Chaos and order in librating quantum planar elastic pendulum

We investigate, numerically and analytically, the quantum mechanics of planar elastic pendulum in libration. The classical phase space of the system consists of both regular Kolmogorov-Arnold-Moser trajectories and chaotic trajectories. We systematically study the former by using the canonical perturbation theory along with the Einstein-Brillouin-Keller quantization rule and the latter using out-of-time-ordered-correlator among other tools and techniques, like the Husimi distribution and energy level spacing distribution. We find that the deviations in the energy level statistics, due to the presence of nearly degenerate levels, disappear by considering the states with same symmetry. We also propose an eight-point-out-of-time-ordered-correlator whose growth rate between the exponential decay time (around which nonlinearity sets in) and the Ehrenfest time (after which quantum effect washes out the classical growth) can be used to relate the quantum correlator to the second largest Lyapunov exponent of the corresponding classical system.