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We introduce a magnetic clock approach to measure the traversal and reflection times of an electromagnetic wave through a slab, when multiple reflections are taken into account. The traversal time is a complex quantity whose real component is proportional to the Faraday rotation and whose imaginary component is proportional to the degree of ellipticity. We conclude that the complex traversal time found for an electron by several different methods is not a consequence of the quantum nature of the particle, but due to the wavelike character of the entity involved. PACS numbers: 41.20.Jb, 42.25.p The question of the time spent by a particle in a given region of space has often been discussed in the literature [1 -5]. There are two types of difficulties. First, the particle may have to tunnel through a barrier, in which case it has an imaginary wave vector and we do not know what is the analog of the classical velocity. Second, the final transmission amplitude is the superposition of different paths, along different trajectories or due to multiple rejections, corresponding to different traversal times. The problem has been approached from many different points of view, as shown in the recent review on the subject by Landauer and Martin [1].The most direct method to calculate the traversal time of a particle in a barrier would be to follow the behavior of a wave packet and determine the delay due to the barrier, but this type of approach is beset with difficulties.For example, an emerging peak is not necessarily related to the incident peak in a causative way [6]. Physically more significant is the time during which a transmitted particle interacts with the barrier, as measured by some physical clock which can detect the particle's presence within the barrier. One of the principal approaches to this problem is to utilize the Larmor precession frequency of the spin, produced by a weak magnetic field acting within the barrier region, as first proposed by B az'[7]. The amount of precession clocks the characteristic tunneling time~T, the so-called Biittiker-Landauer time [8,9]. Sokolovski and Baskin [10] obtained a complex traversal time, with the Feynman path-integral technique. They define a functional that measures the time spent by a Feynman path in a region and then sum this quantity over all possible paths with the weighting e' ", where 5 is the action. Which of the two components of this complex time is the most relevant depends on the experiment, but it is often the modulus of the complex time that is the magnitude directly related to the experimental measurements.The question of the traversal time of light through a given region is equally important, but it has been seldom referred to in the literature. The advances in femtosecond technology and optoelectronics, in general, increase the inherent importance of the problem. Measurements of single photon tunneling times, using a two-photon interferometer, had to be interpreted with the existing electron theories due to the lack of a proper theory for electromagne...

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We argue that for granular metals a sizable fraction of the grains becomes charged because the energy fluctuation of the highest-occupied level of each grain, as predicted by random matrix theory, is larger than the charging energy. We have computed the ground state density of states and the degree of ionization of granular metals. The density of states shows a Coulomb gap around the Fermi energy, produced by the long-range part of the Coulomb interactions, which should dominate transport properties at low temperatures.PACS numbers: 71.70. Ms, 71.25.Mg, 71.55.Jv The effects of intragrain and intergrain electronelectron Coulomb interactions are bound to be very important in granular metals (GM), mainly in the dielectric regime [1]. Intragrain Coulomb interactions are responsible for the charging energy of the grains and for the possible formation of a Hubbard-type gap. On the other hand, in the dielectric regime the long-range intergrain Coulomb interaction could produce a Coulomb gap, a decrease around the Fermi level in the single-particle density of states (DOS) [2].The T l/2 law of variable range hopping conductivity has been extensively found in GM and has been interpreted as a manifestation of the Coulomb gap [3]. Photoemission experiments and tunnel conductivity measurements of GM in the dielectric regime have also been interpreted in terms of the Coulomb gap [4][5][6]. Sheng [7] has given an alternative explanation of both this type of behavior for the dc conductivity and of the DOS in terms of the distribution of charging energies. Pollak and Adkins [8] have argued against this explanation and claimed that the Coulomb model rather than the Hubbard model best represents GM.For the Coulomb gap to exist in GM in the dielectric regime a significant portion of the grains must be charged in the ground state. Whether this is the case and, if so, what the ionization-producing mechanism is remains to be clarified, and it is this problem we wish to address in this Letter.The accepted belief, at least until recently, is that the only two relevant energies are the charging energy of the grains and the interlevel spacing 8 near the highestoccupied level, both due to the microscopic size of the grains. If this were so, the vast majority of the grains would be neutral, since the charging energy is much bigger than this interlevel spacing, and no Coulomb gap could exist.Shklovskii and Efros [1] and more recently Pollak and Adkins [8] have proposed that variations in the "work function" associated with the different crystallographic faces of the grains could be of the same order of magnitude as the charging energies and thus be responsible for their ionization. Chui [9] has claimed that a great variety in grains size is responsible for overcoming the charging energies.In this Letter, we argue that the large variation in energy of the highest-occupied level of neutral grains, due to their random surfaces and small sizes, is the most important disorder energy of the problem which ionizes a large portion of the grains. We...

We calculate numerically the transmission coefficient and the traversal time for finite Gaussian wavepackets as a function of their widths.We consider electromagnetic waves crossing a slab and a periodic structure.