Geometric Amplification via Non-Hermitian Berry Phase

Despite their apparent simplicity, coupled oscillators exhibit surprisingly complex phenomena. Two notable examples are Berry phase (a geometric or topological aspect of the oscillators’ memory) and non-Hermiticity (the often counterintuitive impact of dissipation), both of which possess rich mathematical structures. Here, we demonstrate that combining Berry phase and non-Hermiticity leads to a fundamentally new form of amplification. Specifically, we show that this combination allows a lossy oscillator system to be converted into one with gain via slow modulation of its parameters. This is distinct from other amplification mechanisms, as it results specifically from the complex-valued Berry phase that is unique to non-Hermitian systems. We show that this mechanism produces continuous, useful gain in an optomechanical system, and that similar results can be realized in a very wide range of settings.

💡 Research Summary

The paper introduces a novel amplification mechanism that arises from the complex-valued Berry phase in non‑Hermitian systems, termed “geometric amplification.” In a conventional Hermitian setting the Berry phase is purely real and only affects the phase of a wavefunction, but in a non‑Hermitian system the Berry phase acquires an imaginary component that directly modifies the amplitude of the state. The authors demonstrate this effect experimentally using an optomechanical platform: a silicon nitride membrane supporting two mechanical vibrational modes is placed inside a high‑finesse optical cavity. Two laser tones drive the cavity; by adjusting their powers, detunings, and especially the relative phase θ₁₂(t) of the beat note, the authors can trace arbitrary closed loops C in a five‑dimensional control‑parameter space (P₁, P₂, detuning δ, small offset η, and θ₁₂). The effective non‑Hermitian Hamiltonian for the two modes can be written as H(t)=B⁰(t)·σ + i B¹(t)·σ, where the real vector B⁰ and the imaginary vector B¹ each trace a trajectory in ℝ³ as the loop is traversed.

To quantify the geometric phase, the authors measure the full propagator U(T) after a loop of duration T. The matrix element U₂₂ is fitted to exp