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Physical Review A

Insights into quantum tunneling via a phase-space approach

Chen Chen, Shuyu Zhou

Published Mar 29, 2024• 1 citation
Publisher

Abstract

Quantum tunneling, as a quintessential quantum phenomenon, has been investigated in detail both theoretically and experimentally. Still, the physical picture of the tunneling process is not intuitive, leading to some confusion and paradoxes. In this paper, we have tried to gain insight into quantum tunneling by a phase-space approach. For this purpose, we scrutinize the evolution of the Wigner distribution during tunneling and derive the energy and momentum spectra by integrating over a segment of phase space. In this way, some of the difficulties and paradoxes in tunneling probability, energy conservation, and tunneling time are given a clearer interpretation. Negative probabilities in the Wigner distribution play a key role in the tunneling process, and the origin of negative probabilities clearly indicates that the volatility of matter underlies the various exotic phenomena involved.

DOI
10.1103/physreva.109.032227

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Quantum tunneling, as a quintessential quantum phenomenon, has been investigated in detail both theoretically and experimentally. Still, the physical picture of the tunneling process is not intuitive, leading to some confusion and paradoxes. In this paper, we have tried to gain insight into quantum tunneling by a phase-space approach. For this purpose, we scrutinize the evolution of the Wigner distribution during tunneling and derive the energy and momentum spectra by integrating over a segment of phase space. In this way, some of the difficulties and paradoxes in tunneling probability, energy conservation, and tunneling time are given a clearer interpretation. Negative probabilities in the Wigner distribution play a key role in the tunneling process, and the origin of negative probabilities clearly indicates that the volatility of matter underlies the various exotic phenomena involved.
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.
Quantum tunnelling is a macroscopically measurable phenomenon that distinguishes quantum from classical physics. We study it through the Wigner quasiprobability dis- tribution and its negativity, which serves as a quantifier of non-classicality of quantum states. We model the tunnelling particle as a Gaussian wave packet evolving in one- dimensional single or double-barrier potentials. We find that the negativity is strongly enhanced at energies near the potential maxima, which correspond to classical unstable equilibrium states. For the double-barrier potentials, the negativity is further accentuated and long-lived due to the presence of quasibound resonant states temporarily trapped in- side the central well. This gives us insight that is invisible in the plane-wave limiting analysis.
Asymptotic time evolution of a wave packet describing a non-relativistic particle incident on a potential barrier is considered, using the Wigner phase-space distribution. The distortion of the trasmitted wave packet is determined by two time-like parameters, given by the energy derivative of the complex transmission amplitude. The result is consistent with various definitions of the tunneling time (e.g. the Büttiker-Landauer time, the complex time and Wigner's phase time). The speed-up effect and the negative dispersion are discussed, and new experimental implications are considered.
Phase space is the unity of position and momentum configuration space. It allows for an effective description of dynamical systems and is particularly useful when it comes to studying chaos theory and statistical mechanics. After the advent of quantum physics early in the 20th century, E. Wigner [91], J. E. Moyal [62] and H. J. Groenewold [31] introduce a quantum theory in phase space. Despite the apparent added complexity of the mathematics involved in this new framework, the underlying classical and quantum equations show many similarities. The probability distribution in classical physics becomes the Wigner distribution, a probability distribution usually featuring negative values. In 2013, O. Steuernagel and D. Kakofengitis, inspired by the work of H. Bauke [7] and E. Wigner [91], identified the quantum analogue of the classical phase space flow: the Wigner current J [83]. This Wigner current allows the visualisation of quantum dynamics through a quantum fluid dynamics perspective in phase space. This thesis is written by collection of five articles. They are prefaced by an introduction into the basics of quantum phase space theory and its link with both classical phase space dynamics and the standard Schrödinger approach, followed by the articles published during this PhD. Article 1 shows the importance of the integral form of the Wigner current. We use it to derive the Ehrenfest’s theorem, as well as to refute some propositions made within the community. Article 2 shows that, using the Wigner current, an Eulerian and Lagrangian point of view do not always give the same results for the quantum case. We demonstrate that the negativities of the Wigner distribution, sign of quantumness of the system, are created by the Wigner velocity field singularities. The Wigner velocity field is the quantum analogue of the classical phase space velocity field. In Article 3, we see that even though Wigner distributions of quantum systems feature spotty structures which saturate on scales ɑZ [97], the construction of a superoscillating Wigner distribution allows one to generate much smaller structures, of the order of ɑZ /α with α a positive constant potentially very large. In Article 4, we introduce the concept of quantum shear suppression in phase space. The Wigner current features an effective quantum “viscosity”, suppressing classical dynamics fine details. This viscosity is the mechanism by which the Zurek scale is enforced dynamically onto the state in phase space. In Article 5, we apply the previous ideas to Kerr-type oscillators. Its Wigner current is derived, and using it we show that its values are conserved on a ring during the time evolution of the Kerr oscillator. The shear suppression is also studied.
We develop an effective quantum force to propagate trajectories in Wigner phase space. In this way, the quantum entanglement between trajectories is represented by the effective quantum force, which is related to the variance of the momentum distribution. Reaction probabilities are obtained for three model potentials. For a cubic potential, the reaction probabilities are found to continuously grow in time long after the classical reaction probabilities have become stationary. This growth is due to tunneling and is in good agreement with numerically exact quantum results. The variance of the momentum distribution is updated at each time step of the trajectories, which is the key to the increase of the reaction probabilities at longer times and allows a nice interpretation of tunneling.
We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.