Introduction

Dimensionality of a physical system is an essential determinant of its properties and of the dynamic processes that it can support. The three spatial dimensions in the physical world facilitate a rich variety of phenomena observable in condensed matter systems. However, theory has progressed in describing higher dimensional configurations that exhibit complexity well beyond what can be expected to be found in naturally occurring materials, and with dynamics that have no lower-dimensional counterparts. Nevertheless, a way to experimentally realize and explore models that exceed the apparent geometrical dimensionality of a system is provided by the synthetic dimension approach1, which was pioneered in the cold atom physics discipline2,3.

Simulating lattice structures and dynamic phenomena in the optical domain has emerged as a fruitful research field, with a particular focus given to topological photonics4. In addition to being a platform to explore fundamental phenomena, realizing photonic analogues of systems such as the 2D Harper-Hofstadter model, a lattice configuration associated with the 2D quantum Hall effect, exhibits application potential as a non-magnetic approach to enact optical isolation5. Investigations of this phenomenon exemplify how photonic synthetic dimensions enable the realization of higher-dimensional concepts in lower-dimensional devices—while landmark early demonstrations relied on extended 2D arrays of optical cavities6, a synthetic dimension approach was able to demonstrate the Hall ladder edge state component of the 2D Harper-Hofstadter model in a single resonant cavity by using two independent non-spatial variables of the photon7.

Photonic synthetic dimensions8,9 can be constructed by harnessing various properties of light such as polarization10, optical angular momentum11, time bins12, spatial mode structure13, and frequency14. A well-developed example of the latter approach are the discrete equidistantly spaced frequency modes in a ring resonator, that are conceptually straightforward to couple and partition into lattices by dynamically modulating the cavity15. Key milestones in ring cavity synthetic frequency dimension investigations include tight-binding model lattice14, 2D Hall ladder7, non-Hermitian band structure16, frequency dimension boundary17, and Su-Schrieffer-Heeger (SSH) model18 demonstrations. These early results were achieved primarily using benchtop-scale fiber loop setups with around 10 m length scale optical paths. This experimental platform, while benefiting from being straightforward to assemble from high performance off-the-shelf telecommunication wavelength components and instruments, is limited in its scalability due to susceptibility to interference from environmental perturbations. In light of this, miniaturization and on-chip integration for scalability has been identified among the most important challenges for synthetic dimension photonics19.

Efforts towards on-chip monolithic integration explored modeling thin-film lithium niobate20 as well as silicon21 based resonant electro-optic frequency combs as representing a tight-binding lattice model. Both of these photonic circuit platforms offer different advantages and challenges. For instance, silicon photonics has reached considerable maturity and possess a high refractive index contrast, allowing for a higher degree of miniaturization and integration. However, high-speed phase modulation in silicon typically relies on the plasma dispersion effect in doped waveguide sections, which, in addition to introducing significant propagation losses, is associated with spurious intensity modulation that can compromise the pseudo-Hermiticity of conventional balanced optical systems. Viability of alternative DC Kerr effect-based modulation in silicon22 is yet to be shown for synthetic dimension devices. Conversely, lithium niobate on insulator (LNOI)23, on the other hand, is at a comparatively earlier stage of technological development, however, it offers a broad spectral transparency ranging from visible to mid-infrared wavelengths, highly efficient, low loss and broadband electrooptic modulation, strong quadratic and cubic optical nonlinearities, as well as optical poling capabilities for quasi-phase matching. Furthermore, optical propagation losses reported in state-of-the-art optical lithium niobate23 are notably lower than those in silicon. These properties make LNOI particularly well suited for photon wavelength conversion, that is at the core of synthetic frequency dimension state coupling.

We report a synthetic frequency dimension realization on an integrated LNOI racetrack resonator equipped with a broadband traveling wave electrooptic modulator. By driving the device with a radio frequency (RF) signal oscillating at the cavity free-spectral range (FSR) of Ω1FSR = 9.52 GHz, a frequency dimension lattice spanning up to a 500 GHz bandwidth was observed. Tuning the modulation signals to rates multiple of the FSR value enabled the realization of different range couplings between disparate ring modes, and the corresponding synthetic band structures were confirmed by detecting the time-resolved ring transmittance. Arbitrarily reconfigurable frequency dimension lattices ranging in complexity from the basic 1D tight binding model and 2D triangular ladder, to a 3D chiral tube configuration are demonstrated on the same device by controlling the strength of concurrent 1×FSR, 2×FSR, and 3×FSR RF modulation signals. Furthermore, the band structure engineering and induction of non-reciprocity and photonic gauge potentials are realized by tuning the relative phase of the dynamic modulation signals. Advantages and limitations of the electronic modulation approach for on-chip synthetic frequency lattice generation and probing are explored and the unique opportunities presented by the LNOI platform to enable high-dimensional long-range state connectivity are discussed.

