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arXiv:2104.06144

Decoding electron tunnelling delay time by embracing wave-particle duality

Chuncheng Wang, Xiaokai Li, Xiwang Liu, Jie Li, Shengpeng Zhou, Yizhang Yang, Xiaohong Song, Jing Chen, Weifeng Yang, Dajun Ding

Published Apr 13, 2021• 2 citations

Abstract

Tunnelling lies at the heart of quantum mechanics and is a fundamental process in attosecond science, molecular biology, and quantum devices. Whether tunnelling takes time and how a microscopic particle transits through a barrier have been debated since the early days of quantum mechanics. The time required for an electron to tunnel through an atomic potential barrier has been measured with attosecond angular streaking (attoclock), and a recent work on the hydrogen atom claimed that electron tunnelling is instantaneous. However, the time required for Rb atoms to tunnel through an optical potential barrier has been measured to be on the order of milliseconds with a recent Larmor clock measurement. The essence of electron and atom tunnelling is identical, but the reason for the contradictory conclusions remains unknown. Here, we demonstrate that the sub-barrier potential interaction is the root of the nonzero tunnelling delay time. We reveal that the wave-particle duality of a tunnelling electron must be fully taken into account when decoding the tunnelling delay time from the attoclock measurements. Based on energy-resolved attoclock measurements, we show that the tunnelling delay time of an electron ranges from 24 to 58 attoseconds, and the counterintuitive result that an electron with a lower energy may spend less time in the barrier is consistent with the velocity-dependent tunnelling time of atoms in Larmor clock measurements. Our results unify the tunnelling time of microscopic particles by highlighting the classically forbidden sub-barrier potential interactions of matter waves.

DOI: 10.48550/arxiv.2104.06144

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Tunnelling lies at the heart of quantum mechanics and is a fundamental process in attosecond science, molecular biology, and quantum devices. Whether tunnelling takes time and how a microscopic particle transits through a barrier have been debated since the early days of quantum mechanics. The time required for an electron to tunnel through an atomic potential barrier has been measured with attosecond angular streaking (attoclock), and a recent work on the hydrogen atom claimed that electron tunnelling is instantaneous. However, the time required for Rb atoms to tunnel through an optical potential barrier has been measured to be on the order of milliseconds with a recent Larmor clock measurement. The essence of electron and atom tunnelling is identical, but the reason for the contradictory conclusions remains unknown. Here, we demonstrate that the sub-barrier potential interaction is the root of the nonzero tunnelling delay time. We reveal that the wave-particle duality of a tunnelling electron must be fully taken into account when decoding the tunnelling delay time from the attoclock measurements. Based on energy-resolved attoclock measurements, we show that the tunnelling delay time of an electron ranges from 24 to 58 attoseconds, and the counterintuitive result that an electron with a lower energy may spend less time in the barrier is consistent with the velocity-dependent tunnelling time of atoms in Larmor clock measurements. Our results unify the tunnelling time of microscopic particles by highlighting the classically forbidden sub-barrier potential interactions of matter waves.

Tunnelling is one of the most paradigmatic and evocative phenomena of quantum physics, underlying processes such as photosynthesis and nuclear fusion, as well as devices ranging from SQUID magnetometers to superconducting qubits for quantum computers. The question of how long a particle takes to tunnel, however, has remained controversial since the first attempts to calculate it, which relied on the group delay. It is now well understood that this delay (the arrival time of the transmitted wave packet peak at the far side of the barrier) can be smaller than the barrier thickness divided by the speed of light, without violating causality. There have been a number of experiments confirming this, and even a recent one claiming that tunnelling may take no time at all. There have also been efforts to identify another timescale, which would better describe how long a given particle spends in the barrier region. Here we present a direct measurement of such a time, studying Bose-condensed

^{87}

Rb atoms tunnelling through a 1.3-

μ

m thick optical barrier. By localizing a pseudo-magnetic field inside the barrier, we use the spin precession of the atoms as a clock to measure the time it takes them to cross the classically forbidden region. We find a traversal time of 0.62(7) ms and study its dependence on incident energy. In addition to finally shedding light on the fundamental question of the tunnelling time, this experiment lays the groundwork for addressing deep foundational questions about history in quantum mechanics: for instance, what can we learn about where a particle was at earlier times by observing where it is now?

How much time does it take for a particle to tunnel through a barrier? This question continues to baffle up to this very day, one of the reasons being the non-existence of an energy–time commutator analogous to the position–momentum commutation relation. Instead of worrying about the definition of an operator one may consider time as a parameter in the time-dependent Schrödinger equation and study the time evolution of a wavepacket appropriately scattered on a potential barrier. Using this approach, one may formally define a tunnelling flight time that may be measured in a time-of-flight experiment. The resulting tunnelling flight time either vanishes or is very small. The implications of this observation with respect to recent hydrogen and helium atom attosecond photoionization experiments are discussed. The vanishing or small flight time does not contradict the finite time measured in Larmor clock experiments, where the tunnelling particle affects an external degree of freedom whose dynamics induced by the tunnelling may be interpreted in terms of a time scale. Flight times are also relevant to quantum reflection where coherences due to resonance scattering are not well accounted for by Wigner's phase time delay.

A clear consensus on how long it takes a particle to tunnel through a potential barrier has never been so urgently required, since the electron dynamics in strong-field ionization can be resolved on attosecond time-scale in experiment and the exact nature of the tunneling process is the key to trigger subsequent attosecond techniques. Here a general picture of tunneling time is suggested by introducing a quantum travel time, which is defined as the ratio of the travel distance to the expected value of the velocity operator under the barrier. Specially, if applied to rectangular barrier tunneling, it can retrieve the Büttiker-Landauer time

τ_{BL}

in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein

τ_{BL}

can not be well defined. In the case of strong-field tunneling process, with the help of the newly defined time, the tunneling delay time measured by attoclock experiment can be interpreted as a travel time spent by the electron to tunnel from a point under the barrier to the tunnel exit. In addition, a peculiar oscillation structure in the wavelength dependence of tunneling delay time in deep tunneling regime is observed, which is beyond the scope of adiabatic tunneling picture. This oscillation structure can be attributed to the interference between the ground state tunneling channel and the excited states tunneling channels.

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}

Rb 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\%

of transmitted atoms tunneled through the barrier, spending an average of

0.59(2)

ms inside. This is

0.11(3)

ms faster than atoms traversing the same barrier with energy close to the barrier's peak and

0.21(3)

ms faster than when the atoms traverse a barrier with

23\%

less energy.

Quantum tunneling, a quintessential quantum phenomenon, challenges classical intuitions by allowing particles to pass through seemingly insurmountable barriers. This study delves into the intricate dynamics of electron tunneling, focusing on the critical role of sub-barrier interactions and complex time processes. Through a combination of experimental measurements and simulations based on Feynman’s path integral of complex time trajectories, we reveal that the sub-barrier potential interaction introduces an asymmetric electron wave packet probability distribution along the real time axis and a marked delay at the tunneling exit. The energy-dependent behavior of this delay in electronic tunneling agrees with that observed in the previous atomic tunneling studies, suggesting a universal mechanism underlying quantum tunneling across different scales. Our findings underscore the necessity of incorporating both the wave and particle aspects of quantum entities to fully comprehend tunneling dynamics, which holds promise for an intuitive interpretation of complex-number quantum processes.

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.