Recently, experimental measurements of the Larmor time of incident wave packets have been conducted, once again drawing attention to such a time. These experiments measured the tunneling time of Bose-condensed $^{87}\mathrm{Rb}$ atoms, with theoretical analyses employing free particles. In this paper, we investigate the tunneling process of wave packets, which, unlike free particles with a definite incident energy, have a continuous distribution of energies. We show how the Larmor time ${\ensuremath{\tau}}_{z(y)}$ for a wave packet tunneling through a barrier depends on the barrier width $d$ , the barrier's characteristic wave vector ${k}_{0}$ , the incident wave packet's central wave vector ${k}_{c}$ , and its width $\ensuremath{\sigma}$ in wave-number space. When the barrier width $d$ is small, we find that ${\ensuremath{\tau}}_{z(y)}$ varies monotonically. For wide barriers ($d>2.0\ensuremath{\pi}$), we find an anomaly where wave packets with lower incident energy or higher barriers have shorter tunneling times, which is consistent with recent experimental results. Additionally, for narrow wave packets in wave-number space ($\ensuremath{\sigma}<\ensuremath{\gamma}$), where $\ensuremath{\gamma}$ is the width of the fitted Fano peak, the results are generally consistent with those for plane waves. However, as the wave packet widens in wave-number space, the maximum points of ${\ensuremath{\tau}}_{z(y)}$ will undergo a nonlinear decrease and $\ensuremath{\sigma}$ significantly affects the maximum Larmor time.Recently, experimental measurements of the Larmor time of incident wave packets have been conducted, once again drawing attention to such a time. These experiments measured the tunneling time of Bose-condensed $^{87}\mathrm{Rb}$ atoms, with theoretical analyses employing free particles. In this paper, we investigate the tunneling process of wave packets, which, unlike free particles with a definite incident energy, have a continuous distribution of energies. We show how the Larmor time ${\ensuremath{\tau}}_{z(y)}$ for a wave packet tunneling through a barrier depends on the barrier width $d$, the barrier's characteristic wave vector ${k}_{0}$, the incident wave packet's central wave vector ${k}_{c}$, and its width $\ensuremath{\sigma}$ in wave-number space. When the barrier width $d$ is small, we find that ${\ensuremath{\tau}}_{z(y)}$ varies monotonically. For wide barriers ($d>2.0\ensuremath{\pi}$), we find an anomaly where wave packets with lower incident energy or higher barriers have shorter tunneling times, which is consistent with recent experimental results. Additionally, for narrow wave packets in wave-number space ($\ensuremath{\sigma}<\ensuremath{\gamma}$), where $\ensuremath{\gamma}$ is the width of the fitted Fano peak, the results are generally consistent with those for plane waves. However, as the wave packet widens in wave-number space, the maximum points of ${\ensuremath{\tau}}_{z(y)}$ will undergo a nonlinear decrease and $\ensuremath{\sigma}$ significantly affects the maximum Larmor time.

Physical Review ASep 19, 20250 citations

Transmission and reflection times for scattering of wave packets off tunneling barriers

E. H. Hauge, J. P. Falck, Tor A. Fjeldly

The proper basis for the calculation of transmission and reflection times for wave packets scattered off arbitrary tunneling structures in one dimension is considered. With packets narrow in wave-number space, we demonstrate that the classic phase times are indeed correct to lowest order. Explicit, general expressions for the leading correction terms for finite wave packets are given. The physics associated with these corrections is discussed. We also consider the dwell time, as it is currently defined, and derive a general relation between this dwell time and the phase times. This relation shows when the dwell time can and cannot be used. Finally, we discuss wave packets transmitted from narrow resonances, and derive an explicit, exact formula for the tunneling time with resonance transmission from a symmetric double barrier. Comparison with earlier approximate results is made.The proper basis for the calculation of transmission and reflection times for wave packets scattered off arbitrary tunneling structures in one dimension is considered. With packets narrow in wave-number space, we demonstrate that the classic phase times are indeed correct to lowest order. Explicit, general expressions for the leading correction terms for finite wave packets are given. The physics associated with these corrections is discussed. We also consider the dwell time, as it is currently defined, and derive a general relation between this dwell time and the phase times. This relation shows when the dwell time can and cannot be used. Finally, we discuss wave packets transmitted from narrow resonances, and derive an explicit, exact formula for the tunneling time with resonance transmission from a symmetric double barrier. Comparison with earlier approximate results is made.

