Propagation-Distance Limit for a Classical Nonlocal Optical System

TL;DR

This work translates quantum-speed-limit concepts to classical, highly nonlocal optical beams mapped onto a reversed harmonic oscillator, deriving distance-domain MT and ML bounds via the Bures angle with a closed-form propagator. The authors obtain analytic expressions for $\langle H\rangle$ and $\Delta H$ for displaced Gaussian inputs and show that the propagation-distance limit $z_{\mathrm{PDL}}$ remains finite despite exponential transverse spreading, with MT typically setting the bound. They demonstrate this with numerical illustrations and introduce a millimetre-scale PDL beam shaper that achieves picosecond switching and supports high-precision refractive-index and temperature sensing, linking universal QSL geometry to practical photonics. The results establish a rigorous bridge between quantum-inspired dynamical constraints and classical optical applications, offering a framework for ultrafast reshaping, switching, and metrology in nonlocal media.

Abstract

We derive closed-form analog quantum-speed-limit (QSL) bounds for highly nonlocal optical beams whose paraxial propagation is mapped to a reversed (inverted) harmonic-oscillator generator. Treating the longitudinal coordinate $z$ as an evolution parameter (propagation distance), we construct the propagator, evaluate the Bures distance, and obtain analytic Mandelstam--Tamm and Margolus--Levitin bounds that fix a propagation-distance limit $z_{\mathrm{PDL}}$ to reach a prescribed mode distinguishability. This distance-domain constraint is the classical optical analogue of the minimal orthogonality time in quantum mechanics. We then propose a compact self-defocusing PDL beam shaper that achieves strong transverse-mode conversion within millimeter scales. We further show that small variations in refractive index, beam power, or temperature shift $z_{\mathrm{SL}}$ with high leverage, enabling speed-limit-based metrology with index sensitivities down to $10^{-7}$ RIU and temperature resolutions of order $1$ mK. The results bridge distance-domain QSL geometry and practical photonic applications.

Figures (1)

• Figure 1: Propagation distance limit for the inverted-oscillator analogue: (a) $z_{ML}$ and (b) $z_{MT}$ versus propagation distance $z$ (evolution parameter) and the amplitude parameter $|a|^{2}$.