The phase delay and its complex time: From stationary states up to wave packets

Complex time is often invoked about tunneling effect where the classical phase delay is completed with a crucial filter effect. Usually the complex times are obtained by considering the flux–flux correlation function, but this can be obtained by a very simple approach using the search of the maximum of the generalized complex phase function, including the amplitude of the wave function. Various aspects of the phase delay are presented in the case of wave packets impinging on simple or resonant quantum barriers.Formal links with the classical mechanics give birth to quasi-trajectories of the quantum particle, totally compatible with the quantum mechanics.

► The stationary phase method is extended in including the variations of the spectra.
► The complex phase delay leads to a complex trajectory inside and out-side the barrier.
► Examples of quasi-trajectories are given in case of different quantum barriers.
► Phase delays are specified for resonant tunneling or above-barrier wave-packets.
► The coherence between the quasi-trajectories and quantum mechanics is shown.