Applications of fidelity measures to complex quantum systems

TL;DR

This paper explores how fidelity measures can assess the stability and complexity of quantum systems, especially cold atoms and the quantum kicked rotor, by analyzing static and dynamical fidelities and their applications.

Contribution

It introduces and compares static and dynamical fidelity measures, demonstrating their effectiveness in detecting avoided crossings and understanding dynamical regimes in quantum systems.

Findings

Static fidelity detects avoided crossings effectively.

Dynamical fidelity characterizes different dynamical regimes.

Fidelity measures provide insights into quantum system complexity.

Abstract

We revisit the fidelity as a measure for the stability and the complexity of the quantum motion of single and many-body systems. Within the context of cold atoms, we present on overview of applications of two fidelities which we call static and dynamical fidelity, respectively. The static fidelity applies to quantum problems which can be diagonalized since it is defined via the eigenfunctions. In particular, we show that the static fidelity is a highly effective practical detector of avoided crossings characterizing the complexity of the systems and their evolutions. The dynamical fidelity is defined via the time-dependent wave functions. Focussing on the quantum kicked rotor system, we highlight a few practical applications of fidelity measurements in order to better understand the large variety of dynamical regimes of this paradigm of a low-dimensional system with mixed regular-chaotic phase space.