Results

Device design

The essential properties a synthetic frequency dimension device should possess include a set of low loss modes that can act as lattice sites for photons, and a perturbation mechanism that enables particle hopping between different lattice sites. A modulator-equipped ring resonator is a particularly expedient platform to explore dynamics along the frequency axis14. If the group velocity dispersion in the waveguide forming the ring is negligibly small, ring cavity mode spacing is equidistant, allowing for a set type of perturbation to act uniformly along the relevant span of the lattice and drive cascaded photon hopping. The perturbation mechanisms typically leverage electrooptic modulation20,21, although nonlinear wavelength conversion mechanisms are gaining prominence as a way to enact extended range coupling between spectrally distant modes24.

Images of our integrated add-drop racetrack resonator device fabricated on a X-cut LNOI chip platform are shown in Fig. 1a. The cavity span lengthwise was 6.8 mm with a total circumference of l ≈ 14.1 mm, which was selected to ensure a sufficiently small 9.5 GHz FSR that can be bridged by RF electronic modulation. This integrated realization represents a three order of magnitude decrease in optical path length compared to typical 13.5 m optical fiber loop setups14, albeit at the cost of a corresponding increase in the relevant process rates from megahertz to gigahertz scale. Sub-1 dBcm−1 propagation loss waveguides are fabricated by using a strip-loading approach, in which mode confinement is established by a 1 µm wide ridge patterned in a 300 nm thick silicon nitride film covering the likewise 300 nm thickness LiNbO3 layer25,26, resulting in a stack sketched in Fig. 1b. This enables the localization of most of the mode energy in the lithium niobate thin film without having to conduct technologically demanding and complementary metal-oxide-semiconductor (CMOS) process incompatible LiNbO3 pattern transfer. Figure 1c shows higher magnification images of fabricated waveguide sections.

Fig. 1: Imaging of the LNOI synthetic frequency dimension device.
figure 1

a Micrographs of the LNOI chip and a highlighted individual racetrack resonator with a traveling wave RF electrode. b Sketch of the SiN loaded waveguide modulator section. c Magnified micrographs of the highlighted device sections - traveling-wave modulator, Euler bend coupling section, and scanning electron image of the SiN strip waveguide loading.

Sufficiently high modulation bandwidths and a flat frequency response for inter-mode coupling was ensured by using a traveling-wave electrooptic phase modulator, patterned in a ground-signal-ground (GSG) coplanar waveguide (CPW) RF transmission line configuration27 as a 300 nm thickness gold layer along the entire length of one side of the racetrack loop to minimize the switching voltage Vπ. To this end the RF electrode with a 6 µm gap between the center signal electrode and ground planes are oriented perpendicular to the LiNbO3 crystal Z-axis to make use of the highest r33 Pockels tensor component, yielding a VπL = 8 V·cm. The simulated 3 dB electrooptic response bandwidth of the traveling-wave phase shifter is in excess of 100 GHz—well beyond the modulating frequencies used in our experiments—as illustrated by Supplementary Fig. 1. In addition, electrooptic modulators enabled by LNOI exhibit considerably lower optical propagation losses than the free carrier injection/depletion based silicon photonics p-i-n junction modulators21.

