Larmor and delay times in transient quantum tunnelling

Abstract We derive a time-dependent solution for the Schrödinger-Pauli equation within a quantum shutter model to analyse cut-off plane waves scattered by a finite-range potential under a constant magnetic field. Exact analytical solutions for the time-dependent spinor components are derived for the low-energy plane-wave case, enabling the exploration of dynamical effects within the transmission region of a potential barrier. We demonstrate that the spinor probability density exhibits a diffraction in time pattern, with peaks shifted by the presence of the magnetic field relative to the field-free case. This shift introduces a delay time, which we analyse as a function of the Larmor frequency, with tunable peak positions controlled by the field intensity. We further explore the dynamical features of the spin expectation values 〈 S k 〉, which exhibit unique diffraction in time behaviour and converge to stationary values in the long-time asymptotic regime. Additionally, we show that measuring the average total spin of the system offers an alternative approach for calculating the dynamical delay time. For infinitesimally small magnetic fields, we define a characteristic dynamical Larmor timescale associated with the evolution of 〈 S y 〉, enabling the recovery of the Büttiker Larmor time, τ y , at low energies in the long-time limit. These findings demonstrate that magnetic fields can precisely manipulate tunnelling timescales in quantum systems, providing a means to control matter wave dynamics.