Complex-gauge control of anomalous Floquet corner responses in a non-Hermitian physical-synthetic photonic lattice

Abstract: We propose a non-Hermitian Floquet photonic lattice formed by a physical resonator coordinate and a synthetic frequency coordinate. A two-step modulation protocol realizes a chiral walk in this physical-synthetic plane, with a real synthetic flux controlling loop interference and imaginary gauge fields controlling non-reciprocal envelopes. We show that anomalous corner pairs at quasienergies zero and (π/T) exhibit three distinct layers of physics. A non-Bloch higher-order construction predicts whether the (0/π) corner pair exists under open boundaries. The imaginary gauge fields select where the right eigenmodes accumulate. The real flux controls the local interference matrix element that determines whether the doubled-period optical response is visible. As a result, the same topological coexistence sector can be bright, skin-dark, or flux-dark in a local optical measurement. We further show that the complex gauge can tune an exceptional point of the two-period corner propagator. At this point the anomalous response keeps its doubled-period sign alternation, but its envelope becomes algebraic because of a Jordan block. These results provide a photonic route to separate topological existence, skin-selected localization, optical visibility, and defective two-period dynamics in a non-Hermitian synthetic dimension.

• The paper demonstrates a two-step driven protocol using complex gauge fields to control chiral Floquet dynamics and anomalous corner responses.
• It employs a non-Bloch formalism to link topological invariants with eigenmode localization and flux-driven interferometric effects.
• Experimental diagnostics via local cross amplitude and interference measurements are proposed to reveal skin effects and exceptional point phenomena.

Complex-Gauge Control of Anomalous Floquet Corner Responses in a Non-Hermitian Physical-Synthetic Photonic Lattice

Introduction

This work presents a comprehensive analysis of non-Hermitian Floquet photonic lattices operating in a combined physical-synthetic space, integrating resonator arrays and synthetic frequency dimensions. The core of the proposal is a two-step driven protocol with tunable complex (real and imaginary) gauge fields, enabling control over chiral Floquet dynamics, skin effects, higher-order topology, and exceptional point (EP) phenomena. The study systematically addresses the interplay between topology, non-Hermitian skin localization, flux-induced interferometric effects, and dynamical non-diagonalizability, with a focus on observable consequences in photonic synthetic dimension platforms.

Model Construction and Floquet Protocol

The lattice comprises a 1D array of optical resonators indexed by a physical coordinate $x$ and a synthetic coordinate $w$ constructed from a ladder of discrete frequencies. Each site $(x, w)$ hosts two internal modes ($A, B$), yielding a non-Hermitian Floquet system with the following ingredients:

• Step 1: Non-reciprocal synthetic-frequency drift proportional to $\sigma_z$, with Peierls phase $\phi_w$ (real gauge) and asymmetry $\gamma_w$ (imaginary gauge).
• Step 2: Flux-threaded internal-state mixing along $x$, governed by synthetic flux $\Phi$ and physical axis non-reciprocity $\gamma_x$.
• The evolution operator over one period is $w$0.

Chiral symmetry is secured via anti-commuting internal matrices, protecting anomalous Floquet corner multipliers at quasienergies $w$1 and $w$2.

Corner Modes, Interference, and Visibility

Corner-localized states at quasienergies $w$3 and $w$4 underpin an interference mechanism yielding a stroboscopic doubled-period ($w$5) optical response. The observable is not the static eigenmode intensity, but the local, lock-in-extracted interference between the two corner modes, detected via a balanced analyzer in the synthetic-physical corner windows. The cross amplitude $w$6 (right eigenmode overlap) is the relevant quantity for optical visibility. This construction highlights that eigenmode co-localization is necessary but not sufficient for a strong $w$7 signal—internal-mode coherence and flux-dependent interference are critical.

Non-Bloch Higher-Order Topology

The bulk-corner correspondence for these higher-order topological modes is established via a non-Bloch formalism. In open, non-Hermitian systems, generalized Brillouin zones (GBZs) are parameterized by complex variables $w$8 and $w$9. The GBZ radii are analytically given by $(x, w)$0, $(x, w)$1 for the nearest-neighbor limit. Topological invariants are extracted as windings of $(x, w)$2 (off-diagonal blocks after chiral rotation) along these GBZ contours. The higher-order invariant is

$(x, w)$3

where $(x, w)$4 and $(x, w)$5 are edge and synthetic direction non-Bloch invariants, respectively.

