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An Entropic Formulation of Tunneling Time

D. Demir, Tuğrul Güner

Published Dec 14, 2015• 1 citation
OpenAlex

Abstract

Quantum tunneling governs numerous phenomena in biology, chemistry, physics and technology. Tunneling time, formulated in various different forms due to the absence of a time operator in quantum theory, has been measured recently in experiments based on the attoclock in ultrafast laser ionization of Helium atoms [A.~Landsman {\it et al.}, Optica {\bf 1}, 343 (2014)]. The experiment performs a refined measurement with which no tunneling time formula in the literature exhibits adequate congruence. Here we show that, entropic considerations lead to a real tunneling time which shows remarkable agreement with the experimental data and stays always subluminal. Indeed, with phase space volume setting the number of microstates for a single evanescing particle in a state of definite momentum consistent with energy conservation, one is naturally led to a statistical description for quantum tunneling in which thermal energy sets the tunneling time. This entropic tunneling time is rather general and might also be extended to photon and phonon tunneling phenomena.

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Quantum tunneling governs numerous phenomena in biology, chemistry, physics and technology. Tunneling time, formulated in various different forms due to the absence of a time operator in quantum theory, has been measured recently in experiments based on the attoclock in ultrafast laser ionization of Helium atoms [A.~Landsman {\it et al.}, Optica {\bf 1}, 343 (2014)]. The experiment performs a refined measurement with which no tunneling time formula in the literature exhibits adequate congruence. Here we show that, entropic considerations lead to a real tunneling time which shows remarkable agreement with the experimental data and stays always subluminal. Indeed, with phase space volume setting the number of microstates for a single evanescing particle in a state of definite momentum consistent with energy conservation, one is naturally led to a statistical description for quantum tunneling in which thermal energy sets the tunneling time. This entropic tunneling time is rather general and might also be extended to photon and phonon tunneling phenomena.
Time it takes to travel from one position to another, devoid of any quantum mechanical description, has been modeled variously, especially for quantum tunneling. The model time, if universally valid, must be subluminal, must hold everywhere (inside and outside the tunneling region), must comprise interference effects, and must have a sensible classical limit. Here we show that the quantum travel time, hypothesized to emerge with the state vector, is a function of the probability density and probability current such that all the criteria above are fulfilled. We compute it inside and outside a rectangular potential barrier and find physically sensible results. Moreover, we contrast it with recent ionization time measurements of the He\rm HeHe as well as the Ar\rm ArAr and Kr\rm KrKr atoms, and find good agreement with data. The quantum travel time holds good for stationary systems, and can have applications in numerous tunneling-driven phenomena.
Abstract With the development of attosecond science, tunneling time can now be measured experimentally with the attoclock technique. However, there are many different theoretical definitions of tunneling time and no consensus has been achieved. Here, we bridge the relationship between different definitions of tunneling time based on a quantum travel time in one-dimensional rectangular barrier tunneling problem. We find that the real quantum travel time t Re is equal to the Bohmian time t Bohmian, which is related to the resonance lifetime of a bound state. The total quantum travel time τ t can perfectly retrieve the transversal time t x and the Büttiker–Landauer time τ BL in two opposite limits, regardless of the particle energy.
Tunneling is one of the most bizarre phenomena in quantum mechanics. An attempt to understand it led to the next natural question of how long does a particle need to tunnel a barrier. The latter gave rise to several definitions such as the phase, dwell, Larmor and traversal times among others. A short review of the evolution of these time concepts, followed by an account of experiments involving field-induced tunnel ionization and electron tunneling in a solid state junction is presented here. Whereas the former experiments use sophisticated techniques involving femtosecond laser pulses and determine the tunneling time by mapping the angle of rotation of the field vector to time, like the hands of a watch, the latter provides a simpler method through the measurement of current-voltage characteristics of the junction.
The quantum tunneling time remains the subject of heated debate, and one of its most curious features is faster-than-light or superluminal tunneling. Our tunnel-ionization model of the time-delay, presented in previous work, shows good agreement with the attoclock measurement in the adiabatic and nonadiabatic field calibrations, which also enables the determination of the barrier time-delay. In the present work, we show that the tunnel-ionization for H-like atoms with large nuclear charge can be superluminal (quantum superluminality), which in principle can be investigated experimentally using the attoclock scheme. We discuss the quantum superluminality in detail for the different regimes of the tunnel-ionization. Our result shows that quantum tunneling faster-than-light is indeed possible, albeit only under somewhat extreme conditions.
Abstract Quantum tunnelling is a fundamental phenomenon in quantum physics, yet the question of how long it takes for a particle to tunnel through a barrier remains a topic of ongoing debate. Recent advances in attoclock experimental techniques in strong-field physics have renewed interest in this topic. While various definitions of tunneling time have been proposed, a consensus has yet to be reached. In this study, we adopt a definition of tunnelling time based on the ratio of the barrier width to the expected value of the velocity operator within the barrier. This approach reduces to the Büttiker-Landauer time (τBL\tau_{BL}τBL​) in the case of an opaque barrier and offers a clear physical interpretation for very thin barriers, where τBL\tau_{BL}τBL​ may not be well-defined. Additionally, by utilizing a virtual detector at the tunnel exit, we calculate the time delay between the peak of the electric field and the moment of ionization for long wavelengths ($\geq$6 μ\muμm). Our findings indicate that the time delay stabilizes around 190 as, remaining independent of the laser wavelength. Furthermore, through our quantum tunnelling time definition, this time delay can be interpreted as the time required for the electron to tunnel from a point within the barrier to the exit.
Attosecond science, beyond its importance from application point of view, is of a fundamental interest in physics. The measurement of tunneling time in attosecond experiments offers a fruitful opportunity to understand the role of time in quantum mechanics. In the present work, we show that our real T-time relation derived in earlier works can be derived from an observable or a time operator, which obeys an ordinary commutation relation. Moreover, we show that our real T-time can also be constructed, inter alia, from the well-known Aharonov–Bohm time operator. This shows that the specific form of the time operator is not decisive, and dynamical time operators relate identically to the intrinsic time of the system. It contrasts the famous Pauli theorem, and confirms the fact that time is an observable, i.e., the existence of time operator and that the time is not a parameter in quantum mechanics. Furthermore, we discuss the relations with different types of tunneling times, such as Eisenbud–Wigner time, dwell time, and the statistically or probabilistic defined tunneling time. We conclude with the hotly debated interpretation of the attoclock measurement and the advantage of the real T-time picture versus the imaginary one.