The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question has gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach using the truncated Wigner approximation, which accounts for nonvanishing quantum fluctuations that are essential for the dynamics. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows us to identify a Lyapunov exponent ${\lambda }_L$. In particular, we find that ${\lambda }_L$ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator.

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• Nonequilibrium & irreversible thermodynamics
• Quantum aspects of black holes
• Quantum chaos
• Quantum foundations
• Quantum statistical mechanics
• Quantum transport
• Semiclassical physics
• Chaos & nonlinear dynamics
• Semiclassical methods