Realization of all-band-flat photonic lattices

Flatbands lead to localization, high density of states (DOS), and non-trivial topology, attracting increasing interest in electronic materials, atomic physics, and photonic lattices. In strongly correlated electronic systems, the small energy width and high DOS of flat bands facilitate the observation of many-body physics such as fractional quantum Hall effect, ferromagnetism, and superconductivity. Flatbands also have promising applications in photonic systems: the zero group velocity in flatbands can be used to achieve slow light, enhanced light-matter interaction, and dispersionless image transmission. Systematic methods have been developed to generate flatbands in one and two dimensions. In particular, by carefully engineering the hopping strengths between lattice sites, it is possible to realize all-band-flat (ABF) lattices, which balance the trade-off between band flatness and useful bandwidth by turning all bands flat to utilize the full lattice energy spectra. ABF lattices provide a platform to investigate Aharonov-Bohm caging, compact localized states, and non-linear and quantum caging. Although finite ABF lattices have been theoretically proposed by finely tuning coupling strengths, experimental realization has been limited by achievable coupling ranges and resonator dissipation, preventing all eigenstates from being grouped into flatbands. In this Letter, the authors experimentally realize ABF honeycomb lattices of microwave resonators by engineering coupling strengths to mimic the Fock-state lattices (FSLs) of a three-mode Jaynes-Cummings model. They precisely control spatially varying couplings to group all eigenstates into flatbands, achieving high DOS at discrete energies and going beyond the perturbative regime of strain engineering. They map state distributions and selectively excite eigenmodes of different chiralities, demonstrating a scalable method for ABF lattices and a platform for topological transport studies in photonic lattices.

The work builds on extensive literature across flatband physics in electrons, atoms, and photonics, where flatbands yield localization, high DOS, and non-trivial topology. Prior approaches in photonics have employed synthetic magnetic fields and lattice interference to create a limited number of flatbands, and strained hexagonal lattices have been used to synthesize pseudo-magnetic fields yielding photonic Landau levels; however, these typically access only the first few levels near Dirac points in the perturbative regime. Theoretical proposals indicate that all-band-flat spectra can be realized in finite lattices via carefully engineered, site-dependent hopping strengths, and one-dimensional Glauber-Fock photonic lattices mimicking Fock-state couplings have been demonstrated experimentally. The present study extends these ideas to two-dimensional honeycomb lattices, leveraging FSL-inspired coupling profiles to realize ABF spectra experimentally, overcoming previous limitations in coupling range and losses that hindered full ABF realization.

Design principle and theory: The authors implement a finite honeycomb lattice of microwave resonators with site-dependent nearest-neighbor coupling strengths designed to emulate Fock-state lattices (FSLs) of a three-mode Jaynes-Cummings model. Resonators are labeled A_ijk and B_ijk for the A and B sublattices with indices i, j, k along lattice directions e1=(√3/2, -1/2), e2=(√3/2, 1/2), e3=(0,1), satisfying i + j + k + (l/2) = N with l = -1 for A and l = +1 for B. At the triangular boundary one of i, j, k becomes zero, analogous to the vacuum state in FSLs. The finite lattice contains (N+1)^2 sites. The coupling between A_ijk and B_{i-1,j,k} is set to √i t0, and similarly for the j and k directions, introducing square-root scaling characteristic of bosonic annihilation operators. This yields a tight-binding Hamiltonian with analytically solvable eigenenergies E_m = ±√3 m t0 (m = 0, 1, …, N) with degeneracy N − m + 1, grouping all eigenstates into N+1 flatbands. Hardware implementation: The lattice is constructed from aluminum coaxial cavity resonators operating at a TM hexapole mode around 12.002 GHz, with fields confined inside the cavities to minimize radiation. Each resonator couples to its three nearest neighbors via short rectangular waveguides (WGs) connected through three wall openings with widths d1, d2, d3; a fourth opening with width d4 compensates frequency shifts to align all cavity resonances. The evanescent coupling between adjacent resonators is controlled by WG widths. Two-resonator simulations establish the coupling–width relation t = (4.4 d2^2 − 37.8 d2 + 89.8) MHz, enabling site-by-site design of the target coupling map extending beyond the perturbative regime. The minimum coupling strength t0 ≈ 69 MHz is set much larger than the resonance linewidths to ensure well-resolved coupling-induced splittings. Simulation: Full-wave electromagnetic simulations are performed in CST Microwave Studio, modeling metals as perfect electric conductors (PEC). Discrete ports are used as excitations, and simulation regions are enclosed by open-space boundaries. Experimental setup and measurements: Resonators are machined from 6061 aluminum (CNC). Measurements are conducted with a Ceyear-3672C vector network analyzer. 3-mm monopole antennas are used for local excitation and detection of fields at lattice sites. For selective excitation of degenerate eigenmodes, three monopole antennas with equal amplitudes and controlled relative phases (using adjustable attenuators and phase shifters) are driven simultaneously to match the spatial phase patterns of target eigenmodes. Spatial distributions of flatband modes are imaged via site-resolved reflection coefficients R(r, ν_m) at flatband eigenfrequencies. Data and code supporting plots are available at Figshare (DOIs provided by the authors).

