Quantum speed limit as a sensitive probe of Planck-scale effects

TL;DR

The paper investigates how a quadratic generalized uncertainty principle modifies quantum speed limits (MT and ML) by embedding a $p^{4}$-type deformation into the Hamiltonian. It derives closed-form, first-order QSL corrections for a particle in a box, coherent and squeezed harmonic-oscillator states, and reveals a universal scaling with the effective Hilbert-space size $N_{\mathrm{eff}}$, enabling Planck-scale signatures to be amplified in large systems. The authors map QSL deviations to bounds on the dimensionless GUP parameter $\beta_0$, recapitulating existing experimental bounds and proposing concrete heavy-oscillator metrology pipelines (back-action-evading readout and stroboscopic tomography) to tighten these bounds by orders of magnitude. Collectively, the work shows that QSL-based timing provides a practical, near-term route to test minimal-length physics on quantum-optical and optomechanical platforms, with explicit guidance for experimental implementation.

Abstract

Many quantum-gravity scenarios predict a minute modification of the canonical commutator, known as the generalized uncertainty principle (GUP), whose low-energy signatures are, in principle, accessible to state-of-the-art laboratory tests. We compute first-order minimal-length corrections to the quantum speed limit (QSL) for three cases: uniform superpositions in an infinite square well, coherent harmonic-oscillator states, and squeezed-oscillator states. We identify a universal amplification law: for any pure state, the fractional shift of either speed limit scales linearly with $β$ and algebraically with the state's effective Hilbert-space size. As the effective Hilbert-space dimension can be exceedingly large, the associated minimal-length signatures are amplified by several orders of magnitude. Using high-precision matter-wave timing data, we set a direct bound on the GUP parameter $β$, which quantifies minimal-length quantum-gravity effects. Our analysis indicates that phase-locked, short-time overlap fits on kilogram-scale optical-spring modes can tighten this bound by orders of magnitude. We outline two implementable measurement pipelines -- continuous back-action-evading single-quadrature readout and stroboscopic, phase-locked pulsed tomography -- that exploit this leverage, making QSL-based timing a practical, near-term probe of minimal-length physics on quantum-optical and optomechanical platforms.

Figures (6)

• Figure 1: QSL $\tau_{\min}$ for a particle in an infinite square well, with equally occupied energy levels. The dashed curve shows the undeformed limit $\tau_{\min}^{(0)}$, while the solid curve includes corrections from the quadratic GUP with $\beta=1.0\times10^{-2}$.
• Figure 2: QSL $\tau_{\min}(M,t)$ as a function of both the number of equally occupied energy levels $M$ and evolution time $t$. The gray mesh represents the undeformed result $\tau_{\min}^{(0)}(M,t)$, while the colored mesh (viridis colormap) shows the GUP-corrected values at $\beta=1.0\times10^{-2}$. The visible separation between these surfaces clearly illustrates the acceleration of quantum dynamics due to minimal-length effects.
• Figure 3: Impact of quantum gravity on MT and ML bounds for coherent states. The plot shows QSL surfaces versus evolution time $t$ and mean photon number $\alpha_0^2$. The orange and blue surfaces show the MT and ML bounds without GUP corrections ($\beta = 0$). The gray and green surfaces include GUP corrections ($\beta = 0.08$). GUP corrections shorten the minimal evolution time, clearly accelerating quantum evolution.
• Figure 4: Minimal-length acceleration of quantum evolution. This figure presents MT and ML bounds for a fixed Bures angle $\mathcal{L} = 1$. Panels (a) and (b) display the MT and ML bounds for GUP-modified coherent states, plotted against mean photon number $n = \alpha_{0}^{2}$. Panels (c) and (d) show the same bounds for GUP-modified squeezed vacuum states versus the squeezing parameter $r$. In each panel, black curves show results without deformation ($\beta = 0$), and colored curves include the quartic $p^{4}$ corrections at $\beta = 0.1$. For both state types, minimal-length effects shorten the GUP time.
• Figure 5: Acceleration of quantum dynamics in squeezed vacuum states from minimal-length effects. This figure plots the MT and ML bounds against the squeezing parameter $r$ and evolution time $t$. As shown, the GUP-corrected bounds (gray for MT, green for ML; $\beta=0.08$) fall below the corresponding undeformed surfaces (orange for MT, blue for ML).