Hartman Effect Research Articles

Relativistic tunneling through two successive barriers

We study the relativistic quantum mechanical problem of a Dirac particle tunneling through two successive electrostatic barriers. Our aim is to study the emergence of the so-called generalized Hartman effect, an effect observed in the context of nonrelativistic tunneling as well as in its electromagnetic counterparts and which is often associated with the possibility of superluminal velocities in the tunneling process. We discuss the behavior of both the phase (or group) tunneling time and the dwell time, and show that in the limit of opaque barriers the relativistic theory also allows the emergence of the generalized Hartman effect. We compare our results with the nonrelativistic ones and discuss their interpretation.

Dirac equation studies in the tunneling energy zone

We investigate the tunnelling zone V0 < E < V0+m for a one-dimensional potential within the Dirac equation. We find the appearance of superluminal transit times akin to the Hartman effect.

Time evolution of electromagnetic wave packets through superlattices: Evidence for superluminal velocities

We study the space-time evolution of electromagnetic wave packets through optical superlattices. We present rigorous analytical solutions describing the multiple-scattering processes of Gaussian wave packets defined in the band gap and in the resonant energy regions. Following their space-time evolution, we obtain the Maxwell equations prediction for the time spent inside the superlattice. From a close and careful observation of the reflected and transmitted parts of Gaussian packets in a photonic band gap, we conclude unambiguously that the superluminal transmission and the Hartman effect are inherent properties of the electromagnetic theory. It is also shown that the theoretical predictions for the time spent inside an optical superlattice are in good agreement with the experimental results and the phase time predictions.

PHASE TIME FOR A TUNNELING PARTICLE

We study the nature of tunneling phase time for various quantum mechanical structures such as networks and rings having potential barriers in their arms. We find the generic presence of the Hartman effect, with superluminal velocities as a consequence, in these systems. In quantum networks, it is possible to control the "super arrival" time in one of the arms by changing the parameters on another arm which is spatially separated from it. This is yet another quantum nonlocal effect. Negative time delays (time advancement) and "ultra Hartman effect" with negative saturation times have been observed in some parameter regimes. In the presence and absence of Aharonov-Bohm (AB) flux, quantum rings show the Hartman effect. We obtain the analytical expression for the saturated phase time. In the opaque barrier regime, this is independent of even the AB flux thereby generalizing the Hartman effect. We also briefly discuss the concept of "space collapse or space destroyer" by introducing a free space in between two barriers covering the ring. Further, we show in presence of absorption that the reflection phase time exhibits the Hartman effect in contrast to the transmission phase time.

Hartman effect in a Kane-type semiconductor quantum ring

The Hartman effect for a tunnelling particle implies that group delay time is independent of the opaque barrier width. In the present study, the tunnelling delay time in the transmission mode is studied taking into account the real band structure of an InSb-type semiconductor quantum ring and compared with that of a parabolic band structure. The system considered in this study consists of a circular loop in the presence of Aharonov–Bohm flux. It is shown that while tunnelling through an opaque barrier, the group delay time for a given incident energy becomes independent of the barrier thickness as well as the magnitude of the flux.

Delay times and detector times for optical pulses traversing plasmas and negative refractive media

We show that arrival times for electromagnetic pulses measured through the rate of absorption in an ideal impedance matched detector are equivalent to the arrival times using the average flow of optical energy as proposed by Peatross [Phys. Rev. Lett. 84, 2370 (2000)]. We then investigate the transport of optical pulses through dispersive media with negative dielectric permittivity and negative refractive index choosing the geometry such that no resonant effects come into play. For evanescent waves, the definitions of the group delay, and the reshaping delay get interchanged in comparison to propagating waves. The total delay times for the evanescent waves can be negative in an infinite plasma medium even for broadband pulses. The total delay time is, however, positive for broadband pulses in the presence of an interface when the radiation is detected outside the plasma. We find evidence of the Hartman effect for pulses when the distance traversed in the plasma is much smaller than the free space pulse length. We also show that for a negative refractive index medium (NRM) with epsilon(omega)=mu(omega) the reshaping delay for propagating waves is identically zero. The total delay time in NRM is otherwise dominated by the reshaping delay time, and for broadband pulses in NRM the total delay time is subluminal.

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.

Confronting the Hartman effect with data from frustrated total internal reflection (FTIR)

Over 40 years ago, Hartman noted that the tunneling time τ of a particle through a barrier becomes independent of width for thick barriers. Lately, the Hartman effect has been seen as a support for superluminal tunneling time. By interpreting the reflection and transmission amplitudes in terms of multiple reflection series, we show that τ is linear in barrier width for thin barriers and may be associated with actual traversal time; for thick barriers, τ saturates to the Hartman value because of the suppression of all but the first term of the series due to the smallness of the tunneling factor. For large widths, τ cannot be identified with the propagation time but may be associated with a time to penetrate to a characteristic depth into the barrier, which is independent of width. We discuss data from frustrated internal reflection experiments, which support this view.

Dynamic pattern of finite-pulsed beams inside one-dimensional photonic band gap materials

The dynamics of two-dimensional electromagnetic (EM) pulses through one-dimensional photonic crystals (1DPC) has been theoretically studied. Employing the time expectation integral over the Poynting vector as the arrival time [Phys. Rev. Lett. 84, 2370, (2000)], we show that the superluminal tunneling process of EM pulses is the propagation of the net forward-going Poynting vector through the 1DPC, and the Hartman effect is due to the saturation effect of the arrival time (smaller and smaller time accumulated) of the net forward energy flow caused by the interference effect of the forward and the backward field (from the interfaces of each layer) happened in the region before the 1DPC and in the front part of the 1DPC.