Physical review. B, Condensed matterSep 15, 1987212 citations

Average transmission times for the tunneling of wave packets

Jose Tadeu Lunardi, Luiz A. Manzoni, Andrew T. Nystrom +1 more

Journal of Russian Laser ResearchSep 1, 20113 citations

Larmor clock reexamined

J. P. Falck, E. H. Hauge

The Larmor clock, a thought experiment designed by Baz' to measure scattering times, is reexamined in the context of one-dimensional tunneling by wave packets, narrow in wave-number space, through arbitrary static barriers. It is shown that the Larmor clock is, in general, inaccurate when used on a single Fourier component, i.e., on a stationary state. A reliable Larmor clock depends for its operation on properly designed wave packets. Precise conditions for its reliability are given. Thus, the apparent conflict between previous Larmor clock results for tunneling times, and results arrived at by other methods, is removed. When properly set, the Larmor clock shows the classic phase times. Stationary-state calculations of Larmor and dwell ``times'' are shown to represent coherent superpositions of, in general, widely different collision events. The relative merits of the Larmor clock and the center-of-gravity clock are discussed. Finally, it is shown that, for an arbitrary symmetric potential, the reflection and transmission times coincide.The Larmor clock, a thought experiment designed by Baz' to measure scattering times, is reexamined in the context of one-dimensional tunneling by wave packets, narrow in wave-number space, through arbitrary static barriers. It is shown that the Larmor clock is, in general, inaccurate when used on a single Fourier component, i.e., on a stationary state. A reliable Larmor clock depends for its operation on properly designed wave packets. Precise conditions for its reliability are given. Thus, the apparent conflict between previous Larmor clock results for tunneling times, and results arrived at by other methods, is removed. When properly set, the Larmor clock shows the classic phase times. Stationary-state calculations of Larmor and dwell ``times'' are shown to represent coherent superpositions of, in general, widely different collision events. The relative merits of the Larmor clock and the center-of-gravity clock are discussed. Finally, it is shown that, for an arbitrary symmetric potential, the reflection and transmission times coincide.

Physical review. B, Condensed matterAug 15, 1988109 citations

Tunneling Time and Wave Packets Evolution

Pedro Pereyra

Graduate texts in physicsJan 1, 20260 citations

^{87}\mathrm{Rb}

atoms, with theoretical analyses employing free particles. In this paper, we investigate the tunneling process of wave packets, which, unlike free particles with a definite incident energy, have a continuous distribution of energies. We show how the Larmor time ${\ensuremath{\tau}}_{z(y)}$ for a wave packet tunneling through a barrier depends on the barrier width

d

, the barrier's characteristic wave vector

{k}_{0}

, the incident wave packet's central wave vector

{k}_{c}

, and its width $\ensuremath{\sigma}$ in wave-number space. When the barrier width

d

is small, we find that ${\ensuremath{\tau}}_{z(y)}$ varies monotonically. For wide barriers ($d>2.0\ensuremath{\pi}$), we find an anomaly where wave packets with lower incident energy or higher barriers have shorter tunneling times, which is consistent with recent experimental results. Additionally, for narrow wave packets in wave-number space ($\ensuremath{\sigma}<\ensuremath{\gamma}$), where $\ensuremath{\gamma}$ is the width of the fitted Fano peak, the results are generally consistent with those for plane waves. However, as the wave packet widens in wave-number space, the maximum points of ${\ensuremath{\tau}}_{z(y)}$ will undergo a nonlinear decrease and $\ensuremath{\sigma}$ significantly affects the maximum Larmor time.