Interference and dispersion effects on tunneling time

We study the interplay between pulse width, interference and tunneling for a wave packet incident upon a barrier and, within the context of tunneling time, we offer a complementary insight into the origin of the Hartman effect. We find that interference together with momentum spread lower (increase) the transmission (reflection) tunneling time thereby `breaking the symmetry between transmission and reflection times'. But, within the limits of our method, we are unable to confirm that negative tunneling time can be obtained.     

Only above barrier energy components contribute to barrier traversal time

A time of arrival operator across a square potential barrier is constructed. The expectation value of the barrier time of arrival operator for a sufficiently localized incident wave packet is compared with the expectation value of the free particle time of arrival operator for the same wave packet. The comparison yields an expression for the expected traversal time across the barrier. It is shown that only the above barrier components of the momentum distribution of the incident wave packet contribute to the barrier traversal time, implying that below the barrier components are transmitted without delay. This is consistent with the recent experiment in attosecond ionization in helium indicating that there is no real tunneling delay time [P. Eckle et al., Science 322, 1525 (2008)].     

Quantum tomography as a new approach to simulating quantum processes

A new method is proposed for ab initio calculations of nonstationary quantum processes on the basis of a probability representation of quantum mechanics with the help of a positive definite function (quantum tomogram). The essence of the method is that an ensemble of trajectories associated with the characteristics of the evolution equation for the quantum tomogram is considered in the space where the quantum tomogram is defined. The method is applied for detailed analysis of transient tunneling of a wave packet. The results are in good agreement with the exact numerical solution to the Schrödinger equation for this system. The probability density distributions are obtained in the coordinate and momentum spaces at consecutive instances. For transient tunneling of a wave packet, the probability of penetration behind the barrier and the time of tunneling are calculated as functions of the initial energy.     

Transmission of Correlated Gaussian Packets Through a Delta-Potential

We study the evolution of the most general initial Gaussian packet with nonzero correlation coefficient between the coordinate and momentum operators in the presence of a repulsive delta-potential barrier, using the known exact propagator of the time-dependent Schrödinger equation. For the initial packet localized far enough from the barrier, we define the transmission coefficient as the probability of discovering the particle in the whole semi-axis on the other side of the barrier. It appears that the asymptotical transmission coefficient (calculated in the large time limit) depends on two dimensionless parameters: the normalized ratio of the potential strength to the initial mean value of momentum and the ratio of the initial momentum dispersion to the initial mean value of momentum. For small values of the second parameter, the result is reduced to the well-known formula for the transparency of the delta barrier, obtained in the plane-wave approximation by solving the stationary Schrödinger equation. For large values of the second parameter, the transmission coefficient can be much larger than that calculated in the plane-wave approximation. For a fixed initial spread of the packet in the coordinate space, the initial correlation coefficient influences the transparency of the barrier only indirectly, through the increase in the initial momentum dispersion.     

Inelastic resonant tunneling

A general expression for the resonant contribution to a tunneling current has been obtained and analyzed in the tunneling Hamiltonian approximation. Two types of resonant tunneling structures are considered: structures with a random impurity distribution and double-barrier structures, where the resonant level results from size quantization. The effect of temperature on the current-voltage curves of tunneling structures is discussed. The study of the effect of potential barrier profile on the d ² I/dV ² line shape is of interest for experiments in inelastic tunneling spectroscopy. Various experimental situations where the inelastic component of the tunneling current can become comparable to the elastic one are discussed. Fiz. Tverd. Tela (St. Petersburg) 40, 1151–1155 (June 1998)     

Stationary phase method and delay times for relativistic and non-relativistic tunneling particles

The stationary phase method is frequently adopted for calculating tunneling phase times of analytically-continuous Gaussian or infinite-bandwidth step pulses which collide with a potential barrier. This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. After reexamining the above-barrier diffusion problem, we notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. Using a recently developed procedure - multiple wave packet decomposition - for some specifical colliding configurations, we demonstrate that the analytical difficulties arising when the stationary phase method is applied for obtaining phase (traversal) times are all overcome. In this case, we also investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstrate that these two distinct transit time definitions are explicitly connected. The traversal times are obtained for a symmetrized (two identical bosons) and an antisymmetrized (two identical fermions) quantum colliding configuration. Multiple wave packet decomposition shows us that the phase time (group delay) describes the exact position of the scattered particles and, in addition to the exact relation with the dwell time, leads to correct conceptual understanding of both transit time definitions. At last, we extend the non-relativistic formalism to the solutions for the tunneling zone of a one-dimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation where the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation based on the non-relativistic tunneling dynamics is revisited. Lessons concerning the dynamics of relativistic tunneling and the mathematical structure of its solutions suggest revealing insights into mathematically analogous condensed-matter experiments using electrostatic barriers in single- and bi-layer graphene, for which the accelerated tunneling effect deserves a more careful investigation.     

