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

Physical Review A

Time dependence of the probability density in the transient regime for tunneling

Gastón García‐Calderón, Jorge Villavicencio

Published Jun 15, 2001• 47 citations
Publisher

Abstract

An exact analytical solution to the time-dependent Schr\"odinger equation with cutoff wave initial conditions is used to investigate the fast tunneling response of a rectangular potential barrier. We find that just across the tunneling region, the probability density exhibits at short times a transient behavior that may be characterized by a peak ${t}_{p}$ and a width $\ensuremath{\Delta}t.$ We show that ${t}_{p}$ provides the earliest tunneling response of the system and that the top-barrier S-matrix poles play an important role in the process. As a function of the barrier width, ${t}_{p}$ exhibits two regimes. Along the first regime, ${t}_{p}$ remains almost a constant; as the barrier width increases, a second regime appears where ${t}_{p}$ grows linearly with the barrier width.

DOI
10.1103/physreva.64.012107

Related papers

Transient Effects in Time-Dependent Tunneling

G. Garc�a-Calder�n

An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.

physica status solidi (b)•Apr 1, 2002•1 citation

Early times in tunneling

Jorge Villavicencio

Exact analytical solutions of the time-dependent Schrodinger 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 peakp and a width ��. Numerical results for other tunneling times, as the phase-delay time, fall within ��. The Buttiker traversal time is the closest top. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.

•Aug 3, 2000•0 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

Transient tunneling effects of resonance doublets in triple barrier systems

Roberto Romo, Jorge Villavicencio, Gastón García‐Calderón

Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr\"odinger equation with a cutoff wave initial condition. We derive a two-level formula for incidence energies E near the first resonance doublet of the system. Based on that expression we find that the probability density along the internal region of the potential is governed by three oscillation frequencies: one of them refers to the well known Bohr frequency, given in terms of the first and second resonance energies of the doublet, and the two others represent a coupling with the incidence energy E. This allows us to manipulate the above frequencies to control the tunneling transient behavior of the probability density in the short-time regime.

arXiv:quant-ph/0210009•Physical review. B, Condensed matter•Jul 16, 2002•20 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

Transient time-domain resonances and the time scale for tunneling

Gastón García‐Calderón, Jorge Villavicencio

Transient time-domain resonances found recently in time-dependent solutions to Schr\"odinger's equation are used to investigate the issue of the tunneling time in rectangular potential barriers. In general, a time-frequency analysis shows that these transients have frequencies above the cutoff frequency associated with the barrier height, and hence correspond to nontunneling processes. We find, however, a regime characterized by the barrier opacity, where the peak maximum ${t}_{\mathrm{max}}$ of the time-domain resonance corresponds to under-the-barrier tunneling. We argue that ${t}_{\mathrm{max}}$ represents the relevant tunneling time scale through the classically forbidden region.

arXiv:quant-ph/0311175•Physical Review A•Nov 17, 2003•11 citations
An exact analytical solution to the time-dependent Schr"odinger equation with cutoff wave initial conditions is used to investigate the fast tunneling response of a rectangular potential barrier. We find that just across the tunneling region, the probability density exhibits at short times a transient behavior that may be characterized by a peak tp{t}_{p}tp​ and a width \ensuremathΔt.\ensuremath{\Delta}t.\ensuremathΔt. We show that tp{t}_{p}tp​ provides the earliest tunneling response of the system and that the top-barrier S-matrix poles play an important role in the process. As a function of the barrier width, tp{t}_{p}tp​ exhibits two regimes. Along the first regime, tp{t}_{p}tp​ remains almost a constant; as the barrier width increases, a second regime appears where tp{t}_{p}tp​ grows linearly with the barrier width.
An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.
Exact analytical solutions of the time-dependent Schrodinger 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 peakp and a width ��. Numerical results for other tunneling times, as the phase-delay time, fall within ��. The Buttiker traversal time is the closest top. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.
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.
Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr\"odinger equation with a cutoff wave initial condition. We derive a two-level formula for incidence energies E near the first resonance doublet of the system. Based on that expression we find that the probability density along the internal region of the potential is governed by three oscillation frequencies: one of them refers to the well known Bohr frequency, given in terms of the first and second resonance energies of the doublet, and the two others represent a coupling with the incidence energy E. This allows us to manipulate the above frequencies to control the tunneling transient behavior of the probability density in the short-time regime.
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.
Transient time-domain resonances found recently in time-dependent solutions to Schr\"odinger's equation are used to investigate the issue of the tunneling time in rectangular potential barriers. In general, a time-frequency analysis shows that these transients have frequencies above the cutoff frequency associated with the barrier height, and hence correspond to nontunneling processes. We find, however, a regime characterized by the barrier opacity, where the peak maximum tmax{t}_{\mathrm{max}}tmax​ of the time-domain resonance corresponds to under-the-barrier tunneling. We argue that tmax{t}_{\mathrm{max}}tmax​ represents the relevant tunneling time scale through the classically forbidden region.