How fast is a quantum jump?

Highlights

• We propose a way to estimate the duration of a single atomic transition.

• The action of the zero-point field of Compton's frequency on the electron is central.

• Our theoretical result is expressed in terms of universal constants.

• The estimated time is on the order of attoseconds, in line with recent experiments.

Abstract

A proposal is put forward for an estimate of the duration of a transition between atomic states. The proposal rests on the consideration that a resonance of the atomic electron with modes of the zero-point radiation field of Compton's frequency is at the core of the phenomenon. The theoretical result, given essentially by the expression ${(\alpha {\omega }_C)}^{-1}$, where α is the fine structure constant and ${\omega }_C$ the Compton angular frequency for the electron, lies well within the range of the recently experimentally estimated values of the order of attoseconds (${10}^{-18}$ s).

Introduction

It is in a way puzzling that the physics involved in atomic quantum jumps (or single atomic transitions) has remained in almost complete darkness, the more so considering the crucial role spectroscopy has played for more than a century, and the impressive advances in both theoretical and experimental quantum physics. Attention to this intriguing subject has presumably been hindered for a long time by the masterful dogma of the instantaneous character of atomic transitions postulated by Bohr [1]—and bitterly opposed by Schrödinger [2], [3]. One can still come across articles negating quantum jumps—and any other kind of discontinuities, for that matter (e.g. [4])—or taking them as a sudden increase of our knowledge of the system (e.g., [5], [6]) rather than a physical phenomenon.

For atomic and molecular spectroscopists it is clear that quantum jumps exist; this is part of their daily bread. Most spectroscopists are also aware that the time involved in a transition is finite but very short; so short indeed that the Franck-Condon principle applies, which sets an upper limit to their duration, on the order of femtoseconds (${10}^{-15}$ s). Because atomic transitions are so fast, up to recently they were considered “instantaneous”, this term being taken by some in the rigorous sense, and by others as meaning “in an unmeasurably (or unobservably) short time”. This picture, however, is changing thanks to recent calculational and experimental work, notably using attosecond spectroscopy applied to photoionization [7], [8]. Photoionization experiments in bulk materials are known to involve electronic correlations, which makes it difficult to ascertain the time it takes for one atom to lose one electron. With this caveat, chronoscope measurement of the times involved in the photoelectric effect assigns to the primary photoexcitation process a duration on the order of ${10}^{-17}$ s [9]. Further, although not directly comparable to (natural) atomic transitions, recent experimental work with an artificial atom (a superconducting circuit consisting of two hybridized qubits on a chip) in which a quantum jump is intercepted and reverted by means of an electric pulse, seems to confirm Schrödinger's intuition that the evolution of the jump itself is continuous and needs a finite time to take place [10]. This is in line with Schulman's definition of “jump time” as the time scale such that perturbations occurring at intervals of this duration affect the transition [11]. Based on his definition, Schulman's own estimate made in terms of the “Zeno time” (related to the second moment of the Hamiltonian) and the natural lifetime, results however in a much shorter time than the experimental estimates, as short as ${10}^{-20}$ s for atomic transitions.

The various computational and experimental estimates have contributed to establish the existence of (finite-time) quantum jumps, and have apparently set tighter bounds on their duration. The basic physics behind the process, however, has not been clarified, so the question remains: what is it that determines the duration of a transition?

In the present work we attempt to throw light on this question via a theoretical analysis that does not rely on specific experimental settings. We do so by invoking the existence of the electromagnetic zero-point radiation field (zpf) and applying the conventional approach followed in stochastic electrodynamics (sed) to the specific problem of the dynamics of the electron during a transition. We start by recalling Schrödinger's work on the zitterbewegung as a rapid oscillation of the Dirac electron, and appeal to sed to identify it as a result of its resonance with the Compton-frequency components of the zpf. The stationary solution of the equation of motion for the rapidly oscillating electron corresponds to the zitterbewegung; the transient solution, in its turn, describes the dynamics of the transition between states. The decay time associated with the transient solution, which we propose to take as an approximate measure of the transition time, is expressed in terms of universal constants and its value is of the order of 10⁻¹⁸ s.