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Proceedings of the Samahang Pisika ng Pilipinas

Quantum traversal time through a double well

Dean Alvin L. Pablico, Eric A. Galapon

Published May 17, 2019• 0 citations
OpenAlex

Abstract

The problem of quantum traversal time through two successive potential barriers introduces the concept of generalized Hartmann effect (GHE) in quantum mechanics. It suggests that the tunneling time through two opaque barriers is independent of both barrier width and barrier separation. Here, we consider instead the quantum traversal time through two successive potential wells using the time of arrival (TOA) operator approach. With the hypothesis that the support of the incident wave packet does not extend inside the well region, it is found that the expectation value of the traversal time is independent of the separation distance between the two wells but dependent to the width of each well.

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The generalized Hartmann effect (GHE) predicts a strict inequality between the traversal times across a contiguous and a separated double-barrier system. This is compared to the implications of the time-of-arrival (TOA) operator approach to barrier traversal time [E. A. Galapon, Phys. Rev. Lett. 108, 170402 (2012)]. It is shown that, for initial wave packets with compact supports in the far incident side of the barrier system, the expectation value of the traversal time is independent of the separation between the barriers. On the other hand, for wave packets with supports extending inside the first barrier, the contribution of the barrier separation to the traversal time exponentially increases with the barrier height. Our result shows that if the support of the incident wave packet is far from the barrier region, the GHE inequality is violated. However, if the support of the wave packet extends inside the barrier region, the GHE inequality is consistent with the TOA operator approach, but only when the particle's incident energy is very small.

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•Mar 10, 2000•0 citations
The problem of quantum traversal time through two successive potential barriers introduces the concept of generalized Hartmann effect (GHE) in quantum mechanics. It suggests that the tunneling time through two opaque barriers is independent of both barrier width and barrier separation. Here, we consider instead the quantum traversal time through two successive potential wells using the time of arrival (TOA) operator approach. With the hypothesis that the support of the incident wave packet does not extend inside the well region, it is found that the expectation value of the traversal time is independent of the separation distance between the two wells but dependent to the width of each well.
The generalized Hartmann effect (GHE) predicts a strict inequality between the traversal times across a contiguous and a separated double-barrier system. This is compared to the implications of the time-of-arrival (TOA) operator approach to barrier traversal time [E. A. Galapon, Phys. Rev. Lett. 108, 170402 (2012)]. It is shown that, for initial wave packets with compact supports in the far incident side of the barrier system, the expectation value of the traversal time is independent of the separation between the barriers. On the other hand, for wave packets with supports extending inside the first barrier, the contribution of the barrier separation to the traversal time exponentially increases with the barrier height. Our result shows that if the support of the incident wave packet is far from the barrier region, the GHE inequality is violated. However, if the support of the wave packet extends inside the barrier region, the GHE inequality is consistent with the TOA operator approach, but only when the particle's incident energy is very small.
We consider the quantum traversal time of an incident wave packet across a potential well using the theory of quantum time of arrival (TOA) operators. This is done by constructing the corresponding TOA operator across a potential well via quantization. The expectation value of the potential-well TOA operator is compared to the free-particle case for the same incident wave packet. The comparison yields a closed-form expression of the quantum well traversal time which explicitly shows the classical contributions of the positive and negative momentum components of the incident wave packet and a purely quantum-mechanical contribution significantly dependent on the well depth. An incident Gaussian wave packet is then used as an example. It is shown that for shallow potential wells, the quantum well traversal time approaches the classical traversal time across the well region when the incident wave packet is spatially broad and approaches the expected quantum free-particle traversal time when the wave packet is localized. For deep potential wells, the quantum traversal time oscillates from positive to negative, implying that the wave packet can be advanced or delayed.
The mechanism of superluminal traversal time is investigated from the viewpoint of interference between multiple finite wave packets, due to the multiple reflections inside the well or barrier. In the case of potential well traveling that is classically allowed, each of the successively transmitted constituents is delayed by a subluminal time. When the thickness of the well is much smaller in comparison with a characteristic length of the incident wave packet, the reshaped wave packet with superluminal traversal time in transmission maintains the pro. le of the incident wave packet. In the case of potential barrier tunneling that is classically forbidden, though each of the successively transmitted constituents is delayed by a time that is independent of the barrier thickness, the interference between multiple transmitted constituents explains the barrier thickness dependence of the traversal time for thin barriers and its barrier thickness independence for thick barriers. This manifests the nature of the Hartman effect. Copyright (c) EPLA, 2008.
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.
We study the phenomenon of one-dimensional non-resonant tunneling through two successive (opaque) potential barriers, separated by an intermediate free region , by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called "Hartman effect"), but also on the width: so that the effective velocity in the region , between the two barriers, can be regarded as practically infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunneling experiment herein considered due to the known formal identity between the Schroedinger and the Helmholtz equation.
We study the phenomenon of one-dimensional non-resonant tunneling through two successive potential barriers, separated by an intermediate free region R, by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called 'Hartman effect'), but also on the R width: so that the effective velocity in the region R, between the two barriers, can be regarded as infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunnelling experiment herein considered due to the formal identity between the Schroedinger and the Helmholtz equation.