On the Time Dependence of Adiabatic Particle Number

TL;DR

This paper investigates how to define and compute a meaningful, time-dependent particle number in quantum field systems under time-dependent perturbations, using optimal truncation of the divergent adiabatic expansion to reveal universal behavior.

Contribution

It introduces a method to define a consistent time-dependent particle number during non-equilibrium processes by optimal truncation of the adiabatic expansion, unifying multiple approaches.

Findings

Optimal truncation yields a universal time dependence of particle number.

The approach clarifies quantum interference effects during perturbations.

Multiple definitions of adiabatic particle number are shown to be equivalent.

Abstract

We consider quantum field theoretic systems subject to a time-dependent perturbation, and discuss the question of defining a time dependent particle number not just at asymptotic early and late times, but also during the perturbation. Naively, this is not a well-defined notion for such a non-equilibrium process, as the particle number at intermediate times depends on a basis choice of reference states with respect to which particles and anti-particles are defined, even though the final late-time particle number is independent of this basis choice. The basis choice is associated with a particular truncation of the adiabatic expansion. The adiabatic expansion is divergent, and we show that if this divergent expansion is truncated at its optimal order, a universal time dependence is obtained, confirming a general result of Dingle and Berry. This optimally truncated particle number provides a clear picture of quantum interference effects for perturbations with non-trivial temporal sub-structure. We illustrate these results using several equivalent definitions of adiabatic particle number: the Bogoliubov, Riccati, Spectral Function and Schrodinger picture approaches. In each approach, the particle number may be expressed in terms of the tiny deviations between the exact and adiabatic solutions of the Ermakov-Milne equation for the associated time-dependent oscillators.