Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Causality, 'superluminality', and reshaping in undersized waveguides

D. Sokolovski, Elena V. Akhmatskaya

Published Dec 31, 2012• 0 citations
OpenAlex

Abstract

We analyse the reshaping mechanism leading to apparently 'superluminal' advancement of a pulse traversing an undersized section of a waveguide. For frequencies below the first inelastic threshold (cut off one), there are only evanescent modes in the narrow region, and the problem becomes similar to quantum mechanical tunnelling across an effective rectangular 'barrier'. In the coordinate representation, the barrier is shown to act as an effective beamsplitter, recombining envelopes of the freely propagating pulse with various spacial shifts. Causality ensures that none of the constituent envelopes are advanced with respect to free propagation, yet the resulting pulse is advanced due to a peculiar interference effect, similar to the one responsible for 'anomalous' values which occur in Aharonov's 'weak measurements'. In the momentum space, the effect is understood as a bandwidth phenomenon, where the incident pulse probes local, rather than global, analytical properties of the transmission amplitude T(p). The advancement is achieved when T(p) mimics locally an exponential behaviour, similar to the one occurring in Berry's 'superoscillations'.

Related papers

Causality, apparent “superluminality,” and reshaping in barrier penetration

D. Sokolovski

We consider tunneling of a nonrelativistic particle across a potential barrier. It is shown that the barrier acts as an effective beam splitter which builds up the transmitted pulse from the copies of the initial envelope shifted in the coordinate space backward relative to the free propagation. Although along each pathway causality is explicitly obeyed, in special cases reshaping can result an overall reduction of the initial envelope, accompanied by an arbitrary coordinate shift. In the case of a high barrier the delay amplitude distribution (DAD) mimics a Dirac $\ensuremath{\delta}$ function, the transmission amplitude is superoscillatory for finite momenta and tunneling leads to an accurate advancement of the (reduced) initial envelope by the barrier width. In the case of a wide barrier, initial envelope is accurately translated into the complex coordinate plane. The complex shift, given by the first moment of the DAD, accounts for both the displacement of the maximum of the transmitted probability density and the increase in its velocity. It is argued that analyzing apparent ``superluminality'' in terms of spacial displacements helps avoid contradiction associated with time parameters such as the phase time.

arXiv:1002.1673•Physical Review A•Apr 30, 2010•15 citations

Physical mechanism for apparent superluminality in barrier tunneling

Herbert G. Winful

Tunneling without distortion is not a propagation phenomenon but a quasistatic process in which the slowly varying envelope of the input pulse modulates the amplitude of a standing wave in the barrier. Because the envelope has a bandwidth that is small compared to the cutoff frequency of the barrier it does not propagate; it merely stands and waves. The envelope of the reflected and transmitted fields can adiabatically follow the envelope of the input pulse with only a small delay proportional to the energy stored in the barrier. The Hartman effect, in which the delay becomes independent of barrier length, is explained on the basis of saturation of stored energy with increasing length. Because the delay is not a propagation delay, it should not be linked to a velocity of propagation. True evanescent waves have zero speed and not infinite velocity as often asserted.

IEEE Journal of Selected Topics in Quantum Electronics•Jan 1, 2003•22 citations

Theoretical evidence for the superluminality of evanescent modes

Zhiyong Wang, Cai-Dong Xiong

Though both theoretical and experimental investigations have revealed the superluminal behavior of evanescent electromagnetic waves, there are many disputes about the physical meaning and validity of such superluminal phenomenon, which is due to the fact that the traditional investigations are based on the theory of tunneling time and concerned with the problem of what the group velocity of evanescent waves means. In this paper, by studying the quantum probability amplitude for photons to propagate over a spacelike interval along an undersized waveguide, we present theoretical evidence for such superluminality.

arXiv:quant-ph/0612102•Physical Review A•Apr 11, 2007•15 citations

Superluminal delays of coherent pulses in nondissipative media: A universal mechanism

Yonathan Japha, Gershon Kurizki

We identify the universal mechanism that is responsible for superluminal (faster-than-light) traversal times as well as the narrowing of wave packets transmitted through various nondissipative media. This mechanism is shown to be predominantly destructive interference between successive wave-packet components traversing all accessible causally retarded paths. It strongly depends on wave-packet coherence and width, and can cause superluminal traversal not only in evanescent-wave ``tunneling'' but also in allowed propagation. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Jan 1, 1996•92 citations

Why do superluminal pulses become subluminal once they go far enough?

Lipsa Nanda, Harshawardhan Wanare, S. Anantha Ramakrishna

Anomalous dispersion is at the heart of superluminal pulse propagation in dispersive media. Most earlier analyses of these effects have been confined to discussions of pulse shapes or group velocity and group-velocity dispersion, while experiments typically measure the absorbed energy. We investigate the transport of energy in causal dispersive media including superluminal effects, via the time moments of the Poynting vector. We find that superluminal and negative traversal times are invariably accompanied by an initial temporal compression of the pulse. We predict a crossover distance at which superluminal behavior becomes subluminal even when the spectral width of the initial pulse lies well within the anomalous dispersion region. This effect is generic and we offer a theoretical basis for the experimental results of Talukder et al. [Phys. Rev. Lett. 86, 3546 (2001)]. As the pulse propagates, a strong spectral modification due to dissipation (gain) results in the dominance of normally dispersive spectral components that suffer from lower dissipation (higher gain) and is responsible for this crossover.

