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Physical Review A

Larmor time of a bound electron wave packet tunneling through a barrier

Lijuan Jia, Haijun Xing, Libin Fu

Published Jun 7, 2022• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.105.062804

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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.
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 87Rb^{87}\mathrm{Rb}87Rb 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 ddd, the barrier's characteristic wave vector k0{k}_{0}k0​, the incident wave packet's central wave vector kc{k}_{c}kc​, and its width $\ensuremath{\sigma}$ in wave-number space. When the barrier width ddd 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.
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.
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.
We study the time required for a wave packet to tunnel beyond a square barrier, or to be reflected, by envisaging a physical clock which ticks only when the particle is within the barrier region. The clock consists in a magnetic moment initially aligned with the xxx axis which in the barrier region precesses around a constant magnetic field aligned with the zzz axis, the motion being in the yyy direction. The values of the xxx and yyy components of the magnetic moment beyond or in front of the barrier allow to assign a tunneling or reflection time to every fraction of the packet which emerges from the barrier and to calculate tunneling times τT,xτ_{\rm T,x}τT,x​ and τT,yτ_{\rm T,y}τT,y​ and reflection times τR,xτ_{\rm R,x}τR,x​ and τR,yτ_{\rm R,y}τR,y​. The times τT,xτ_{\rm T,x}τT,x​ and τT,yτ_{\rm T,y}τT,y​ (τR,xτ_{\rm R,x}τR,x​ and τR,yτ_{\rm R,y}τR,y​) are remarkably equal, and independent of the initial position (in front of the barrier) of the packet.
The Larmor precession of a neutral spinning particle in a magnetic field confined to the region of a one-dimensional rectangular barrier is investigated for both a nonrelativistic and a relativistic incoming particle. The spin precession serves as a clock to measure the time spent by a quantum particle traversing a potential barrier. With the help of a general spin coherent state it is explicitly shown that the precession time is equal to the dwell time in both the nonrelativistic and relativistic cases. We also present a numerical estimation of the precession time showing an apparent superluminal tunneling.
Baz' and Rybachenko have proposed the use of the Larmor precession as a clock to measure the time it takes a particle to traverse a barrier. An applied magnetic field is confined to the barrier. The spin of the incident particles is polarized perpendicular to this field. The extent of the Larmor precession occurring during transmission is used as a measurement of the time spent traversing the barrier. However, the particles tunneling through an opaque barrier also acquire a spin component parallel to the field since particles with spin parallel to the field have a higher transmission probability than particles with spin antiparallel to the field. Similar effects are actually used to polarize electrons and neutrons. An interpretation of this experiment compares the results with an approach which determines the traversal time by studying transmission of particles through a time-modulated barrier. This leads to three characteristic times describing the interaction of particles with a barrier. A dwell time measures the average time interval during which a particle interacts with the barrier whether it is reflected or transmitted at the end of its stay, a traversal time measures the time interval during which a particle interacts with the barrier if it is finally transmitted, and a reflection time measures the interaction time of a reflected particle.