Experimental characterization

SiN-loaded waveguide racetrack resonator cavities shown in Fig. 1 were fabricated using electron beam lithography, whereas the modulator electrodes were patterned using a maskless laser lithography tool. Chip-to-fiber optical interfacing was realized by way of grating couplers, each with an approximate 4.5 dB insertion loss. The experimental optical transmittance spectral plot, measured on the resonator drop-port and shown in Fig. 2a, indicates that the fabricated device exhibits a 9.52 GHz FSR, approximately matching the design value. The bus waveguides interfacing with the ring were in the undercoupled regime, resulting in a drop-port extinction ratio of 14 dB. The quality factor of the loaded cavity was approximately Q = 180·103 in the λ = 1555 nm wavelength range used in our experiments. This corresponds to a photon decay rate of 2γ = 1.26 GHz. Optical losses are attributable to a combination of three primary factors: (i) absorption in the plasma-enhanced chemical vapor deposition coated SiN due to residual Si-H bonds, (ii) sidewall roughness of the etched SiN waveguides, and (iii) evanescent interaction between the waveguide mode and the metal electrode.

Fig. 2: Modulator-equipped ring cavity characterization.
figure 2

a Drop-port transmittance spectrum in the λ = 1555 nm wavelength region, encompassing a 20 FSR interval with an approximate 9.52 GHz mode spacing. b Optical frequency comb spectrum of the same LNOI ring resonator device pumped with 16 dBm power on-resonance optical excitation and RF modulation at FSR-matched 9.52 GHz frequency and 27 dBm on-chip power (OSA - Optical Spectrum Analyzer). Inset shows a schematic of the device.

The ring cavity resonant modes can be interpreted as frequency dimension lattice sites for photons. Simulating hopping of particles along such a lattice requires a wavelength conversion mechanism. This can be achieved by harmonic phase-modulation of the resonator cavity using the linear electro-optic effect, which is equivalent to a cascading sequence of sum- and difference-frequency generation interactions of an optical pump and an RF signal. Matching the modulation frequency to a multiple of the ring cavity optical roundtrip rate ensures that the modulation sideband intensity is resonantly enhanced. Figure 2b shows the output of the ring resonator device when it is resonantly pumped with 16 dBm optical input power and the modulator is driven by a 9.52 GHz 1×FSR frequency 27 dBm power RF signal. This enables random walk-type hopping of photons along the frequency lattice as they gradually propagate from the pump site m. The 500 GHz span of the comb-like spectrum indicated that an extended lattice was established, with up to 51 sites occupied by photons. The 2.3 dB spectral power drop at each line down the cascade is attributable to optical losses in the ring cavity and is representative of the dissipative nature of the continuously pumped system.

Versatile 1D tight-binding model lattice coupling

A key advantage of the synthetic dimension approach is that inter-state coupling is facilitated by an extrinsic perturbation, and therefore can be controlled in both extent and strength with a great deal of flexibility. The broadband nature of the traveling-wave electrooptic modulator in our ring resonator makes it possible to couple not only the Ω1 = 9.52 GHz spaced nearest-neighbor cavity modes, but also second- and third-nearest by respectively driving the device with 2×FSR Ω2 = 19.04 GHz and 3×FSR Ω3 = 28.56 GHz matched harmonic RF signals of the form V(t) = Vncos(Ωnt). Steady-state device output spectra, obtained at moderate 10 dBm on-resonance optical pumping and 20 dBm RF drive, are shown in Fig. 3a. In each case an extended photon random walk over a 400–600 GHz span lattice segment is observed. Insets illustrate how the ring resonator modes are partitioned by different frequency range couplings into unconnected 1D sublattices, only one of which in 2×FSR and 3×FSR cases is subject to pumping and populated by photons.

Fig. 3: Steady-state and time-resolved response of the cavity to different FSR range modulation.
figure 3

a Optical frequency comb output of the resonantly pumped resonator driven at 1×FSR (red lines), 2×FSR (green lines), and 3×FSR (blue lines) RF frequencies, representing different partitioning of the frequency lattice, sketched in the inset. Gray-shaded bars represent 1D tight-binding model simulations. b Corresponding time-resolved modulated ring cavity transmittance spectra.

