# Time-dependent relaxation of observables in complex quantum systems

## TL;DR

This paper investigates how observables relax over time in complex quantum systems, linking wave function spread to survival probability and demonstrating differences between chaotic and regular systems.

## Contribution

It provides a unified analysis of relaxation dynamics in both chaotic and regular quantum systems, highlighting the role of wave function complexity and survival probability.

## Key findings

- Survival probability determines wave function spread in chaotic systems.
- Relaxation timescales can differ significantly from survival probability predictions.
- Non-chaotic states exhibit longer relaxation times than chaotic models suggest.

## Abstract

We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. We show that the spread of the wave function in the Hilbert space is determined by the survival probability which is known to have pre-exponential, exponential, and long-term power-law limiting behaviors. This result relies on complexity of the wave functions and thus is generic to many systems. In the chaotic limit modeled by the Gaussian Orthogonal Ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general, using realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. An example of the non-chaotic limit of coherent and squeezed states provides an additional illustration.