# Early-Time Exponential Instabilities in Non-Chaotic Quantum Systems

## TL;DR

This paper shows that quantum mechanics can induce early-time exponential instabilities in classically non-chaotic systems, challenging the classical-quantum correspondence in dynamics using the out-of-time-ordered correlator (OTOC).

## Contribution

It demonstrates that quantum systems can exhibit exponential growth in the OTOC even when their classical counterparts are non-chaotic, revealing a new quantum instability phenomenon.

## Key findings

- OTOC shows exponential growth in non-chaotic models at early times.
- Classical Lyapunov exponents remain zero, but quantum rates depend on Planck's constant.
- Quantum instabilities appear before quantum interference effects dominate.

## Abstract

The vast majority of dynamical systems in classical physics are chaotic and exhibit the butterfly effect: a minute change in initial conditions can soon have exponentially large effects elsewhere. But this phenomenon is difficult to reconcile with quantum mechanics. One of the main goals in the field of quantum chaos is to establish a correspondence between the dynamics of classical chaotic systems and their quantum counterparts. In isolated systems in the absence of decoherence, there is such a correspondence in dynamics, but it usually persists only over a short time window, after which quantum interference washes out classical chaos. We demonstrate that quantum mechanics can also play the opposite role and generate exponential instabilities in classically non-chaotic systems within this early-time window. Our calculations employ the out-of-time-ordered correlator (OTOC) -- a diagnostic that reduces to the Lyapunov exponent in the classical limit, but is well defined for general quantum systems. Specifically, we show that a variety of classically non-chaotic models, such as polygonal billiards, whose classical Lyapunov exponents are always zero, demonstrate a Lyapunov-like exponential growth of the OTOC at early times with Planck's-constant-dependent rates. This behavior is sharply contrasted with the slow early-time growth of the analog of the OTOC in the systems' classical counterparts. These results suggest that classical-to-quantum correspondence in dynamics is violated in the OTOC even before quantum interference develops.