The photon dynamics in our device can be described by using a tight-binding model Hamiltonian by reinterpreting photons circulating in a ring cavity as lattice site-bound particles, and the electrooptic effect induced three-wave-mixing of optical field and different frequency and strength RF modulating signals as particle hopping to other lattice sites with rates that are defined by proximity21. Neglecting pumping and loss, in a 1D case such a Hamiltonian has the form:

$$H={\sum}_{m}\varepsilon {\hat{b}}_{m}^{{{\dagger}} }{\hat{b}}_{m}-{\sum}_{m}{J}_{\eta }\left({\hat{b}}_{m}^{{{\dagger}} }{\hat{b}}_{m+\eta }+{\hat{b}}_{m+\eta }^{{{\dagger}} }{\hat{b}}_{m}\right),$$
(1)

where \({b}_{m}\) is an operator for particle creation at site m, ε is the on-site potential, \({J}_{\eta }\) is the hopping rate, and index η represents the hopping distance. The optical pumping and dissipation in the ring resonator can be accounted for by using mode-dependent loss rate γm, and continuous monochromatic light injection fm(t) eiωt terms5. The equation of motion for the cavity field in the \( {\beta }_{{m}}(t)\equiv \langle {\hat{b}}_{{m}}(t)\rangle\) rotating frame is given by28:

$$i\frac{\partial }{\partial x}{\beta }_{m}=-{J}_{\eta }{\beta }_{m+\eta }-{J}_{\eta }^{* }{\beta }_{m-\eta }-i{\gamma }_{m}+{f}_{m}\left(t\right).$$
(2)

Gray bars in the three different coupling range cases in Fig. 3a represent steady-state simulations according to Eq. (2), and show that the 1D tight-binding model is in close accordance with experimental observations, assuming γ/J1 ≈ 1.4, γ/J2 ≈ 2.1 and γ/J3 ≈ 4.2 loss to coupling strength ratios.

The dynamics of the modulated ring cavity in 1×FSR, 2×FSR, and 3×FSR coupling range cases are revealed by measuring time-resolved spectral transmittance 2D plots encompassing one device FSR, shown in Fig. 3b. They directly illustrate how the harmonically driven phase modulator periodically changes the cavity optical path length, and therefore the resonance frequency, and show the slight drop-off in modulation strength achievable at higher RF driving rates.

Band structures of 1D lattices

The range of allowed energy states in a system as a function of the wavenumber k, plotted as band structures, provide a descriptive overview of the expected behavior of a synthetic dimension lattice just as in condensed materials. For a photonic frequency dimension device, the way to experimentally investigate the synthetic band structure was demonstrated on a macroscale fiber loop setup14 and was shown to relate to the time-resolved transmittance of a modulated ring cavity. Indeed, the reciprocal space for a frequency dimension is time, and the Brillouin zone can be represented as a single optical round-trip period, which for our LNOI ring cavity is T ≈ 100 ps. The energy axis is represented by optical pump frequency detuning Δω.

Figure 4a shows the experimentally measured synthetic frequency dimension band structures for our LNOI integrated device, representing 1×FSR, 2×FSR, and 3×FSR range 1D lattice coupling cases, acquired using moderate RF modulation strengths. Three Floquet bands, matching three ring cavity free-spectral ranges, are shown on each energy axis. Conversely, when the steady-state optical transmittance spectra of a dynamically modulated ring resonator are acquired, the result is equivalent to time-averages of the band structure. This represents the photonic density of states of a synthetic dimension lattice14, which for the three different modulation frequency cases are plotted next to band structures in Fig. 4a. Details of the measurement setup are provided in Supplementary Fig. 2.

Fig. 4: Synthetic band structure measurements of 1D tight-binding models LNOI ring cavity.
figure 4

a Synthetic frequency dimension bands, as well as time-averaged transmission corresponding to density of states, measured for 1×FSR (J1 [GHz] – red lines/markers), 2×FSR (J2 [GHz] – green lines/markers) and 3×FSR (J3 [GHz] – blue lines/markers) coupling range cases. Superimposed plots are 1D tight binding model dispersion curve εk = 2Jncosn fits. b Coupling strength-related fitting parameter \({J}_{\eta }\) dependence on on-chip RF power. Curves represent 9.52 GHz, 19.04 GHz, and 28.56 GHz FSR-matched modulation frequencies. Curves in (a) represent highest modulation power points in (b).

Nearest-, next-nearest-, and second-next-nearest-coupling lattice cases differ in their band structure representations by the number of driving modulation swings that re-occur within one optical cavity mode round-trip. This is in accordance with the 1D tight-binding model dispersion relationship:

$${\varepsilon }_{k}=2{J}_{{{{{n}}}}}\cos (k{n\varOmega }_{{{\rm{R}}}}),$$
(3)

where k is the quasi-momentum, \({J}_{n}\) denotes the coupling strength, and ΩR is the cavity optical round-trip rate that is matched by an integer multiple modulation frequency for the respective lattice hopping range indexed by n. Such dispersion curve plots are overlaid onto the experimental band structures in Fig. 4a and exhibit a strong agreement. While Fig. 4 only depicts the band structure plots measured under strongest modulation conditions for each driving RF signal, additional plots for lower coupling strengths \({J}_{n}\) all the way down to zero (flat bands) are given in Supplementary Figs. 35.

