Chapter 19: Time-Delay, Reflection Time Operator and Minimum Tunneling Time

Daniel Bakucz Canário, Michael Klaiber, Karen Z. Hatsagortsyan +1 more

The problem of time delay in tunneling ionization is revisited. The origin of time delay at the tunnel exit is analyzed, underlining the two faces of the concept of the tunneling time delay: the time delay around the tunnel exit and the asymptotic time delay at a detector. We show that the former time delay, in the sense of a delay in the peak of the wave function, exists as a matter of principle and arises due to the sub-barrier interference of the reflected and transmitted components of the tunneling electronic wave packet. We exemplify this by describing the tunneling ionization of an electron bound by a short-range potential within the strong-field approximation in a ``deep tunneling'' regime. If sub-barrier reflections are extracted from this wave function, then the time delay of the peak is shown to vanish. Thus, we assert that the disturbance of the tunneling wave packet by the reflection from the surface of the barrier causes a time delay in the neighborhood of the tunnel exit.

Eli Pollak

How much time does it take for a particle to tunnel through a barrier? This question continues to baffle up to this very day, one of the reasons being the non-existence of an energy–time commutator analogous to the position–momentum commutation relation. Instead of worrying about the definition of an operator one may consider time as a parameter in the time-dependent Schrödinger equation and study the time evolution of a wavepacket appropriately scattered on a potential barrier. Using this approach, one may formally define a tunnelling flight time that may be measured in a time-of-flight experiment. The resulting tunnelling flight time either vanishes or is very small. The implications of this observation with respect to recent hydrogen and helium atom attosecond photoionization experiments are discussed. The vanishing or small flight time does not contradict the finite time measured in Larmor clock experiments, where the tunnelling particle affects an external degree of freedom whose dynamics induced by the tunnelling may be interpreted in terms of a time scale. Flight times are also relevant to quantum reflection where coherences due to resonance scattering are not well accounted for by Wigner's phase time delay.

Παναγιωτησ Τροχουτσοσ

The present work consists of eight chapters. The first chapter gives a brief historical overview of the impasses that Classical Physics found in the late 19th and early 20th centuries, and the efforts made in the framework of Old Quantum theory to overcome it, ultimately leading to the foundation of Quantum Engineering in the 1920s. The second chapter shows the ratio of the Schrodinger equation to the classical wave differential equation, the concept of wave function, the physical importance of probability density, continuity equation and probability current density, while indicating the conditions to be met by the wave function. The third chapter examines square dynamic wells as well as the rectangular dynamic barrier, through the study of which the tunnel phenomenon emerges. The reflecting and transit coefficients are also calculated, while the tunnel effect is also being examined in relation to the principle of uncertainty. The fourth chapter presents semi-classical approaches to the solution of the Schrodinger equation, with an emphasis on the WKB approach. The power of the WKB approach is analysed, and applied to a dynamic well, as well as in the case of tunneling through a dynamic barrier, by calculating the transit coefficient. The fifth chapter analyses the time delay in the tunnel phenomenon and examines the concept of minimum transit time. The sixth chapter refers to natural systems where the tunnel phenomenon occurs, such as cold emission of electrons from metals, a-decay and nuclear fusion. Finally, due to the purely undulating nature of the phenomenon, cases of observation are mentioned macroscopicly, such as the case of total internal reflection. The seventh chapter presents a lesson plan for the teaching of the phenomenon in Secondary Education, using information and communication technologies in the educational process, and finally the main conclusions presented resulting from the previous study.