Context-Dependent Time-Energy Uncertainty Relations from Projective Quantum Measurements
arXiv:2507.23059 · doi:10.1103/f9s1-32kz
Abstract
We introduce a general framework for defining context-dependent time distributions in quantum systems using projective measurements. The time-of-flow (TF) distribution, derived from population transfer rates into a measurement subspace, yields a time--energy uncertainty relation of the form , where quantifies net population transfer. This bound applies to arbitrary projectors under unitary dynamics and reveals that time uncertainty is inherently measurement-dependent. We demonstrate the framework with two applications: a general time-of-arrival (TOA)-energy uncertainty relation and a driven three-level system under detuned coherent driving. The TF framework unifies timing observables across spin, atomic, and matter-wave systems, and offers an experimentally accessible route to probing quantum timing in controlled measurements.
7 pages + 2 pages supplementary material. 1 figure + 1 table
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