achieve the frequency conversion. We take the dynamic phase modulation applied in a ring resonator as an example to demonstrate the underlying mechanism^[73]. We consider the model of the modulated ring resonator as shown in Fig. 2(d), where the ring supports a set of resonant frequency modes at ωn=ω0+nΩR,(2)where ω0 is the reference frequency, ΩR=2πvg/L is the free spectral range (FSR) of the ring, vg is the group velocity of light propagating inside the ring resonator, L is the length of the ring, and n is an integer. The electric field of light that propagates inside the ring can be expanded in the basis of the resonant modes^[74]: E(t,r⊥,z)=∑nEn(t,z)En(r⊥)eiωnt,(3)where z is the azimuthal position along the propagation direction, r⊥ is the position perpendicular to the propagation direction, En(r⊥) is the modal profile, and En(t,z) is the modal amplitude component at the frequency ωn. The phase modulation dynamically causes a variation in the permittivity of the material: δϵ(t)=δ(r)cos(ΩMt+ϕM),(4)where δ(r) is the modulation profile generated by the modulator, ΩM is the modulation frequency, and ϕM is the modulation phase. The time-periodic variation of the permittivity changes the phase of the light when the light passes through the modulation part at the location labeled by zM, which can be described by^[65]E(t+,r⊥,zM)=E(t−,r⊥,zM)eiαcos(ΩMt+ϕM),(5)where t±=t+0± and α is the modulation amplitude. Under the slowly varying envelope approximation, Eq. (5) can be expressed using the Jacobi-Anger expansion in terms of modal amplitude En(t,zM) as $$\begin{gathered} En(t+,zM)=J0(α)En(t−,zM) \\ +∑q>0iqJq(α)[En+q(t−,zM)e−iq(Δt−+ϕM)+En−q(t−,zM)eiq(Δt−+ϕM)], \end{gathered}$$(6)where Δ=ΩM−ΩR is the frequency detuning, and Jq is the qth order Bessel function of the first kind. To see how such an operation leads to an analog of the tight-binding model, we assume that the modulation is weak. In this weak modulation limit α→0, Jq is very close to zero for |q|>1, and J0(α)≈1, J1(α)≈α/2. The variation of the optical field is then expressed as δEn(t,zM)=iα2[En+1(t,zM)e−i(Δt+ϕM)+En−1(t,zM)ei(Δt+ϕM)].(7)The accumulation of the optical field variation after propagating over one loop inside the ring is written by En(t+TR)−En(t)=δEn(t),(8)where TR=2π/ΩR is the round-trip propagation time. We define t=τ+T·TR, where τ is the fast time variable in one round-trip and T is the discrete round-trip number^[75]. Substituting the expression of t into Eq. (8), we obtain En[τ+(T+1)·TR]−En(τ+T·TR)=δEn(τ+T·TR).(9)If we further take the approximation that δEn/TR=∂En/∂T as the time interval, the variation of the optical field can be expressed in the form of a differential equation: En[τ+(T+1)·TR]−En(τ+T·TR)TR=∂En(τ+T·TR)∂T.(10)Here T can be treated as a continuous variable. Finally, by substituting Eq. (7) into Eq. (10), one can obtain the relationship between the variation of the optical field at ωn and two nearby modes at ωn±1 in the following equation: TR∂En∂T=iα2[En+1(t,zM)e−i(Δt+ϕ)+En−1(t,zM)ei(Δt+ϕ)].(11)Equation (11) can be regarded as the equation of motion of the photon wave packet component at each resonant mode, dictated by the corresponding Hamiltonian H=g∑n[ei(Δt+ϕM)cn+1†cn+e−i(Δt+ϕM)cn†cn+1],(12)where g=−α/2TR is the coupling coefficient. We therefore see that the modulated ring model in Fig. 2(d) results in a lattice Hamiltonian along the frequency axis of light [see Fig. 2(e)], which indeed provides a photonic analogue of the tight-binding model of the electron. More importantly, such a lattice model lies on a frequency dimension, which is virtually compared to its spatial counterparts^[76–78].

The 1D Hamiltonian in Eq. (12) contains many parameters or “knobs” to design the property of the lattice in the synthetic frequency dimension using, for example, the detuning, modulation phase, and even the modulation profile. First, we introduce the way for constructing the high dimension in the synthetic frequency dimension by adding multiple modulation frequencies in the modulation profile. In particular, the dynamic modulation can not only provide nearest-neighbor coupling between the resonant frequency modes in the ring resonator, but also create long-range coupling by implementing an additional modulation frequency that is an integer multiple of the FSR of the ring (NΩR). We assume both modulations are resonant for simplicity here. The transmission coefficient that incorporates such a two-modulation-frequency scheme becomes^[79]T=ei[αcos(ΩRt)+α′cos(NΩRt)],(13)where α′ is the corresponding modulation amplitude of the additional modulation frequency. The transmission coefficient in Eq. (13) can lead to the system Hamiltonian^[79]H=∑ng(cn+1†cn+cn†cn+1)+g′(cn+N†cn+cn†cn+N),(14)where g′ is the coupling coefficient relevant to α′. One notices that the Hamiltonian in Eq. (14) describes the physical dynamics that the field component of light at the frequency ωn can hop to components at modes with ωn±1 and ωn±N simultaneously. This hopping picture corresponds to an analog to a lattice in two dimensions where the photon can move along two directions. Such fact can be seen if we divide the integer n into two integers, n=qx+qyN, where qx=mod(n−n0,N)+1, qy=(n−n0−qx+1)/N, n0 is a reference index, and the 1D Hamiltonian in Eq. (14) is then written as H=∑qx,qy(gcqx+1,qy†cqx,qy+g′cqx,qy+1†cqx,qy+h.c.)+∑qy(gcqx=1,qy+1†cqx=N,qy+h.c.).(15)Equation (15) describes a two-dimensional (2D) lattice with a twisted boundary^[79]. Higher-dimensional lattices are possible to be studied in the synthetic frequency dimension by incorporating more long-range modulation frequencies^[79–81], and multi-dimensional information can be solely encoded in the frequency variable^[80,82].

Another aspect that one may find from the Hamiltonian in Eq. (12) is its capability for introducing the gauge degree of freedom for photons. It is well known that the gauge potential plays a vital role in controlling the motion of the charged particles through electric fields and magnetic fields. In fact, the quantum mechanical way to account for the effect of electromagnetic fields on the electron is to couple it to the gauge potential. In contrast to electrons, photons, as neutral particles, cannot be directly controlled through the electric and magnetic fields. Nevertheless, time modulation gives a possible solution for inducing the effective gauge potential that can couple to photons, inspired by similar ideas introduced in atomic systems^[23,24,83]. We again take a look at the Hamiltonian in Eq. (12) and note that the hopping phase reads as Φ=Δt+ϕ, which may be used to construct the effective gauge potential. To see this fact, we consider a 2D lattice with the Hamiltonian^[8]H=g∑⟨m,n⟩[e−iΦmn(t)cm†cn+eiΦmn(t)cn†cm],(16)where ⟨m,n⟩ labels the nearest-neighbor sites, and Φmn(t) is the hopping phase between sites m and n. The gauge potential and the hopping phase have the following relationship^[84]: ∫mnA·dr=Φnm(t),(17)where A is the effective gauge potential. The effective magnetic field B can be obtained from the effective gauge potential A^[85]: B=1S∮plaquetteA·dr,(18)where S is the area of the plaquette. For example, the position-dependent hopping phase Φnm=nϕ along the l^nm direction generates the effective magnetic field B=ϕz^, where z^ is the unit vector perpendicular to the plane of the lattice. The effective electric field E for photons can also be constructed from the effective gauge potential^[86]: E=−∂A∂t=−l^nm∂Φnm(t)∂t.(19)Therefore, by carefully designing the hopping phase Φ between different frequency modes in the synthetic frequency dimension, one can manipulate the frequency detuning Δ to construct the effective electric field along the frequency axis of light^[73,87,88] and design distribution of constant phase parts to generate an effective magnetic field in space including the synthetic frequency dimension^[70,89].

The discussions on the synthetic frequency dimension are focused on the Hermitian lattices. It turns out that the synthetic frequency dimension can be used to design and explore non-Hermitian physics in a quite straightforward way. The idea for exploring non-Hermitian physics in the synthetic frequency dimension has started by introducing the amplitude modulation in ring resonators in a theoretical proposal to produce skew-Hermitian or anti-Hermitian couplings^[90], and later has been implemented in experiments including both phase and amplitude modulations^[40]. To briefly introduce the process of constructing a general non-Hermitian model using both the phase modulator and amplitude modulator, we consider the corresponding transmission^[40]: T=eiαcos(ΩMt)−βsin(ΩMt),(20)where α is the modulation amplitude from the phase modulation, while β is the modulation amplitude from the amplitude modulation. Such a transmission coefficient can be used to derive a Hamiltonian in the weak modulation limit as H=∑nκ+cn+1†cn+κ−cn†cn+1,(21)with κ±=−α/2TR∓β/2TR. One can see that the Hamiltonian in Eq. (21) is non-Hermitian for β≠0. Therefore, based on the choice of both the phase modulation and the amplitude modulation parts to design specific non-Hermitian Hamiltonians, Wang et al. generated arbitrary one-band topological windings^[40] and topological two-band complex-energy braiding^[41]. These experimental demonstrations highlight the versatility of the ring resonator system in building synthetic frequency dimension models that support arbitrarily complex band structures.

The approach to analyzing the synthetic frequency lattice based on the tight-binding Hamiltonian provides a simple approach to solving many problems. However, such a model is limited to the weak coupling regime. It also cannot account for the variation in the propagation of light at different points within the circumference of the ring. To provide a better theoretical method, let us get back to Maxwell’s equations: ∇×E=−∂∂tμH,(22)∇×H=∂∂tεE,(23)∇·E=0,(24)∇·H=0,(25)where E, H, ε, and μ are the electric field, magnetic field intensity, permittivity, and permeability of the medium, respectively. Putting operator ∇× on the two sides of Eq. (22), one can obtain ∇×(∇×E)=−∂∂t(μ∇×H).(26)Substituting Eq. (23) into Eq. (26), one gets ∇×(∇×E)=−∂2∂t2(μεE).(27)The left side of Eq. (27) is ∇×(∇×E)=∇(∇·E)−∇2E=−∇2E.(28)Substituting Eq. (28) into Eq. (27), one can obtain Maxwell’s wave equation for the electric field: ∇2E−με∂2∂t2E=0.(29)

In photonic structures such as waveguides, the optical field E characterized in one dimension under the slowly varying envelope approximation can be expanded as E=E(t,z)e−iωt+ikz,(30)where z is the propagation direction, ω is the frequency of the light, and k is the wave vector. Substituting Eq. (30) into Eq. (29), one gets the wave equation for the slowly varying envelope of the modal amplitude: ∂∂zE(t,z)+1vg∂∂tE(t,z)=0,(31)where vg=1/με is the group velocity of light in the medium.

The wave equation in Eq. (31) can hold the universal solution E(t−z/vg,z), meaning that the field does not change its amplitude while it propagates inside the waveguide. One can then derive the transfer-matrix method, which has been successfully applied to calculate the band structure of systems such as the three-dimensional (3D) screw dislocation^[91] and photonic Weyl points in synthetic frequency dimensions^[92]. In Ref. [92], the band structures calculated from the tight-binding model and the transfer-matrix method have been compared. Results show that the overall shapes of the band structures from the two models are quite similar, but there are subtle differences indicating that the transfer-matrix method may present more detail of the system than the tight-binding model. In the following, we introduce the transfer-matrix method.