Physical Review A•Apr 29, 2009•13 citations

Evanescent-mode propagation and quantum tunneling

Axel Enders, G. Nimtz

The tunneling of particles is in direct analogy with the transmission of evanescent electromagnetic waveguide modes as has been shown quite recently. We compare experimental data of an electromagnetic wave packet traversing an evanescent waveguide region with theoretical values derived for particle tunneling through a rectangular potential barrier. The transmission time was deduced by transformation of the experimental frequency data to the time domain. The data are in agreement and reveal superluminal wave-packet velocities for opaque evanescent regions.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 1, 1993•206 citations
We analyse the reshaping mechanism leading to apparently 'superluminal' advancement of a pulse traversing an undersized section of a waveguide. For frequencies below the first inelastic threshold (cut off one), there are only evanescent modes in the narrow region, and the problem becomes similar to quantum mechanical tunnelling across an effective rectangular 'barrier'. In the coordinate representation, the barrier is shown to act as an effective beamsplitter, recombining envelopes of the freely propagating pulse with various spacial shifts. Causality ensures that none of the constituent envelopes are advanced with respect to free propagation, yet the resulting pulse is advanced due to a peculiar interference effect, similar to the one responsible for 'anomalous' values which occur in Aharonov's 'weak measurements'. In the momentum space, the effect is understood as a bandwidth phenomenon, where the incident pulse probes local, rather than global, analytical properties of the transmission amplitude T(p). The advancement is achieved when T(p) mimics locally an exponential behaviour, similar to the one occurring in Berry's 'superoscillations'.
We consider tunneling of a nonrelativistic particle across a potential barrier. It is shown that the barrier acts as an effective beam splitter which builds up the transmitted pulse from the copies of the initial envelope shifted in the coordinate space backward relative to the free propagation. Although along each pathway causality is explicitly obeyed, in special cases reshaping can result an overall reduction of the initial envelope, accompanied by an arbitrary coordinate shift. In the case of a high barrier the delay amplitude distribution (DAD) mimics a Dirac $\ensuremath{\delta}$ function, the transmission amplitude is superoscillatory for finite momenta and tunneling leads to an accurate advancement of the (reduced) initial envelope by the barrier width. In the case of a wide barrier, initial envelope is accurately translated into the complex coordinate plane. The complex shift, given by the first moment of the DAD, accounts for both the displacement of the maximum of the transmitted probability density and the increase in its velocity. It is argued that analyzing apparent ``superluminality'' in terms of spacial displacements helps avoid contradiction associated with time parameters such as the phase time.
Tunneling without distortion is not a propagation phenomenon but a quasistatic process in which the slowly varying envelope of the input pulse modulates the amplitude of a standing wave in the barrier. Because the envelope has a bandwidth that is small compared to the cutoff frequency of the barrier it does not propagate; it merely stands and waves. The envelope of the reflected and transmitted fields can adiabatically follow the envelope of the input pulse with only a small delay proportional to the energy stored in the barrier. The Hartman effect, in which the delay becomes independent of barrier length, is explained on the basis of saturation of stored energy with increasing length. Because the delay is not a propagation delay, it should not be linked to a velocity of propagation. True evanescent waves have zero speed and not infinite velocity as often asserted.
Though both theoretical and experimental investigations have revealed the superluminal behavior of evanescent electromagnetic waves, there are many disputes about the physical meaning and validity of such superluminal phenomenon, which is due to the fact that the traditional investigations are based on the theory of tunneling time and concerned with the problem of what the group velocity of evanescent waves means. In this paper, by studying the quantum probability amplitude for photons to propagate over a spacelike interval along an undersized waveguide, we present theoretical evidence for such superluminality.
We identify the universal mechanism that is responsible for superluminal (faster-than-light) traversal times as well as the narrowing of wave packets transmitted through various nondissipative media. This mechanism is shown to be predominantly destructive interference between successive wave-packet components traversing all accessible causally retarded paths. It strongly depends on wave-packet coherence and width, and can cause superluminal traversal not only in evanescent-wave ``tunneling'' but also in allowed propagation. \textcopyright{} 1996 The American Physical Society.
Anomalous dispersion is at the heart of superluminal pulse propagation in dispersive media. Most earlier analyses of these effects have been confined to discussions of pulse shapes or group velocity and group-velocity dispersion, while experiments typically measure the absorbed energy. We investigate the transport of energy in causal dispersive media including superluminal effects, via the time moments of the Poynting vector. We find that superluminal and negative traversal times are invariably accompanied by an initial temporal compression of the pulse. We predict a crossover distance at which superluminal behavior becomes subluminal even when the spectral width of the initial pulse lies well within the anomalous dispersion region. This effect is generic and we offer a theoretical basis for the experimental results of Talukder et al. [Phys. Rev. Lett. 86, 3546 (2001)]. As the pulse propagates, a strong spectral modification due to dissipation (gain) results in the dominance of normally dispersive spectral components that suffer from lower dissipation (higher gain) and is responsible for this crossover.
The tunneling of particles is in direct analogy with the transmission of evanescent electromagnetic waveguide modes as has been shown quite recently. We compare experimental data of an electromagnetic wave packet traversing an evanescent waveguide region with theoretical values derived for particle tunneling through a rectangular potential barrier. The transmission time was deduced by transformation of the experimental frequency data to the time domain. The data are in agreement and reveal superluminal wave-packet velocities for opaque evanescent regions.