Decoherence produces coherent states: an explicit proof for harmonic chains

TL;DR

This paper proves that infinite harmonic oscillator systems naturally evolve into Gaussian states, called coherent states, under broad conditions, suggesting that such states are prevalent in physical systems due to decoherence.

Contribution

The paper provides an explicit proof that decoherence leads to the formation of coherent states in harmonic chains, extending the CLT to Wigner distributions and identifying conditions for this to occur.

Findings

Wigner distributions become Gaussian over time under general conditions.

Dispersive waves are essential for the formation of coherent states.

A special class of functions allows CLT to hold even with negative probability densities.

Abstract

We study the behavior of infinite systems of coupled harmonic oscillators as t->infinity, and generalize the Central Limit Theorem (CLT) to show that their reduced Wigner distributions become Gaussian under quite general conditions. This shows that generalized coherent states tend to be produced naturally. A sufficient condition for this to happen is shown to be that the spectral function is analytic and nonlinear. For a rectangular lattice of coupled oscillators, the nonlinearity requirement means that waves must be dispersive, so that localized wave-packets become suppressed. Virtually all harmonic heat-bath models in the literature satisfy this constraint, and we have good reason to believe that coherent states and their generalizations are not merely a useful analytical tool, but that nature is indeed full of them. Standard proofs of the CLT rely heavily on the fact that probability densities are non-negative. Although the CLT generally fails if the probability densities are allowed to take negative values, we show that a CLT does indeed hold for a special class of such functions. We find that, intriguingly, nature has arranged things so that all Wigner functions belong to this class.