Operator scrambling, hypersensitivity, and quantum Lyapunov spectrum*

This work presents a model study of the quantum inverted harmonic oscillator (IHO), an archetype example for quantum chaotic systems. It is shown that the IHO exhibits hypersensitivity to perturbations, i.e., the Loschmidt echo decays as a double exponential. A quantum Lyapunov spectrum has been introduced to detect the exponential operator growth in chaotic systems. The IHO is demonstrated to possess a quantum Lyapunov spectrum that is identical to the classical one.