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.
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.
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.
How much time does a tunneling particle spend in a barrier? A Larmor clock, one proposal to answer this question, measures the interaction between the particle and the barrier region using an auxiliary degree of freedom of the particle to clock the dwell time inside the barrier. We report on precise Larmor time measurements of ultracold ^{87}Rb atoms tunneling through an optical barrier, which confirm longstanding predictions of tunneling times. We observe that atoms generally spend less time tunneling through higher barriers and that this time decreases for lower energy particles. For the lowest measured incident energy, at least 90% of transmitted atoms tunneled through the barrier, spending an average of 0.59±0.02 ms inside. This is 0.11±0.03 ms faster than atoms traversing the same barrier with energy close to the barrier's peak and 0.21±0.03 ms faster than when the atoms traverse a barrier with 23% less energy.
How much time does a tunneling particle spend in a barrier? A Larmor clock, one proposal to answer this question, measures the interaction between the particle and the barrier region using an auxiliary degree of freedom of the particle to clock the dwell time inside the barrier. We report on precise Larmor time measurements of an ultra-cold gas of
87Rb atoms tunneling through an optical barrier. The data capture distinctive features that confirm longstanding predictions of tunneling times. In particular, we demonstrate that atoms spend less time tunneling through higher barriers and that this time decreases for lower energy particles. For the lowest measured incident energy, at least
90% of transmitted atoms tunneled through the barrier, spending an average of
0.59(2)ms inside. This is
0.11(3)ms faster than atoms traversing the same barrier with energy close to the barrier's peak and
0.21(3)ms faster than when the atoms traverse a barrier with
23% less energy.