Although time is not an quantum-mechanical operator, novel attosecond techniques can measure very fundamental time delays. Using the tunneling time as an example, this tutorial should give a better insight on current issues and hot topics.
This article addresses topics regarding time measurements performed on quantum systems. The motivation is linked to the advent of 'attophysics' which makes feasible to follow the motion of electrons in atoms and molecules, with time resolution at the attosecond (1 as = 10 −18 s) level, i.e. at the natural scale for electronic processes in these systems. In this context, attosecond 'time-delays' have been recently measured in experiments on photoionization and the question arises if such advances could cast a new light on the still active discussion on the status of the time variable in quantum mechanics. One issue still debatable is how to decide whether one can define a quantum time operator with eigenvalues associated to measurable 'time-delays', or time is a parameter, as it is implicit in the Newtonian classical mechanics. One objective of this paper is to investigate if the recent attophysics-based measurements could shed light on this parameter-operator conundrum. To this end, we present here the main features of the theory background, followed by an analysis of the experimental schemes that have been used to evidence attosecond 'time-delays' in photoionization. Our conclusion is that these results reinforce the view that time is a parameter which cannot be defined without reference to classical mechanics.
Tunneling and tunneling time are hot debated and very interesting issues because of their fundamental role in the quantum mechanics. The measurement of the tunneling time in today's attosecond and strong field (low-frequency) experiments, despite its controversial discussion offers a fruitful opportunity to understand the time measurement and the role of time in quantum mechanics. In previous work Kullie (2015 Phys. Rev. A 92 , 052118), we suggested a model and derived a simple relation to calculate the tunneling time, which showed a good agreement with the experimental result for He-atom. In the present work we analyze and discuss our model considering the experimental result for H-atom, which is obtained recently by Satya Sainadh et al (2017 arXiv: 1707.05445 ). In the tunneling region, we find that our model shows a good agreement with their experimental result (similar to the He-atom in our previous work), and with the accompanied time-dependent Schrödiger equation simulations. However, Sainadh et al use a different picture of the tunneling, in which tunneling time becomes an imaginary quantity. Whereas our model represents a real tunneling time picture, precisely a delay time with respect to the ionization time at the atomic field strength. Moreover, even though the above-threshold-ionization is beyond the tunneling regime, we still see that the actual ionization time is related to our model. However, crucial points arise and keep some questions open especially on the experimental side.
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.
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.
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.
Theoretical models often fail to describe the dynamics due to numerical limitations, and we need to develop better approximate models for the ultrafast dynamics on an atomic scale, where quantum mechanics becomes essential. Novel time-resolved attosecond streaking techniques such as energy streaking and the attoclock (i.e. angular streaking) are currently being applied in an attempt to answer a very fundamental question in quantum mechanics: how fast can light remove a bound electron from an atom or a solid?