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.     

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.     

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).     

Photon tunneling in a frustrated total internal reflection structure with a lossy indefinite metamaterial barrier

We study the phenomenon of photon tunneling through a frustrated total internal reflection structure with a dispersive lossy indefinite metamaterial barrier. The tunneling coefficient, lateral shift, and tunneling time for different incident light waves through the barrier are obtained by employing the stationary-phase approximation. The properties of tunneling time and lateral shift are discussed for different metamaterial parameters. It is shown that negative lateral shift and tunneling time can appear in the cutoff and always cutoff metamaterials. The lateral shift can be enhanced by adjusting the light frequency, the barrier thickness, and the permittivity or permeability of the indefinite metamaterial. In addition, it is found that the lateral shift can be resonantly amplified due to the weak loss of the indefinite metamaterial slab. The Hartman effects of the lateral shift and tunneling time are also demonstrated. PACS 42.50.Ct; 42.25.Bs; 73.43.Jn; 73.40.Gk; 42.70.Qs     

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.     

Tunneling of Dirac fermions through magnetic barriers in graphene

We have studied the tunneling of Dirac fermions through magnetic barriers in graphene. Magnetic barriers are produced via delta function-like inhomogeneous magnetic fields in which Dirac fermions in graphene experience the tunneling barrier in the real sense in contrast to Klein paradox caused by electrostatic barriers. The transmission through the magnetic barriers as functions of incident energy and angle of incoming fermions shows characteristic oscillations associated with tunneling resonances. We have also found the confined states in the magnetic barrier region which turn out to correspond to the total internal reflection in the usual optics.     

Quasi-distribution of tunneling time

Using real time Feynman histories, a quasi-distribution of tunneling time Q(τ) is introduced. For the tunneling time of resident time type, an explicit expression for Q is shown for square barriers. Q becomes oscillatory as the barrier becomes opaque. Some well-known tunneling times fall within the range of τ where Q takes non-negligible values. The formal “average” and the “variance” of the tunneling time are found to be related to known tunneling times. It is thus demonstrated that the quasi-distribution extracts the temporal information about tunneling from real time Feynman histories.     

Hartman effect in presence of Aharanov-Bohm flux

The Hartman effect for a tunnelling particle implies the independence of group delay time on the opaque barrier width with superluminal velocities as a consequence. This effect is further examined on a quantum ring geometry in the presence of Aharonov-Bohm flux. We show that while tunnelling through an opaque barrier, the group delay time for given incident energy becomes independent of the barrier thickness as well as the magnitude of the flux. The Hartman effect is thereby extended beyond one dimension in the presence of Aharonov-Bohm flux.     

New approach to the quantum tunneling process: Characteristic times for transmission and reflection

In [1] we have demonstrated that scattering of a quantum particle on a one-dimensional potential barrier should be considered as a combined process involving two alternative elementary transmission and reflection processes. For symmetric potential barriers, we have found solutions of the Schrödinger equation which describe the transmission and reflection processes in all stages of scattering. The present work studies time aspects of both processes. The local and asymptotic group tunneling times, dwell time, and Larmor tunneling time are determined for each process. Among these time characteristics, the group tunneling times should be considered as auxiliary. As to the dwell and Larmor tunneling times, they are the best estimates (of the expected values) of times the quantum particle in stationary and localized nonstationary states dwells in the barrier region. Moreover, the Larmor time is simply the dwell time averaged over the corresponding ensemble of particles. This characteristic can be measured experimentally and hence the suggested model of scattering can be verified.     

Questions concerning the generalized Hartman effect

With reference to a particle tunneling through two successive barriers, it seems to have been generally accepted that the tunneling time does not depend on the separation distance between the barriers. This phenomenon has been called the generalized Hartman effect. In this Letter, we point out a lack of mathematical rigor in the reasoning by which this effect was deduced about ten years ago. A mathematically rigorous treatment shows us that the tunneling time does indeed depend on the length of the free space between the barriers.     

Spin-dependent delay time and Hartman effect in asymmetrical graphene barrier under strain

We study the spin-dependent tunneling time, including group delay and dwell time, in a graphene based asymmetrical barrier with Rashba spin–orbit interaction in the presence of strain, sandwiched between two normal leads. We find that the spin-dependent tunneling time can be efficiently tuned by the barrier width, and the bias voltage. Moreover, for the zigzag direction strain although the oscillation period of the dwell time does not change, the oscillation amplitude increases by increasing the incident electron angle. It is found that for the armchair direction strain unlike the zigzag direction the group delay time at the normal incidence depends on the spin state of electrons and Hartman effect can be observed. In addition, for the armchair direction strain the spin polarization increases with increasing the RSOI strength and the bias voltage. The magnitude and sign of spin polarization can be manipulated by strain. In particular, by applying an external electric field the efficiency of the spin polarization is improved significantly in strained graphene, and a fully spin-polarized current is generated.     

Time domain de Broglie wave interferometry along a magnetic guide

Propagation and tunneling of light through photonic barriers formed by thin dielectric films with continuous curvilinear distributions of dielectric susceptibility across the film, are considered. Giant heterogeneity-induced dispersion of these films, both convex and concave, and its influence on their reflectivity and transmittivity are visualized by means of exact analytical solutions of Maxwell equations. Depending on the cut-off frequency of the film, governed by the spatial profile of its refractive index, propagation or tunneling of light through such barriers are examined. Subject to the shape of refractive index profile the group velocities of EM waves in these films are shown to be either increased or deccreased as compared with the homogeneous layers; however, these velocities for both propagation and tunneling regimes remain subluminal. The decisive influence of gradient and curvature of photonic barriers on the efficiency of tunneling is examined by means of generalized Fresnel formulae. Saturation of the phase of the wave tunneling through a stack of such films (Hartman effect), is demonstrated. The evanescent modes in lossy barriers and violation of Hartman effect in this case is discussed.     