The coupling strengths \({J}_{n}\) scale with RF driving amplitude, as shown in Fig. 4b, and this scaling remains consistent between different modulation frequencies. This illustrates the broadband nature of the LNOI traveling-wave modulator. However, the maximum coupling strengths achievable for 2×FSR (J2 = 0.526 GHz) and 3×FSR (J3 = 0.425 GHz) range interactions are nevertheless lower than for 1×FSR (J1 = 0.759 GHz) due to a drop in modulating signal power that reaches the chip, likely due to a combination of factors that include efficiency loss of the broadband RF amplifier towards its 40 GHz bandwidth limit and increasing RF cable losses at high frequencies. It must be noted that the maximum modulation strengths given in Fig. 4 are limited by the RF amplifier power saturation level, and considerably higher coupling strengths can be realized on LNOI due to its high RF power handling capability, with 27 dBm (0.5 W) successfully tested on our devices as shown in Fig. 2b. This is markedly higher than was observed for a Si photonic synthetic frequency dimension device, which capped at approximately 18 dBm21.

Lattice engineering using synchronized coupling

The range of phenomena that can be simulated in a 1D lattice is rather limited, typically involving some type of random walk or Bloch oscillation behavior20,29. A considerably richer set of effects are possible in higher-dimensional systems; however, their realization requires a commensurate increase in the interconnectivity between sites in a lattice. Perturbation-induced coupling underlying the synthetic dimension approach is unique in its versatility in simulating complex lattices that can be changed and reconfigured in real-time.

Our modulator-equipped LNOI ring resonator can be made to host an effectively N-dimensional synthetic frequency lattice by simultaneously establishing lattice coupling over N distinct mode spacings. The degree of connectivity available to each individual frequency mode scales as 2 N. Dispersion relations for a tight-binding model with a complex coupling configuration can be expressed as:

$${\varepsilon }_{k}={\sum }_{n}^{N}2{J}_{{{{{n}}}}}\cos \left({kn}{\varOmega }_{{{\rm{R}}}}+{\phi }_{n}\right),$$
(4)

where \({J}_{n}\) is the coupling strength parameter for lattice hopping range indexed by n, whereas ϕn is the hopping phase a photon accrues as it translates through this lattice connection. This hopping phase is key in controlling photon transport through lattices, as it allows the bosonic emulation of gauge potentials like the electric and magnetic fields. Inducing an electric field analog along a synthetic frequency dimension is possible even in the 1D case by straightforward RF modulation frequency detuning from the optical round-trip rate ΩR30. On the other hand, an effective magnetic field relies on interference of photons with different phases accumulated via alternate propagation routes31, hence requires at least quasi-2D lattice connectivity.

Conceptually and technically simplest is the 2D case when nearest and next-nearest lattice site coupling is used, as summarized in Fig. 5a–d. By applying a composite RF modulation voltage signal of the form Vm(t) = VΩcos(ΩRt) + V2Ωcos(2ΩRt + ϕ2), illustrated in Fig. 5a, comprised of 1×FSR (ΩR = 9.52 GHz) and 2×FSR (2ΩR = 19.04 GHz) frequency components, ring resonator modes are partitioned as shown in Fig. 5b into a triangular ladder lattice, sketched in Fig. 5c. Such a lattice structure was shown to exhibit an effective gauge potential in both fiber loop14 and integrated21 synthetic frequency dimension devices by controlling the relative phase ϕ2 of the two modulating signals, which manifests as an alternating magnetic flux piercing the triangular plaquettes.