Time and energy dependent dynamics of the STM tip — graphene system

Probability current and probability density of wave packets was calculated by solving the three dimensional time-dependent Schrödinger equation for a local potential model of the scanning tunneling microscope (STM) tip — graphene system. Geometrical and electronic structure effects of the three dimensional tunneling process are identified by studying three models of increasing complexity: a jellium half space, a narrow jellium sheet, and a local one electron pseudopotential. It was found that some of the key characteristics of the STM tip — graphene tunneling process are already present at the simple jellium models. In the STM tip — jellium half space system the direction of the momentum does not change during the tunneling event, hence this setup is characterised by introducing an effective distance. For the STM tip — narrow jellium sheet system the direction of the momentum is changed from vertical to horizontal during the tunneling event. The wave packet preferentially tunnels into the bound state of the jellium sheet. For the atomistic model of the graphene sheet an anisotropic spreading of the wave packet was found for hot electrons. This may open new opportunities to build carbon based nanoelectronic devices.     

Features of the tunneling of a system with an internal degree of freedom through a potential barrier

The tunneling of a quantum system with an internal degree of freedom through a potential barrier is considered. Based on the exact numerical solution to the nonstationary Schrödinger equation, the tunneling of a model two-particle system through a potential barrier is studied and the dependences of the tunneling transparency of the barrier on the parameters of the wave packet that describes the system at the initial moment are obtained. A sharp increase in the tunneling probability related to the formation of a long-lived quasibound state of the system in the barrier region is demonstrated. A simple analytical model of the tunneling of a system with an internal degree of freedom that allows for a qualitative interpretation of the main features of the tunneling is constructed.     

Tunneling by an electron packet with an initially sharp wavefront

We have studied the time behavior of electron wavepackets traversing one-dimensional potentials. These packets are described as plane wave states that have been cut off to give a sharp initial wavefront. We find that the shift, or delay time, of the main part of the pulse is comparable to that of a Gaussian wavepacket with the same momentum, and that both kinds of pulses have the same broad-pulse (sharp-momentum) limit. This is shown to hold for a general class of potentials. We show explicitly that the steepest descents calculation described by Stevens does not lead to a finite tunneling velocity. An exact expression is given for the packet transmitted through a delta-function barrier, which suggests a new interpretation of the tunneling velocity that has been obtained in other calculations.     

Current-voltage characteristics of tunnel junctions between superconductors with anisotropic pairing

Charge transfer in tunnel junctions between superconductors with anisotropic singlet pairing is considered theoretically on the basis of the Eilenberger equations for the quasiclassical Green’s functions. New singularities of the current-voltage characteristics, which are characteristic of the case of anisotropic pairing, are treated analytically assuming that the electrons reflect specularly from the boundaries of the tunnel barrier. All four contributions to the tunneling current are investigated. Two of them describe Josephson tunneling, and the other two contributions correspond to the quasiparticle current (the last term appears only for a variable voltage). Different dependences of the order parameter on the momentum directions in the interior of the superconductors and different orientations of the crystal axes relative to the junction plane are considered. The results of numerical calculations of the current-voltage characteristics for several particular cases are presented. Zh. éksp. Teor. Fiz. 111, 1120–1146 (March 1997)     

Salecker-Wigner-Peres clock and average tunneling times

The quantum clock of Salecker-Wigner-Peres is used, by performing a post-selection of the final state, to obtain average transmission and reflection times associated to the scattering of localized wave packets by static potentials in one dimension. The behavior of these average times is studied for a Gaussian wave packet, centered around a tunneling wave number, incident on a rectangular barrier and, in particular, on a double delta barrier potential. The regime of opaque barriers is investigated and the results show that the average transmission time does not saturate, showing no evidence of the Hartman effect (or its generalized version).     

Underbarrier interference

Quantum tunneling through a two-dimensional static barrier becomes unusual when a momentum of an electron has a tangent component with respect to a border of the prebarrier region. If the barrier is not homogeneous in the direction perpendicular to tunneling a fraction of the electron state is waves propagating away from the barrier. When the tangent momentum is zero a mutual interference of the waves results in an exponentially small outgoing flux. The finite tangent momentum destroys the interference due to formation of caustics by the waves. As a result, a significant fraction of the prebarrier density is carried away from the barrier providing a not exponentially small penetration even through an almost classical barrier. The total electron energy is well below the barrier.     

Wave function mapping of self-assembled quantum dots by capacitance spectroscopy

We use frequency-dependent capacitance–voltage spectroscopy to study the dynamic charging of self-assembled InAs quantum dots. With increasing frequency, the AC charging becomes suppressed, beginning with the low-energy states. By applying an in-plane magnetic field, we generate an additional magnetic confinement that alters the tunneling barrier and hence the charging dynamics. In traveling through the potential barrier, the electrons acquire an additional momentum k₀, proportional to the magnetic field B. As the tunneling is enhanced, when k₀ matches the maximum of the electronic wave function Ψ (in momentum representation), we are able to map out the shape of Ψ by varying B.     

