Hartman Effect Research Articles

Time delay of reflected and transmitted laser pulse at laser action on a dense plasma slab

The time delay phenomenon of reflected and transmitted laser pulse under the action of s-polarized laser radiation on a supercritical plasma slab is considered. The dependence of the time delay and peak intensity of reflected and transmitted signals on the slab thickness, plasma density, and angle of laser radiation incidence is investigated. It is shown that with increasing thickness of the plasma slab, the saturation of the transmitted signal delay time (Hartman effect) does not occur due to the exponential decrease in its intensity. The tunneling speed of the laser pulse energy through the slab of supercritical plasma is calculated, and it is shown that at large angles of laser radiation incidence, it can exceed the speed of light in vacuum.

Superluminality in parity-time symmetric Bragg gratings

Parity-time ( PT ) symmetric Bragg gratings (PTBGs) possess unique features compared to traditional ones. For example, the photonic bandgap of a PTBG can be modified and even closed when the PT symmetry evolves from an exact phase to a broken one, and the complex reflection coefficient of a PTBG is sensitive to the direction of incidence. In this article, we reveal how the superluminal effects of transmission behave following the modified band structure of PTBGs. The superluminality of the directionally sensitive reflection is also discussed. We then investigate the Hartman effect and argue that, to account for the superluminal effects in PTBGs, a directionally sensitive dwell time should be applied. This study offers unique insights into the mechanisms of superluminality and group delay of light in non-Hermitian open systems and contributes to the advancement of PTBGs, which can eventually become an indispensable platform for probing some of the exotic properties of optical wave phenomena and light–matter interactions.

Tunnel Effect for Ultrasonic Waves in Tapered Waveguides

Traversal time in the tunneling effect for ultrasonic waves in tapered waveguides is derived considering its analogy with quantum and electromagnetic wave tunneling. If, as traversal time, the so-called phase time is considered, the ultrasonic wave packet shows the equivalent in acoustics of superluminality, i.e., the derived velocity, crosses the limit of bulk transverse ultrasonic waves in the medium of the waveguide that is the equivalent of c in the quantum and electromagnetic cases. The graphs clearly illustrating this so-called Hartman effect are obtained confirming the experimental results in the three different fields.

Time delay and nonadiabatic calibration of the attoclock. Multiphoton process versus tunneling in strong field interaction

The measurement of the tunneling time delay in attosecond experiments, termed attoclock, sparked a heated debate about tunneling time and the role of time in quantum mechanics. We show a model that explains the experimental result of the attoclock in the nonadiabatic field calibration, where we find good agreement with the experimental data of Hofmann et al. (J. Mod. Opt. 66, 1052, 2019). We confirm our result with the numerical integration of the time-dependent Schrödinger equation. We also found that at a field strength F≤Fa (Fa is the atomic field strength), the model always predicts a time delay with respect to the quantum (lower) limit τa at F=Fa. For an adiabatic tunneling, the saturation at the limit (F=Fa) explains the well-known Hartman effect or Hartman paradox.

Wigner time delay and Hartman effect in quantum motion along deformed Riemannian manifolds

Elastic scattering of a wave can be quantified by a shift in the phase with respect to the phase of the incoming wave. A qualitative measure of the time during which the effect occurs is given by the Wigner time delay. The tunneling time is known to saturate with increasing tunneling barrier width (Hartman effect). Here, we analyze the elastic quantum mechanical scattering in a deformed one-dimensional Riemannian manifold, particularly with respect to the Wigner time delay and conclude on the Hartman effect. It is shown that scattering due to local curvature variations results in imperfect conduction and leads to a Wigner time delay that at low energies, is in variance with the classical time delay that is inferred from the arc length. At moderate and high energies, however, classical and quantum time delays coincide.

