Geometric scaling limits of phase-only control in multimode coherent systems
Control of coherent waves is often restricted to phase-only actuation in multimode systems, yet the resulting physical limits remain poorly understood. Here, we show that restricting control to relative phases confines dynamics to a compact manifold whose geometry produces isolated stationary interference basins with robustness governed by local curvature. Imperfections act as smooth perturbations that soften basin structure without eliminating stationary states. This geometry imposes a universal scaling constraint: although the number of stationary states increases with system dimensionality, achievable localization contrast degrades through leakage into uncontrolled degrees of freedom. Experimentally, we demonstrate this in a telecom-wavelength multimode photonic lantern, where coarse phase-only scans directly map stable interference basins, reveal efficiency-stability trade-offs, and identify robust operating regimes without transmission-matrix reconstruction, adaptive optimization, or system inversion. The framework establishes a practical calibration-free approach for constrained multimode coherent control and applies broadly to optical, microwave, acoustic, and finite-dimensional quantum systems.