The light propagation inside a ring resonator follows the same way in the waveguide, except for the periodic condition imposed by the length of the ring E(t,z+L)=E(t,z). The ring resonator supports a set of resonant frequency modes ωn due to the boundary condition. The modal amplitude with frequency ωn at position j and at time ti is Ej,n(ti). The transfer process in the ring is Ej+1,n(ti+1)=eiδωδteiknδzEj,n(ti),(32)where δz=vgδt, δt=2π/(JΩR), J is the total discrete position number in the ring (the total propagation steps), δω is the frequency detuning, and kδz satisfies the resonant condition eiknL=ei2πn. Next, we consider the dynamic electro-optic modulation (EOM) process inside the ring, which provides the connectivity between nearest-neighbor frequency modes. We assume that the EOM is placed at the position with j=1 and gets modulated by the external signal αcos(ΩMt), where ΩM is the modulation frequency and is set to be equal to the FSR of the ring (ΩM=ΩR). The modulation process leads to Ej=1,n(t+)=eiαcos(ΩRt)Ej=1,n(t−),(33)where the exponent part in Eq. (33) can be expressed using the Jacobi-Anger expansion^[92]eiαcos(ΩRt)=∑q=−∞+∞iqJq(α)eiqΩRt.(34)

Substituting Eq. (34) into Eq. (33), one can obtain the transfer relation between different frequency modes: Ej=1,n(t+)=∑q=−∞+∞Ej=1,n+q(t−)iqJq(α).(35)All optical components include every resonant frequency mode at each position j: Ψ(ti)=[E1,1(ti),E2,1(ti),…EJ,1(ti)⏟n=1,E1,2(ti),…EJ,2(ti)⏟n=2,…]T.(36)The transfer process in Eq. (32) and the modulation process in Eq. (35) can then be generalized into an evolution function: Ψ(ti+1)=eiδωδtHΨ(ti),(37)where H is the matrix form of the transfer relationship between optical components. In order to obtain the band structure between δω and reciprocal lattice vector (kf), one needs to apply the steady-state condition, assume the optical field does not vary with time, i.e., Ψ(ti)=Ψ(ti+1), and then perform Fourier transformation Ej,kf=∑nEj,ne−ikfnΩR. The eigenfunction equation is e−iδωδtΨkf=HkfΨkf.(38)

As a simple example, we consider a single-ring case with J=2. The corresponding Hamiltonian is Hkf=(0,eiπeiπeiαcos(kfΩR),0).(39)The wave function is written as Ψkf=(E1,kf,E2,kf). The two eigenvalues for Eq. (39) read as {Eig1=eiπeiα2coskfΩR,Eig2=eiα2coskfΩR.(40)Taking the natural logarithm of eigenvalues in Eq. (40) and using δω1(2)δt=iln[Eig1(2)], one can then obtain δω: {δω1ΩR=−[1+α2πcos(kfΩR)],δω2ΩR=−α2πcos(kfΩR).(41)One can use band structures in Eq. (41) within δω∈[−ΩR/2,ΩR/2] to analyze the property of the system.

For the ring coupled with an external waveguide, we can write the input-output formalism^[79]: Ej′,n(t+)=1−γ2Ej′,n(t−)−iγEnin(t−),(42)Enout(t+)=1−γ2Enin(t−)−iγEj′,n(t−),(43)where Enin (Enout) is the amplitude of the input (output) light with frequency ωn in the external waveguide, γ is the coupling strength, and we assume here the external waveguide is coupled at the position j′. Equations (42) and (43) can also be rewritten as a scattering matrix: S=(1−γ2,−iγ−iγ,1−γ2).(44)The scattering matrix in Eq. (44) not only can describe the coupling between the ring and the external waveguide, but also can describe the coupling between two rings. By adding the scattering matrix properly into Eq. (37), one may compute the resulting band structure of a specific model in the synthetic frequency dimension^[93].

Last but not least, the wave equation for the optical field in Eq. (31) can also be used to perform the direct simulation, together with the modulation process in Eq. (35) and the input-output process or the coupling between rings in Eq. (44). Such simulation procedure can be used to study the exact output spectrum of the field from rings and also show the dynamics of a model in the synthetic frequency dimension^[91–93].

After many theoretical proposals developed to realize various simulations and functionalities with the synthetic frequency dimension, experimental efforts have been made in a number of photonic platforms. One such critical platform is a fiber-based ring resonator system incorporating electro-optic modulators^{[38,40,41,80,82,88,94–100]}. The length of the fiber ring typically is ∼10m in experiments [see Fig. 3(a)], which corresponds to an FSR of the order ∼20MHz. For the input and output processes, an external waveguide accompanied by a directional coupler is required to couple light into or out of the main cavity. The frequency of the input laser can be flexibly adjusted to selectively excite the desired eigenenergy^[38]. In addition, the off-resonant excitation can also be used to achieve a direct band structure measurement by analyzing the output optical field information^{[38,40,41,82,94,96,97,99,100]}. The ring resonator system enables the flexible ability to explore photonic analogs of condensed matter physics effects. For example, the naturally existing clockwise and counter-clockwise modes in the ring can be used to mimic a pseudo-spin degree of freedom^[38]. The modulation phase controls the photon’s gauge degree of freedom and can be directly engineered to mimic the magnetic field acting on electrons^[97,100]. The modulation frequency can be tailored to mimic an electric field by controlling the frequency detuning between the modulation frequency and the FSR of the ring (Δ=ΩM−ΩR)^[95]. Additionally, adding an amplitude modulator makes it possible to realize non-Hermitian coupling^[40,41,82]. Apart from this, many schemes have been proposed to design boundaries for frequency lattices, which further builds upon the spatial lattice analogy^[97,101].

The properties of the synthetic frequency lattice can be obtained from the optical field information inside the ring. There is a convenient way to directly measure the optical field information in the ring, by coupling an external waveguide to the main ring, where the field can leak out of the ring. This enables the dynamic evolution of the field inside the ring to be visualized. Furthermore, it has been observed that the reciprocal lattice vector (kf) of the frequency lattice is the time (τ) within each round trip. By reconstructing the experimental time-resolved transmission with a simultaneous sweep of off-resonant excitation detuning (δω=ωin−ωn), the band structure of the corresponding frequency lattice in a single ring can be obtained. The transmission of the output has a relationship between δω and τ as follows: Tout(τ=kf,δω)=γloss2[δω−gcos(ΩRkf−Δτ−ϕM)]2+γloss2,(45)where γloss is the loss in the ring and ϕM is the modulation phase. For a simple demonstration, we set Δ=0 and ϕM=0. The transmission in Eq. (45) can reach its maximum at δω=gcos(ΩRkf), and thus presents a cosine-like pattern, which is the band structure for a 1D synthetic frequency lattice^[94,97]. With this useful approach, one may observe band structures for different models constructed in synthetic frequency dimensions.

Besides the flexible-fiber-based ring resonator system that constructs the synthetic frequency dimension, research is also being conducted with integrated on-chip nanophotonic resonators, which are scaled-down versions of fiber-based rings together with state-of-the-art integrated lithium niobate electro-optic modulators [see Fig. 3(b)]^[102]. This system may not only provide a way to build synthetic frequency lattices but also shows potential for multifunctional on-chip devices^[103–107]. In one outstanding example, Hu et al. experimentally realized a synthetic frequency dimension with an on-chip electro-optic frequency comb, demonstrating a scheme to construct higher-dimensional synthetic lattices in a single ring resonator^[108]. The length of the on-chip ring is of the order of ∼10mm, which corresponds to an FSR of the order of ∼10GHz. Consequently, the modulation frequency is also of the order of ∼10GHz. Balčytis et al. realized the synthetic frequency lattice on a silicon CMOS (complementary metal-oxide semiconductor) platform, which enables research in synthetic frequency dimensions on mature technology^[109]. Moreover, Zhao et al. achieved a single layer of a matrix-vector multiplier in synthetic frequency dimensions through integrated cavity acousto-optics, which holds promise for scalable analog optical computing^[110]. Miniaturization of the on-chip ring provides a condition to manipulate photons in the quantum regime. Javid et al. combined electro-optic modulation and spontaneous parametric down-conversion to study on-chip entangled quantum states in synthetic frequency dimensions^[111], highlighting the ability to perform large-scale analog quantum simulations and computations within the time-frequency domain. Microwave photons (∼2π×155MHz) in synthetic frequency dimensions have also been demonstrated in a superconducting resonator^[112] and a chain of modulated cavities^[113], showcasing the applicability of the concept to different frequency regimes. More recent work in parametrically driven superconducting resonators has realized few-mode lattices of the Creutz ladder and the bosonic Kitaev-Majorana chain in microwave frequency dimensions^[114,115].

The above-introduced schemes to construct the synthetic frequency lattice typically require the waveguide to form a loop/ring, where the resonant frequency modes naturally serve as the lattice sites. However, one can build a synthetic frequency lattice without using a ring, by taking an open waveguide and applying traveling wave modulation^{[81,116–128]}. In such platforms as shown in Fig. 3(c), the traveling-wave modulation induces a variation of the material’s refractive index as nt=αcos(ΩMt−qMz+ϕM),(46)where ΩM, qM, ϕM, and α are the modulation frequency, modulation wave vector, modulation phase, and modulation amplitude, respectively. The frequency of the input laser is ω0, which corresponds to the site (ω0,q0) along the dispersion curve of the waveguide. Once the optical field inside the waveguide is under traveling-wave modulation, its frequency and wave vector experience a discrete variation (ω0±nΩM,q0±nqM) with n being an integer. This forms a discrete lattice along the frequency dimension [see Fig. 3(c)]. With the phase-matching condition being satisfied, the amplitude of the n-th mode an(z) can be obtained by solving the coupled-mode equations i∂an(z)∂z=C[eiϕMan−1(z)+e−iϕMan+1(z)],(47)where C=Δnk0/2 is the coupling coefficient and k0 is the vacuum wave vector. The Bloch mode in the frequency lattice has the form an=a0exp(ikω·nΩM)exp(ikzz), where kz is the propagation constant along the z direction and kω is the Bloch wave vector. The dispersion relation for the frequency lattice is then kz=−2Ccos(kωΩM−ϕM). One can incorporate a wave vector mismatch δq of the traveling-wave modulation to induce an effective force F=−δq along the frequency dimension. Although such platforms do not necessitate alignment of resonances in the ring, careful design of the traveling-wave modulation is required. Moreover, it is also challenging to extend the spatial dimension by coupling multiple waveguides or realize specific band excitations through the choice of the input frequency of light. Nevertheless, this platform can also be potentially scaled down with integrated photonics to form functional and tunable on-chip devices.

The traveling-wave modulation process performed in the waveguide above is tunable, but this is also possible in a waveguide with χ(3) nonlinearity. Cross-phase modulation of a probe signal and a sinusoidally shaped pump in a highly nonlinear fiber was used in experiments to observe spectral Bloch oscillations^[129–131], and the quantum Hall effects have been studied in one-dimensional four-waveguide arrays including the synthetic frequency dimension^[132]. On the other hand, the nonlinear four-wave mixing process has been explored to construct the frequency-domain photonic lattices^[133–136]. A pair of pumps with their frequencies separated by the spectral distance ΩR can lead to a hopping of the signal field from its original frequency to frequencies separated by ΩR through four-wave mixing [see Fig. 3(d)]. Multiple pairs of pumps with frequency differences of multiples of ΩR, i.e., nΩR, can then be applied to introduce long-range coupling between frequency components spanning the interval nΩR. The Hamiltonian for the signal carrying different frequency modes that propagate along the nonlinear waveguide under the phase-matching condition is H=∑m∑nCnam†am+n+h.c.,(48)where m is the index labeling the frequency component and Cn is the corresponding coupling coefficient, determined by four-wave mixing: Cn=2χP∑mAmAm−n*,(49)where P is the average pump power, χ is the effective nonlinearity, and Am is the complex amplitude of one of the pump fields. The experimental realization of the spectral photonic lattice used a co-propagating signal and multiple pumps in a nonlinear medium for the tunable, long-range, and complex coupling coefficients in the synthetic frequency lattice^[133]. Wang et al. further utilized the nonlinear four-wave-mixing platform to study topological effects in multidimensional synthetic chiral-tube lattices^[135], illustrating the capability to study topological phenomena using nonlinear optical effects.

