Time Dependent Floquet Theory and Absence of an Adiabatic Limit

TL;DR

This paper investigates the behavior of quantum systems under time-periodic fields, revealing that the adiabatic limit does not exist for wave functions but quasienergies become time-independent, challenging conventional assumptions.

Contribution

It demonstrates the non-existence of an adiabatic limit for wave functions in large systems and introduces a modified adiabatic theorem for quasienergies.

Findings

Wave functions become highly irregular as system size increases.

Perturbation theory's convergence radius approaches zero.

Quasienergies become independent of time in the adiabatic limit.

Abstract

Quantum systems subject to time periodic fields of finite amplitude, lambda, have conventionally been handled either by low order perturbation theory, for lambda not too large, or by exact diagonalization within a finite basis of N states. An adiabatic limit, as lambda is switched on arbitrarily slowly, has been assumed. But the validity of these procedures seems questionable in view of the fact that, as N goes to infinity, the quasienergy spectrum becomes dense, and numerical calculations show an increasing number of weakly avoided crossings (related in perturbation theory to high order resonances). This paper deals with the highly non-trivial behavior of the solutions in this limit. The Floquet states, and the associated quasienergies, become highly irregular functions of the amplitude, lambda. The mathematical radii of convergence of perturbation theory in lambda approach zero. There is no adiabatic limit of the wave functions when lambda is turned on arbitrarily slowly. However, the quasienergy becomes independent of time in this limit. We introduce a modification of the adiabatic theorem. We explain why, in spite of the pervasive pathologies of the Floquet states in the limit N goes to infinity, the conventional approaches are appropriate in almost all physically interesting situations.