Quantum reflection of a Bose-Einstein condensate with a dark soliton from a step potential

We study dynamical behaviors of a Bose-Einstein condensate (BEC) containing a dark soliton reflected from potential wells and potential barriers, respectively. The orientation angle of the dark soliton and the width of the potential change play key roles on the reflection probability R_s. Variation of the reflection probability with respect to the orientation angle θ of the dark soliton can be well described by a cosine function R_s~cos[λ(θ-π/2)], where λ is a parameter determined by the width of the potential change. There are two characteristic lengths which determine the reflection properties. The dependence of the reflection probability on the width of the potential change shows distinct characters for potential wells and potential barriers. The length of the dark soliton determines the sensitive width of potential wells, whereas for potential barriers, the decay length of the matter wave in the region of the barrier qualifies the sensitive width of the barrier. The time evolution of the density profiles of the system during the reflection process is studied to disclose the different behaviors of matter waves in the region of the potential variation.     

Resonance energy level selection in symmetrical rectangular quadruple-barrier structures

Resonant tunneling is studied theoretically in symmetrical rectangular quadruple-barrier structures by taking into account mass difference between the well layer and the barrier layer. Analytical expressions for the transmission coefficient and the resonance condition are derived. Especially, it is obtained that two independent resonance energies can be determined analytically as a function of the outer and the central well widths, which may be very useful for resonant tunneling device fabrications. Furthermore, the variation of the resonance energy with the outer barrier width is investigated. It is found that the resonance level is rarely affected with a small change of the outer barrier width. It is derived analytically that the transmission peak is decreased to less than unity when the outer barrier width becomes thinner than a critical value.     

Magnetotransport in a graphene monolayer with two tunable magnetic barriers

The magnetotransport property for a monolayer graphene with two turnable magnetic barriers has been investigated by the transfer-matrix method. We show that the parameters of barrier height, width, and interval between two barriers affect the electron wave decaying length, which determine the conductance with parallel or antiparallel magnetization configuration, and consequently the tunneling magnetoresistance (TMR) for the system. Interestingly, a graphene attached by two barriers with different heights can produce a resonant TMR peak at low energy region one order of magnitude larger than that for the system with two same height barriers because that the asymmetry of magnetic barriers block the electron transmission in the case of antiparallel magnetization configuration. The results obtained here may be useful in understanding of electron tunneling in graphene and in designing of graphene-based nanodevices.     

Giant proximity effect in cuprate superconductors

Using an advanced molecular beam epitaxy system, we have reproducibly synthesized atomically smooth films of high-temperature superconductors and uniform trilayer junctions with virtually perfect interfaces. We found that supercurrent runs through very thick barriers. We can rule out pinholes and microshorts; this "giant proximity effect" (GPE) is intrinsic. It defies the conventional explanation; it might originate in resonant tunneling through pair states in an almost-superconducting barrier. GPE may also be significant for superconducting electronics, since thick barriers are easier to fabricate.     

The effect of non-constant effective tunneling mass and asymmetry for resonant tunneling in double-barrier structures

We discuss the transmission coefficient τ_d in non-repetitive, one-dimensional, rectangular double-barrier structures without simplifications such as strongly attenuating barriers, strong localization, or overall constant effective tunneling mass of the electron. For resonance τ_d=1, we obtain two non-approximative conditions which require different resonance energies of the tunneling electron than previously reported in the literature. In fact, the resonance peaks are shifted to higher energy levels in the order of the width of the peaks due to the effect of non-constant tunneling mass. We investigate the dependence of the resonance condition and the shape of the resonance peaks in regard to perturbation of the electron energy, the gap width as well as the barrier width and height. Resonance is stable for variation of the barrier width but sensitive for variation of the barrier height and the gap width. Received: 9 December 1998 / Accepted: 5 January 1999 / Published online: 31 March 1999     

含单负介质层受阻全内反射结构的光子隧穿现象研究

利用稳定相位理论得到了光子穿越含单负介质层受阻全内反射结构的隧穿时间以及光子穿透光学势垒后产生的横向位移.分析结果表明，当势垒为单负介质时，光子隧穿可能表现出负的隧穿时间和负的横向位移.隧穿时间和横向位移存在Hartman效应，使得光子隧穿过程具有超光速性质.此外，基于TM波和TE波通过负介电常数介质和负磁导率介质势垒产生的横向位移的方向正好相反，得到了一种有效的区分两类单负介质的方法. 关键词： 光子隧穿 负折射 单负介质 超光速     

Single-photon tunneling in photonic crystals with deep defect states

A new design for a single-photon tunneling device based on tunneling of light through a gap in a photonic crystal linear waveguide containing a deep defect state is suggested. Unlike recently observed single-photon tunneling through nanometer scale nonlinear pinholes in a thick gold film, in the device described here the photon current should be strictly conserved. By appropriate design of the barrier width, defect level position in the bandgap, and defect spatial position in the barrier, a desired photon tunneling rate can readily be engineered.