# Out-of-time-order correlators in finite open systems

## TL;DR

This paper investigates the behavior of out-of-time-order correlators in finite open quantum systems, revealing exponential saturation in discrete systems and analyzing decay times and saturation values, especially in two-level systems.

## Contribution

It provides a microscopic calculation of OTOC decay times and saturation values in open quantum systems, highlighting differences between quantum-chaotic and discrete systems.

## Key findings

- OTOCs saturate exponentially in discrete energy level systems.
- Decay times are linked to inelastic transitions and dephasing.
- Some OTOCs are immune to dephasing, affecting their decay behavior.

## Abstract

We study out-of-time order correlators (OTOCs) of the form $\langle\hat A(t)\hat B(0)\hat C(t)\hat D(0)\rangle$ for a quantum system weakly coupled to a dissipative environment. Such an open system may serve as a model of, e.g., a small region in a disordered interacting medium coupled to the rest of this medium considered as an environment. We demonstrate that for a system with discrete energy levels the OTOC saturates exponentially $\propto \sum a_i e^{-t/\tau_i}+const$ to a constant value at $t\rightarrow\infty$, in contrast with quantum-chaotic systems which exhibit exponential growth of OTOCs. Focussing on the case of a two-level system, we calculate microscopically the decay times $\tau_i$ and the value of the saturation constant. Because some OTOCs are immune to dephasing processes and some are not, such correlators may decay on two sets of parametrically different time scales related to inelastic transitions between the system levels and to pure dephasing processes, respectively. In the case of a classical environment, the evolution of the OTOC can be mapped onto the evolution of the density matrix of two systems coupled to the same dissipative environment.