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arXiv:2207.13130 · Physical Review A

Numerical quantum clock simulations for measuring tunneling times

Fumika Suzuki, William George Unruh

Published Apr 20, 2023• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.107.042216

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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.
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.
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.
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.
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 87^{87}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%90\%90% of transmitted atoms tunneled through the barrier, spending an average of 0.59(2)0.59(2)0.59(2)ms inside. This is 0.11(3)0.11(3)0.11(3)ms faster than atoms traversing the same barrier with energy close to the barrier's peak and 0.21(3)0.21(3)0.21(3)ms faster than when the atoms traverse a barrier with 23%23\%23% less energy.
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 question of how long a tunneling particle spends in the forbidden region of a barrier has been a perplexing puzzle in foundational quantum mechanics for many decades. The Larmor time, one definition of tunneling time, uses an auxiliary degree of freedom of the tunneling particles to clock the time spent inside the barrier. Recently, we made our first measurement of the Larmor time for Bose-condensed 87Rb atoms tunneling through an optical barrier.1 Here, we report on follow up measurements with improved precision in the measured times and also study the time for the reflected atomic cloud. We observe significant discrepancies between our results and a simple theory based on weak measurement. We discuss our findings, hypothesize explanations for our results, and suggest future studies.