Massive coherent equipartition of light by the geometric phase of null space

Light source is a foundational to photonic science and technology. However, a significant challenge remains in generating and distributing coherent light from a single on-chip source with high phase stability across multiple channels. Integrated lasers typically operate independently, and conventional splitters (e.g., multi-mode interferometers) do not guarantee the phase coherence required for advanced applications. Here, we report a purely geometric scheme for achieving massive equipartition of coherent light on a photonic chip by leveraging the geometric phases of a null space spanned by degenerate states with zero eigenvalue. The evolution of the null space maps to real-space rotation described by the special orthogonal group SO(N), thus enabling precise and scalable control over light distribution by engineering the system parameters. We experimentally realize up to one-to-nine equipartition of light on a waveguide array fabricated on a glass-based photonic chip. The framework can be upscaled for one-to-N light distribution. This work establishes a versatile and scalable platform for integrated coherent light sources, paving the way for integrated photonic applications such as quantum photonics and optical computing.

💡 Research Summary

The paper introduces a fundamentally new method for distributing coherent light from a single on‑chip source to many output channels while preserving phase stability. Conventional splitters such as multimode interferometers (MMIs) or directional couplers can split power but do not guarantee a fixed relative phase among the outputs, which limits their use in quantum photonics, coherent optical computing, and other applications that require a well‑defined global phase.

The authors base their scheme on a real‑valued, chiral‑symmetric Hamiltonian H that also respects parity‑time (PT) symmetry. For an N‑dimensional system, the null space (kernel) of H—spanned by eigenvectors with zero eigenvalue—has dimension M = N − rank(H). Because all modes in the null space have zero eigenenergy, they acquire no dynamical phase during propagation; only a geometric (Berry) phase can accumulate. When a set of controllable parameters λ (e.g., inter‑waveguide spacings) is varied slowly along a closed loop, the Berry connection Aij(λ)=⟨ψi|∂λψj⟩ forms a traceless, real antisymmetric matrix. Consequently, the evolution operator U = 𝒫 exp(−∮A·dλ) belongs to the special orthogonal group SO(M). In other words, the null‑space state vector undergoes a pure real‑space rotation, and the rotation angle is directly proportional to the solid angle Ω enclosed by the loop on the parameter sphere. By designing Ω, one can prescribe any desired power‑splitting ratio and relative phase among the M output channels.

Experimentally, the team fabricated waveguide arrays in Corning 7980 glass using femtosecond laser direct writing. The coupling coefficients κ1, κ2, κ3 between neighboring waveguides were modulated by varying the center‑to‑center spacing from 8 µm to 24 µm, thereby tracing a closed trajectory on a unit 2‑sphere in parameter space. In a four‑waveguide device (M = 2) they realized a solid angle Ω = π/2, achieving a 1→2 equipartition with less than 3 % intensity variation and negligible phase error. In a five‑waveguide cluster (M = 3) they set Ω = π, producing a 1→5 split where two of the outputs are out‑of‑phase by π, exactly as predicted by the geometric‑phase theory.

To reach higher fan‑out, the authors cascaded basic units: a 1→5 splitter followed by four identical 1→2 splitters, yielding a 1→9 distribution. Measured intensity variation across the nine outputs was under 5.6 %, and the relative phases remained locked, confirming the coherence of the process. A modular design for 1→41 splitting is also presented, illustrating how arbitrary N can be achieved by combining 1→5, 1→3, and 1→2 blocks.

Key insights include: (1) exploiting a degenerate zero‑energy subspace eliminates dynamical phase noise; (2) the SO(M) rotation framework provides a compact mathematical description that scales naturally with the number of output channels; (3) solid‑angle engineering offers an intuitive, geometric control knob for arbitrary power‑splitting ratios and phase relationships.

The approach, however, has practical constraints. The adiabatic condition requires propagation lengths of several centimeters, increasing the device footprint. Precise control of the coupling trajectory is essential; fabrication tolerances, temperature drift, or material inhomogeneities can alter the solid angle and thus the splitting fidelity. Losses or strong nonlinearities could break the null‑space protection, degrading phase coherence.

Future work suggested by the authors includes active tuning (electro‑optic or thermo‑optic) for real‑time correction of the coupling path, non‑adiabatic shortcuts to reduce device length, integration with active gain media for on‑chip laser sources, and exploration of topological or quantum‑metric effects to accelerate the geometric evolution. If these challenges are addressed, the technique promises a versatile platform for large‑scale coherent photonic networks, quantum‑light distribution, high‑power beam combining, and optical computing architectures that demand strict phase coherence across many channels.