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

arXiv:quant-ph/0303133 · Physical Review A

Tunneling dynamics in relativistic and nonrelativistic wave equations

F.S. Delgado, Juan Gonzalo Muga, A. Ruschhaupt, Gastón García‐Calderón, Jorge Villavicencio

Published Sep 5, 2003• 29 citations
arXivHTMLPDF

Abstract

We obtain the solution of a relativistic wave equation and compare it with the solution of the Schr\"odinger equation for a source with a sharp onset and excitation frequencies below cutoff. A scaling of position and time reduces to a single case all the (below cutoff) nonrelativistic solutions, but no such simplification holds for the relativistic equation, so that qualitatively different ``shallow'' and ``deep'' tunneling regimes may be identified relativistically. The nonrelativistic forerunner at a position beyond the penetration length of the asymptotic stationary wave does not tunnel; nevertheless, it arrives at the traversal (semiclassical or B\"uttiker-Landauer) time $\ensuremath{\tau}.$ The corresponding relativistic forerunner is more complex: it oscillates due to the interference between two saddle-point contributions and may be characterized by two times for the arrival of the maxima of lower and upper envelopes. There is in addition an earlier relativistic forerunner, right after the causal front, which does tunnel. Within the penetration length, tunneling is more robust for the precursors of the relativistic equation.

DOI
10.1103/physreva.68.032101

Related papers

Time scale of forerunners in quantum tunneling

Gastón García‐Calderón, Jorge Villavicencio, F.S. Delgado +1 more

The forerunners preceding the main tunneling signal of the wave created by a source with a sharp onset or by a quantum shutter have been generally associated with over-the-barrier (nontunneling) components. We demonstrate that, while this association is true for distances that are larger than the penetration length, for smaller distances the forerunner is dominated by under-the-barrier components. We find that its characteristic arrival time is inversely proportional to the difference between the barrier energy and the incidence energy, a tunneling time scale different from both the phase time and the B\"uttiker-Landauer time.

arXiv:quant-ph/0206020•Physical Review A•Oct 30, 2002•13 citations

Delay time and tunneling transient phenomena

Gastón García‐Calderón, Jorge Villavicencio

Analytic solutions to the time-dependent Schr\"odinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity $\ensuremath{\alpha},$ we find that the probability density exhibits two evolving structures. One refers to the propagation of a forerunner related to a time domain resonance [Phys. Rev. A 64, 0121907 (2001)], while the other consists of a semiclassical propagating wave front. We find a regime where the forerunners are absent, corresponding to positive time delays, and show that this regime is characterized by opacities $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}.$ The critical opacity ${\ensuremath{\alpha}}_{c}$ is derived from the analytical expression for the delay time, which reflects a link between transient effects in tunneling and the delay time.

arXiv:quant-ph/0210008•Physical Review A•Sep 16, 2002•19 citations

Front propagation in evanescent media

Μ. Büttiker, H. Thomas

We investigate the time evolution of waves in evanescent media generated by a source within this medium and observed at some distance away from the location of the source. The aim is to find a velocity which describes a causal process and is thus, for a medium with relativistic dispersion, limited by the velocity of light. For a source with a sharp onset in time, the wave function consists of a forerunner generated by the onset of the source, and of a monochromatic front. The forerunner is dominated by a frequency which decreases with time, and the monochromatic front carries the oscillation frequency of the source into the evanescent medium. For a medium with Schrödinger-like dispersion the velocity of the front is infinite and the monochromatic front propagates with a velocity which is in agreement with the traversal time for tunneling. In the relativistic case the forerunners travel with the velocity of light and the velocity the monochromatic front is smaller than the velocity of light and only for special energies equal to the velocity of light. For sources with a sharp onset, the forerunners are not attenuated and in magnitude far exceed the monochromatic front. This renders the detection of the monochromatic front difficult. To avoid the different behavior of forerunners and monochromatic fronts, sources which are frequency-band limited can be considered or a short-time Fourier transform of the field at the observation point can be taken. Both discussions suggest that the traversal time can be determined only up to a factor of 𝒪(1).

Annalen der Physik•Dec 1, 1998•21 citations

Early times in tunneling

Gastón García‐Calderón, Jorge Villavicencio

Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak $τ_p$ and a width $Δτ$. Numerical results for other tunneling times, as the phase-delay time, fall within $Δτ$. The Büttiker traversal time is the closest to $τ_p$. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.

arXiv:quant-ph/0008014•Aug 3, 2000•0 citations

Quantum-wave evolution in a step potential barrier

Jorge Villavicencio, Roberto Romo, Sukey Sosa y Silva

By using an exact solution to the time-dependent Schr\"odinger equation with a point source initial condition, we investigate both the time and spatial dependence of quantum waves in a step potential barrier. We find that for a source with energy below the barrier height, and for distances larger than the penetration length, the probability density exhibits a forerunner associated with a nontunneling process, which propagates in space at exactly the semiclassical group velocity. We show that the time of arrival of the maximum of the forerunner at a given fixed position inside the potential is exactly the traversal time $\ensuremath{\tau}.$ We also show that the spatial evolution of this transient pulse exhibits an invariant behavior under a rescaling process. This analytic property is used to characterize the evolution of the forerunner, and to analyze the role played by the time of arrival ${3}^{\ensuremath{-}1/2}\ensuremath{\tau}$ found recently by Muga and B\"uttiker [Phys. Rev. A 62, 023808 (2000)].