Besides the four introduced typical platforms to build the synthetic frequency dimensions in photonics, there are also other potential platforms that may be used. For instance, an optical microdisk-resonator supports resonant modes spaced with an interval Ωdisk=c/(ndiskR), where c is the light speed in the vacuum, ndisk is the effective refractive index of the microdisk, and R is the radius of the microdisk. A continuous-wave pump laser can excite the microdisk utilizing Kerr nonlinearity to introduce the coupling between different resonant modes. By doing so, one can construct a synthetic frequency lattice therein and similarly use the phase of the pump laser to engineer an effective gauge potential^[137]. A Raman medium undergoing molecular modulation also shows the potential to construct synthetic frequency dimensions^[138]. This leads to a naturally asymmetric synthetic lattice model but also discusses a different direction of research at the interface of topological photonics and nonlinear optics^[139].

Here, we give some outstanding examples of exploring various physical phenomena from different designs of photonic lattices with synthetic frequency dimensions. The most striking feature of studying physics using the synthetic frequency dimension is that the corresponding band structure of the synthetic lattice model can be directly visualized by measuring the transmission [see Eq. (45)] through the ring resonator in experiments. Dutt et al. performed the first experiment in a fiber-based ring resonator to measure the band structure of a 1D lattice^[94]. The modulation frequency is set to the FSR of the ring and the measured band structure presents a cosine-like pattern, a typical band shape associated with a 1D tight-binding lattice [see Fig. 4(a)]. They further included the long-range coupling by adding a signal with the modulation frequency 2ΩR. They introduced an effective gauge flux by tuning modulation phases, creating a nonreciprocal band structure with broken time-reversal symmetry. Later on, Li et al. measured the so-called dynamic band structure using non-zero frequency detuning Δ≠0 in the modulation signal, tracking the dynamic motion of the band structure with the form^[88]δω(t)=gcos(ΩRkf−Δt).(50)

The frequency detuning Δ here plays the role of an effective electric force along the synthetic frequency dimension, as summarized in Fig. 4(b). In contrast to the static band structure which is time-independent, the dynamic band structure varies with time. Therefore, in experiments, one can slowly scan the injected frequency of the input field to exactly capture the moving band structure at each time slice. Doing so provides a novel way to capture more delicate information of a dynamic system and shows the potential to design on-chip functionality with time-dependent Hamiltonians^[140].

Besides the above simple models, synthetic frequency dimensions also give an approach to explore nontrivial physics in photonics. For instance, it has been well known that the realization of the quantum Hall effect in condensed-matter physics requires a strong magnetic field and a low temperature^[141]. However, in photonics, these two conditions may be relaxed. Fang et al. theoretically pointed out that the effective magnetic field for photons can be created by designing non-reciprocal hopping phases^[85,142]. The theoretical proposal was then realized in synthetic frequency dimensions with a similar idea for constructing the effective magnetic flux. In the experiment, Dutt et al. utilized the clockwise- and counterclockwise-propagating modes in a single-ring resonator^[38] to mimic the pseudo-spin degree of freedom and used auxiliary waveguides to couple these two modes. Utilizing the individual frequency axes of each pseudo spin and electro-optic modulation inside the ring, they built a two-legged ladder lattice model with an effective magnetic flux therein. Quantum Hall associated physical phenomena including spin-momentum locking and topological chiral one-way edge currents were observed in this single-photonic-cavity experiment [see Fig. 4(c)]. Ye et al. then used two coupled rings to build such two-legged ladder models but introduced gain/loss, where they observed the breakdown of chirality^[100]. The design of synthetic frequency dimensions using two rings can provide more flexibility in construction of lattice models. For instance, strongly coupled rings can induce resonance splitting resulting in a photonic molecule^[143–145]. Based on the symmetry along the frequency axis of light, Li et al. constructed a Su-Schrieffer-Heeger (SSH) model^[99]. Different from the SSH counterparts in real space^{[77,146–148]}, the synthetic frequency SSH lattice supports band structures including wave-function interference of the eigenstate information, which exhibit different patterns for the trivial and nontrivial cases, as shown in Fig. 4(d). One can then use this unique feature to achieve direct extraction of the topological Zak phase in experiments. Another way to obtain the topological invariant is using the mean-chiral displacement in the driven-dissipative lattice^[149]. As a direct step forward from this experiment, Qiao et al. explored phenomena associated with the ultrastrong coupling regime in this synthetic SSH configuration, where the edge states of the 1D Floquet SSH lattice were observed at 0 and π energy bandgaps^[150]. Research on the photonic molecule platform can also be extended to study the wave-function tomography of the generalized SSH model including long-range couplings and the Haldane model^[151,152]. On the other hand, in a proposed system including an auxiliary ring, Yu et al. showed the direct extraction of the topological invariant in a nontrivial Hamiltonian in quench dynamics, from the collected output optical field solely in the time dimension^[75]. Two rings of different lengths have also been used to construct the stub lattice (1D Lieb lattice), where a gapped flat band, mode localization effect, and flat-to-non-flat-band transition were observed in experiments^[96].

Although non-Hermiticity can be introduced into a photonic model by tuning the gain and loss distribution^[153–156], it has been observed that non-Hermiticity in the synthetic frequency dimension can also be realized through incorporating both amplitude modulation and phase modulation in the ring resonator. Wang et al. constructed a synthetic frequency lattice in one ring resonator imposing both amplitude modulation and phase modulation on the ring to form a 1D Hamiltonian^[40]: H=∑m,n(κ+man+m†an+κ−man†an+m),(51)where the coupling coefficient κ±m=Cmexp(±iαm)±Δmexp(±iβm), Cm (Δm) is the phase (amplitude) modulation strength, αm (βm) is the modulation phase induced by the phase (amplitude) modulation process, and m is the coupling order. Non-zero amplitude modulation with Δm≠0 induces non-Hermiticity in the Hamiltonian H≠H†. The expression of the detected signal I(kf,δω) is I(kf,δω)∝1[Re(E(kf))−δω]2+[Im(E(kf))]2,(52)where Re(E(kf)) (Im(E(kf))) is the real (imaginary) part of the eigenenergy of the Hamiltonian in Eq. (51). For a given kf, the detected signal I has a Lorentzian function shape along the detuning axis δω. Therefore, through fitting one can obtain Re(E(kf)) and Im(E(kf)) in experiments [see Fig. 4(e)]. By plotting the extracted kf-dependent Re(E(kf)) and Im(E(kf)) on the complex plane, one can visualize winding features if the corresponding Hamiltonian is nontrivial. Wang et al. further used two ring resonators to study the topological complex-energy braiding. The Hamiltonian of their system is^[41]H=∑ngan†bn+gbn†an−iγan†an+κan+1†an+κan†an+1+(C−Δ)an+m†an+(C+Δ)an†an+m,(53)where an† (an) and bn† (bn) are the creation (annihilation) operators in ring A and ring B, γ is the loss in ring A, and κ, g, and C±Δ are the coupling coefficients. The real and imaginary parts of eigenenergies of the Hamiltonian in Eq. (53) on the complex 3D space [Re(E),Im(E),kf] represent the unlink, unknot, Hopf link, and trefoil diagrams when specific parameters are chosen, as shown in Fig. 4(f).

The synthetic frequency lattice may further be enriched by including nonlinearity in the lattice model^[157–159]. Tusnin et al. theoretically explored the nonlinear effects induced by χ(2) and χ(3) susceptibilities in the ring resonator^[160], and later Englebert et al. experimentally investigated dissipative solitons in synthetic frequency dimensions, where long-living Bloch oscillations are guaranteed by the Kerr-type nonlinearity and dissipative structures (solitons), as shown in Fig. 4(g)^[161]. Javid et al. utilized electro-optic modulation and spontaneous parametric down-conversion to construct a 2D synthetic frequency lattice, where nonlinearity in the form of parametric down-conversion plays the role of expansion of the frequency dimension^[111]. In this example, the quantum correlation in the synthetic space has been shown experimentally [see Fig. 4(h)].

All previous examples focus on the model in one dimension, i.e., a singular frequency dimension. It is useful to extend the model to a higher-dimensional space. To achieve higher dimensions, one can build a synthetic space by combining the synthetic frequency dimension and a geometric dimension^[162]. For example, Yu et al. theoretically studied the Lieb lattice including the frequency axis of light using a 1D array consisting of two types of rings and explored the isolated photonic flat band^[163]. The four-dimensional Hall effect has been theoretically proposed in a 3D geometric structure by adding frequency as the additional fourth dimension^[71]. Moreover, Lin et al. constructed a 3D screw dislocation model with 2D ring arrays^[91]. On the other hand, one can also combine synthetic frequency dimensions with other synthetic dimensions (which we shall introduce in later sections) to create a higher-dimensional synthetic space. For example, Yuan et al. theoretically investigated the photonic gauge potential in synthetic frequency and orbital angular momentum dimensions^[164]. Furthermore, synthetic frequency dimensions can also be combined with the time dimension in a nonintuitive way to build 2D models^[124,165]. Last but not the least, it has also been theoretically proposed^[79] and experimentally demonstrated^[80,82] that one can fold the frequency axis of light by introducing long-range coupling between resonant modes at multiples of the FSR to construct 2D or 3D models. In particular, traveling-wave modulation in waveguides using incommensurable frequencies was proposed to construct higher-dimensional lattices^[81].

The concept of synthetic frequency dimensions not only can be used to build different lattice models for simulating various physical phenomena but also the resulting dynamics from the evolution of the Hamiltonian or the steady-state output optical field distribution may find many applications. Here, we briefly list several aspects of the synthetic frequency dimension applications.

One of the earliest goals for developing synthetic frequency dimensions was for quantum simulations, especially for nontrivial physics, as well as for exploring many-body physics^[70,166]. To serve such a purpose, the local interaction between photonic modes is required. In this sense, large nonlinearity is usually desired for achieving the photon-photon interaction in synthetic frequency lattices. Previous experiments have demonstrated dissipative solitons in a synthetic frequency lattice built in a Kerr-type nonlinear material, showing the existence of classical light-light interactions^[161]. Moreover, nonlinearity in the form of spontaneous parametric down-conversion in on-chip platforms shows the potential to execute photon-photon interactions from the generated quantum entangled photon pairs^[111]. By introducing nonlinearity into the synthetic frequency lattice, it has also been theoretically pointed out that many-body physics may be explored, which could lead to frequency entanglement for photons^[166,167]. Moreover, it has also been observed that if an atom, or quantum emitter, is added into a ring resonator that supports synthetic frequency dimensions, an artificial giant atom is induced that can interact with the photon via multiple quantum channels, due to the fact that all modes in the synthetic frequency lattice can interact with the atom even with frequency detunings^[168–170]. The research that follows this line can lead to quantum optics in the synthetic frequency dimension, which could later contribute to quantum simulations of many-body physics with photonic technologies.

Another direct output from experiments with synthetic frequency dimensions is frequency comb generation, previously overlooked in this context. Developments in integrated photonics such as high-efficiency and broadband on-chip electro-optic frequency comb generators in a coupled-resonator platform^[171] and integrated actively mode-locked lasers on thin-film lithium niobate^[172] have provided related experimental platforms. From the perspective of fundamental physics, Li et al. used two rings with different lengths to build a synthetic moiré frequency superlattice and demonstrated frequency comb generation with mode spacing reduction^[173]. In addition, non-Hermitian physical dynamics in the synthetic frequency lattice may also be used to control comb generation^[174]. From the perspective of a new platform, Heckelmann et al. have recently taken the synthetic frequency space to demonstrate quantum walk comb generation in a fast gain laser, which produces a low-noise, nearly flat broadband comb^[49]. Such a system brings the new active lattice model into the synthetic frequency dimension, which may not only be used to develop the quantum walk comb laser^[175], but also bring novel comb generations such as developing the low-threshold comb source at telecommunication wavelengths^[176] and providing the comb spectral shaping through phase engineering in the synthetic lattice^[177]. Moreover, this active system also provides a way for measuring the dynamics of the wave evolution in the synthetic lattice, important for performing quantum simulations with synthetic dimensions, where as an example, Dikopoltsev et al. experimentally demonstrated the quench dynamics of Wannier-Stark states in the active synthetic frequency space^[178], highlighting the ability to generate coherent multi-frequency comb sources. On the other hand, Flower et al. demonstrated topological frequency combs in a 2D resonator lattice that hosts edge states, by manipulating spectral domain information using nonlinearity^[179]. From the perspective of quantum applications, Pontula et al. theoretically studied the mechanism to shape above-threshold frequency combs that exhibit tunable bright squeezing and quantum correlations in a multimode nonlinear cavity through dissipation and Bloch mode engineering^[180]. Different from conventional comb generation experiments, these works specifically use physical phenomena in the synthetic lattice to design a platform to generate tunable frequency combs with desired spectra.

