Relativistic Quantum-Speed Limit for Gaussian Systems and Prospective Experimental Verification

Abstract

Timing and phase resolution in satellite QKD, kilometre-scale gravitational-wave detectors, and space-borne clock networks hinge on quantum-speed limits (QSLs), yet benchmarks omit relativistic effects for coherent and squeezed probes. We derive first-order relativistic corrections to the Mandelstam-Tamm and Margolus-Levitin bounds. Starting from the Foldy-Wouthuysen expansion and treating $-p^4/(8m^3{c}^2)$ as a harmonic-oscillator perturbation, we propagate Gaussian states to obtain closed-form QSLs and the quantum Cramér-Rao bound. Relativistic kinematics slow evolution in an amplitude- and squeezing-dependent way, increase both bounds, and introduce an $\hat{I}{\mu }^2{t}^2$ phase drift that weakens timing sensitivity while modestly increasing the squeeze factor. A single electron ($\hat{I}\mu \approx 1.5\times 10^{-10}$) in a $5.4\mathrm{T}$ Penning trap, read out with $149\mathrm{GHz}$ quantum-limited balanced homodyne, should reveal this drift within $\sim 15\min$ -- within known hold times. These results benchmark relativistic corrections in continuous-variable systems and point to an accessible test of the quantum speed limit in high-velocity or strong-field regimes.

16(9+7) PAGES, 4 figures