Laplace–Floquet Theory for Quantum–Classical Transduction: A Unified Framework for Energy–Flow Shaping and Harmonic Remapping

Abstract: Emerging hybrid quantum-classical systems such as quantum base stations and quantum-enhanced sensors for future wireless networks, require a signal-processing stage that converts quantum-derived classical observables into waveforms compatible with existing communication infrastructure. This transduction stage must simultaneously shape energy flow through resonant structures and translate harmonic content between incommensurate timing domains, two operations that existing frameworks treat separately: Laplace-domain methods handle energy storage and stability but assume time-invariance, while Floquet and Fourier methods handle periodic modulation but discard transient dynamics. Neither alone captures the coupled physics of resonant transduction under periodic modulation. This paper addresses that gap by developing a unified Laplace-Floquet framework. The central result (Equation 7) is a harmonic transfer relation that retains both Laplace-domain pole structure (encoding causality, damping, and transient response) and Floquetdomain harmonic coupling (encoding deterministic frequency translation under periodic modulation). From this single relation, the paper derives closed-form conditions for causality, BIBO stability, parametric stability margins, and noise propagation across harmonic channels. A numerical illustration demonstrates the framework on a representative transduction scenario, quantifying how resonant and modulation parameters jointly determine stability margins and noise gain. The framework thereby provides the analytical machinery needed to co-design the resonant and modulation parameters of quantum-classical interfaces, directly supporting the engineering of quantum-enabled network infrastructure.