Fig. 5: Band-structure engineering using different RF modulation-induced frequency dimension coupling configurations.
figure 5

ad 2D triangular ladder lattice constructed by 1×FSR (red lines) and 2×FSR (green lines) coupling. eh Quasi-3D chiral ribbon established by 1×FSR (red lines) and 3×FSR (blue lines) coupling. il 3D tube-lattice connected using 1×FSR (red lines), 2×FSR (green lines), and 3×FSR (blue lines) coupling. In each case, (a, e, i) panels show the modulating signal components with phase shift notation, (b, f, j) ring frequency mode coupling and (c, g, k) effective tight-binding lattice structure schematics, as well as (d, h, l) experimental synthetic band structures. Variation of modulation phase delays ϕn induces gradual band structure changes. Superimposed red curves are the respective tight-binding model dispersion curve εk(k) fits.

This effective gauge potential is directly observable in the experimental band structure plots in Fig. 5d, where one half of the FSR spectral span and one Brillouin zone is shown for clarity. Overlaid fits of Eq. 4 show that the dispersion curves are in agreement with experimental data, assuming 1×FSR and 2×FSR coupling strengths of J1 = 0.51 GHz and J2 = 0.28 GHz, with a J2/J1 ≈ 0.5 ratio. The plots show band structures that represent different effective gauge potentials, produced by the tunable phase offset ϕ2 between the two synchronized modulation signals. Of note are the ϕ2 ≈ π/2 and 3π/2 phase offset cases, in which the band structure is asymmetric relative to the k = 0 point on the quasi-momentum/time axis, indicating that these modulation regimes induce time-reversal symmetry breaking for photons.

An alternative 2D coupling scheme can be realized by combining 1×FSR (ΩR = 9.52 GHz) and 3×FSR (3ΩR = 28.56 GHz) modulation, as summarized in Fig. 5e–h. Here the nearest-neighbor coupling induced chain of frequency states can be interpreted as being wound up into a spiral, every third site on which is further connected to each other by the longer range coupling term along three parallel directions. The overall lattice structure, despite being fundamentally 2D, is reminiscent of a quasi-3D chiral ribbon. Fitting using Eq. 4 shows that 1×FSR and 3×FSR tight-binding model coupling strengths were J1 = 0.50 GHz and J3 = 0.18 GHz, yielding a J3/J1 ≈ 0.4 ratio which was primarily limited by the lower achievable on-chip power of higher frequency signals due to the frequency dependent nature of the RF amplifier gain and RF cable loss. The modulation phase offset term, therefore the synthetic gauge potential, in this case can be taken as ϕ3, again assuming a 1×FSR phase term ϕ1 as a zero-reference point. Band structure plots in Fig. 5h reveal similar control over the synthetic magnetic field analog that can be leveraged to facilitate time-reversal symmetry breaking.

The quasi-3D tight-binding model lattice, shown in Fig. 5i–l, was realized by simultaneously combining all three modulation signals − 1×FSR (ΩR = 9.52 GHz), 2×FSR (2ΩR = 19.04 GHz), and 3×FSR (3ΩR = 28.56 GHz). The lattice structure associated with this coupling configuration is reminiscent of the ribbon in the 1×FSR and 3×FSR case of Fig. 5g, with additional hopping channels added to form a chiral tube-like structure. The tight-binding model dispersion curve fitting shows a match with experimental plots for J1 = 0.34 GHz, J2 = 0.24 GHz, and J3 = 0.21 GHz coupling strength values. With two relative modulation signal phase values ϕ2 and ϕ3 (each defined relative to ϕ1 = 0), a higher degree of control over the synthetic frequency dimension band structure can be exerted. For example, dispersion plots in Supplementary Fig. 6, acquired by fixing ϕ1 = ϕ2 = 270°, remain non-reciprocal through the entire ϕ3 tuning range. By manipulating both ϕ2 and ϕ3 relative phase components, full control over a magnetic field analogous photonic gauge potential can be exerted in a LNOI integrated device.

Steady state optical frequency comb measurements indicate that the LNOI ring resonator device is responsive to even higher 4×FSR (4ΩR = 38.08 GHz) RF modulation (shown in Supplementary Fig. 7), making lattice models with higher dimensionality possible. However, time-resolved band structure acquisition with comparative fidelity shown in Figs. 4 and 5 was not possible due to bandwidth limitations of the oscilloscope used in our experiments.

