A unified invariant-based framework for the synthesis and implementation of two-mode SU(2) transformations in integrated quantum photonics

Prescribed two-mode SU(2) transformations are fundamental building blocks of integrated quantum photonics, motivating inverse-design methodologies capable of synthesizing coherent transformations. Existing invariant-based approaches formulate inverse design around modal-power evolution for selected inputs, leaving the relative transformation phase required for coherent operation unconstrained. Here, we reformulate inverse design from state-trajectory engineering to operator synthesis by treating a prescribed two-mode SU(2) transformation as the design objective, from which the invariant boundary conditions and the corresponding coupled-waveguide Hamiltonian are systematically derived. Full-wave simulations demonstrate that representative shortcut-to-adiabaticity and STIRAP devices do not reproduce the prescribed Hadamard transformation for coherent superposition inputs with varying relative phases, whereas the proposed device simultaneously reproduces the target modal amplitudes and transformation phases, achieving an average state fidelity of 0.977 (minimum 0.971). As a proof of concept, the framework is validated through the design and full-wave simulation of a compact passive two-waveguide CMOS-compatible beam splitter on a Si 3 N 4 /SiO 2 platform operating at 1.55 μm. Under the investigated \(\pm 20\,\mathrm{nm}\) fabrication perturbations, the proposed devices exhibit maximum power-splitting and relative-phase errors of 6% and 5 ∘ , respectively. This framework establishes a systematic route from a prescribed SU(2) operator to a physically realizable passive photonic device, providing compact building blocks for programmable linear-optical quantum information processing.