Quantum temporal probabilities in tunneling systems

Abstract

We study the temporal aspects of quantum tunneling as manifested in time-of-arrival experiments in which the detected particle tunnels through a potential barrier. In particular, we present a general method for constructing temporal probabilities in tunneling systems that (i) defines 'classical' time observables for quantum systems and (ii) applies to relativistic particles interacting through quantum fields. We show that the relevant probabilities are defined in terms of specific correlations functions of the quantum field associated with tunneling particles. We construct a probability distribution with respect to the time of particle detection that contains all information about the temporal aspects of the tunneling process. In specific cases, this probability distribution leads to the definition of a delay time that, for parity-symmetric potentials, reduces to the phase time of Bohm and Wigner. We apply our results to piecewise constant potentials, by deriving the appropriate junction conditions on the points of discontinuity. For the double square potential, in particular, we demonstrate the existence of (at least) two physically relevant time parameters, the delay time and a decay rate that describes the escape of particles trapped in the inter-barrier region. Finally, we propose a resolution to the paradox of apparent superluminal velocities for tunneling particles. We demonstrate that the idea of faster-than-light speeds in tunneling follows from an inadmissible use of classical reasoning in the description of quantum systems.

Path of a tunneling particle

“…where γ = λa quantifies the opacity of the barrier and = (i) Since β k (a) = 0, a detector placed at the barrier's exit, records τ 1 = t ph (k 0 ) for all values of γ and . This is compatible with the idea that the phase time is the time it takes for a particle to cross the barrier, at least as far as time-of-arrival measurements are concerned [1].…”

“…This apparent superluminality is an artifact of the chain of approximations involved in deriving (45), and, in particular, the consideration of non-relativistic particles. The probability densities in the QTP method are obtained from the correlation functions of a local relativistic quantum field theory that fully respect causal propagation of signals [1].…”

“…2mV 0 < 1. Interestingly, dS/dD ≥ 0 for all D ∈ [−1,1] and for all values of γ and . There is an one-to-one correspondence between S and D, hence, Eq.…”

Time of arrival and localization of relativistic particles

“…For a derivation, see Refs. [14,16,21], and also the appendix. This construction originates from the treatment of spacetime coarse-grainings in the decoherent histories approach to quantum mechanics [28,33].…”

Multi-time measurements in Hawking radiation: information at higher-order correlations

“…In recent years, we developed a new method [6][7][8][9] for describing QFT measurements that shares the above property with Glauber's theory. We call this method the Quantum Temporal Probabilities (QTP) method, as its original motivation was to provide a general framework for temporally extended quantum observables [26][27][28].…”