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arXiv:1608.03492 · Physical Review A

Tunneling in attosecond optical ionization and a dynamical time operator

M. Bauer

Published Aug 29, 2017• 15 citations
arXivHTMLPDF

Abstract

The conundrum parameter-operator of time in quantum mechanics, as well as the time-energy uncertainty relation and the tunneling delay time, have recently been addressed in attosecond optical ionization experiments. Dirac's formulation of the electron's relativistic quantum mechanics (RQM) allows the introduction of a dynamical self-adjoint time operator, representing an internal time of the system. Its relation to the parametric (laboratory) time is derived. Its relevance to the tunneling measurements in these experiments is exhibited, namely, as the time it takes the wave packet to cross the exit point of the barrier, thus justifying considering this time operator an additional observable in RQM.

DOI
10.1103/physreva.96.022139

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The conundrum parameter-operator of time in quantum mechanics, as well as the time-energy uncertainty relation and the tunneling delay time, have recently been addressed in attosecond optical ionization experiments. Dirac's formulation of the electron's relativistic quantum mechanics (RQM) allows the introduction of a dynamical self-adjoint time operator, representing an internal time of the system. Its relation to the parametric (laboratory) time is derived. Its relevance to the tunneling measurements in these experiments is exhibited, namely, as the time it takes the wave packet to cross the exit point of the barrier, thus justifying considering this time operator an additional observable in RQM.
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.
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 τBLτ_{BL}τBL​ in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein τBLτ_{BL}τBL​ 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.
Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.
Attosecond science is of a fundamental interest in physics. The measurement of the tunneling time in attosecond experiments, offers a fruitful opportunity to understand the role of time in quantum mechanics (QM). We discuss in this paper our tunneling time model in relation to two time operator definitions introduced by Bauer and Aharonov–Bohm. We found that both definitions can be generalized to the same type of time operator. Moreover, we found that the introduction of a phenomenological parameter by Bauer to fit the experimental data is unnecessary. The issue is resolved with our tunneling model by considering the correct barrier width, which avoids a misleading interpretation of the experimental data. Our analysis shows that the use of the so-called classical barrier width, to be precise, is incorrect.
Abstract Tunneling time in attosecond and strong-field experiments is one of the most controversial issues in current research, because of its importance to the theory of time, the time operator and the time–energy uncertainty relation in quantum mechanics. In Kullie (2015 Phys. Rev. A 92 052118 ) we derived an estimation of the (real) tunneling time, which shows an excellent agreement with the time measured in attosecond experiments, our derivation is found by utilizing the time–energy uncertainty relation, and it represents a quantum clock. In this work, we show different aspects of the tunneling time in attosecond experiments, we discuss and compare the different views and approaches, which are used to calculate the tunneling time, i.e. Keldysh time (as a real or imaginary quantity), Mandelstam–Tamm time, the classical view of the time measurement and our tunneling time relation(s). We draw some conclusions concerning the validity and the relation between the different types of the tunneling time with the hope that they will help to answer the question put forward by Orlando et al (2014 J. Phys. B 47 204002 , 2014 Phys. Rev. A 89 014102 ): tunneling time, what does it mean? However, as we will see, the important question is a more general one: how to understand the time and the measurement of the time of a quantum system? In respect to our result, the time in quantum mechanics can be, in more general fashion, classified in two types, intrinsic dynamically connected, and external dynamically not connected to the system, and consequently (perhaps only) classical Newtonian time remains as a parametric type of time.
A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.