Tunneling as a quantum mechanical effect takes place in a classically forbidden region between two regions of classically allowed motion. While the term “dynamical tunneling” typically refers to tunneling of quantum states across dynamical barriers in classical phase space [1], the original problem simply intended tunneling across a potential barrier. Both types of tunneling are addressed in this chapter, with major focus on situations in which external forces make the studied systems intrinsically time-dependent and allow for a dynamical control of tunneling through potential barriers or across band gaps which are dynamically explored by the system.
The interaction between incident and target particles is inferred from the numbers and angular distribution of scattered particles and from their energy distribution. It is important to recognize that a scattering region, as represented for example by a rectangular barrier, is a region where, in classical terms, forces are applied to the incident particles. A full analysis of scattering events must, in principle, embody the description of a localized free particle that travels toward a center of force and is scattered by it. Quantum-mechanically, that implies a theoretical analysis in terms of packet states, with an explicit account of their change of position with time. One of the most striking effects described by quantum mechanics is the transmission of incident particles across a potential barrier whose height is greater than the particle energy. Classically, these particles lack the energy to get over the barrier, yet for quantum-mechanical systems some fraction of incident particles are transmitted.
Time it takes to travel from one position to another, devoid of any quantum mechanical description, has been modeled variously, especially for quantum tunneling. The model time, if universally valid, must be subluminal, must hold everywhere (inside and outside the tunneling region), must comprise interference effects, and must have a sensible classical limit. Here we show that the quantum travel time, hypothesized to emerge with the state vector, is a function of the probability density and probability current such that all the criteria above are fulfilled. We compute it inside and outside a rectangular potential barrier and find physically sensible results. Moreover, we contrast it with recent ionization time measurements of the
He as well as the
Ar and
Kr atoms, and find good agreement with data. The quantum travel time holds good for stationary systems, and can have applications in numerous tunneling-driven phenomena.