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.
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.
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.
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.
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.
Quantum transit times have a long history of inequivalent definitions, including phase times, dwell times, and quantum-clock constructions. In this context we revisit the Salecker--Wigner--Peres stationary quantum clock as a phase-sensitive scattering observable, with clock time defined by the energy derivative of the transmission phase shift across the interaction region. For real compactly supported one-dimensional potentials, we show that the raw stationary Peres clock generically contains a universal 1/E continuum-edge term whose coefficient is fixed by low-energy scattering data. For the attractive square well, this threshold singularity is inherited from the vanishing exterior momentum and the associated scattering matching, rather than from resonant delay itself. We derive the exact stationary clock time for the square well and introduce a new threshold-subtracted clock observable. Away from exceptional zero-energy tuning, the subtraction removes the universal low-energy term and isolates the resonant contribution. Comparison with the dwell time and the transmission Wigner phase delay shows that the threshold-subtracted clock acquires the expected local Lorentzian form near isolated transmission resonances. Near the continuum edge, if ε denotes the detuning from threshold, the resonant peak grows only as ε−1/2, whereas the unsubtracted threshold background grows as ε−3/2. A symmetric barrier--well--barrier cavity and a numerical asymmetric two-step attractive well provide complementary controls. The result is a new threshold-subtracted stationary-clock candidate that separates universal threshold kinematics from pole-sensitive resonant delay.
The Larmor time of an incident wave packet has recently been measured experimentally [R. Ramos, D. Spierings, I. Racicot, and A. M. Steinberg, Nature (London) 583, 529 (2020)]. It differs essentially from previous well-studied free-particle cases, whereas the unique properties of wave packets are unrevealed in their theoretical analysis and state-of-the-art Larmor clock studies. In this paper, we study the Larmor time of a bound two-component electron wave packet and supplement the theoretical lack of wave-packet tunneling. We find that the spin shows a pure Larmor precession in the plane perpendicular to the field without a rotation parallel to the field. The Larmor time is defined in the limit of vanishing field where the precession angle is proportional to the field. For relatively weak fields, this precession angle will respond to the field approximately linearly. For the wave-packet tunneling case, we propose an effective scheme that utilizes the fidelity of the tunneled wave packets to calibrate an approximate linear-response region. Furthermore, we show that in the approximate linear-response region the change in the fidelity can be approximately regarded as the result of the accumulated displacement in phase space. This result reveals the inner mechanism of wave-packet tunneling. Our findings may have implications for future experiments of the Larmor clock.
We numerically study two methods of measuring tunneling times using a quantum clock. In the conventional method using the Larmor clock, we show that the Larmor tunneling time can be shorter for higher tunneling barriers. In the second method, we study the probability of a spin-flip of a particle when it is transmitted through a potential barrier including a spatially rotating field interacting with its spin. According to the adiabatic theorem, the probability depends on the velocity of the particle inside the barrier. It is numerically observed that the probability increases for higher barriers, which is consistent with the result obtained by the Larmor clock. By comparing outcomes for different initial spin states, we suggest that one of the main causes of the apparent decrease in the tunneling time can be the filtering effect occurring at the end of the barrier.