Synthetic Frequency Dimension in Photonics

• Synthetic frequency dimension is a framework where discrete frequency modes act as lattice sites via dynamic modulation, enabling higher-dimensional emulation in lower-dimensional setups.
• It employs controlled coupling through electro-optic, acousto-optic, or magnetic modulation to implement tight-binding Hamiltonians and synthetic gauge fields in photonic and magnonic systems.
• Applications include topological photonics, quantum information processing, and optical computing, with experimental advances revealing robust edge states and many-body effects.

Synthetic frequency dimension refers to the use of discrete frequency modes of a photonic, magnonic, or related wave system as lattice sites along an artificial dimension orthogonal to its physical degrees of freedom. This concept leverages dynamic modulation—typically via electro-optic, acousto-optic, or magnetic means—to induce controlled coupling among these modes, allowing researchers to sculpt higher-dimensional lattice Hamiltonians in systems whose real-space structure is of lower dimension.

1. Principle of Synthetic Frequency Dimensions

The synthetic frequency dimension arises by mapping the frequency (or spectral) modes of a system—such as the resonant modes of a ring resonator or magnonic ring—to the sites of a mathematical lattice. Dynamic modulation (e.g., at frequency Ω matching or harmonically related to the mode spacing FSR) is engineered to induce hopping between modes with indices $m$, $m+1$, etc., analogous to nearest- or long-range neighbor hopping in lattice models (Yuan et al., 2015, Dutt et al., 2019, Xu et al., 11 Aug 2024). The amplitude and phase of the modulation, and the choice of modulation harmonics, together define the topology and couplings in this lattice.

This procedure transforms a one-dimensional (physical) structure into a two- or higher-dimensional synthetic lattice. For example, in a one-dimensional array of ring resonators, the physical axis ($l$) and the frequency comb index ($m$) together constitute a two-dimensional lattice (Yuan et al., 2015). In a single ring with multiple modulation frequencies, a multidimensional synthetic lattice with complex connectivity can be synthesized (Cheng et al., 2023).

2. Mathematical Formulation and Hamiltonians

The canonical model of a synthetic frequency dimension in a modulated ring resonator is a tight-binding Hamiltonian where frequency mode creation/annihilation operators $a_m^{(\dagger)}$ or $b_m^{(\dagger)}$ are used:

$$H = \sum_{m} \omega_m a_m^\dag a_m + \sum_{m} [ g a_{m+1}^\dag a_m e^{i\phi} + h.c. ],$$

where $g$ is the modulation-induced hopping amplitude and $\phi$ is the phase imparted by the modulator (Yuan et al., 2021). In arrays, spatial coupling is included:

$$H = \sum_{l,m} \omega_m a_{l,m}^\dag a_{l,m} + \sum_{l,m} \left[ \kappa (a_{l,m}^\dag a_{l+1,m} + h.c.) + 2g \cos(\Omega t + \phi_l)(a_{l,m}^\dag a_{l,m+1} + h.c.) \right],$$

which, under a rotating-wave approximation, maps to a two-dimensional lattice with Peierls phase $e^{i\phi_l}$ realizing an effective gauge potential (Yuan et al., 2015).

More generally, every element of the coupling matrix, including long-range hopping and non-Hermitian or non-Abelian gauge structure, can be synthesized by engineering the Fourier expansion of the modulation waveform, leading to Hamiltonians describing square, honeycomb, or ladder lattices with arbitrary connectivity and synthetic magnetic/electric fields (Cheng et al., 2023, Yu et al., 2021, Cheng et al., 1 Jun 2024, Wang et al., 20 Nov 2024).

3. Physical Implementation Strategies

(a) Photonic ring and waveguide systems:

Equally spaced frequency modes (FSR) are modulated via integrated or fiber electro-optic modulators, with modulation harmonics generating nearest- and longer-range couplings in frequency. On-chip implementations using silicon, lithium niobate, or silicon nitride platforms afford scalability and integration with other photonic components (Yuan et al., 2015, Yuan et al., 2021, Balčytis et al., 2021, Wang et al., 20 Nov 2024).

