Time Operator, Real Tunneling Time in Strong Field Interaction and the Attoclock

Abstract

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.

Timelessness Strictly inside the Quantum Realm

“…Ossama Kullie offers an alternative view but it might be somewhat doubted as it is based on a specific time—an energy uncertainty relation. In fact, his account might still be consistent with the suggestions made here by considering properly the three separate phases of preparation, tunneling, and measurement [ 92 , 93 , 138 , 139 ]. The collapse of the wavefunction effected by the measurement, and, somewhat symmetrically, preparation, certainly are not instantaneous but are associated with some finite duration.…”

“…Long known general findings fit nicely: the entropy in an isolated quantum system is constant, and that time is not a simple quantum observable was pointed out already by Wolfgang Pauli [79]. Time cannot be described by a Hermitian operator; but see, e.g., [80,81]. A recent proposal of an evolving block universe acknowledges the omnipresent daily use of classical boundary conditions; the past is fixed and the future is described as open; the future simply does not exist at any experienced point in time yet (and thus, photons cannot emanate from there) [63].…”

“…Long known general findings fit nicely: the entropy in an isolated quantum system is constant, and that time is not a simple quantum observable has been pointed out already by Wolfgang Pauli [ 91 ]. Time cannot be described by a Hermitian operator; for an alternative view under specific conditions, see, e.g., [ 92 , 93 ].…”

“…Inherently in QM, when there is a finite probability density attached to one side of the barrier, there is some at the other side, too. Since the discovery of QM, the question has been raised as to what time the tunneling process would take, or whether it would take any time at all, and whether any delay might be understood as a transit time [ 92 , 93 , 133 ]. Introducing dissipation, the predicted as well as the observed group delay increases linearly with the barrier length as expected for classical propagation [ 134 , 135 ].…”

“…Time–energy uncertainty relations have been discussed already by the founding fathers of quantum physics and ever since. The only thing clear so far is that time–energy uncertainty is not at the same level as others, e.g., position–momentum, and there are many facets to the topic [ 92 , 93 , 122 , 138 , 139 , 142 , 144 ]. The very basis of the difficulties with any time–energy uncertainty relation might actually consist of real time being only defined at/by durable CM recording points.…”