Lower bound of group velocity imposed by non-linearities induced by external sources in photonic and phononic crystals with losses

In the present work, it has been studied how the group velocity, in waves propagating trough photonic and phonic crystals with losses, is affected by non-linearities induced by external sources. The work takes as starting point the mathematical approach exposed in the article called ‘Effect of loss on the dispersion relation of photonic and phononic crystals’ of V Laude et al, where it is considered, as a plausible hypothesis, that the non-linearities are only induced by external mechanisms and not by the modes propagating through the structure. The research has been focused on some points of the band structure that are of great interest: frozen modes and in the middle of bandgap. A strong dependency of the lower bound of group velocity has been found as a function of points considered. As a main result, it has been observed that the non-linear effects induced and material losses considered can have opposite effects on the group velocity, observing cases where the lower bound of group velocity is reduced below the limits established by the work of V Laude et al.On the other hand, it has been found that the non-linearities limit the group velocity in the middle of bandgap, disappearing the classical Hartman effect.

Phase time delay caused by quantum effects in nearby plasmonic nanostructures of a one-dimensional photonic crystal

A one-dimensional photonic crystal (1DPC) incorporated with a defect layer containing a four-level double V-type quantum system adjacent to a plasmonic nanostructure is employed to investigate the Hartman effect. The study involves the interaction of two orthogonal circularly polarized laser beams with the defect layer, possessing identical frequencies but vary in phase and electric field amplitude. The defect layer exhibits quantum system adjacent to plasmonic nanostructure and field interaction phenomena like optical transparency, nonzero dispersion with zero absorption, gain without inversion, and others related effects. By manipulating the phase of the driving fields and probe detuning, the 1DPC can function as either a positive index material (PIM) or a negative index material (NIM), correlating to the normal and anomalous dispersion of the defect layer, respectively. The positive and negative Hartman effects for PIM and NIM, respectively, can be observed by adjusting the relative phase with respect to the driving fields. Our suggested approach might be used in optical memory, all-optical switching, all-optical routing, and interferometry.

Does the Hartman effect exist in triangular barriers

We study the phase, Larmor and dwell times of a particle scattered off triangular barriers (TBs). It is interesting that the dependences of dwell, reflective phase and Larmor times on the wave number, barrier width and height for a pair of mirror-symmetric (MS) exact triangular barriers (ETBs) are quite different, as the two ETBs have quite distinct scattering surfaces. In comparison, the dependence of the transmitted phase or Larmor times is exactly the same, since the transmitted amplitudes are the same for a pair of MS TBs. We further study the Hartman effect by defining the phase and Larmor velocities associated with the phase and Larmor times. We find no barrier width saturation effect for the transmitted and reflected times. This is indicated by the fact that all the velocities approach finite constants that are much smaller than the speed of light in vacuum for TBs with positive-slope impact faces. As for ETBs with vertical left edges, the naive velocities seem to also indicate the absence of the Hartman effect. These are quite distinct from rectangular barriers and may shed new light on the clarification of the tunneling time issues.

Effect of the Rashba interaction on the tunneling time and Hartman effect in an 8-Pmmn borophene superlattice

The spin-dependent group delay time and Hartman effect as well as the valley/spin polarization in an 8-Pmmn borophene superlattice under Rashba interaction are investigated theoretically, by using the stationary phase and the transfer matrix approaches. The group delay time depends on the spin degree of freedoms, and can be effectively controlled by changing the direction of superlattice, incident electron angle and Rashba strength. Both the valley and spin polarization reveal a strong dependence on the number of the superlattice barriers. Furthermore, group delay time oscillates as the width of the potential barriers increases, but in special conditions, the dependence on the width of the potential barriers will disappear. Interestingly, by increasing the angle of the direction of the superlattice the Hartman effect can be observed for most electron incidence angles. Our study show that, the 8-Pmmn borophene superlattice can be useful for future electronics and spintronics applications.

Evanescent-field Mach-Zehnder interferometer

We propose an evanescent-field interferometer in the Mach-Zehnder configuration, which can provide an interferometric measure of the phase of evanescent fields and can be used to quantify the Hartman effect. The phase difference for such tunneled fields saturates to $\ensuremath{\pi}/2$, thus confirming the classic Hartman effect. An analytical model of the proposed evanescent-field interferometer is presented and validated by numerical simulations based on the finite-difference time-domain technique. In the numerical treatment, the requisite elements of the interferometer, such as beam splitters and mirrors, are realized through carefully designed cavities in a two-dimensional photonic crystal system. The underlying cavity modes are further engineered using a weak perturbation to spatially maneuver the evanescent field along preferred directions. The evanescent-field interferometer offers multiple avenues of introducing phase delay and has higher sensitivity that is spatially delocalized in comparison to the conventional propagating-field interferometers.