This invariant predicts the existence of zero/$(x, w)$6 corner pairs but does not determine their spatial or synthetic localization under non-Hermitian conditions, nor the visibility of their interference in any local measurement.

Localization, Skin Effect, and Flux Control

The complex gauge allows decoupling between topological quantum numbers, skin-selected localization, and interference visibility:

• Skin-Dark Regime: Imaginary gauge fields ($(x, w)$7, $(x, w)$8) induce directional accumulation of right eigenmodes (the non-Hermitian skin effect). A topological $(x, w)$9 corner pair may exist, but if the two modes are selected at spatially separated corners, their interference vanishes in any fixed local window, and the $A, B$0 response is exponentially suppressed.
• Flux-Dark Regime: Even if both modes accumulate in the same corner, destructive interference of flux-dependent paths can suppress the local cross amplitude $A, B$1. Thus, local corner weights alone do not guarantee strong synthetic-frequency interference.
• Bright Regime: Coexistence of topological modes, co-localization, and constructive interference result in a strong local $A, B$2 response.

This differentiation leads to the central conclusion that topological existence does not necessarily imply optical observability—an essential consideration for experimental diagnostics.

Exceptional Points and Defective Dynamics

The two-period ($A, B$3) propagator, projected onto the biorthogonal $A, B$4 corner subspace, is used to investigate EP physics. Unlike the one-period Floquet operator (where eigenvalues $A, B$5 cannot coalesce while retaining their labeling), $A, B$6 folds both sectors to $A, B$7. The effective $A, B$8 non-Hermitian propagator generally takes the form

$A, B$9

where $\sigma_z$0, $\sigma_z$1, $\sigma_z$2, $\sigma_z$3 encapsulate spectrally and spatially projected parameters.

A second-order EP occurs when $\sigma_z$4, leading to coalescence of the two eigenvectors and the emergence of a Jordan block structure. The observable signature is a defective $\sigma_z$5 response: the existing $\sigma_z$6 subharmonic oscillation acquires an algebraically growing envelope (proportional to $\sigma_z$7 in the $\sigma_z$8-th period), distinguishable from the bounded or beating behavior in non-defective regimes. The phase rigidity, extracted from the left and right eigenvector overlap, serves as the diagnostic for the occurrence of the EP.

This EP-driven mechanism highlights that the doubled-period signal's anomalous sign alternation is not generated at the EP, but the EP modifies the dynamical envelope of a pre-existing topological $\sigma_z$9 corner pair.

Experimental Implications and Diagnostic Framework

All theoretically derived observables—the non-Bloch topological phase, skin-chosen localization, local interference visibility, and defective $\phi_w$0 dynamics—are accessible in state-of-the-art dynamically modulated photonic resonator arrays with synthetic frequency dimensions. Physical and synthetic coordinates can be resolved by spatial and spectral detection, while internal-mode interferometry measures the balanced intensity required for the local cross amplitude. Asymmetric conversion and modulation phase engineering realize the required real and imaginary gauge fields.

The study prescribes the following diagnostic approach for such systems:

1. Compute the non-Bloch higher-order invariant to predict zero/$\phi_w$1 corner existence.
2. Analyze open-boundary eigenmode spatial/synthetic accumulation to identify skin-selected localization.
3. Extract the local right-right cross amplitude $\phi_w$2 in the relevant detection window to determine $\phi_w$3 interference visibility.
4. Project the two-period propagator to identify and characterize EPs and associated defective signals.

The separation of these observables substantiates that observing an anomalous Floquet corner signal cannot be attributed to topological existence alone.

Conclusion

This work systematically elucidates the mechanisms that govern the existence, spatial/synthetic localization, and observability of anomalous Floquet corner responses in non-Hermitian physical-synthetic lattices with complex gauge fields. The analysis demonstrates the nontrivial consequences of separating non-Bloch higher-order topology from non-Hermitian localization and interference effects and shows how complex gauge tuning enables control over topological visibility and dynamical defectiveness. These results not only clarify the theoretical landscape of non-Hermitian higher-order Floquet phases but also provide a direct pathway for experimental exploration in modulated photonic synthetic dimension platforms.