Sub-Planck Structure, Decoherence, and Many-Body Environments

TL;DR

This paper examines how sub-Planck structures influence decoherence, showing that many-body environments are essential for quantum decoherence, especially when the number of degrees of freedom is large.

Contribution

It challenges Zurek's claim about state orthogonality in systems with few degrees of freedom and demonstrates the importance of many-body environments for decoherence.

Findings

Orthogonality result holds only for large degrees of freedom.

Many-body environments are crucial for effective quantum decoherence.

Sub-Planck structures influence the sensitivity of quantum states to perturbations.

Abstract

In a recent paper [Nature 412, 712 (2001)], Zurek has argued that (1) time evolution typically causes chaotic quantum systems to generate structure that varies on the scale of phase-space volume elements of size $(\hbar^2/A)^d$, where A is a classical action characteristic of the state and d is the number of degrees of freedom, and that (2) this structure implies that a small change in a phase-space coordinate X by an amount $\delta X \sim \hbar X/A$ generically results in an orthogonal state. While we agree with (1), we argue that (2) is not correct if the number of degrees of freedom is small. Our arguments are based on the Berry-Voros ansatz for the structure of energy eigenstates in chaotic systems. We find, however, that (2) becomes valid if the number of degrees of freedom is large. This implies that many-body environments may be crucial for the phenomenon of quantum decoherence.