Stellar Braid Monodromy of Finite-Rank Non-Gaussian Photonic States
Finite-rank non-Gaussian bosonic states admit a holomorphic description in the Bargmann representation: after a zero-free Gaussian factor is separated off, their non-Gaussian structure is encoded by a finite stellar divisor. This article introduces a topological refinement of stellar rank for regular parameterized families of such states. Rather than only counting the zeros of the stellar divisor, we follow their motion under deformations of the state and record the associated braid monodromy. In the finite-Fock chart, a regular degree-r stellar state is represented by a monic polynomial with r simple zeros. The regular stratum is biholomorphic to the unordered configuration space of r points in the complex plane, and its fundamental group is the Artin braid group on r strands. Thus braid monodromy is an intrinsic invariant of loops in the regular finite-rank stellar state space. We then extend the construction to admissible finite stellar divisors of the form E_tau,mu(z) P(z); the zero-free Gaussian parameters form a contractible fiber over the same configuration-space base. Experimentally motivated finite-Fock families, especially the cubic subspace spanned by the first four Fock states, provide concrete laboratories, while trinomial slices yield explicit discriminants and local half-twists. The resulting invariant is post-tomographic and applies to preparation loops and parameterized families; it complements Wigner negativity, stellar rank, approximate stellar rank, and other scalar diagnostics of non-Gaussianity.