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

TL;DR

This work addresses how special-relativistic corrections modify quantum-speed limits for Gaussian probes used in metrology. By applying the Foldy-Wouthuysen expansion and adding the leading quartic kinetic term $-rac{p^{4}}{8 m^{3} c^{2}}$, the authors derive closed-form first-order corrections to the Mandelstam-Tamm and Margolus-Levitin bounds for both coherent and squeezed states, and obtain the corresponding relativistic Quantum Cramér-Rao bounds. The key result is an $oldsymbol{\epsilon^{2} t^{2}}$ phase drift that degrades timing sensitivities while modestly increasing squeeze factors, with explicit expressions for the phase readout in balanced-homodyne detection. They propose a feasible Penning-trap experiment, using a quantum-limited 149 GHz balanced microwave homodyne setup, to observe the drift within about 15 minutes of averaging, thereby providing a practical test of relativistic QSLs in high-velocity or strong-field regimes.

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 Cramér-Rao bound. Relativistic kinematics slow evolution in an amplitude- and squeezing-dependent way, increase both bounds, and introduce an $ε^{2} t^{2}$ phase drift that weakens timing sensitivity while modestly increasing the squeeze factor. A single electron ($ε\approx 1.5\times 10^{-10}$) 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.

Figures (4)

• Figure 1: First-order relativistic modification of MT and ML bounds for coherent states. Surfaces give the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) bounds as functions of the evolution time $t$ and the mean photon number $\alpha_{0}^{2}$. Blue and orange sheets are the non-relativistic limits ($\epsilon=0$); green and grey include the first-order relativistic correction $\epsilon=0.08$. Throughout the plotted window $\mathrm{MT}^{(\epsilon)}$ (grey) forms the true quantum speed limit.
• Figure 2: First-order relativistic modification of MT and ML bounds for squeezed vacua. Same conventions as Fig. \ref{['fig:plot1']}, but with the squeezing parameter $r$ on the secondary axis $(0\le r\le0.4)$. First-order relativistic corrections ($\epsilon=0.08$) lift both ML (green) and MT (grey) bounds over the entire domain, and the gap between corrected and uncorrected surfaces widens monotonically with $r$, signalling the enhanced sensitivity of highly squeezed states to relativistic effects.
• Figure 3: Relativistic enhancement of the optical squeeze factor. Squeeze-factor (SF) surfaces, defined by $\mathrm{SF}=-10\log_{10}\bigl[\Delta^{2}X_{s}/\Delta^{2}X_{c}\bigr]$, are plotted versus $|\alpha|^{2}$ and the squeezing parameter $r$ for four relative phases $\theta$ between displacement and squeezing. Blue sheets: non-relativistic ($\epsilon=0$); green sheets: first-order relativistic correction ($\epsilon=0.08$). Relativity lifts the SF across the full domain ($0\le r\le1.8,\;0\le|\alpha|^{2}\le3$), with a phase-dependent magnitude.
• Figure 4: BHD scheme for experimentally testing the relativistic corrections to QSLs.