Discussion

We designed and fabricated an LNOI integrated synthetic frequency dimension device and experimentally demonstrated its capability of simulating a 3D tight-binding lattice on what in real space terms is conceptually a zero-dimension structure. The broadband and efficient electrooptic modulation in thin-film lithium niobate enabled versatile coupling between 9.52 GHz FSR-spaced ring cavity modes. Although our SiN-loaded LNOI waveguide device exhibits higher propagation loss compared to state-of-the-art demonstrations using an etched LNOI waveguide platform20, moderate quality factors can be preferrable when direct electronic detection of modulated cavity transmission transients for synthetic dimension band structure reconstruction is required. Such experiments are generally limited by photodetector and oscilloscope bandwidths, which can be exceeded if pulses transmitted by a driven cavity become too short, resulting in signal distortion. Similarly, our inter-mode hopping range was capped to under n = 3 by the limitations of RF drive and detection electronics instead of intrinsic device characteristics. Direct time-resolved band structure measurements confirmed different tight-binding model lattice connectivity schemes, including triangular ladder, chiral ribbon, and tube structures. In addition, full control over a synthetic analog to a magnetic field gauge potential is shown in these frequency space lattices. While it must be stressed that non-trivial synthetic gauge are not expected in the single-band lattice configurations shown in this work, such modulated ring resonator devices on the LNOI platform represent promising building blocks for higher complexity integrated circuits for both fundamental modeling of topological and quantum systems, as well as for technologically relevant uses in optical isolation5 and photonic computation32.

The foremost advantage of photonic integration in enabling the synthetic frequency dimension approach is the substantially higher complexity of circuits in which photons can be made to maintain coherence against environmental perturbations, which greatly limit the scalability of analogous fiber loop setups14. Also worth mentioning are the roughly three order of magnitude shorter chip-bound optical paths that allow for substantial device miniaturization and broaden the range of potential application areas. However, the reduced device scale comes with additional challenges, primarily related to the commensurate increase in process rates. Indeed, fiber loop synthetic frequency devices driven at megahertz frequencies14 need to be driven at gigahertz frequencies once practicably realized in an integrated format21. Given that widely accessible RF signal generation and detection equipment generally has bandwidths not exceeding 65 GHz, full potential for exceptional levels of synthetic dimension connectivity exhibited by high-performance electrooptic material platforms like lithium niobate with reported modulator bandwidths beyond 100 GHz27, remain untapped.

Beyond optimization of technical approaches, further development of ring resonator based integrated synthetic frequency dimension devices is likely to move along at least three directions: (i) ring cavity mode engineering and synthetic dimension boundaries, (ii) higher complexity lattice models in multi-ring configurations, and (iii) harnessing nonlinear effects. Boundaries are essential to topological physics due to the bulk-edge correspondence. While natural in real space, abrupt terminations are not typical for modes in a resonant cavity. Furthermore, one of the limitations in the possible mode partitioning demonstrated in this work is that all couplings proceed as uninterrupted chains. This makes the attainable latices have a chiral character. However, if, for example, every third mode could be disrupted in a way that would make coupling to it inefficient, applying synchronized 1×FSR and 3×FSR signals would result in a square synthetic frequency lattice akin to the Hofstadter model7. Proposals of creating mode splitting defects in a ring resonator by strongly coupling to an auxiliary ring have been reported using fiber loop experiments17 as well as LNOI-based on-chip devices33,34. Furthermore, mode splitting in symmetrical pairs of strongly coupled modulated ring resonators has been harnessed to realize the staggered Su–Schrieffer–Heeger configuration18. Conversely, a pair of two weakly coupled modulated rings without substantial mode hybridization is a step towards an extended 1D array realization of the 2D Harper-Hofstadter model5. Lastly, in addition to efficient electrooptic response, the LNOI platform possesses a strong quadratic χ(2) nonlinearity, which, together with periodically-poled lithium niobate-enabled quasi-phase matching, or polling-free modal phase-matching35 and low process threshold due to sub-wavelength scale waveguide mode confinement, shows promise for all-optical long-range frequency dimension state coupling24. The versatility of the LNOI platform is likely to be pivotal in facilitating all these advances and unlocking the full potential of the synthetic dimension approach in an integrated photonic chip format.

