Threshold Crossover of Transmitted Larmor Time in Trapezoidal Barriers: A Quantum-Clock Analysis

The Larmor clock extracts time information from quantum scattering through a weak spin-dependent phase readout. We apply this formulation to the transmitted Larmor time in trapezoidal barriers and examine its geometry-dependent threshold response. The main quantity is the dimensionless peak coordinate xp=Ep/V0, where Ep is the energy of the dominant transmitted-time peak. For a right trapezoidal barrier consisting of a linear ramp and a flat segment, the peak moves from the classically forbidden side (xp<1) to the allowed side (xp>1) as the flat part becomes more important. The crossover is characterized by rc=(l-a)/a, defined through xp=1. For a=2 nm and V0=2 μeV, we obtain rc≃0.428. A symmetric triangle-rectangle-triangle barrier shows a corresponding crossover at sc=(d-a)/(2a)≃0.220. These results demonstrate that the strongest transmitted-channel clock response is controlled by the spatial distribution of the sloped matching regions and the flat phase-accumulating region. The peak coordinate therefore provides a phase-sensitive marker of how this response shifts across the nominal threshold as the barrier geometry is varied.