Synthetic frequency dimensions may also find use in the field of quantum information and optical computing, where the encoding, manipulation, and decoding of information along the synthetic axis of light become important ingredients. Lu et al. reviewed emerging developments towards frequency-bin quantum information processing and networking, showing its unique advantages for multiplexing, interconnects, and high-dimensional communications^[181]. Other than that, implementation of optical computing in synthetic frequency dimensions offers an integrated photonic architecture to achieve essential applications spanning quantum information processing, classical signal processing, and neural networks^[182,183]. For example, Buddhiraju et al. proposed a scheme to realize arbitrary linear transformations for photons in the frequency synthetic dimension, which uses ring resonators to implement tunable couplings between multiple frequency modes in a single waveguide^[184]. Fan et al. theoretically and experimentally demonstrated convolution processing in photonic synthetic frequency dimensions with one ring resonator incorporating a phase and an amplitude modulator, which lays the foundation to perform artificial-intelligence applications on compact devices^[98,185].

In addition to this, many physical phenomena including unidirectional frequency conversion^[70], non-Hermitian effects^[40], and higher-dimensional topology^[91] have been well studied theoretically and experimentally in synthetic frequency dimensions, which may trigger future applications on the integrated platforms with different functionalities. For example, Yu et al. theoretically showed the integration of multiple functions in the synthetic frequency dimension on an array of ring resonators^[89]. With respect to non-Hermiticity, the anti-PT symmetry and the PT symmetry in synthetic frequency dimensions can lead to pulse-shortening in mode-locked lasers^[90,186,187]. Moreover, the theoretical design of the 2D non-Hermitian skin effect in a synthetic photonic lattice would enable programmable light propagation and frequency conversion in an array of ring resonators^[188].

Current prosperous research in synthetic frequency dimensions shows it is a powerful tool for not only quantum simulations but also photonic emulator designs based on the photon’s frequency degree of freedom. More possibilities can be further explored in this space, including non-Abelian physics^[189–192], mirror-induced reflection^[193], time reflection and refraction^[194], topological spin pumps^[195], synthetic Floquet lattices^[196–198], and mode-locked topological insulator lasers^[199]. In particular, recent work theoretically predicted that homogeneous non-Abelian lattice gauge potentials may induce Dirac cones^[189], which was also confirmed by the experiment^[190]. Therefore, synthetic frequency dimensions show tremendous potential for fundamental exploration as well as on-chip applications.

As another very important degree of freedom for photons, the orbital angular momentum (OAM), carrying the spatial information of light, attracts fundamental research interest in the structured light manipulation^[200–204] and has broad applications from quantum information^[205,206] to optical and photonic communications^[205,207]. It has been observed that OAM can also be used to construct a synthetic dimension in photonics. In this section, we will introduce the theoretical description of a synthetic OAM lattice model, and recent experimental progress.

Theoretical works^[208,209] indicate that a main cavity coupled with an auxiliary cavity can be used to construct the synthetic OAM dimension [see Fig. 5(a)], where light with the OAM information passes through spatial light modulators multiple times to form a lattice structure. Coupling between sites is achieved by converting a portion of the energy from its original OAM mode into a mode with a lower (higher) OAM number l [see Fig. 5(b)]. The corresponding tight-binding model reads as^[53,208,209]H=κ∑l(eiφcl†cl−1+e−iφcl†cl+1),(54)where κ is the coupling coefficient between two nearby OAM modes, φ is the hopping phase that is determined by the propagation length inside the auxiliary cavity, and cl† (cl) is the creation (annihilation) operator for the lth OAM mode. Although the model in Eq. (54) only includes nearest-neighbor coupling, long-range couplings may also be achieved by engineering multiple auxiliary cavities accommodating suitable spatial light modulators^[210].

The tight-binding model can provide a clear way to reveal the intrinsic physics in the synthetic lattice with different OAM modes. It is also important to show the ability to measure the corresponding band structure and evolution dynamics of the system. One can use the input and output formalism of light with OAM following the same procedure as that for the synthetic frequency dimension in Eq. (44) to model the coupling between different cavities and construct high-dimensional synthetic OAM lattices. Different from the synthetic Bloch momentum of the frequency dimension, which is the time taken by light to circulate inside the ring for one round trip, here the Bloch momentum for the synthetic OAM lattice is the azimuthal angle ϕ (considering the fact that the wave function for light-carrying OAM has the expression eilϕ)^[14]. Through measuring the transmitted intensity along the azimuthal angle ϕ, it has been proposed that one can reconstruct the band structure of the synthetic OAM lattice^[14,208].

Based on the framework introduced above, a lot of physical models based on the cavity structure have been established. Zhou et al. used a single degenerate optical cavity and proposed to demonstrate 1D topological physics in the synthetic OAM lattice [see Fig. 5(c)], where they designed a pinhole structure in the cavity to form a sharp boundary along the OAM dimension so that the edge state associated with the SSH model can emerge^[53]. Luo et al. used a 1D array of optical cavities to study 2D topological physics [see Fig. 5(d)], with one dimension being the OAM dimension and the other being the spatial dimension^[208]. By engineering a hopping phase along the OAM dimension, it is possible to realize an effective gauge field for photons. Consequently, topologically protected edge states for photons in the synthetic OAM space have thus formed, and the projected band structure versus the hopping phase displays a Hofstadter-Harper butterfly pattern [see Fig. 5(d)]. Furthermore, Sun et al. used a 2D geometric structure to construct a 3D synthetic lattice [see Fig. 5(e)], where the extra synthetic dimension is OAM. Particularly, they used this synthetic 3D structure to demonstrate Wely semimetal phases^[211]. Besides the above-introduced fundamental physical effects studied in synthetic OAM lattices, Luo et al. showed potential applications in optical communication and quantum information by theoretically proposing a topological photonic OAM switch^[212]. They also illustrated synthetic OAM lattice applications involving quantum memory and optical filters by designing the hopping phase and coupling between two cavities^[209].

As side notes, it has also been observed that one may build a 2D synthetic space in a single cavity using OAM and frequency dimensions^[164,213]. In these proposals, the mechanisms for constructing the synthetic OAM dimension using an auxiliary cavity with spatial light modulators and the synthetic frequency dimension using dynamic modulation are combined in the single main cavity. Therein, an effective magnetic flux is naturally induced in the 2D synthetic space due to the linear dependence between the additional propagation phase in the auxiliary cavity for each mode and its carrier frequency. Apart from forming a cavity to build the synthetic OAM dimension, it has also been observed that one may establish the OAM dimension by arranging sequences of Q-plates (consisting of anisotropic liquid crystal molecules) and wave plates without using the cavity. This structure can then be used to design quantum walks in the OAM space and study the physics therein^[214]. In a different way, theoretical and experimental efforts have been devoted to designing effective magnetic fields for the structured modes in free space by utilizing an intracavity astigmatic mode converter to generate photonic Landau levels corresponding to a series of exotic OAM laser modes^[215].

Although the original theoretical proposal for constructing the OAM dimension is based on one main cavity being coupled to an auxiliary cavity containing spatial light modulators, experimental verification is not easy with such a complicated configuration. However, Yang et al. devised a simplified design where they utilized a single Q-plate inside a cavity to vary the OAM number of the light by one. By doing so, and further including the right-handed and left-handed circular polarizations of light, they demonstrated the construction of the synthetic OAM lattice in the experiment^[216]. They then constructed a topological photonic model and measured the topological invariant by analyzing the band structure of the synthetic lattice therein [see Fig. 6(a)]. As a step further, the same authors also showed that this experimental platform can be used to study non-Hermitian physics once loss is added to the system. In experiments, they explored exceptional points (EPs) in the synthetic OAM dimension [see Fig. 6(b)]^[217], pointing to potential applications in EP-based sensing. Liao et al. used a similar experimental setup and realized a sharp boundary in a synthetic OAM lattice by drilling a pinhole in the cavity. Unique features associated with the boundary state, dynamic moving of the edge modes, as well as spectrum discretization are observed in experiments^[218].

In a different experimental platform, Cardano et al. demonstrated a discrete quantum walk in a lattice along the synthetic OAM dimension, and showed that the OAM degree of freedom can be utilized as a versatile photonic platform to perform quantum simulation tasks^[219]. Furthermore, they used probability distribution moments to reveal the topological quantum transition^[220] and the mean chiral displacement method to measure Zak phases and topological invariants in the synthetic OAM lattices [see Fig. 6(c)]^[221]. Apart from the 1D synthetic OAM space, Wang et al. also constructed a 2D quantum walk with one spatial dimension and one OAM dimension in the experiment [see the left panel in Fig. 6(d)], where they observed topologically protected bound states but with vanishing Chern numbers [see the right panel in Fig. 6(d)]^[214].

We have seen that the basic strategy to construct a synthetic dimension is to find a set of discrete states, for example, using the degrees of freedom from light, and then introducing the appropriate coupling mechanism to connect these states to form a regular structure. It has also been noted that one may use the spatial dimension to construct the synthetic dimension^[37]. The key idea is to use the modal dimension of supermodes propagating in a spatially distributed array of waveguides and connecting these discrete modal states through the evanescent-wave coupling via the design of waveguide spacings. This idea is interesting as there is no additional dimension added to the physical space if we convert one spatial dimension to one synthetic modal dimension. Nevertheless, the advantage is obvious once it is noted that the coupling between modal states can be engineered in a complex way and long-range coupling can also be induced.

We next discuss the way to introduce the connectivity between these discrete eigenstates by designing the oscillation of the array of waveguides [see Fig. 7(b)], where the oscillation of waveguides is restricted within the z–y plane and light propagates along the z direction. The variation amplitude of the oscillation is Rsin(Ωz+ϕ),(60)where R is an amplitude, Ω is the frequency that is equal to the eigenvalue interval, and ϕ is the hopping phase. The equation to describe the coupling between oscillatory waveguides is i∂∂zcn(z)=pneiφncn+1+pn−1eiφn−1cn−1,(61)where cn is the amplitude at nth waveguide, pn is the coupling strength between the n-th and (n+1)th waveguides, φn=dy,nk0ΩRcos(Ωz+ϕ), dy,n is the distance between the n-th waveguide and (n+1)th waveguide that is tailored according to the coupling strength between the two waveguides, and k0=2πn0/λ is the wavenumber in the waveguide. A static waveguide array (R=0) supports the discrete modal states without the connectivity. It has been found that for small R, the oscillation gives perturbation, which introduces the connectivity between modal states and hence constructs the synthetic modal dimension in such a curved array of waveguides^[37]. The construction of the synthetic modal dimension uses the array of waveguides along the y direction. Copying such an array of waveguides over the x-axis can build the 2D lattice with one spatial dimension and one modal dimension [see Fig. 7(c)]. Experimental efforts have used such a construction to demonstrate the first photonic topological insulator in 1 real-space +1 synthetic-space dimension^[37].

Lustig et al. first experimentally realized a photonic topological insulator in synthetic spatial-modal space^[37] based on the framework introduced in the previous part. The effective magnetic-field-induced topological edge state for photons is realized by designing an appropriate hopping phase ϕ in each column [see Fig. 7(d)]. The experimental results show that the light propagates clockwise along the edge of the synthetic lattice, wherein the light with the lowest modal mode moves towards the left direction along the spatial dimension, and then the modal mode of the light starts to convert towards higher modal numbers when light reaches the spatial boundary [see the lower panel in Fig. 7(e)]. Such a topologically protected edge state cannot be seen in the original waveguide arrays in 2D spatial space [see the upper panel in Fig. 7(e)], as the edge state is embedded in the bulk in real space, which has the potential to find possible applications in topological insulator lasers. In this setup, the anticlockwise edge state can also be achieved by exciting the lower bandgap. In contrast, once the effective magnetic field is tuned to zero, the topological edge state disappears and the light in the synthetic lattice penetrates into the bulk^[37].