Spin-dependent tunnelling time in phosphorene superlattice

ABSTRACT We study theoretically the Hartman effect and spin-dependent group delay time in a strained phosphorene superlattice in the presence of Rashba spin–orbit interaction (RSOI). It is found that the total group delay time depends on the spin degree of freedom. Thus, spin filtering can be obtained in the time domain by means of the phosphorene superlattice. The total group delay time can be efficiently controlled by the angles of incidence of electrons, strength of RSOI and number of superlattice barriers. Particularly, the magnitude of the total group delay time can be tuned via the direction and strength of strain. Therefore, in phosphorene superlattice the total group delay time can be efficiently adjusted mechanically. Moreover, in a strained phosphorene superlattice the superluminal tunnelling or Hartman effect can be observed even for a normal incident.

Controllable Hartman effect by vortex beam in a one dimensional photonic crystal doped by graphene quantum dots

The Hartman effect is studied in a one dimensional photonic crystal doped with graphene quantum dots. It is shown that the Hartman effect can be switched from negative to positive by increasing the Rabi-frequency of the controlling field and also by manipulating the relative phase of the applied fields. The effect of the vortex beam on the Hartman effect is also presented. We show that the orbital angular momentum (OAM) and the azimuthal phase of the vortex beam do not affect the probe filed transmission while they change the Hartman effect from positive to negative.

Propagation of the Weyl spinor packet through a potential barrier

We investigate the propagation of the Weyl spinor wave packet through a rectangular potential barrier. The coefficients of the reflection and transition are calculated. It is discovered that the Hartman effect, known in nonrelativistic case, is absent. The spinor could not exceed speed of light in the quantum tunneling. This is due to the simple dispersion relation for the relativistic massless particle.

An exact tunneling model and its application to transmission and reflection delay times

Electron emission into a nanogap is not instantaneous, which presents a difficulty in simulating ultra-fast behavior using particle models. A method of approximating the transmission and reflection delay (TARD) times of a wave packet interacting with barriers described by a delta function, a metal–insulator–metal (MIM, rectangular) barrier, and a Fowler Nordheim (FN, triangular) barrier is given and has application to simulation. It is based on the superposition of a finite number of exact basis states obtained from Schrödinger’s equation, analogous to how quantum carpets are simulated. As a result, it can exactly and uniquely follow exponentially small tunneling currents. A Bohm-like trajectory is obtained from the time evolution of the density: it shows delay in both the transmitted and reflected packets that can be simply evaluated. The relations to prior studies of the analytic [Formula: see text]-function barrier and the Wigner distribution function (WDF) methods are described. A comparison of the TARD times is contrasted to alternate times in the Büttiker–Landauer (BL) and McColl–Hartman (MH) times; the MH approach is further reformulated explicitly in terms of Gamow factors to consider how the McColl–Hartman effect is to be related, particularly in the case of the FN barrier of field emission.

Spin dependent Goos–Hänchen-like effect in a strained phosphorene superlattice

Herein, we present a theoretical study on the Goos–Hänchen (GH)-like effect in a phosphorene superlattice in the presence of strain and an external Rashba spin-orbit interaction (ERSOI). The spin-up and spin-down electrons have different GH shifts and their separation can be tuned via the angle of incidence of the electrons, ERSOI strength, direction strain and strength of strain. The results show that the GH shift exhibits an oscillatory evolution with the ERSOI strength, barrier width, and strain strength. In addition, the GH shift can have negative as well as positive values, and the magnitude of the spin-dependent GH shift can be efficiently controlled by the number of superlattice barriers and mechanically. Interestingly, in a strained phosphorene superlattice under special conditions, the GH shift of the electrons with and without spin-flipping increases linearly by the barrier width, which can be related to the Hartman effect.

