Do Bloch waves interfere with one another ?

TL;DR

This paper demonstrates that Bloch states with different wavevectors cannot interfere due to a superselection rule stemming from discrete translation symmetry, with implications for quantum interference and Floquet systems.

Contribution

It reveals a superselection rule preventing superposition of Bloch states with different wavevectors and extends this concept to periodically driven quantum systems.

Findings

Bloch states with different wavevectors cannot be linearly superposed.

A topological origin of the superselection rule is identified.

Superselection forbids coherent superposition of Floquet states with different quasienergies.

Abstract

Here we show that two Bloch states, which are energy eigenstates of a quantum periodic potential problem, with different wavevectors can not be linearly superposed to display quantum interference of any kind that captures the relative phase between them. This is due to the existence of a superselection rule in these systems, whose origin lies in the discrete translation symmetry. A topological reason leading to such a superselection is found. A temporal analogue of this superselection rule in periodically driven quantum systems is also uncovered, which forbids the coherent superposition of any two quasi-periodic Floquet states with different quasienergies.