Erratum: Apparent superluminality and the generalized Hartman effect in double-barrier tunneling [Phys. Rev. E72, 046608 (2005)

Erratum: Apparent superluminality and the generalized Hartman effect in double-barrier tunneling [Phys. Rev. E<b>72</b>, 046608 (2005)

Subluminal and superluminal pulse propagation in inhomogeneous media of nonspherical particles

We study the pulse propagation through a metal/dielectric composites of nonspherical particles enclosed by two gold mirrors. To account for the shape effect, we first adopt Maxwell–Garnett type approximation to obtain the effective dielectric function of composites. Based on the group index, phase time and pulse shape calculations, we find that the particles' shape (characterized by the depolarization factor) plays an important role in determining the subluminal and superluminal pulse propagations through the system. When the inclusions' shape is not spherical, it is possible to observe significant superluminal behavior of the pulse propagation, although the volume fraction is the same. The shape-dependent critical volume fraction is predicted, above which superluminal propagation appears. Furthermore, the Hartman effect in such a system is also investigated.

Photon tunneling in a frustrated-total-internal-reflection structure composed of a single negative material

The tunneling time and lateral shift of photon tunneling in a frustrated-total-internal-reflection structure composed of single negative material is derived by employing stationary-phase approximation. It is found that the tunneling time and the lateral shift are negative when the barrier is a single negative material. On account of the Hartman effect of the tunneling time and the lateral shift, the photon tunneling shows the superluminal property. An effective approach to distinguish ε-negative material and μ-negative material is proposed based on the fact that TE- and TM-polarized incident beams experience opposite lateral shift.

Origin of the Hartman effect

For thin barriers, we show that the tunneling time τ is linear in the width and gives the traversal time. For thick barriers, τ saturates through the suppression of all but the first reflection or transmission term and gives the time to penetrate to a characteristic depth. We discuss data from frustrated internal reflection experiments.

Apparent superluminality and the generalized Hartman effect in double-barrier tunneling

Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.

Tunnelling time of a gaussian wave packet through two potential barriers

Abstract The resonant and non-resonant dynamies of a Gaussian quantum wave packet travelling through a double barrier system is studied as a function of the initial characteristics of the spectrum and of the parameters of the potential. The behaviour of the tunnelling time shows that there are situations where the Hartman effect occurs, while, when the resonances are dominant, and in particular for b&gt;π/Δk (b being the inter-barrier distance and Δk the spectrum width), the tunnelling time becomes very large and the Hartman effect does not take place.

Spin-dependent delay time and the Hartman effect in tunneling through diluted-magnetic-semiconductor/semiconductor heterostructures

We investigate spin-dependent group delay and dwell time in diluted-magnetic-semiconductor/semiconductor (DMS/S) heterostructures, where the $sp\text{\ensuremath{-}}d$ exchange interaction gives rise to a giant spin splitting when an external magnetic field applied along the growth direction of the heterostructures. It is found that both the group delay and the dwell time strongly depend not only on the incident energy and the structural configuration, but also on spin orientations. In the spectra of both the group delay and the dwell time, there are some sharp peaks with larger peak-to-valley ratios for spin-up electrons through the DMS/S heterostructures with a single or double DMS layers, while the curves become more smoothed for spin-down ones through the same heterostructure. The difference of the group delay or of the dwell time between spin-up and spin-down cases reaches its maximum when resonant tunneling occurs. The numerical results indicate that the spin-up and spin-down processes are separated on the time scales. Further, for spin-down electrons, the group delay can be negative at low energy under some magnetic fields.

Concave and convex photonic barriers in gradient optics

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.

Tunneling time for phonons: dependence on the system's size

AbstractWe discuss the physical conditions under which the tunneling time for long‐wavelength phonons through semiconductor heterostructures is independent of the system's size, i.e. the effect equivalent to the Hartman effect for electrons. An analysis of the vibrational energy stored in the system is shown to be relevant for this study. In fact, the phonon Hartman effect is explained on the basis of saturation of the stored vibrational energy. Both acoustic and optical phonons are considered. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Hartman effect and nonlocality in quantum networks

We study the phase time for various quantum mechanical networks having potential barriers in their arms to find the generic presence of Hartman effect. In such systems it is possible to control the ‘super arrival’ time in one of the arms by changing parameters on another, spatially separated from it. This is yet another quantum nonlocal effect. Negative time delays (time advancement) and ‘ultra Hartman effect’ with negative saturation times have been observed in some parameter regimes.

Negative Hartman effect in one-dimensional photonic crystals with negative refractive materials

The Hartman effect inside the one-dimensional photonic crystals (1DPC's) composed of negative index materials (NIM's) is always negative and is reversed to the Hartman effect inside the 1DPC's composed of positive index materials (PIM's). By calculating the phases of Fourier components of a pulse accumulated inside the 1DPC's of NIM's and the evolution of the pulse inside the 1DPC's of NIM's, the origin of the negative phase time is explained. The evolution of the electromagnetic fields inside the 1DPC's of NIM's is time reversal with conjugate to that inside the 1DPC's of PIM's for real spectral pulses. An example for the practical applications to obtain the negative phase time is illustrated.