Tunneling time and energy uncertainty of surface-state electrons

Yucel, S.; Andrei, Eva Y.

doi:10.1103/physrevb.46.2448

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…This will have significant effect on the tunneling process itself if the particle could interact with some additional degree of freedom within the barrier. This explains the appearance of this time scale in interacting tunneling models [11,13,14,38]. On the other hand, increasing the potential barrier further, the uncertainty is dominated by the energy width of the state φ (x), and hence independent of u (e.g.…”

Section: Arrival Time In Tunneling Problemsmentioning

confidence: 94%

“…In Russian literatures it has been known as the Keldysh time [12,38] and is approximately given as ∆τ ∼ d/κ where κ = 2(u − ε) is the magnitude of the imaginary momentum under the barrier. Our result are also in agreement with the observations of Yucel and Andrei that this is the time scale that should appear in the energy-time uncertainty principle [38]. The identification of the traversal time with the variance of probability density of the dwell time has been also found by Olkhovsky [32].…”

Section: Arrival Time In Tunneling Problemsmentioning

confidence: 99%

“…Most importantly, the traversal time dominates the uncertainty whenever the the energy of the state moves under the barrier and the transmission is principally given by |t| ∼ exp{−d 2(u − ε)}, where d is the barrier width and u its heigh. In Russian literatures it has been known as the Keldysh time [12,38] and is approximately given as ∆τ ∼ d/κ where κ = 2(u − ε) is the magnitude of the imaginary momentum under the barrier. Our result are also in agreement with the observations of Yucel and Andrei that this is the time scale that should appear in the energy-time uncertainty principle [38].…”

Section: Arrival Time In Tunneling Problemsmentioning

confidence: 99%

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Time operators in stroboscopic wave-packet basis and the time scales in tunneling

Bokes

2011

Phys. Rev. A

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We demonstrate that the time operator that measures the time of arrival of a quantum particle into chosen state can be defined as a self-adjoint quantum-mechanical operator using periodic boundary conditions on applied to wavefuncions in energy representation. The time becomes quantized into discreet eigenvalues and the eigenstates of the time operator, the stroboscopic wavepackets introduced recently [Phys. Rev. Lett. 101, 046402 (2008).] form orthogonal system of states. The formalism provides simple physical interpretation of the time-measurement process and direct construction of normalized, positive definite probability distribution for the quantized values of the arrival time. The average value of the time is equal to the phase time but in general depends on the choise of zero time eigenstate, whereas the uncertainity of the average is related to the traversal time and is independent of this choise. The general fromalism is applied to a particle tunneling through resonant tunneling barrier in 1D.

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Section: Arrival Time In Tunneling Problemsmentioning

confidence: 94%

Section: Arrival Time In Tunneling Problemsmentioning

confidence: 99%

Section: Arrival Time In Tunneling Problemsmentioning

confidence: 99%

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Time operators in stroboscopic wave-packet basis and the time scales in tunneling

Bokes

2011

Phys. Rev. A

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“…The above results (18)(19)(20)(21) could be directly obtained by applying the SPM to the wave functions ψ f (x, t) expressed in Eq. (15), where the stationary phases for each decomposed wave component f would be given by…”

Section: The Above Barrier Diffusion Problemmentioning

confidence: 99%

“…On the theoretical front, people have tried to introduce quantities that have the dimension of time and can somehow be associated with the passage of the particle through the barrier or, strictly speaking, with the definition of the tunneling time. Since a long time these efforts have led to the introduction of several time definitions [6,11,18,19,20,21,22,23,24,25], some of which are completely unrelated to the others, which can be organized into three groups: (1) The first group comprises a time-dependent description in terms of wave packets where some features of an incident wave packet and the comparable features of the transmitted packet are utilized to describe a delay as tunneling time [1]. (2) In the second group the tunneling times are computed with basis on averages over a set of kinematical paths, whose distribution is supposed to describe the particle motion inside a barrier, i.…”

Section: Limitations On the Tunneling Time Analysismentioning

confidence: 99%

Recovering the Stationary Phase Condition for Accurately Obtaining Scattering and Tunneling Times

Bernardini

2009

Int. J. Mod. Phys. B

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The stationary phase method is often employed for computing tunneling phase times of analytically-continuous gaussian or infinite-bandwidth step pulses which collide with a potential barrier. The indiscriminate utilization of this method without considering the barrier boundary effects leads to some misconceptions in the interpretation of the phase times. After reexamining the above barrier diffusion problem where we notice the wave packet collision necessarily leads to the possibility of multiple reflected and transmitted wave packets, we study the phase times for tunneling/reflecting particles in a framework where an idea of multiple wave packet decomposition is recovered. To partially overcome the analytical incongruities which rise up when tunneling phase time expressions are obtained, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a one dimensional squared potential barrier where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted waves.

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Stationary phase method and delay times for relativistic and non-relativistic tunneling particles

Bernardini

2009

Annals of Physics

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This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. After reexamining the above-barrier diffusion problem, we notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. Using a recently developed procedure -multiple wave packet decomposition -for some specifical colliding configurations, we demonstrate that the analytical difficulties arising when the stationary phase method is applied for obtaining phase (traversal) times are all overcome. In this case, we also investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstrate that these two distinct transit time definitions are explicitly connected. The traversal times are obtained for a symmetrized (two identical bosons) and an antisymmetrized (two identical fermions) quantum colliding configuration. Multiple wave packet decomposition shows us that the phase time (group delay) describes the exact position of the scattered particles and, in addition to the exact relation with the dwell time, leads to correct conceptual understanding of both transit time definitions.At last, we extend the non-relativistic formalism to the solutions for the tunneling zone of a onedimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation where the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation based on the non-relativistic tunneling dynamics is revisited. Lessons concerning the dynamics of relativistic tunneling and the mathematical structure of its solutions suggest revealing insights into mathematically analogous condensed-matter experiments using electrostatic barriers in single-and bi-layer graphene, for which the accelerated tunneling effect deserves a more careful investigation.

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Tunneling time and energy uncertainty of surface-state electrons

Cited by 12 publications

References 17 publications

Time operators in stroboscopic wave-packet basis and the time scales in tunneling

Time operators in stroboscopic wave-packet basis and the time scales in tunneling

Recovering the Stationary Phase Condition for Accurately Obtaining Scattering and Tunneling Times

Stationary phase method and delay times for relativistic and non-relativistic tunneling particles

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