Observation of the Decrease of Larmor Tunneling Times with Lower Incident Energy

David C. Spierings and Aephraim M. Steinberg

Phys. Rev. Lett. 127, 133001 – Published 20 September 2021

Abstract

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 $0.59 \pm 0.02 ms$ inside. This is $0.11 \pm 0.03 ms$ faster than atoms traversing the same barrier with energy close to the barrier’s peak and $0.21 \pm 0.03 ms$ faster than when the atoms traverse a barrier with 23% less energy.

Received 1 February 2021
Accepted 12 July 2021
Corrected 28 October 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.133001

Physics Subject Headings (PhySH)

Quantum foundations Quantum measurements Tunneling & traversal time Weak values & weak measurements

General Physics

Corrections

28 October 2021

Correction: The temperature value in the first sentence of the fourth paragraph contained a typographical error and has been replaced.

Authors & Affiliations

David C. Spierings ^* and Aephraim M. Steinberg ^†

Centre for Quantum Information and Quantum Control, Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 1A7, Canada
Canadian Institute for Advanced Research, MaRS Centre, West Tower 661 University Ave., Toronto, Ontario M5G 1M1, Canada

^*Corresponding author. dspierin@physics.utoronto.ca
^†Also at Canadian Institute for Advanced Research, Toronto, Ontario, Canada.

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Vol. 127, Iss. 13 — 24 September 2021

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Images

Figure 1
(a) Illustrations of $| ψ (y) |^{2}$ for a wave of energy $E$ incident on square barriers of two heights. The dwell time in the red (solid) scenario, with a higher barrier, is lower than in the blue (dashed) case, with a lower barrier. (b) The standard Larmor experiment starts with a spin- $1 / 2$ particle (here an electron), polarized along the $x$ axis, incident on a square barrier. A magnetic field, localized to the barrier region and pointing along the $z$ axis, causes the spin to precess while in the barrier. The angles of precession $θ_{y}$ and alignment $θ_{z}$ of the transmitted wave packet determine the Larmor tunneling times when the energy of the wave packet is below the barrier. The right Bloch sphere illustrates the rotations resulting from a Larmor measurement of the scenarios shown in (a).
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Figure 2
The experimental procedure. (a) A cloud of ${}^{87}{Rb}$ atoms is condensed in the $| F = 2, m_{F} = 2 ⟩$ state and confined at the intersection of two optical dipole traps, one of which is elongated and forms a quasi-1D system for the collision. (b) Matter-wave lensing prepares an atomic wave packet narrow in momentum. (c) A variable-duration magnetic field gradient pushes the atomic wave packet toward a 421.38 nm optical barrier. Before reaching the barrier, the spins of the atoms are transferred to the $| F = 2, m_{F} = 0 ⟩$ state via ARP. (d) A two-photon Raman transition between the clock states of the hyperfine manifold implements the Larmor measurement during the collision with the barrier. (e),(f) Well after the collision, the $⟨ S_{x} ⟩$ , $⟨ S_{y} ⟩$ , and $⟨ S_{z} ⟩$ components of the net magnetization vector are obtained by sequential imaging of the hyperfine populations. Measurements of the $⟨ S_{y} ⟩$ and $⟨ S_{z} ⟩$ components are preceded by $π / 2 MW$ rotations about the $z$ and $y$ axes of the Bloch sphere, respectively. ARP transfers the $| F = 1, m_{F} = 0 ⟩$ population to $| F = 2, m_{F} = 0 ⟩$ prior to the second absorption image.
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Figure 3
Typical data from an experimental run. (a) Transmission of a wave packet with average incident velocity $v^{*}$ through barriers of varying heights measures the width of the incident atomic cloud’s velocity distribution. (b) The components of the Bloch vector for the transmitted wave packet are extracted for wave packets with different incident velocities.
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Figure 4
Larmor times. (a) Measured Larmor times $τ_{y}$ (solid markers) and $| τ_{z} |$ (faded markers) for two different barrier heights, $4.71 \pm 0.05 mm / s$ (red) and $4.13 \pm 0.08 mm / s$ (blue), depicted by the vertical dashed lines, versus incident velocity (lower axis) and normalized energy (upper axis). Statistical and systematic errors are given by the bars and rectangular outlines, respectively, which are sometimes smaller than the markers. Colored bands show GP simulations of the Larmor experiment, bounded by the uncertainties in the measured barrier heights and in the velocity width at $v^{*}$ . Inset: Experimental and simulated transmission. Some of the above-barrier data for the lower barrier height set had an atypically large Rabi frequency, $808 \pm 12 Hz$ . While this modifies the transmission, indicated by the faded blue markers and bands, it does not significantly affect these times. (b) Low energy data of (a), now compared to weak-value calculations weighted by the inferred velocity profile of the transmitted wave packet. The shaded regions are bounded as in (a). (c) Monochromatic weak-value calculations of the Larmor times (red) for a Gaussian barrier of height $4.71 mm / s$ and the semiclassical expression [32] used to calibrate the Larmor frequency above the barrier (see the Supplemental Material [30]). Inset: Transmission profile.
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