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

Understand

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.