Lustig et al. further studied 3D topological insulators based on the synthetic space with two spatial dimensions (x–y plane) and one modal dimension^[44]. In the spatial dimension, they used helical waveguides to construct a Floquet lattice, which has the periodicity along with the propagation direction [see Fig. 8(a)]. This 2D lattice is topologically nontrivial through adjusting the relative initial phase offset between the nearest helical waveguides. The synthetic modal dimension is then introduced to the system by replacing each site with three waveguides, whose refractive indices are different from each other [see Fig. 8(b)]. The set of these three waveguides can form a synthetic modal dimension with a limited lattice site number. Nevertheless, such system can still show an experimental feature that is associated with the 3D screw dislocation, which is then used to reveal the topological transport in the strong photonic topological insulators [see Fig. 8(c)] and paves the way towards the potential application based on 3D topology.

In the framework of optical systems, there are other degrees of freedom forming discrete states that can be used to construct a synthetic dimension. The time degree of freedom of light is a key way to construct synthetic lattices, which we will focus on in the next section. Here, we briefly discuss another degree of freedom of light, namely, the polarization degree of freedom of photon pairs^[226], and show how one can use the polarization of light to construct a synthetic dimension^[227].

Light has a polarization degree of freedom, perpendicular to its propagation direction. The difference between the polarization states of light is observed when it propagates through birefringent materials. A direct laser-written waveguide in fused silica is birefringent^[228]. One can use the two modes of polarization to describe light propagation in this birefringent waveguide, where one mode is parallel to the slow principal axis and the other mode is parallel to the fast principal axis. The refractive indices for these two modes are different, which leads to different corresponding on-site potentials. One can use the following Hamiltonian to describe this system as H=βsas†as+βfaf†af,(62)where as(as†) and af(af†) are the annihilation (creation) operators for the modes on the slow principal axis and the fast principal axis, with propagation constants (as the on-site potential) βs and βf, respectively.

The two modes here form a synthetic space of the two polarizations of light in the waveguide. However, the polarization states are discrete, so if the synthetic dimension is desired to be constructed, one needs to introduce connectivity between them. To do so, one can rotate the waveguide using the following procedure^[229]. First, one can assume that the fast principal axis is along the horizontal direction (H) and the slow principal axis is along the vertical direction (V). Next, the waveguide gets rotated clockwise with an angle α [see Fig. 9(a)]. The two polarizations of the light can be considered to be along the horizontal and vertical directions of the waveguide, i.e., H and V. Therefore, the Heisenberg equation of motion for this dynamic process is iddz(aH†aV†)=(β¯+Δcos2α,Δsin2αΔsin2α,β¯−Δcos2α)(aH†aV†),(63)where aH† and aV† are the creation operators for the horizontal polarization mode and the vertical polarization mode, β¯=(βs+βf)/2 is the mean propagation constant, and Δ=(βs−βf)/2 is the birefringence strength. Interestingly, Eq. (63) is mathematically equivalent for two coupled and detuned waveguides. However, for the coupled polarization degree of freedom, the on-site potential Δcos2α and the coupling strength Δsin2α both can be controlled by the birefringence strength. When α=π/4, the on-site potentials for both modes are equal and the coupling strength between the two modes is maximum.

Ehrhardt et al. used the above setup to verify the Hong-Ou-Mandel effect [see Fig. 9(b)] using a correlated photon source, where they can explore quantum optical phenomena from the quantum walks of photons in the synthetic polarization dimension. In particular, they used two and even three waveguides to explore quantum interference of correlated photons on 3D graphs, which may open up avenues for experimental explorations of quantum dynamics, and for emulating many bosonic or fermionic models including further study of the nonlinearities and nontrivial topologies with the graph isomorphism problem on optical platforms^[230].

The construction of the synthetic dimension using the polarization states in the birefringent waveguides gives a continuous model in the time evolution [or the propagation distance; see Eq. (63)]. On the other hand, there are also many works using sets of quarter-wave and half-wave plates to manipulate the polarization states of light in a sequence, where a discrete quantum walk on the two polarization states can be demonstrated^{[221,231–235]}. Although some of these works may not claim the construction of synthetic dimensions, the connectivity is built between discrete polarization states and the unitary conversion processes occur based on a lattice framework, which makes these photonic quantum walks using the polarization degree of light be an important supplement to the synthetic dimension in optical systems. Other than the polarization states, the transverse momentum of light can be engineered to design synthetic dimensions. For example, D’Errico et al. used the 2D momentum lattice to reveal the topological quantum walks^[236] and two correlated photons in quantum walks^[237]. Moreover, the geometrical angular coordinate around a ring resonator has also been used to construct the synthetic dimension, where a combination between synthetic dimensions and effective magnetic fields with local interactions was proposed^[238].

The previous section uses the discrete states of light as the basic components to create the synthetic dimension. As one of the most important parameters for light, the time information has not been discussed in the previous section, despite the volume of research into temporal synthetic dimensions in the past few years^{[39,45,50,51,239–252]} (and a long history of time-bin multiplexing for myriad other applications). Unlike synthetic dimensions employing discrete photonic states, where the models usually can be treated safely as continuous evolutions in time from a corresponding Schrödinger equation, the time dimension here is not straightforward in a similar fashion. In this section, we will see how one uses the temporal ordering of light pulses inside a ring as the basic components to construct the synthetic dimension (i.e., the synthetic time lattice built from the time-multiplexed pulses with a fast integer time variable), where the dynamics of the model evolves in discrete steps in time (a slow integer time variable)^[20,253]. Nevertheless, we shall see that with discrete-time models, one may still be able to simulate various physical phenomena, and more importantly, this kind of system provides a natural platform to emulate quantum walks.

We organize this section as follows: in the theoretical model part (Sec. 3.1), we introduce the model to describe the construction of the synthetic dimension using time-multiplexed pulses. In Sec. 3.2 and Sec. 3.3, we introduce the novel physical phenomena in these synthetic time lattices using single loops and two coupled loops, respectively. Lastly, we briefly summarize this section in Sec. 3.4.

In this subsection, we review the basic idea of constructing the synthetic time lattice using the temporal axis of light based on a time-multiplexed network. This method is developed to facilitate the connectivity between pulses at different arrival times in fiber loop(s) using delay lines or by coupling two loops with a length difference. We start using delay lines in a single loop to construct the synthetic time dimension in the first part and discuss its extension by coupling two loops to form the synthetic time lattice in the second part.

We consider a main fiber loop in Fig. 10(a), where optical pulses can propagate along the main loop. A sequence of pulses circulates inside the loop with equal time intervals between two nearest-neighbor pulses (labeled as δt), where the time degree of freedom of each pulse is specified by the arrival time of the pulse on a reference position inside the main loop. We can use the discrete index n to label each pulse and un to mark the amplitude of the nth pulse. To induce connectivity between the nearest-neighbor pulses, two delay lines are introduced into the main loop. One should then deliberately design the lengths of two delay lines (where the longer one is Ll and the shorter one is Ls) and the length between two joint points inside the main loop as Larc to satisfy the condition Larc−Ls=Ll−Larc=vgδt, where vg is the group velocity for the field in the fiber loop. Then a portion of un in the nth pulse can go ahead of (behind) the (n−1)th pulse [(n+1)th pulse] through the shorter (longer) delay line and contribute to un−1(un+1), and vice versa^[254]. Following such procedure, a tight-binding type connectivity between pulses is built after the fields finish each round trip circulation (labeled by an integer m), where un in the nth pulse at the (m+1)th round trip consists of a combination of the major portion of un and small portions of un−1 and un+1 at the mth round trip [see the lower panel of Fig. 10(a)]. This hence constitutes the most basic construction of a synthetic time lattice in a single loop. In principle, the connectivity can be manipulated arbitrarily to be a long range of order N by introducing suitable delay lines that satisfy Larc−Ls=Ll−Larc=Nvgδt. Moreover, the synthetic lattice evolves with discrete time-steps, quantified by the number of round trips for the pulse circulation.

A potentially simpler design for constructing the synthetic time lattice (or time-multiplexed mesh lattice) is to use two loops with different lengths coupled by a 50/50 coupler, as shown in Fig. 10(b). In this setup, again the sequence of pulses gets circulated, now in two loops, and one can use the arrival time to label pulses in each round trip. We assume the time for a pulse to complete one round trip in the short loop as Ts and the time for that in the long loop as Tl. The time difference is set to be Tl−Ts=2ΔT, and the average time is (Tl+Ts)/2=T, so Ts and Tl can be re-written as Ts=T−ΔT and Tl=T+ΔT. We denote a pulse at the nth arrival time in the short loop as unm and a pulse at the nth arrival time in the long loop as vnm for the mth round trip. Hence the time that a pulse arrives at the coupler is defined as t=mT+nΔT. Now two pulses unm and vnm arriving at the coupler together can contribute to the pulse circulating in the short loop for the next round trip with the time ΔT ahead (i.e., the pulse spends time Ts for a round trip), which gives un−1m+1. This process can be expressed as^[8,20]un−1m+1=12(unm+ivnm).(64)Similarly, the same two pulses can contribute to the pulse circulating in the long loop instead, which makes ΔT behind as the pulse spends time Tl for the next round trip, which leads to vn+1m+1 by vn+1m+1=12(iunm+vnm)eiϕ(n).(65)Here ϕ(n) is the modulation phase that can be imprinted from the phase modulator in the long loop. The lower panel in Fig. 10(b) shows the hopping diagram described in Eqs. (64) and (65), forming the discrete m-n mesh grid, which creates the synthetic time lattice model.

There are many interesting phenomena that can be observed in the time-multiplexed network in a single loop, using the concept of the synthetic time lattice. As examples, we discuss achievements in photonic emulation of the Ising machine, topological physics observed with dissipative photonics, efforts towards quantum computation, and realization of quantum walks.

The Ising model, which models magnetic interactions in spin-chains, has been proposed to tackle quadratic unconstrained binary optimization (QUBO) problems by mapping the binary variables of interest to individual spins on a chain and finding the ground state of the corresponding Ising Hamiltonian. This method can have potential applications in drug discovery^[255] and artificial intelligence^[256]. Classical annealing methods have already shown great promise for utilizing Ising machines as hardware solvers^[257]. However, quantum annealing, which takes advantage of quantum tunneling and superposition, shows great potential to speed up the optimization process^[257]. Marandi et. al. therefore have used the synthetic time dimension framework for the construction of the coherent Ising machine in the experiment^[258], which we briefly introduce here.

The standard Hamiltonian of the Ising model is^[258]H=−∑ijNJijσiσj,(66)where σi represents the ith z projection of the spin, and Jij is the coupling between spin σi and σj. The photonic emulation of the Ising model is applied in a time-multiplexed network as shown in Fig. 11(a), where four sequenced pulses with the same time intervals are circulating inside the fiber loop and represent four spins [see Fig. 11(b)]. The binary phase of the degenerate optical parametric oscillator (OPO) above the threshold^[259] mimics the spin up and spin down states, and they are coupled to each other below the threshold to realize a superposition of different spin configurations. The couplings Jij between each spin [see Fig. 11(c)] are tailored by controlling the three delay lines. More flexible connectivities beyond the nearest neighbor have been obtained using a measurement-feedback architecture for nearly arbitrary amplitudes of the Jij coupling. When the frustrated Ising the spin model is considered, the couplings are set to be out-of-phase, Jij<0. Ground energy of the Hamiltonian in Eq. (66) corresponds to a set of eigenstate distributions, which are given by the alignment of the spin patterns, and the Ising Hamiltonian’s energy spectrum is thus mapped onto the loss of each coupled OPO state. In order to obtain the ground energy, one can gradually increase the OPO gain by increasing the pump field, until the OPO gain balances out minimum loss, which corresponds to the minimum energy of the Ising problem [see Fig. 11(d)], and collapses the superposition of degenerate OPOs to a spin configuration corresponding to this ground state. Hence, at this stage, only the ground states from the Ising Hamiltonian in Eq. (66) can stably exist in the system and then be observed.