Methods

Device fabrication

Ring resonator integrated photonic circuits were made on NanoLN X-cut LNOI wafer chips with 300 nm thickness lithium niobate thin film and a 4.7 µm buried oxide layer on a 500 µm Si carrier, purchased from Jinan Jingzheng Electronics Co., Ltd. The likewise 300 nm thickness near-stoichiometric silicon nitride optical loading film was deposited on the wafer using reactive sputtering. Loaded waveguide structure exposure was performed using a Raith EBPG5000plusES electron beam lithography tool, and the resultant pattern was transferred across the entire depth of the SiN film using an Oxford Instruments PLASMALAB100 ICP380 reactive ion etcher. The modulating coplanar waveguide (CPW) RF electrodes were created by way of a lift-off process. The design was patterned onto the photonic waveguide circuit using Heidelberg Instruments Maskless Aligner MLA 150 laser lithography tool. The 10 nm Ti adhesion, and 300 nm Au conductive layer metal film deposition was performed using a Kurt J. Lesker eKLipse electron beam evaporator physical vapor deposition system.

Steady state characterization

Drop port transmittance spectra measurements were done using a telecommunications C-band Agilent 8164 A excitation laser, equipped with an 81635 A optical power meter module. Light was coupled into and out of the LNOI device plane using grating couplers by way of a fiber array. Steady-state frequency comb spectra were detected using a Finsair WaveAnalyzer(TM) 1500 S optical spectrum analyzer. RF driving for electrooptic characterization of the device was performed using continuous wave sinusoidal signals supplied by an Anritsu MG3694A generator with a 40 GHz maximum output frequency and 12 dBm maximum power, further boosted using a CTT Inc. ASN/105-2942 narrowband amplifier. Conversely, multi-dimensional lattice coupling experiments used an Keysight M8195A 65GSa/s arbitrary waveform generator, connected to a SHF Communication Technologies AG SHF 830 P 14 dB gain 35 kHz–40 GHz frequency range broadband amplifier. Electronic interfacing with the traveling wave modulator metal contact pads on the integrated device was done using a pair of GSG RF probes, which were connected to the opposing ends of the 6.4 mm long CPW representing RF input and output. The input probe was connected to the amplified RF signal generator, whereas the output of the traveling wave electrode was terminated with a 50 Ω terminator to minimize reflections.

Band structure measurements

Dispersion relationships in a frequency dimension system are plotted against a reciprocal space quasi-momentum represented by time. This means that band structures are directly resolvable by probing the time-dependent spectral distribution of optical power in the ring resonator as it undergoes modulation. The measurement setup included a tunable software-controlled Yenista Optics TUNICS T100S-HP fiber laser, running at an around λ = 1550 nm wavelength and 13 dBm output power, for probing the ring resonator mode structure. At resonance, the cavity output was around −20 dBm and had to be boosted by up to 20 dB using a PriTel PMFA-20-IO erbium doped fiber amplifier. Spontaneous emission noise from this amplifier was suppressed using a Finsair WaveShaper 4000 s programmable optical filter to limit the output spectrum to a ΔλBP = 2 nm pass band. This range was selected as a minimum that permits scanning the probing laser over a several FSR wavelength range without clipping the modulation sidebands and distorting the signal. Optical-to-electronic conversion was performed using a Finisar XPDV3120R High-speed photodetector with an up to 70 GHz bandwidth. Signal acquisition was done using a Keysight DSO-Z504A oscilloscope with an up to 33 GHz bandwidth RF detector. Radio frequency signal for modulating the ring resonator device was supplied by a Anritsu MG3694A signal generator for single harmonic drive case, and a Keysight M8195A 65GSa/s arbitrary waveform generator for supplying multiple FSR-matched frequencies, which also permitted electronic control over the relative phase delays between signal harmonics. The power of RF signals in both cases was boosted using a SHF Communication Technologies AG SHF 830 P broadband RF amplifier. In addition, an auxiliary Anritsu MG3694A signal generator was used to supply the optical round-trip rate matched synchronization signal to the oscilloscope and was phase-locked to the RF drive generators using their internal 10 MHz clocks. Band structure measurements were performed by automatically scanning the probing laser wavelength in fixed increments over a pre-determined optical frequency range and collecting a sequence of time-resolved transmittance oscilloscope traces. Arranging these traces in sequence produces a 2D transmittance plot against time and frequency, which can be interpreted as the synthetic frequency dimension band structure. A schematic of the measurement setup is provided in Supplementary Fig. 2.