Superluminal tunnelling times as weak values

Abstract

We consider the tunnelling particle as a pre- and post-selected system and prove that the tunnelling time is the expectation value of the position of a ‘clock’ degree of freedom weakly coupled to it. Such a value, called a ‘weak value’, typically falls outside the eigenvalue spectrum of the operator. The appearance of unusual weak values has been associated with a unique interference structure called ‘superoscillations’ (band-limited functions which on a finite interval, approximate functions with spectra well outside their band). It is proposed that superoscillations play an important role in the interferences which give rise to superluminal effects. To demonstrate that, we consider a certain simple tunnelling barrier which allows a wave packet to travel in zero time and negligible distortion, a distance arbitrarily longer than the width of the wave packet. The peak is shown to result from a superoscillatory superposition at the tail. Similar reasoning applies to the dwell time. For this system, both the Wigner time (related to the group velocity) and a clock time correspond to superluminal velocities.     

Continuous input nonlocal games

We present a family of nonlocal games in which the inputs the players receive are continuous. We study three representative members of the family. For the first two a team sharing quantum correlations (entanglement) has an advantage over any team restricted to classical correlations. We conjecture that this is true for the third member of the family as well.     

Hybrid AlGaAs/GaAs/AlGaAs nanowires with a quantum dot grown by molecular beam epitaxy on silicon

Data on the growth features and physical properties of GaAs inserts embedded in AlGaAs nanowires grown on Si(111) substrates by Au-assisted molecular beam epitaxy are presented. It is shown that by varying the growth parameters it is possible to form structures like quantum dots emitting in a wide wavelength range for both active and barrier regions. The technology proposed opens up new possibilities for the integration of direct-band III–V materials on silicon.     

Dynamic neural cortical ensembles model

One-layered model of cortical neurons as a set of overlapping ensembles, each with a structure similar to Hopfield network, is proposed. Ensemble equilibrium equation is solved and formulas for connections weights calculation for given set of attractors are obtained. Concept of dynamic attractors that consists of consequent recalling of stored patterns with moving activity through the network is introduced. Role of dynamic attractors in long-term memory is discussed and mechanism for memory recovery after destruction of some neurons is proposed. Results of experiments on associative memory recovery after partial removal of neurons are shown.     

Sampling functions in nonuniform sampling of a periodic signal

Relations for sampling functions in nonuniform sampling of a trigonometric polynomial and its derivatives are obtained.     

Generalized Catalan numbers in problems of processing of random discrete images

The notion of the Catalan numerical sequence is extended. Problems leading to generalized Catalan numbers are formulated and solved.     

Analytical computer calculations in problems with random division of an interval

The novelty and originality of the presented studies consists of the fact that the rigorous mathematical proof of one of relationships that describe the probability of an error-free reading of random discrete images contained in them was previously found by the authors using specially created programs for computer analytical calculations.