(b) Magnonic systems:

Spin-wave modes in magnetic ring resonators are modulated through voltage control of magnetic anisotropy, inducing tight-binding dynamics among frequency modes in the linear spin-wave regime—a setting labeled magnonics in synthetic dimensions (Xu et al., 11 Aug 2024).

(c) Hybrid and higher-order platforms:

Polarization multiplexed systems, dissipative and nonlinear (Kerr or $\chi^{(3)}$) resonators, and multi-ring arrays extend synthetic frequency concepts to emulate spin, multi-band, and non-Abelian models (Cheng et al., 1 Jun 2024, Yuan et al., 2019, Tusnin et al., 2020).

4. Topological Photonics and Synthetic Gauge Fields

Synthetic frequency dimensions allow physical realization of lattice gauge theories and topological phases beyond conventional spatial constraints. Through spatially engineering modulation phases across an array—e.g., setting $\phi_l = l\pi/2$—a uniform synthetic magnetic field is realized in the (space, frequency) lattice, leading to nontrivial Chern bands, Landau levels, and topologically protected edge states localized along the frequency axis (Yuan et al., 2015, Lin et al., 2018, Yuan et al., 2021). These edge states can mediate one-way propagation in frequency (i.e., sideband climbing/descending), robust to losses and disorder. The extension to non-Abelian gauge fields allows photonic simulation of Yang–Mills Hamiltonians, with Dirac cones at time-reversal invariant points and spin-dependent transport (Cheng et al., 1 Jun 2024).

Non-Hermitian or dissipative modulation enables the observation of skin effects, point-gap topology, and other open system topological phenomena in frequency space (Cheng et al., 2023).

5. Experimental Probes and Band Structure Spectroscopy

Measurement of band structures in the synthetic frequency dimension is performed via time-resolved transmission. Due to the periodicity of modulation, the time within one modulation period maps to the quasimomentum of the synthetic lattice, enabling a direct "read out" of the Floquet or tight-binding bands—a salient distinction from spatial lattices that require spatial or reciprocal space scanning (Dutt et al., 2019, Balčytis et al., 2021). Sweeping the laser detuning provides access to the band energy axis, and extending modulation waveforms allows reconstruction of multi-dimensional Brillouin zones (Cheng et al., 2023).

Shaping the modulation permits direct synthesis of scattering/convolution matrices (with desired translation invariance or more elaborate structure), making it possible to implement arbitrary linear or convolutional processing in frequency (Buddhiraju et al., 2020, Fan et al., 2023).

6. Nonlinearity, Many-Body Effects, and Quantum Information

Tailored nonlinear media integrated with dynamically modulated resonators bring local photon–photon interactions into the synthetic frequency dimension. With appropriate dispersion engineering, spatially local (on-site) interactions such as those of the Bose–Hubbard model can be realized, circumventing the inherently long-range nature of standard optical nonlinearity (Yuan et al., 2019, Wang et al., 11 Jun 2024). This allows for the paper of photon blockade, quantum chaos, localization, and many-body phenomena using frequency–space analogs of conventional condensed-matter models, including the extraction of spectral statistics (e.g., spectral form factor, level spacing distributions) using photonic correlation measurements.

Such systems support the generation of highly controllable frequency-entangled photon states and open a route toward scalable quantum information processing using frequency-encoded qubits and qudits (Wang et al., 11 Jun 2024, Zhao et al., 2021).

