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

TL;DR

This paper derives relativistic corrections to quantum speed limits for Gaussian systems, showing how high velocities and strong fields affect quantum evolution and proposing an experimental test with trapped electrons.

Contribution

It provides the first derivation of relativistic quantum speed limits for Gaussian states, including closed-form bounds and an experimental proposal.

Findings

Relativistic effects slow quantum evolution and increase speed limits.

Relativistic kinematics introduce a phase drift that affects timing sensitivity.

An experiment with a trapped electron can detect these relativistic quantum effects within minutes.

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 $-p^{4}/(8 m^{3} c^{2})$ as a harmonic-oscillator perturbation, we propagate Gaussian states to obtain closed-form QSLs and the quantum Cram\'er-Rao bound. Relativistic kinematics slow evolution in an amplitude- and squeezing-dependent way, increase both bounds, and introduce an $\epsilon^{2} t^{2}$ phase drift that weakens timing sensitivity while modestly increasing the squeeze factor. A single electron ($\epsilon \approx 1.5\times 10^{-10}$) in a $5.4\,\mathrm{T}$ Penning trap, read out with $149\,\mathrm{GHz}$ quantum-limited balanced homodyne, should reveal this drift within $\sim 15\,\mathrm{min}$ -- 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.