• All-band-flat spectrum: By engineering coupling strengths to follow √i, √j, √k scaling, the finite honeycomb lattice realizes N+1 flatbands with analytically predicted eigenenergies E_m = ±√3 m t0 and degeneracies N − m + 1, grouping all eigenstates into flatbands and yielding high DOS at discrete energies.
• Practical coupling engineering: A calibrated relation between waveguide width and coupling, t = (4.4 d2^2 − 37.8 d2 + 89.8) MHz, enables precise realization of the target coupling map. The minimum coupling strength is t0 ≈ 69 MHz, set well above cavity linewidths to ensure strong, controllable coupling.
• Imaging of flatband eigenmodes: Spatial mode profiles for Landau levels m = 0, 1, 6, 8, 9 show distinct distributions. The m = 0 level exhibits sublattice polarization (nonzero local DOS only on A sites) and confinement within the lattice incircle, attributed to nonperturbative strain inducing a semimetal–insulator transition at the incircle (Lifshitz topological edge). As m increases from 0 to N/2, modes expand toward edges, then contract toward the center as m approaches N; distributions preserve C3 symmetry.
• Level-specific localization: Different Landau levels preferentially occupy different regions: for m = 1, A-site weights peak near the three corners; for m = 6, along the edges; for m = 8, annular distribution with a central node; for m = 9 (single mode), strong central localization. Higher Landau levels extend beyond the incircle with near-equal sublattice populations.
• Selective excitation of degenerate chiral modes: Within a given Landau level, degenerate eigenmodes characterized by chirality C are selectively excited using three-site driving with tailored relative phase φ. In the 7th Landau level, the C = 0 mode is efficiently excited with φ = 0 near the center, while the C = 2 mode is efficiently excited with φ = 2π/3 away from the center, consistent with the eigenmode phase winding (e.g., 4π around the center for C = 2). Phase maps measured across the lattice match the targeted eigenmode patterns.
• Performance metrics: The precision of coupling control yields a fidelity exceeding 0.85 for most Landau levels (as defined and evaluated in the Supplementary Information), indicating high band flatness and accurate mode realization over a large usable bandwidth.

The study demonstrates that precise, spatially varying nearest-neighbor couplings in a finite honeycomb lattice can emulate Fock-state lattice physics and realize an all-band-flat spectrum, thereby overcoming the limitations of perturbative strain engineering that typically yields only a few flat Landau levels near Dirac points. By grouping all eigenstates into flatbands, the platform simultaneously achieves high band flatness and uses the full spectral bandwidth. The observed spatial mode structures, including sublattice polarization of the m = 0 level within the incircle and the systematic evolution of localization with m, are consistent with nonperturbative strain-induced band topology, including a Lifshitz edge separating semi-metallic and insulating regions. The ability to selectively excite degenerate eigenmodes by matching multi-site phase patterns reveals a practical handle analogous to momentum selection in infinite systems, enabling control over transport and localization in flatband photonics. Compared with ABF proposals relying on discrete translational symmetry, this finite, scalable approach avoids non-flat edge modes and can be applied to arbitrary lattice sizes, expanding the toolkit for flatband-enabled photonic devices and topological transport studies.

By emulating Fock-state lattice couplings in a finite honeycomb array of microwave resonators, the authors realize all-band-flat photonic lattices where every eigenstate belongs to a flatband with analytically predictable energies and degeneracies. They image and control flatband eigenmodes, demonstrating sublattice-polarized zero modes confined within an incircle and selective excitation of degenerate chiral modes via multi-antenna phase control. The approach goes beyond perturbative strain engineering, is scalable to arbitrary sizes without non-flat edge modes, and provides a versatile platform for exploring flatband physics and topological photonic transport. Future directions include incorporating Kerr and second-order nonlinearities to study nonlinear localization, breathing dynamics between flatbands, enhanced harmonic generation, and soliton phenomena; extending the method to ABF photonic waveguides for dispersionless imaging; and leveraging finite-size ABF lattices in nanoscale laser architectures for high-purity, configurable emission.