Dissipation is ubiquitous in all areas of physics^[260,261], and is typically a deleterious effect in the study of particle transport dynamics, but can be a resource in nonlinear physics for creating optical solitons. Besides this, dissipation can also be used to realize exceptional points in non-Hermitian Hamiltonians, offering possible improvements for optical sensing^[28]. Recently, Leefmans et al. revealed that dissipation can also be utilized to engineer the topological nature of a system^[244], using a time-multiplexed photonic network and introducing dissipative coupling in a synthetic time lattice.

In the experimental setup [see Fig. 11(e)], a single loop, where a sequence of pulses circulates inside, is used to construct a lattice model in the synthetic time dimension. The delay lines are also introduced in the loop contributing to a dissipative coupling between lattice sites. The modulators are added in the delay lines, which are set to design the coupling strengths and phases. In such a configuration, one can construct the dissipative SSH lattice model, where the Hamiltonian is purely imaginary, so the topological invariants now are calculated from topologically nontrivial imaginary bands. In experiments, one can observe the diffusion of the pulses where the energy couples into the initially unoccupied lattice sites to form a topologically trivial structure [see Fig. 11(f)]. On the other hand, if the topologically nontrivial model is constructed, the distribution of the pulses in the synthetic time lattice is consistent with the theoretical topologically protected edge state distribution [see Fig. 11(g)].

The synthetic time lattice configured here can also be expanded to two dimensions by incorporating multiple delay lines. For example, the same loop is coupled with four delay lines, two of which are introduced additionally to provide the long-range coupling, in which way the lattice in the time dimension can be folded to effectively a 2D lattice. In the experiment, a 4×10 Harper-Hofstadter (HH) lattice is then constructed in the synthetic space together with the effective magnetic field induced by setting suitable modulation phases on each pulse traveling in the delay lines. The corresponding topologically protected edge state is then observed in the experiments, where the distribution of pulses shows localization behavior along the boundary of the synthetic lattice [see Fig. 11(h)]. The demonstration of topological phenomena with purely dissipative Hamiltonians opens a novel direction to understanding topological phases, which has led to further studies including the creation of a temporally mode-locked laser^[262] and the generalization of the topological laser proposal in a dissipative time-frequency synthetic space^[165].

Quantum computation has been proposed as an alternative computational paradigm with potential applications in combinatorial optimization, factoring large integers, and quantum simulation^[263–266]. Photonics can provide alternative ways for the quantum computation process^[5,6]. However, operations on optical qubits need to be conducted mid-flight, which typically requires sequential optical components. These components make photonic quantum circuits requiring massive resource overheads, especially to achieve fault-tolerance^[267], and are hard to be integrated in photonic chips^[268]. Bartlett et al. proposed a scalable architecture for a deterministic photonic quantum computer using the synthetic time dimension^[241], with the construction steps briefly summarized below.

Single photons propagating in the clockwise (CW) and counterclockwise (CCW) modes of the loop constitute the photonic qubit basis |↻⟩ and |↺⟩. The scattering unit consisting of the π/4 phase shifter, 50:50 beam splitter, an atom, and a cavity is coupled to the loop through two optical switches, as illustrated in Fig. 11(i). The atom has a Λ-shaped three-level energy structure, where two ground states |g0⟩ and |g1⟩ represent the atomic qubit basis, with one of the Λ-level transitions resonant with the cavity [see Fig. 11(j)]. In this configuration, when the photonic qubit scatters with the atom, the photonic qubit and the atomic qubit are then entangled. After the scattering process is completed, a rotation operation between states |g0⟩ and |g1⟩ is applied using external coherent lasers, and the measurement of the atomic state in the |g0⟩ and |g1⟩ basis is also performed. Due to the entanglement between the atomic and photonic qubits, the operation and the measurement on the atom can be teleported to the photonic qubit. An arbitrary single-qubit gate can be designed by configuring a three-step rotation scheme, which forms an arbitrary rotation in the Bloch sphere, providing a direct visualization of the subsequent transformation on the photonic qubits [see Fig. 11(k)]. More importantly, a two-qubit controlled Z gate can be deterministically implemented by directing the two pulses corresponding to the qubits successively on the cavity-coupled atom^[269]. The prepared photonic qubit can then be read out using the swap operation, thus satisfying all of the criteria for deterministic photonic quantum computation in the synthetic time dimension^[241]. This scheme has the potential to be significantly more scalable by re-using the same atom or quantum emitter in the cavity in a time-multiplexed fashion, obviating the need for multiple such identical atoms or emitters, which itself is experimentally challenging. Other quantum computing schemes re-using a single quantum emitter through time-multiplexed pulses for cluster state generation have also been proposed^[270,271].

The classical random walk puts forward the idea that the shift of a walker is dependent on a coin toss. The quantum walk as the quantum counterpart of the classical random walk has aroused broad interest, as it provides insights into a quantum algorithm to speed up search processes^[272–276]. Apart from quantum walks based on the previously introduced OAM framework, the synthetic time dimension provides another experimental candidate to simulate discrete-time quantum walks in photonics^{[21,226,239,249,251,277–280]}. A typical quantum walker can be described by the product of the position Hilbert space Hx and the coin Hilbert space Hc, where the evolution of the wave function under the quantum walk is^[277]|Ψ(t1)⟩=S^C^|Ψ(t0)⟩,(67)where C^ is the coin operation and S^ is the step operation. For the coin operator, the polarization degree of freedom of light is usually a good candidate to perform the coin toss procedure. The position operator S^ that moves the quantum walker according to the result of the coin toss with the expression^[277]S^=∑x∈Z(|x+1,H⟩⟨x,H|+|x−1,V⟩⟨x,V|),(68)where H is the horizontal polarization and V is the vertical polarization, is emulated in the spatial dimension as before. The use of the synthetic time lattice is to replace such movement with the conversion between pulses at different arrival times. There are several important experiments that have achieved this, which we summarize in the following.

Nitsche et al. demonstrated dynamic control of a 1D discrete-time quantum walk by actively adjusting the coin operator, where tunable graph structures can now be observed^[277]. Barkhofen et al. configured a topological structure by designing time-reversal symmetry, particle-hole symmetry, and chiral symmetry into the synthetic time lattice, and topological invariants were measured and edge states were observed during the discrete-time quantum walks^[278]. Lorz et al. studied a four-dimensional coin operator by adding clockwise- and counterclockwise-propagating modes in the loop, where multiple wavefronts are observed in the quantum walk process^[279]. Chalabi et al. used two beam splitters in the loop model to construct a 2D synthetic lattice^[280], where the effective gauge field in the discrete-time multiplexed quantum walk can induce bandgaps and give rise to suppressed diffusion. Besides this, they also showed a particular walk along the domain boundary when two topological edge states with opposite velocities are excited. Moreover, Chalabi et al. engineered synthetic electric fields into the 2D quantum walk, where Bloch oscillations, revival of the optical field, and waveguide photons in nonuniform electric fields were experimentally observed in the time-multiplexed network^[239]. The same group has more recently demonstrated dynamic control of the non-Hermitian skin effect in 2D quantum walks^[252]. Schreiber et al. used the time-multiplexed network to construct a 2D quantum walk, where two-particle dynamics and strong nonlinearities were experimentally obtained^[21].

Non-Hermicity and Anderson localization can also be explored in the time-multiplexed network by incorporating wave plates to introduce polarization-dependent on-site loss and incorporating an electro-optical modulator to induce walker-position-dependent disorder, respectively. Lin et al. thus demonstrated the interplay of non-Hermiticity, disorder, and topology, where they observed the non-Hermitian skin effect, Anderson localization, and disorder-induced topological phase transitions [see Fig. 12(a)]^[246]. The non-Hermicity can be further combined with the quasiperiodicity by constructing an Aubry-André-Harper (AAH) model in the synthetic time dimension. Lin et al. used the polarization- and position-dependent phase operators to fulfill the quasiperiodicity condition, and notably, they experimentally observed the mobility edge as well as the concurrence of global delocalization-localization transitions, the spectral topological transition, and the PT-symmetry breaking transition^[245]. In the non-Hermitian framework, an interesting correspondence between non-Hermitian spectral topology and transient self-acceleration was also experimentally unveiled by Xue et al., where they showed that the 1D self-acceleration is proportional to the spectral area encircled by the complex eigenenergy while the 2D self-acceleration is proportional to the volume encircled by the complex eigenenergy [see Fig. 12(b)]^[251]. The non-Hermicity together with the synthetic magnetic flux can be utilized to engineer the directional flow of light in the 2D square lattice, where Lin et al. used the time-multiplexed 2D quantum walk to reveal the novel phenomena, including the adjustable orientation of the non-Hermitian skin effect by controlling the photon-loss parameters, the suppression of the bulk flow by applying the magnetic confinement, and the Floquet topological edge modes under the impacts from non-Hermiticity and the magnetic flux [see Fig. 12(c)]^[249].

Before ending this part, we want to discuss the distinction between the simulations of quantum walks in the synthetic time dimension and the universal model of quantum walks for quantum computation^[275]. Although classical light is used in experiments, the evolution of light forms the dynamics on a synthetic time lattice following a well-defined evolution equation dependent on the local coin choice, for example, Eq. (68). It then provides the basic framework of discrete-time quantum walks of single particles, and hence leads to an experimental platform in photonics to simulate the dynamics of quantum walks for single particles. However, if one considers the quantum walker for quantum computation, the interaction between walkers has to be added into consideration, which might need to overcome the bigger challenge of realizing strong photon-photon interactions in optical platforms^[281].

In the previous part, we discussed interesting physics explored in the synthetic time lattice using a single loop. Although the models constructed in a single loop are straightforward to be understood using the tight-binding picture as illustrated in Fig. 10(a), the two-loop configuration in Fig. 10(b) has been developed with more broad aspects due to its relatively simpler setup in experiments. Here, we show some outstanding instances of studying various physical models in two loops.

The photonic topological effect can be used to control the flow of light^[282–285], where different platforms use ferrite rods^[282], delay lines^[283], etc., to create the effective magnetic field for photons to break the time-reversal symmetry of the system and then create topologically protected edge states. The time-multiplexed network formed by two loops can provide reconfigurable and scalable synthetic time lattices, which manifests a powerful tool to simulate the photonic topological effect^{[39,45,240,243,248,250,286–289]}, where the phase modulator in the loop provides the effective magnetic field for photons.

Weidemann et al. constructed the prominent 1D topological structure, the SSH model, in the synthetic time dimension based on the two-loop setting, where they displayed the topological funneling of light with the incorporation of the non-Hermitian effect [see Fig. 13(a)]^[39]. With the two-loop setup, they designed the modulation phase to induce effective vector potentials for the construction of the AAH model in the synthetic time dimension, which results in the Floquet Hofstadter butterfly pattern in the energy spectrum [see Fig. 13(b)]^[45]. The topological triple phase transition is also observed in this experiment, where the bulk metal-insulator phase transition, topologically nontrivial phase for the non-Hermitian Floquet model, and parity-time (PT) symmetry breaking simultaneously emerge in experiments by controlling one single parameter [see Fig. 13(c)]^[45]. A method to quantify the topological invariant has also been developed on this platform. For example, Longhi et al. demonstrated the dynamical topological winding of the synthetic lattice by adopting the mean survival time of a pulse to characterize the topological winding of the non-Hermitian system^[288]. Wimmer et al. also used the anomalous displacement of wave packets to obtain the Berry curvature in the mesh lattice based on the two-loop setting^[286]. In addition, Ye et al. used scalar and vector gauge potentials to experimentally design the electric and magnetic Aharonov-Bohm effect in the synthetic time dimension, which shows the potential application in quantum information processing^[248]. Moreover, temporal refraction and reflection has been observed at gauge potential interfaces^[248], including novel phenomena such as temporal Goos-Hänchen shift^[290], and based on moving gauge potential barriers^[291].

