Operator-Shaped Energy Landscapes in Hybrid Field-Programmable Photonic Arrays

This article develops the thermodynamic extension of hybrid field‑programmable photonic arrays (FPHAs), introducing the concept of operator‑shaped energy landscapes. Starting from a hybrid architecture composed of coarse holographic layers, sparse fast correction layers, and butterfly–diagonal–permutation (BDP) meshes, the paper shows how the same operator stack can be interpreted as defining a structured energy functional

whose minima, basins, and attractors encode the computational behavior of the system.

The work establishes:

• a deterministic relaxation model in which the FPHA behaves as a gradient‑flow operator,

• a stochastic relaxation model driven by fluctuations in the fast correction layers,

• a Gibbs stationary distribution under fluctuation–dissipation balance,

• and a polarization‑indexed holographic memory arising from the structure of the coarse layers.

This thermodynamic perspective unifies deterministic optical inference, associative memory, and stochastic sampling within a single operator‑defined landscape. In the low‑noise limit, the system implements a deterministic optical transform; with calibrated noise, it becomes a relaxation or sampling device capable of exploring multiple basins. The fast correction layers therefore serve not only as residual operator refinements but also as controlled stochastic sources enabling thermodynamic photonic computation.

This article is the second in a two‑part series. The companion paper develops the hybrid FPHA architecture and operator‑approximation framework underlying the energy‑landscape formulation.