Dissipative tunneling at finite temperature

A simple theory is presented for the influence of a weakly coupled interaction system on the tunneling of a particle out of a metastable well. It is based on the standard model of momentum and energy transfer to an infinite set of oscillators and is applied to the case of phase tunneling in a Josephson contact. The distribution of the energy transfer and in particular the Debye-Waller factor for elastic processes is determined by the imaginary part of the dielectric function. For small damping γ the main influence of dissipation on the total tunneling probability is contained in a factor exp —AMγ(Δq)². The numerical coefficientA and the distance Δq under the barrier depend on the considered tunneling state andA(T) vanishes at a temperatureT ^* above which classical activation prevails. The tunneling probability of any level is therefore predicted to increase with temperature. In additional general expressions are derived for the correlation functions of a damped quantum oscillator in terms of the classical response of the interaction system.     

Tunneling time in space fractional quantum mechanics

We calculate the time taken by a wave packet to travel through a classically forbidden region of space in space fractional quantum mechanics. We obtain the close form expression of tunneling time from a rectangular barrier by stationary phase method. We show that tunneling time depends upon the width b of the barrier for $b\to \infty$ and therefore Hartman effect doesn't exist in space fractional quantum mechanics. Interestingly we found that the tunneling time monotonically reduces with increasing b. The tunneling time is smaller in space fractional quantum mechanics as compared to the case of standard quantum mechanics. We recover the Hartman effect of standard quantum mechanics as a special case of space fractional quantum mechanics.     

Barrier traversal times using a phenomenological track formation model

A phenomenological model for a measurement of “barrier traversal times” for particles is proposed. Two idealized detectors for passage and arrival provide entrance and exit times for the barrier traversal. The averaged traversal time is computed over the ensemble of particles detected twice, before and after the barrier. The “Hartman effect” can still be found when passage detectors that conserve the momentum distribution of the incident packet are used.     

Average transmission times for the tunneling of wave packets

We consider the average time for a localized wave packet to tunnel through a finite rectangular barrier, as prescribed by the Salecker–Wigner–Peres clock after a post-selection of transmitted final states. We investigate the properties of this time both in the relativistic and nonrelativistic regimes and address the questions of the emergence of the Hartman effect and superluminal tunneling velocities.     

On the Traversal Time of Barriers

Fifty years ago Hartman studied the barrier transmission time of wave packets (J Appl Phys 33:3427–3433, 1962). He was inspired by the tunneling experiments across thin insulating layers at that time. For opaque barriers he calculated faster than light propagation and a transmission time independent of barrier length, which is called the Hartman effect. A faster than light (FTL or superluminal) wave packet velocity was deduced in analog tunneling experiments with microwaves and with infrared light thirty years later. Recently, the conjectured zero time of electron tunneling was claimed to have been observed in ionizing helium inside the barrier. The calculated and measured short tunneling time arises at the barrier front. This tunneling time was found to be universal for elastic fields as well as for electromagnetic fields. Remarkable is that the delay time is the same for the reflected and the transmitted waves in the case of symmetric barriers. Several theoretical physicists predicted this strange nature of the tunneling process. However, even with this background many members of the physics community do not accept a FTL signal velocity interpretation of the experimental tunneling results. Instead a luminal front velocity was calculated to explain the FTL experimental results frequently. However, Brillouin stated in his book on wave propagation and group velocity that the front velocity is given by the group velocity of wave packets in the case of physical signals, which have only finite frequency bandwidths. Some studies assumed barriers to be cavities and the observed tunneling time does represent the cavity lifetime. We are going to discus these continuing misleading interpretations, which are found in journals and in textbooks till today.     

Entangled electron current through finite size normal-superconductor tunneling structures

We investigate theoretically the simultaneous tunneling of two electrons from a superconductor into a normal metal at low temperatures and voltages. Such an emission process is shown to be equivalent to the Andreev reflection of an incident hole. We obtain a local tunneling Hamiltonian that permits to investigate transport through interfaces of arbitrary geometry and potential barrier shapes. We prove that the bilinear momentum dependence of the low-energy tunneling matrix element translates into a real space Hamiltonian involving the normal derivatives of the electron fields in each electrode. The angular distribution of the electron current as it is emitted into the normal metal is analyzed for various experimental setups. We show that, in a full three-dimensional problem, the neglect of the momentum dependence of tunneling causes a violation of unitarity and leads to the wrong thermodynamic (broad interface) limit. More importantly for current research on quantum information devices, in the case of an interface made of two narrow tunneling contacts separated by a distance r, the assumption of momentum-independent hopping yields a nonlocally entangled electron current that decays with a prefactor proportional to r ^-2 instead of the correct r ^-4.Received: 14 June 2004, Published online: 24 September 2004PACS: 74.45. + c Proximity effects; Andreev effect; SN and SNS junctions - 74.50. + r Tunneling phenomena; point contacts, weak links, Josephson effects