A single-shot technique to measure the band structure from the synthetic time lattice has also been developed for observing topological effects, where bandgaps close and reopen by controlling the modulation phase^[243]. The emergent topology in the band closing and reopening process can then be used to realize a time-reversal operation by utilizing the exchange of eigenstates. Based on this mechanism, Wimmer et al. realized the restoration of a pulse sequence in the photonic time lattice^[287]. Adiyatullin et al. used heterodyne measurement to directly visualize the bulk winding bands as well as the bands with anomalous edge states in the synthetic time lattice, where they found Bloch sub-oscillations within the bulk and edge states at the interface^[250,292]. The robustness and vulnerability of the topological anomalous Floquet interface state are also experimentally examined by Bisianov et al^[240].

Besides various topological phenomena based on the dual-loop temporal synthetic dimension that we summarized above, a recent work has also suggested the important potential for using synthetic time lattices to study temporal topology^[293]. Different from spatial topology, this work reports the experimental observation of momentum-gap topology, manifesting as temporal topological boundary states. In addition, with a new direction of time and space-time topology, time topological states can be observed, linked to a time-topological invariant, which then introduces the concept of space-time topology^[294]. Therefore, optical synthetic time lattices may provide a new way for studying the concept of photonic time crystals^[295,296] and time-varying photonics^[297–300].

Non-Hermiticity has been widely explored by the photonics community^[29,301,302]. The introduction of non-Hermicity in a time-multiplexed network reveals numerous novel physical effects^{[20,39,45,51,242,247,253,288,303–307]}, which promotes potential applications in optical sensing and light harvesting. Wimmer et al. utilized the phase gradient generated from the phase modulator to form a linear potential for Bloch oscillations and utilized acousto-optical modulators to generate the effective gain and loss for the system, where they discovered secondary emissions during the Bloch revivals. They also demonstrated reconstruction of the non-Hermitian band structure using Bloch oscillations^[304]. Regensburger et al. explored the synthetic photonic lattice with parity-time symmetry, where power unfolding, secondary emissions, and unidirectional invisibility are experimentally obtained^[20].

Steinfurth et al. utilized amplitude modulators to induce controllable gain and loss of the system, which plays the role of the imaginary part of the lattice potential. In addition, they utilized phase modulators to induce the hopping phase of the light in the mesh lattice in the time dimension, which plays the role of the real part of the underlying lattice potential^[247]. By designing suitable parameters in experiments, they realized shape-preserving beam transmission and non-Hermitian-induced transparency [see Fig. 14(a)]^[247].

The specific design of the intensity modulations and phase modulations can also be used to explore Dirac mass with optical gain and loss included^[51]. Yu et al. used the synthetic time lattice to experimentally reveal phenomena related to Dirac masses. In their design, by carefully tuning the gain/loss parameter g, they build a massive Dirac cone, following the effective Dirac Hamiltonian^[51]Heff=M(g)σ1+12vDkσ2−vD(ϕ−π)σ3,(69)where σ1,2,3 are the Pauli matrices, k is the quasimomentum, ϕ is a tunable phase parameter, M(g)=cosh(g−1) is the real Dirac mass, and vD is the Dirac velocity. The scalar potential barrier formed by appropriately engineered phase modulation is added into the synthetic time lattice, and Klein tunneling of massive Dirac quasiparticles is experimentally observed. Furthermore, with proper design of the gain and loss of the system, the sign flip of the mass is also engineered to realize control of time reflection and refraction [see Fig. 14(b)]^[51].

The dissipation-induced non-Hermicity in time-multiplexed networks in two loops can be further adapted to reveal novel Anderson localization phenomena, where the disorders are introduced from the modulators^[242]. The resulting stochastic dissipation leads to the breakdown of the Hermitian Anderson model, where dynamical delocalization and spectral localization are concurrently observed in the experiment [see Fig. 14(c)]^[242].

Thermodynamics, with temperature as an important physical quantity, is significant in many physical processes including superconductivity^[308,309], photovoltaic module electrical performance^[310], and soil carbon decomposition^[311]. The photonic synthetic time lattice is capable of simulating phenomenon involving thermalization^[50], where Joule expansion effects and isentropic expansions-compressions can be experimentally simulated at a negative temperature. The underlying mapping to the thermodynamic process under negative temperature is summarized as follows.

The two-loop setup for the synthetic time lattice is also capable of simulating quantum walks in experiments. The coupler between two loops can play the role of the coin, and the coupler ratio acts as the probability of getting two states of the coin, while the pulse propagation in short (long) loops corresponds to the left (right) movement of the walker^[314]. Regensburger et al. experimentally used a two-loop setup to reveal the ballistic spreading of discrete-time quantum walks^[253]. Numerous phenomena are shown in the synthetic time dimension based on the two-loop setting, for example, the non-Hermitian-potential-induced invisibility^[315] and unconventional anomalous topology emerging during the discrete quantum walk^[289].

Random walks present completely distinct behaviors for quantum and classical particles, and Longhi et al. used traps in the synthetic lattice to distinguish the difference between these two realms [see Fig. 16(a)]^[316]. With a similar construction to build a synthetic time lattice in two-loop resonators [see Fig. 10(b)], the discrete-time equations for the light dynamic in the loops can be described as^[316]{unm+1=[cos(β)un+1m+isin(β)vn+1m]exp(−iϕnm−γn),vnm+1=[cos(β)vn−1m+isin(β)un−1m],(72)where β defines the coupling strength of the fiber coupler, ϕnm is the modulation phase induced by the phase modulator, and γn is the loss induced by the amplitude modulator that can mimic the trap at site n. The design for the quantum random walk can be realized by setting ϕnm=0, which indicates a coherent walk, while for the classical random walk, the modulation phase is randomly distributed in the range (−π,π), which indicates an incoherent walk. Four traps are distributed in the 1D synthetic time lattice at different sites with loss rates being γn/J=1, 0.4, 0.5, and 0.6, where J=0.5cosβ. The simulation results show different behaviors of a photon in the two situations. For the classical random walker, the photon is annihilated by the dissipative traps and the walk process is terminated [see Fig. 16(b)], while for the quantum random walk, the photon can tunnel out of the traps and continue the walk process [see Fig. 16(c)].

In this section, we give a comprehensive summary of using time-multiplexed pulses to construct the concept of a synthetic time dimension. The arrival time of pulses in the optical loop(s) is used to label each synthetic lattice site while delay lines added in the single loop or two loops with a length difference are used to introduce connectivity between different sites, so the tight-binding lattice model is built along the temporal positions inside each round trip while the model evolves along the multiple round trips. In the single loop framework, we review frontier research progress, such as the Ising model, dissipative photonics, quantum computation, and quantum walks. On the other hand, in the two-loop framework, we discuss some remarkable physical effects therein, e.g., topology, non-Hermitian effects, thermodynamics, and quantum walks vs classical walks. However, there are many other works that we do not get into the details of here, such as Landau-Zener tunneling ^[317,318], sampling problems^[319], quantum state processing^[320], Kapitza light guiding^[321], Tera-sample-per-second arbitrary waveform generation^[322], and quantum walks in the synthetic time dimension based on a sequence of birefringent α-barium borate crystals^[323]. Furthermore, localization features of physical models have been studied in the synthetic time dimension^[324], such as dynamic localization^[325–327], Anderson localization^{[246,328,329]}, and quasicrystals^[245]. Hydrodynamic phenomena^[330] and superfluidity of light^[331] have been studied and trigger further interest to explore the dynamics of fluids of light based on a time-multiplexed mesh lattice. Nonlinearity in the synthetic time lattice^[332,333] has also been explored to seek potential applications in soliton-based gas sensing^[334] and optical neural networks^[254,335]. Some counterintuitive phenomena have been experimentally simulated by taking advantage of this unique platform. For example, Wimmer et al. demonstrated the optical diametric drive acceleration by breaking the action-reaction symmetry^[336]. In general, the synthetic dimension based on time-multiplexed pulses can fulfill promising functionalities covering fundamental physics and possible applications^[252].

The synthetic dimension is widely studied in the atomic systems where atomic states are used to build the extra dimensionality and the connectivity is induced by external lasers and electromagnetic fields (radio frequency and microwave). The field of atomic synthetic dimensions includes many efforts and approaches spanning low-energy atomic states, laser-excited metastable states and Rydberg states, discrete states of atomic motion, and even synthetic dimensions by parametric encoding. In this section, we discuss the former three topics, which mainly focus on the use of a discrete degree of freedom in the cold atom platform. We review the basic conceit underlying each of these approaches, discuss considerations relevant to the study of interactions and many-body physics, and review key progress and results from the past decade. Related efforts on superradiant states of neutral atoms, ultracold molecules, and parametric synthetic spaces will also be touched on here and in later sections.

We take the Rb87 model as an example from Ref. [83] to briefly introduce the idea of using the intrinsic atomic states to build the synthetic dimension. In many ways, the picture of using laser-coupled or Raman-coupled atomic states to encode a synthetic dimension was a natural extension of previous work on spin-orbit coupled quantum gases^[337–341]. As considered in Ref. [83] and illustrated in Fig. 17(a), atomic moving in a hybrid 2D lattice involving real-space and synthetic-space motion may be realized by first confining atoms in a 1D optical lattice, shown along the x-axis with a lattice constant of a=λ/2 and for a lattice laser wavelength of λ=1064nm. A moderate lattice depth of Vlat=5EL (where EL=ℏ2kL2/2M is the lattice recoil energy, with kL=2π/λ the recoil momentum and M the atomic mass) is chosen to validate the tight-binding approximation while still enabling appreciable hopping in real space, characterized by a nearest-neighbor tunneling amplitude of t=0.065EL. Considering the ground hyperfine (F=1) manifold, a small bias magnetic field with strength B0 defines a set of three magnetic sublevels |mF=0,±1⟩ [see Fig. 17(b)] that are Zeeman-split in energy by ℏω=gFμBB0, where μB is the Bohr magneton and gF is the Landé g factor. The MHz-level Zeeman-splitting for Gauss-level B0 fields would far exceed any other energy scales (t, thermal energies, or interaction energies) in the system, but connectivity between the bare atomic states can be realized by Raman transitions via a pair of coherent laser fields, where the laser fields can be expressed via an effective magnetic field^[83]: ΩT=δez+ΩR[cos(2kRx)ex−sin(2kRx)ey],(73)where δ is the detuning, kR=2πcos(θ)/λR is the Raman recoil wave vector, θ is the angle deviating from the x-axis, λR is the wavelength of the Raman lasers, ΩR is the Raman coupling strength, and ℏ=1 is taken. The effective atom-light Hamiltonian is^[83]Hal=ΩT·F=δFz+(F+eikRx+F−e−ikRx)ΩR/2,(74)where F±=Fx±iFy can lift/lower the atomic state |m⟩, e.g., F+|m⟩=gF,m|m+1⟩, where m=−F,…,F is the label of the sublevels and gF,m=F(F+1)−m(m+1). The x-dependent hopping phase can then generate Peierls phases when atoms move along the m direction. The combination of the 1D optical lattice in the x direction and the synthetic dimension along intrinsic atomic states can yield a hybrid 2D lattice [see Fig. 17(c)], and the corresponding 2D Hamiltonian with δ=0 is^[83]H=∑n,m(−tan+1,m†+Ωm−1e−i2kRanan,m−1†)an,m+h.c.,(75)where n is the spatial index, Ωm=ΩRgF,m/2 is the corresponding synthetic hopping strength, and an,m† and an,m are the creation and annihilation operators, respectively. One convenient feature of the synthetic dimension is that “site-resolved” detection along that axis can be achieved relatively simply, using techniques such as the Stern-Gerlach separation of magnetic sublevels.

