Non-Hermitian Skin Effect — Photonic Ring Array

Lattice

Non-reciprocity g 0.60

Coupling J 1.00

Ring sites N 32

Live readouts

Winding number w

Boundary

Closed (PBC)

IPR (localization)

0.031

Skin depth ξ

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Steady-state peak site

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How it works

Each site is a photonic ring resonator coupled to its two neighbours with unequal hopping in each direction — right-hop amplitude t_R = J+g, left-hop t_L = J−g (a Hatano–Nelson chain, realised photonically with unbalanced directional couplers or gain/loss loops). This non-reciprocity makes the lattice Hamiltonian non-Hermitian.

i·da_n/dt = -(t_R·a_{n-1} + t_L·a_{n+1})
t_R = J+g,  t_L = J-g
similarity map:  a_n → β^n a_n,  β = √(t_L/t_R)  restores a Hermitian chain
⇒ every eigenmode decays as β^n along the lattice

Under periodic boundary conditions (the ring closed) the spectrum E(k) = −(t_R e^−ik + t_L e^ik) traces a closed loop in the complex-energy plane with a nonzero winding number w — a point-gap topological invariant with no Hermitian analogue. All eigenstates stay plane waves, equal intensity everywhere.

Cut a single bond and the ring becomes open. The winding number is only defined for a closed loop, but its nonzero value under PBC is exactly the topological signal that under OBC every eigenmode — not just one special mode — collapses exponentially onto the same edge, with localization (skin) length ξ = 1/|ln β|. This "non-Hermitian skin effect" is why the field on screen funnels to the cut and piles up there, regardless of where you start it: this simulator evolves the field and continuously renormalizes it (power iteration), so what you see converge is the dominant eigenmode itself.

g — hopping asymmetry; g=0 is the ordinary Hermitian ring (no skin effect at any boundary).
Cut bond — toggles the single bond between site N−1 and site 0, switching PBC ↔ OBC on the same lattice.
IPR = Σ|a_n|⁴ / (Σ|a_n|²)² — 1/N for a fully delocalized ring, → 1 as the field localizes to one site.

Real-world relevance: this is the mechanism behind "topological funneling of light" (Weidemann et al., Science 2020), demonstrated in a time-multiplexed photonic ring-resonator network — an all-optical way to route arbitrary input light to one fixed output port with no active control.

PBC — delocalized

Field intensity |a_n|² Cut / active bond

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