Unified time analysis of photon and particle tunnelling

A unified time analysis of photon and nonrelativistic particle tunnellings is presented, in which time is regarded as a quantum observable, canonically conjugated to energy. Within this approach, one can introduce self-consistent definitions of the tunnelling times, on the basis of conventional quantum mechanics (or one-dimensional quantum electrodynamics) only. The validity of the Hartman effect [which states the tunnelling duration to be independent of the (opaque) barrier width, with superluminal group velocities of the tunnelling packet as a consequence] is verified for all the known expressions of the mean tunnelling time. However, some noticeable generalizations of (and deviations from) the Hartman effect are, as well, briefly investigated. Moreover, the analogy between particle and photon tunnelling is suitably exploited; on the basis of such an analogy, an explanation of some recent interesting microwave and optical experimental results on tunnelling times is proposed. Attention is devoted, at last, to some aspects of the causality problem for particle and photon tunnelling