Thus, the spin-orbit-coupled atomic gas in a superimposed stationary lattice can naturally be interpreted as a Harper-Hofstadter-like Hamiltonian in a hybrid real/internal space, where θ and the ratio λ/λR define the gauge field (magnetic flux per lattice plaquette). This approach is naturally extensible to large-spin atoms^[342]. Importantly, however, interactions along the synthetic dimension are sensitive to the details of specific scattering lengths for same-spin and mixed-spin configurations. While the interactions are easy to interpret (but nearly all-to-all) for Rb87 and alkaline earth (and alkaline earth-like) atoms with SU(N) symmetry, they can be quite random^[343] and potentially exothermic^[344] for generic multi-level atoms. To note, the use of metastable “clock” states as a small (two-site) synthetic dimension has also been explored^[345–348], but interactions are typically quite complicated in this approach as well^[349,350].

Because of the complicated nature of interactions (either all-to-all or potentially inelastic) in neutral atom synthetic dimension experiments, it was motivated to consider the engineering of synthetic dimensions in systems that are violently inelastic when in contact^[351,352] but that can support strong and coherent interactions even when separated in real space^[353,354]. While originally envisioned in the context of ultracold polar molecules^[353,354], recent development has mainly occurred in the entirely analogous synthetic dimension of dipolar-interacting Rydberg levels of excited atoms^[16]. To note, both of these systems (molecules and Rydberg atoms) host many stable or quasi-stable internal states that can be coupled to one another by relatively strong electric dipole-allowed microwave transitions. Both the Rydberg array and the polar molecule array platform had demonstrated coherent dipole-dipole interactions^[355,356] at the time that the ideas for synthetic dimensions with dipolar spin arrays^[353,354] were developed, and earlier work had demonstrated tight-binding-like dynamics^[357,358] in both systems as well.

We emphasize here that for the synthetic dimensions of (lattice-pinned) molecules and dipolar Rydberg atoms, one can simply think of these systems as dipolar spin models with rather exotic external (transverse and longitudinal) fields relating to the microwaves that induce direct state-to-state transitions to mimic “hopping” in the synthetic dimension. Unlike in the hybrid 2D lattice (real and synthetic dimensions) of spin-orbit-coupled gases described earlier, the excitations in Rydberg synthetic dimension experiments only experience single-particle hopping in the synthetic space, with no hopping of the actual particles in real space (ignoring the cases of itinerant molecules or Rydberg-dressed atoms).

Here, we focus on the case of Rydberg synthetic dimensions, which utilize the valence electron of Rydberg atoms as the constituent particles. In contrast to conventional low-lying atomic states, the Rydberg states refer to the valence electron of atoms being excited about high principal quantum numbers. These states feature a large distance between the electron and the positive atomic core [see Fig. 17(d)]^[359,360], imbuing these states with exaggerated properties (large polarizability, large dipole-dipole interactions) and a lifetime that far exceeds typical laser-excited levels. The arrangement of the valence electron and the atomic core is similar to the hydrogen atom, and the modified Rydberg formula to characterize the energy of such Rydberg levels is En,l=−Ry(n−δl)2,(76)where Ry≈13.6eV is the Rydberg constant, n is the principal quantum number, l is the orbital angular momentum of the state, and δl is the quantum defect that accounts for the Hartree interaction shift of the valence electron with the core electrons^[361,362]. In addition to their principal quantum number n, orbital angular momentum l, and its projection ml, the Rydberg levels are characterized by their total angular momentum J and its projection mJ, and all together the levels are split by fine-structure as well as Zeeman splittings in the presence of a magnetic bias field (considering the case of zero electric fields). Spontaneous decay and blackbody-induced decay and transitions can be mostly ignored on short (few microseconds) timescales, and dipole-allowed microwave transitions between Rydberg levels can be used to explore fundamental transport phenomena in synthetic tight-binding models^[363,364]. In a finite bias field, all state-to-state transitions are associated with a unique microwave frequency, such that all hopping terms in the synthetic dimension can be individually controlled due to their spectroscopic isolation.

We briefly review the general approach to Rydberg synthetic dimensions, as explored in Refs. [360,363–367]. A comb of coherent microwaves that address (either resonantly or with controllable detunings) a number of state-to-state transitions is utilized to engineer the effective connectivity along the synthetic dimension [see, e.g., Fig. 17(e) for the case of an SSH model]. Because each transition is separately controlled, the generic form of the single-particle Hamiltonian along the synthetic dimension is H=∑i=1(−ℏJi,i+1|i⟩⟨i+1|+h.c.)+∑i=1ℏδi|i⟩⟨i|,(77)for a simple one-dimensional geometry of state connectivities (where the Ji,i+1 terms are the tunable inter-state coupling amplitudes, which can have controllable complex phases, δi are effective site energies introduced via transition detunings, and ℏ is Planck’s constant). Because of the large state space available and the ability to drive multi-photon processes, there are many opportunities to engineer longer-range hopping terms and even higher-dimensional lattices within the Rydberg synthetic dimension. The physics of the engineered synthetic lattice, and in particular the phenomena that arise due to dipolar interactions between Rydberg atoms, can be explored either through spectroscopy of the microwave-dressed states as in Ref. [364] or by studying the microwave-driven population dynamics within the set of Rydberg levels (after first populating a particular Rydberg level by laser excitation) as in Refs. [360,363,365–367].

One of the most attractive aspects of the Rydberg synthetic dimensions platform is the existence of strong and stable dipole-dipole interactions that enable the exploration of many-body phenomena. These experiments also benefit from the rapidly expanding capabilities related to the preparation, control, and detection in atom array experiments. At present, however, one of the practical difficulties associated with Rydberg synthetic dimensions is the inability to perform high-fidelity simultaneous detection of many Rydberg levels in an atom-resolved fashion. High-fidelity and atom-resolved detection of individual Rydberg levels has been achieved through standard approaches in tweezer arrays^{[363,366,367]} and, complementing this, fully state-resolved detection via sequential field-induced ionization has been achieved for single atoms^{[360,364,365]}. But the combination of these capabilities for full and high-fidelity state readout is still outstanding.

In the previous two examples, the synthetic dimensions occur in an internal degree of freedom that is almost entirely distinct from (and in addition to) the real-space dynamics, given the huge energy separation between the internal and external degrees of freedom. Those approaches can thus be used to explore higher-dimensional physics by combining real-space and synthetic lattices. A bit different from those approaches, one can also consider using momentum states or generic motional states of atoms as a discrete degree of freedom to form an effective tight-binding model. In this case (similar to the works in photonics^[37]), the synthetic dimension replaces a real dimension, but one still benefits from the ability to induce the connectivity between states in the synthetic dimension through laser-driven transitions (so as to engineer complex hopping terms, e.g.). One approach to engineering synthetic dimensions based on laser-coupled momentum states of neutral atoms was considered by Gadway et al., where the presence of the quadratic free-particle dispersion of massive atoms leads to a unique transition frequency for two-photon Bragg transitions (and Raman-Bragg transitions) between states in the synthetic dimension^[368]. To note, the idea to utilize discrete momentum states of atoms as a landscape for exploring tight-binding transport phenomena was highly influenced by the body of research on kicked rotor experiments with cold atoms^[369–375] and the use of the quadratic dispersion to enable the control of complex hopping was highly influenced by contemporary work on Raman- or modulation-assisted hopping in biased real-space optical lattices^[376–378].

Roughly speaking, the momentum state (or space) lattice (MSL) technique can simply be thought of as connecting a set of discrete plane-wave momentum states of ultracold atoms (as derived from a Bose-Einstein condensate) through a set of coherent two-photon Bragg transitions^[379,380]. Because the atomic waves have mass and the free-particle dispersion is quadratic, the energy difference between adjacent pairs (or nearest neighbors) of momentum states is unique, reflecting the Doppler shift of the Bragg transition. Unlike in kicked rotor experiments, which apply short (in time) but strong lattice pulses to atoms, one can operate at weak Bragg coupling and utilize this momentum-dependence of the Bragg transition frequency to achieve a unique control over all of the site-to-site connectivity terms. This is similar to the case of Rydberg and molecule synthetic dimensions, which were heavily inspired by the momentum lattice technique.

We now review the formalism behind the MSL technique. We consider a pair of counter-propagating laser beams applied to atomic waves having mass M, where the left-traveling field contains a set of discrete frequency components and the right-traveling field contains a single frequency component [see Fig. 17(f)]. For a system of two-level atoms with ground state |g⟩ and excited state |e⟩ [see Fig. 17(g)], the interaction process between light and atoms gives the following Hamiltonian after considering the dipole approximation^[368]: H=p22M+ℏωe|e⟩⟨e|+ℏωg|g⟩⟨g|−d·E,(78)where d=−|e|r is the dipole operator with r being the distance vector, p is the free-particle momentum, ℏωg(ℏωe) is the ground (excited) state energy, and E is the electric field of the driving lasers. The right-traveling field and the left-traveling field have the expressions as^[368]E+(x,t)=E+cos(k+·x−ω+t+ϕ+),(79)E−(x,t)=∑jEjcos(kj·x−ωjt+ϕj),(80)where x is the (1D) position, ω+ and ωj are the frequencies, k+ and kj are the wave vectors of the lasers, ϕ+ and ϕj are the phases of the driving lasers, and E+ and Ej correspond to the amplitudes of the respective components. The detuning between atomic resonance (ωeg=ωe−ωg) and laser frequencies is Δ≡ωeg−ω+≃ωeg−ωj. The resonant Rabi couplings between ground and excited states are Ω+=−⟨e|d·E+|g⟩/ℏ and Ωj=−⟨e|d·Ej|g⟩/ℏ. The Hamiltonian can then be re-written as^[368]$$\begin{gathered} H=p22M+ℏωe|e⟩⟨e|+ℏωg|g⟩⟨g| \\ +ℏ[Ω+cos(kx−ω+t+ϕ+)+∑jΩjcos(−kx−ωjt+ϕj)](|e⟩⟨g|+|g⟩⟨e|). \end{gathered}$$(81)By assuming that the single-photon detuning Δ far exceeds all the single-photon Rabi coupling terms (Ω+ and Ωj), tracing out the excited state, and making a rotating wave approximation, the Hamiltonian in Eq. (81) becomes^[368]$$\begin{gathered} Heff=∑n4n2Er|n⟩⟨n|+ℏ4Δ∑nΩ+Ωne−i(ω+−ωn)tei(ϕ+−ϕn)|n+1⟩⟨n| \\ +ℏ4Δ∑nΩ+Ωnei(ω+−ωn)te−i(ϕ+−ϕn)|n⟩⟨n+1|, \end{gathered}$$(82)where n is an integer to characterize the momenta pn=2nℏkx^ and Er=ℏ2k2/2M is the recoil energy. The introduced states |n⟩ relate to the motional, plane-wave-like states of ground state atoms that have momentum pn. The Hamiltonian can be further transformed into the interaction picture as^[368]$$\begin{gathered} HeffI=∑nℏΩ˜n2ei(2n+1)4Erℏte−i(ω+−ωn)tei(ϕ+−ϕn)|n+1⟩⟨n| \\ +∑nℏΩ˜n2e−i(2n+1)4Erℏtei(ω+−ωn)te−i(ϕ+−ϕn)|n⟩⟨n+1|, \end{gathered}$$(83)where Ω˜n=ΩnΩ+2Δ is the two-photon Rabi coupling. By setting ωn=ω+−(2n+1)4Er/ℏ and neglecting non-resonant components, the Hamiltonian can be simplified to^[368]HeffI=∑n(ℏΩ˜n2ei(ϕ+−ϕn)|n+⟩⟨n|+ℏΩ˜n2e−i(ϕ+−ϕn)|n⟩⟨n+1|).(84)Finally, by denoting the hopping amplitude as tn=ℏΩ˜n2 and the hopping phase as φn=(ϕ+−ϕn), the resulting Hamiltonian in the synthetic momentum lattice reads as^[368]HeffI=∑n(tneiφn|n+1⟩⟨n|+h.c.).(85)We can see that in Eq. (85), the coupling strength and the phase can be individually adjusted (and the ignored detuning terms can be used to engineer arbitrary site-energy landscapes^[381]), which provides high tunability in simulating various models for different aims.