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Journal of Physics A Mathematical and General

On the theory of the Larmor clock and time delay

Philippe Martin, Massimiliano Sassoli de Bianchi

Published Jun 21, 1992• 13 citations
Publisher

Abstract

Using the time-dependent scattering theory the authors prove that, in any spatial dimension and for arbitrary spin, the reading of the Larmor clock agrees with the global Eisenbud-Wigner time delay in the limit of an infinitesimal magnetic field. They show that convergence is also achieved at fixed energy (without oscillating terms) in the limit where the spatial switching on the field occurs on a much larger scale than the de Broglie wavelength of the particle. Finally, they investigate the functioning of the spin clock beyond the linear response regime.

DOI
10.1088/0305-4470/25/12/023

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Using the time-dependent scattering theory the authors prove that, in any spatial dimension and for arbitrary spin, the reading of the Larmor clock agrees with the global Eisenbud-Wigner time delay in the limit of an infinitesimal magnetic field. They show that convergence is also achieved at fixed energy (without oscillating terms) in the limit where the spatial switching on the field occurs on a much larger scale than the de Broglie wavelength of the particle. Finally, they investigate the functioning of the spin clock beyond the linear response regime.
Eisenbud-Wigner-Smith delay and the Larmor time give different estimates for the duration of a quantum scattering event. The difference is most pronounced in the case where the de Broglie wavelength is large compared to the size of the scatterer. We use the methods of quantum measurement theory to analyse both approaches and to decide which one of them, if any, describes the duration a particle spends in the region that contains the scattering potential. The cases of transmission, reflection, and three-dimensional elastic scattering are discussed in some detail.
Abstract We derive a time-dependent solution for the Schrödinger-Pauli equation within a quantum shutter model to analyse cut-off plane waves scattered by a finite-range potential under a constant magnetic field. Exact analytical solutions for the time-dependent spinor components are derived for the low-energy plane-wave case, enabling the exploration of dynamical effects within the transmission region of a potential barrier. We demonstrate that the spinor probability density exhibits a diffraction in time pattern, with peaks shifted by the presence of the magnetic field relative to the field-free case. This shift introduces a delay time, which we analyse as a function of the Larmor frequency, with tunable peak positions controlled by the field intensity. We further explore the dynamical features of the spin expectation values 〈S k 〉, which exhibit unique diffraction in time behaviour and converge to stationary values in the long-time asymptotic regime. Additionally, we show that measuring the average total spin of the system offers an alternative approach for calculating the dynamical delay time. For infinitesimally small magnetic fields, we define a characteristic dynamical Larmor timescale associated with the evolution of 〈S y 〉, enabling the recovery of the Büttiker Larmor time, τ y , at low energies in the long-time limit. These findings demonstrate that magnetic fields can precisely manipulate tunnelling timescales in quantum systems, providing a means to control matter wave dynamics.
Abstract In this paper, we calculate the quantum time delays for a neutral particle scattering off the Earth gravitational linear potential. The quantum time delays are obtained by subtracting the classical returning time (CRT) from the Wigner time, the dwell time and the Larmor time respectively. Different from the conventional definition, our Larmor time is defined by aligning the magnetic field along particle’s direction of motion, and this definition does give reasonable results for motions through a free region and a square barrier. It is worth noting that in the zero magnetic field limit, the Larmor time coincides well with the CRT, which is due to the special shape of linear barrier, and may have some relevance to the weak equivalence principle. It is also found that the classical forbidden region plays an essential role for the dwell time τ DW to match with the CRT, and the difference between the dwell and the phase times, i.e. the self-interference time delay, is barrier shape sensitive and clearly shows the peculiarity of the linear barrier. All the time delays are on the order of sub-millisecond and exhibit oscillating behaviors, signaling the self-interference of the scattering particle, and the oscillations become evident only when the de Broglie wavelength λ k = 2 π / k is comparable to the characteristic length L c = [ 2 m 2 g / ℏ 2 ] − 1 / 3 . If the time delay measurement is experimentally realizable, it can probe the quantum nature for particle scattering off the gravitational potential in the temporal domain.
This paper concerns time-dependent scattering theory and in particular the concept of time delay for a class of one-dimensional anisotropic quantum systems. These systems are described by a Schr\"odinger Hamiltonian $H=\ensuremath{-}\ensuremath{\Delta}+V$ with a potential V(x)V(x)V(x) converging to different limits ${V}_{\ensuremath{\ell}}$ and Vr{V}_{r}Vr​ as $x\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$, respectively. Due to the anisotropy, they exhibit a two-channel structure. We first establish the existence and properties of the channel wave and scattering operators by using the modern Mourre approach. We then use scattering theory to show the identity of two apparently different representations of time delay. The first one is defined in terms of sojourn times while the second one is given by the Eisenbud-Wigner operator. The identity of these representations is well known for systems where V(x)V(x)V(x) vanishes as $\ensuremath{\mid}x\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ $({V}_{\ensuremath{\ell}}={V}_{r})$. We show that it remains true in the anisotropic case ${V}_{\ensuremath{\ell}}\ensuremath{\ne}{V}_{r}$, i.e., we prove the existence of the time-dependent representation of time delay and its equality with the time-independent Eisenbud-Wigner representation. Finally, we use this identity to give a time-dependent interpretation of the Eisenbud-Wigner expression, which is commonly used for time delay in the literature.
The Larmor clock which was proposed to determine tunneling times of particles with spin through barriers is studied. Tunneling of wave packets is numerically derived from the time-dependent Schr\"odinger-Pauli equation. The Larmor clock often does not proceed unidirectionally in time. The asymptotic readings are well defined. For wide wave packets the asymptotic Larmor times agree with the stationary-state predictions; deviations are seen for narrow wave packets. B\"uttiker's dwell time is investigated and its connection with the Larmor times is confirmed in the case of wave packets. \textcopyright{} 1996 The American Physical Society.