7. Device Engineering, Applications, and Future Prospects

Table: Key Features and Device Platforms

Property	Implementation Example	Notable Phenomena
Modulation Frequency	GHz (fiber/fiber, Si/LN ring)	Synthetic gauge fields, combs
On-chip Platform	Si (CMOS), TFLN, Si₃N₄	Band structure, Aharonov–Bohm cages (Wang et al., 20 Nov 2024)
Programmable Coupling	MZI-assisted arrays, modulated MZI	Long-range, cross-coupling, tunable magnetic flux (Wang et al., 20 Nov 2024)
Nonlinearity	Engineered χ^3 (e.g., SiN)	Bose–Hubbard, photon blockade
Hybrid Nodes	Giant atoms (superconducting/photonic)	Photon routing, chiral edge states (Chai et al., 3 Mar 2025, Du et al., 2021)

Applications include:

• Robust, high-efficiency frequency conversion via topological edge states (Yuan et al., 2015).
• Arbitrary linear transformations and convolution processing in photonic frequency combs (Buddhiraju et al., 2020, Fan et al., 2023).
• Optical computing with large-scale matrix–vector multiplication through cavity acousto-optic modulation and coherent frequency mixing (Zhao et al., 2021).
• Programmable photonic simulators of topological, disordered, non-Abelian, and quantum many-body phases (Lin et al., 2018, Cheng et al., 1 Jun 2024, Wang et al., 11 Jun 2024).
• High-dimensional quantum network routing via giant atoms coupled to frequency lattices (Chai et al., 3 Mar 2025, Du et al., 2021).

Looking ahead, advances in device integration (thin-film lithium niobate TFLN, Si CMOS, and hybrid quantum–optical architectures), the exploration of additional degrees of freedom (orbital angular momentum, polarization, pseudospin), and nonlinear or dissipative extensions will expand the reach of synthetic frequency dimension physics, enabling flexible, reconfigurable, and robust devices for both fundamental research and applied information processing.

• (Yuan et al., 2015) Photonic gauge potential in a system with a synthetic frequency dimension
• (Lin et al., 2018) Constructing three-dimensional photonic topological insulator using two-dimensional ring resonator lattice with a synthetic frequency dimension
• (Dutt et al., 2019) Experimental band structure spectroscopy along a synthetic dimension
• (Yuan et al., 2019) Creating locally interacting Hamiltonians in the synthetic frequency dimension for photons
• (Tusnin et al., 2020) Nonlinear states and dynamics in a synthetic frequency dimension
• (Buddhiraju et al., 2020) Arbitrary linear transformations for photons in the frequency synthetic dimension
• (Yu et al., 2021) Simulating graphene dynamics in one-dimensional modulated ring array with synthetic dimension
• (Yuan et al., 2021) Tutorial: synthetic frequency dimensions in dynamically modulated ring resonators
• (Balčytis et al., 2021) Synthetic dimension band structures on a Si CMOS photonic platform
• (Zhao et al., 2021) Scaling optical computing in synthetic frequency dimension using integrated cavity acousto-optics
• (Du et al., 2021) Giant Atoms in a Synthetic Frequency Dimension
• (Dutt et al., 2022) Creating boundaries along a synthetic frequency dimension
• (Long et al., 2022) Time reflection and refraction in synthetic frequency dimension
• (Cheng et al., 2023) Multi-dimensional band structure spectroscopy in the synthetic frequency dimension
• (Fan et al., 2023) Experimentally Realizing Convolution Processing in the Photonic Synthetic Frequency Dimension
• (Cheng et al., 1 Jun 2024) Non-Abelian lattice gauge fields in the photonic synthetic frequency dimension
• (Wang et al., 11 Jun 2024) Few-Body Quantum Chaos, Localization, and Multi-Photon Entanglement in Optical Synthetic Frequency Dimension
• (Xu et al., 11 Aug 2024) Frequency modulation on magnons in synthetic dimensions
• (Wang et al., 20 Nov 2024) Versatile photonic frequency synthetic dimensions using a single Mach-Zehnder-interferometer-assisted device on thin-film lithium niobate
• (Chai et al., 3 Mar 2025) Photon Routing Induced by Giant Atoms in a Synthetic Frequency Dimension