Direct and accurate calculation of dwell times and time delays using quantum trajectories

Among the numerous concepts of time in quantum scattering, Smith's dwell time (Smith, 1960 [7] ) and Eisenbud & Wigner's time delay (Wigner, 1955 [12] ) are the most well established. The dwell time represents the amount of time spent by the particle inside a given coordinate range (typically a potential barrier interaction region), while the time delay measures the excess time spent in the interaction region because of the potential. In this paper, we use the exact trajectory-ensemble reformulation of quantum mechanics, recently proposed by one of the authors (Poirier), to study how tunneling and reflection unfold over time, in a one-dimensional rectangular potential barrier. Among other dynamical details, the quantum trajectory approach provides an extremely robust, accurate, and straightforward method for directly computing the dwell time and time delay, from a single quantum trajectory. The resultant numerical method is highly efficient, and in the case of the time delay, completely obviates the traditional need to energy-differentiate the scattering phase shift. In particular, the trajectory variables provide a simple expression for the time delay that disentangles the contribution of the self-interference delay. More generally, quantum trajectories provide interesting physical insight into the tunneling process.