The quantum Bernoulli map

TL;DR

This paper introduces a quantum analogue of the classical Bernoulli map, demonstrating how quantum decaying states evolve with quasi-fractal shapes influenced by quantum uncertainty, extending classical chaos models into quantum regimes.

Contribution

It defines a quantum Bernoulli map as a projection of the quantum baker map and constructs a spectral representation with quantum decaying states.

Findings

Quantum decaying states exhibit quasi-fractal shapes.

Quantum uncertainty limits the fractal development.

The spectral representation extends classical models into quantum domain.

Abstract

The classical Bernoulli and baker maps are two simple models of deterministic chaos. On the level of ensembles, it has been shown that the time evolution operator for these maps admits generalized spectral representations in terms of decaying eigenfunctions. We introduce the quantum version of the Bernoulli map. We define it as a projection of the quantum baker map. We construct a quantum analogue of the generalized spectral representation, yielding quantum decaying states represented by density matrices. The quantum decaying states develop a quasi-fractal shape limited by the quantum uncertainty.