Unifying framework for amplification mechanisms: Spectral criticality, resonance, and nonnormality

We bring together three key amplification mechanisms in linear dynamical systems: spectral criticality, resonance, and nonnormality. We present a unified linear framework that both distinguishes and quantitatively links these effects through two fundamental parameters: (1) the spectral distance to a conventional bifurcation or to a resonance and (2) a nonnormal index <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>K</a:mi> </a:math> (or condition number <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>κ</b:mi> </b:math> ) that measures the obliqueness of the eigenvectors. Closed-form expressions for the system's response in the form of the variance <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi>v</c:mi> <c:mi>∞</c:mi> </c:msub> </c:math> of the observable responding to both Gaussian noise and periodic forcing reveal a general amplification law <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:msub> <d:mi>v</d:mi> <d:mi>∞</d:mi> </d:msub> <d:mo>=</d:mo> <d:msub> <d:mi>v</d:mi> <d:mn>0</d:mn> </d:msub> <d:mrow> <d:mo>(</d:mo> <d:mn>1</d:mn> <d:mo>+</d:mo> <d:mi mathvariant="script">G</d:mi> <d:mrow> <d:mo>(</d:mo> <d:mi>K</d:mi> <d:mo>)</d:mo> </d:mrow> <d:mo>)</d:mo> </d:mrow> </d:mrow> </d:math> with nonnormal gain <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi mathvariant="script">G</f:mi> <f:mrow> <f:mo>(</f:mo> <f:mi>K</f:mi> <f:mo>)</f:mo> </f:mrow> <f:mo>∝</f:mo> <f:msup> <f:mi>K</f:mi> <f:mn>2</f:mn> </f:msup> </f:mrow> </f:math> represented in universal phase diagrams. By reanalyzing a model of remote earthquake triggering based on breaking of Hamiltonian symmetry, we illustrate how our two-parameter framework significantly expands both the range of conditions under which amplification can occur and the magnitude of the resulting response, revealing a broad pseudocritical regime associated with large <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>κ</h:mi> </h:math> that previous single-parameter approaches overlooked. Similarly, in the non-Hermitian extensions of quantum optics provided by forward four-wave mixing experiments, we show the presence of a counterintuitive gain-from-loss effect that directly manifests nonnormal amplification in a propagating-wave setting. This predicts the possibility to engineer transient optical energy amplification without the need for true lasing or exact <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi mathvariant="script">PT</i:mi> </i:math> -symmetry breaking. Our framework applies to many other physical, natural, and social systems and offers diagnostic tools to distinguish true critical behavior from transient amplification driven by nonnormality.