This paper explores the Larmor time and compares it for various shapes of barriers such as rectangular barrier and triangular barrier, and we arrive at consistent results between the traditionally defined Larmor time and a newly defined Larmor time. Interestingly, we have found some regular sharp jumps (discontinuities) in reflection Larmor times for a given rectangular barrier. With further investigation, it was found that the wave numbers where the Larmor time jumps correlate with those where the reflection probability reaches minimal values. Based on known results for triangular barriers, this paper also presents the Larmor times for positive and negative slope trapezoidal barriers, whose treatments are similar to the positive and negative triangular barriers, respectively, though the trapezoidal barriers exhibit more oscillatory behaviors for transmission times in comparison to their triangular counterparts. Finally, we demonstrate that the transmission and reflection Larmor times for M-shaped barriers exhibit more pronounced peaks, as expected. Also interestingly, some of the sharp jumps in reflection time are associated with sharp variations in the corresponding reflection probabilities, which is similar to the correlation mentioned earlier between the sharp jumps in reflection time and reflection probabilities in wave numbers for rectangular barriers.
In this paper, we explore the Larmor time for three types of trapezoidal barriers, and we find consistent results between the traditionally defined Larmor time and a newly defined one. We confirm that the transmission Larmor time for the trapezoidal barriers also satisfies certain properties of mirror-symmetric barriers. Consistent with our expectations, we also find that: 1. as the barrier height increases, the peak of the Larmor time shifts to the right (higher energy); and 2. as the barrier width increases, the peak becomes larger in coincidence with the classical expectation that a particle needs more time to cross a longer path of the same height/inclination.
We study the phase, Larmor and dwell times of a particle scattered off triangular barriers (TBs). It is interesting that the dependences of dwell, reflective phase and Larmor times on the wave number, barrier width and height for a pair of mirror-symmetric (MS) exact triangular barriers (ETBs) are quite different, as the two ETBs have quite distinct scattering surfaces. In comparison, the dependence of the transmitted phase or Larmor times is exactly the same, since the transmitted amplitudes are the same for a pair of MS TBs. We further study the Hartman effect by defining the phase and Larmor velocities associated with the phase and Larmor times. We find no barrier width saturation effect for the transmitted and reflected times. This is indicated by the fact that all the velocities approach finite constants that are much smaller than the speed of light in vacuum for TBs with positive-slope impact faces. As for ETBs with vertical left edges, the naive velocities seem to also indicate the absence of the Hartman effect. These are quite distinct from rectangular barriers and may shed new light on the clarification of the tunneling time issues.
The Larmor clock extracts time information from quantum scattering through a weak spin-dependent phase readout. We apply this formulation to the transmitted Larmor time in trapezoidal barriers and examine its geometry-dependent threshold response. The main quantity is the dimensionless peak coordinate xp=Ep/V0, where Ep is the energy of the dominant transmitted-time peak. For a right trapezoidal barrier consisting of a linear ramp and a flat segment, the peak moves from the classically forbidden side (xp<1) to the allowed side (xp>1) as the flat part becomes more important. The crossover is characterized by rc=(l-a)/a, defined through xp=1. For a=2 nm and V0=2 μeV, we obtain rc≃0.428. A symmetric triangle-rectangle-triangle barrier shows a corresponding crossover at sc=(d-a)/(2a)≃0.220. These results demonstrate that the strongest transmitted-channel clock response is controlled by the spatial distribution of the sloped matching regions and the flat phase-accumulating region. The peak coordinate therefore provides a phase-sensitive marker of how this response shifts across the nominal threshold as the barrier geometry is varied.
Recent generalizations of B\"uttiker's analysis of the Larmor clock to any region
z1\ensuremath{\le}z\ensuremath{\le}
z2 within an arbitrary one-dimensional barrier lead to two local transmission ``times'' for an incident electron of energy E=${\ensuremath{\Elzxh}}^{2}$
k2/2m. These are conveniently regarded as the real and (minus) the imaginary parts of a complex time ${\ensuremath{\tau}}_{T}^{V}$(k;
z1,
z2). In this paper the properties of ${\ensuremath{\tau}}_{T}^{V}$(k;
z1,
z2) are investigated for the double-rectangular-barrier potential V(z)=
V1\ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z) +
V2\ensuremath{\Theta}(z-b)\ensuremath{\Theta}(d-z). Results are presented for the dependence of the real and imaginary parts of the Larmor-clock transmission time on incident energy and barrier and well widths. The local transmission time ${\ensuremath{\tau}}_{T}^{V}$(k;
z1,
z2) is real for symmetric double barriers when the energy of the incident electron is exactly on resonance. Remarkably, the behavior of the real quantity ${\ensuremath{\tau}}_{T}^{V}$(k;0, z\ensuremath{\ll}a) for this special case provides further evidence for the importance of the imaginary part of the local Larmor-clock transmission time for the corresponding (isolated) single barrier V(z)=
V1\ensuremath{\Theta}(z)\ensuremath{\Theta}(a-z). At resonance, for symmetric double barriers, (real) local transmission speeds
vT(z) can be very much in excess of the speed of light in the well region for z very close to a quasinode of the Schr\"odinger stationary-state wave function. It is proven that
vT(z)\ensuremath{\le}c for a symmetric double barrier at resonance when the Dirac equation is used in place of the Schr\"odinger equation. On the other hand, it is shown that application of the Dirac equation to a single opaque rectangular barrier does not alter the well-known difficulty that a/Re[${\ensuremath{\tau}}_{T}^{V}$(k;0,a)] exceeds c for sufficiently large barrier width a.
The desire to minimize the response time of tunneling devices has led to renewed interest in the unresolved fundamental question How much time, on average, does an incident electron that is ultimately transmitted spend inside a one-dimensional potential barrier? In addition to this transmission time ⊤t ·,there are two other times of interest: the reflection time ⊤r for electrons that are ultimately reflected and the dwell time TD, which is the average time spent inside the barrier regardless of the ultimate fate of the incident particle. An exact expression for TD and the time-modulated barrier, Feynman path, Larmor clock, and wave-packet evolution approaches for calculating ⊤t and ⊤r arc discussed. It is argued that none of these approaches is completely satisfactory because they all lead to transmission and reflection times that either are complex or, if real, do not satisfy: all of a few reasonable (but not universally accepted) criteria, such as consistency with the exact expression for the dwell time.