Abstract We theoretically study the tunneling time by investigating a wave packet of Bose-condensed atoms passing through a square barrier. We find that the tunneling time exhibits different scaling laws in different energy regimes. For negative incident energy of the wave packet, counterintuitively, the tunneling time decreases very rapidly with decreasing incident velocity. In contrast, for positive incident energy smaller than the barrier height, the tunneling time increases slowly and then reaches a maximum, which is in agreement with the Larmor clock experiments. The effect of the barrier width related to the uncertainty principle on the maximum tunneling time is also addressed. Our work provides a general scenario of tunneling time that can be used to understand and explain the controversy over tunneling time.
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 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
d, the barrier's characteristic wave vector
k0, the incident wave packet's central wave vector
kc, and its width $\ensuremath{\sigma}$ in wave-number space. When the barrier width
d 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.
Abstract Tunneling time as an important quantity to characterize tunneling process has not been well defined yet in quantum mechanics. In this paper, we adopt the definition of tunneling time given in [S.-C. Li, New J. Phys. 24, 083033 (2022)] to investigate the time spent by a macroscopic quantum wave packet passing through a Gaussian barrier. We show a general scenario of tunneling times in different energy regimes and find that it has a good agreement with the scenario for a rectangular barrier, which verifies the validity of the above definition of tunneling time for the Gaussian barrier. The influence of both incident energy of wave packet and barrier parameters on tunneling time is discussed thoroughly. The connection between tunneling time and tunneling probability for the Gaussian barrier is established as well. In particular, we use the product of the variances of position and momentum to quantify the deformation of wave packet caused by tunneling process and demonstrate that different behaviors of the variances strongly support our oft-emphasized division of energy regimes.
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.
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
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% 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.
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
87Rb atoms tunnelling through a 1.3-
μm 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?
In this work we present a theoretical model of the tunneling time and the tunneling process (in attosecond experiment for the He atom). Our model is supported with physical reasoning leading to a relation which performs an excellent estimation for the tunneling time in attosecond and strong-field experiments, where we address the important case of the He atom. Our tunneling time estimation is found by utilizing the time-energy uncertainty relation and represents a quantum clock. The tunneling time is also featured as the time of passage through the barrier similar to Einstein's photon-box Gedanken experiment. Our work tackles an important case study for the theory of time in quantum mechanics and is very promising for the search for a (general) time operator in quantum mechanics. The work can be seen as a fundamental step in dealing with the tunneling time in strong-field and ultrafast science and is appealing for more elaborate treatments using quantum wave-packet dynamics and especially for complex atoms and molecules.