arXiv:quant-ph/0210010•Physical Review A•Oct 17, 2002•15 citations

Tunneling time scale of under-the-barrier forerunners

Gastón García‐Calderón, Jorge Villavicencio

Time-dependent analytical solutions to Schrödinger's equation with quantum shutter initial conditions are used to investigate the issue of the tunneling time of forerunners in rectangular potential barriers. By using a time-frequency analysis, we find the existence of a regime characterized by the opacity of the barrier, where the maximum peak of a forerunner measured at the barrier transmission edge $x=L$ corresponds to a genuine tunneling process. The corresponding time scale represents the tunneling time of the forerunner through the classically forbidden region.

arXiv:quant-ph/0210094•Oct 12, 2002•0 citations
We obtain the solution of a relativistic wave equation and compare it with the solution of the Schr"odinger equation for a source with a sharp onset and excitation frequencies below cutoff. A scaling of position and time reduces to a single case all the (below cutoff) nonrelativistic solutions, but no such simplification holds for the relativistic equation, so that qualitatively different shallow'' and deep'' tunneling regimes may be identified relativistically. The nonrelativistic forerunner at a position beyond the penetration length of the asymptotic stationary wave does not tunnel; nevertheless, it arrives at the traversal (semiclassical or B"uttiker-Landauer) time \ensuremathτ.\ensuremath{\tau}.\ensuremathτ. The corresponding relativistic forerunner is more complex: it oscillates due to the interference between two saddle-point contributions and may be characterized by two times for the arrival of the maxima of lower and upper envelopes. There is in addition an earlier relativistic forerunner, right after the causal front, which does tunnel. Within the penetration length, tunneling is more robust for the precursors of the relativistic equation.
The forerunners preceding the main tunneling signal of the wave created by a source with a sharp onset or by a quantum shutter have been generally associated with over-the-barrier (nontunneling) components. We demonstrate that, while this association is true for distances that are larger than the penetration length, for smaller distances the forerunner is dominated by under-the-barrier components. We find that its characteristic arrival time is inversely proportional to the difference between the barrier energy and the incidence energy, a tunneling time scale different from both the phase time and the B\"uttiker-Landauer time.
Analytic solutions to the time-dependent Schr\"odinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity $\ensuremath{\alpha},$ we find that the probability density exhibits two evolving structures. One refers to the propagation of a forerunner related to a time domain resonance [Phys. Rev. A 64, 0121907 (2001)], while the other consists of a semiclassical propagating wave front. We find a regime where the forerunners are absent, corresponding to positive time delays, and show that this regime is characterized by opacities $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}.$ The critical opacity ${\ensuremath{\alpha}}_{c}$ is derived from the analytical expression for the delay time, which reflects a link between transient effects in tunneling and the delay time.
We investigate the time evolution of waves in evanescent media generated by a source within this medium and observed at some distance away from the location of the source. The aim is to find a velocity which describes a causal process and is thus, for a medium with relativistic dispersion, limited by the velocity of light. For a source with a sharp onset in time, the wave function consists of a forerunner generated by the onset of the source, and of a monochromatic front. The forerunner is dominated by a frequency which decreases with time, and the monochromatic front carries the oscillation frequency of the source into the evanescent medium. For a medium with Schrödinger-like dispersion the velocity of the front is infinite and the monochromatic front propagates with a velocity which is in agreement with the traversal time for tunneling. In the relativistic case the forerunners travel with the velocity of light and the velocity the monochromatic front is smaller than the velocity of light and only for special energies equal to the velocity of light. For sources with a sharp onset, the forerunners are not attenuated and in magnitude far exceed the monochromatic front. This renders the detection of the monochromatic front difficult. To avoid the different behavior of forerunners and monochromatic fronts, sources which are frequency-band limited can be considered or a short-time Fourier transform of the field at the observation point can be taken. Both discussions suggest that the traversal time can be determined only up to a factor of 𝒪(1).
Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak τpτ_pτp​ and a width ΔτΔτΔτ. Numerical results for other tunneling times, as the phase-delay time, fall within ΔτΔτΔτ. The Büttiker traversal time is the closest to τpτ_pτp​. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.
By using an exact solution to the time-dependent Schr\"odinger equation with a point source initial condition, we investigate both the time and spatial dependence of quantum waves in a step potential barrier. We find that for a source with energy below the barrier height, and for distances larger than the penetration length, the probability density exhibits a forerunner associated with a nontunneling process, which propagates in space at exactly the semiclassical group velocity. We show that the time of arrival of the maximum of the forerunner at a given fixed position inside the potential is exactly the traversal time $\ensuremath{\tau}.$ We also show that the spatial evolution of this transient pulse exhibits an invariant behavior under a rescaling process. This analytic property is used to characterize the evolution of the forerunner, and to analyze the role played by the time of arrival ${3}^{\ensuremath{-}1/2}\ensuremath{\tau}$ found recently by Muga and B\"uttiker [Phys. Rev. A 62, 023808 (2000)].
Time-dependent analytical solutions to Schrödinger's equation with quantum shutter initial conditions are used to investigate the issue of the tunneling time of forerunners in rectangular potential barriers. By using a time-frequency analysis, we find the existence of a regime characterized by the opacity of the barrier, where the maximum peak of a forerunner measured at the barrier transmission edge x=Lx=Lx=L corresponds to a genuine tunneling process. The corresponding time scale represents the tunneling time of the forerunner through the classically forbidden region.