Non‐Hermitian Hartman Effect

AbstractThe Hartman effect refers to the rather paradoxical result that the time spent by a quantum mechanical particle or a photon to tunnel through an opaque potential barrier becomes independent of barrier width for long barriers. Such an effect, which has been observed in different physical settings, raised a lively debate and some controversies, owing to the correct definition and interpretation of tunneling times and the apparent superluminal transmission. A rather open question is whether (and under which conditions) the Hartman effect persists for inelastic scattering, that is, when the potential becomes non‐Hermitian and the scattering matrix is not unitary. Here, tunneling through a heterojunction barrier in the tight‐binding picture is considered, where the barrier consists of a generally non‐Hermitian finite‐sized lattice attached to two semi‐infinite nearest‐neighbor Hermitian lattice leads. A simple and general condition is derived for the persistence of the Hartman effect in non‐Hermitian barriers, showing that it can be found rather generally when non‐Hermiticity arises from nonreciprocal couplings, that is, when the barrier displays the non‐Hermitian skin effect, without any special symmetry in the system.

Reevaluating the Hartman effect for field emission

An electron wave packet tunneling through a barrier has a transmission (or ``group delay'') time ${\ensuremath{\tau}}_{g}$ that, for a rectangular barrier, is commonly held to become independent of the barrier width $L$ as the width increases (the McColl-Hartman effect). In the present study, it is shown that first, the McColl-Hartman effect for a rectangular barrier is dependent upon $L$ as the Gamow tunneling factor $\ensuremath{\theta}(k)$ vanishes, and ${\ensuremath{\tau}}_{g}$ is only independent of $L$ when $\ensuremath{\theta}(k)$ is large; and second, for a triangular barrier to model field emission, although ${\ensuremath{\tau}}_{g}$ can be large for small field, it vanishes when the energy matches the barrier height.

General(ized) Hartman effect

In this letter we prove explicitly that if the Hartman effect exists for an arbitrary “unit cell” potential, then it also exists for a periodic system constructed using the same “unit cell” potential repeatedly. We further show that if the Hartman effect exists, the tunneling time in the limiting case of a sufficiently thick “unit cell” potential is the same as that of its periodic system. This is true for any arbitrary value of the intervening gap between the consecutive “unit cell” of the periodic system. Thus, the generalized Hartman effect always occurs for a general potential constructed using multiple copies of single potential which shows Hartman effect.

Hartman effect at merging point in graphene under uniaxial strain

The dwell time corresponding to the Klein tunneling of Dirac fermions confined in single-layer graphene in the presence of uniaxial strain was investigated. The numerical results of the tunneling time and the transmission probability for several values of the merging parameter were presented. The effect of barrier's height/length, energy, incident angle and merging parameter on the traversal time was considered. Results showed that the merging parameter and the angle of the incidence were important characters in the existence of the Hartman effect, and therefore Hartman effect was not observed at tunneling of Dirac fermions with different values of merging parameter for δ=0 and δ<0 for normal incident. In the case of δ>0 are equivalent to the gapped graphene under uniaxial strain at the merging point, tunneling time was independent on the barrier thickness.

Valley and spin dependent group delay time in silicene superlattice

In this paper, the valley- and spin-dependent transport properties, the tunneling time and the Hartman effect in a silicene superlattice under extrinsic Rashba spin-orbit coupling is theoretically studied. It is found that the transmission probability and the group delay time depend on the degree of freedom of the valley and spin, and that they have an oscillatory behavior with respect to the extrinsic Rashba spin-orbit interaction (ERSOI). Furthermore, the difference of the group delay time between two inequivalent valleys, and also between spin-up and spin-down can be efficiently tuned via the ERSOI strength and the number of superlattice barriers. Thus, valley-spin filtering can be achieved in the time domain in silicene superlattice with ERSOI. Especially, in contrast to the graphene superlattice, the Hartman effect or the superluminal tunneling can be observed in silicene superlattice even for the normal angle of incidence.