Second-order Perturbative OTOC of Anharmonic Oscillators

The out-of-time-order correlator (OTOC) of simple harmonic oscillator with extra anharmonic (quartic) interaction are calculated by the second quantization method. We obtain the analytic formulas of spectrum, Fock space states and matrix elements of coordinate to the second order of anharmonic interaction. These relations clearly reveal the property that the correction of the interaction is proportional to the quantum number of the energy level, and shows the enhancement for some physical quantities. The analytic results explicitly show that the OTOC is raising in the quadratic power law at early times. Then, we use the formulas to do numerical summation to calculate the OTOC, which shows that, at late times, while the first-order perturbative OTOC is oscillating as that in a simple harmonic oscillator, the second-order perturbative OTOC saturates to a constant value. We compare it with $2\langle x^2\rangle_T\langle p^2\rangle_T$, which is associated with quantum chaotic behavior in systems that exhibit chaos, and discuss the validity of the second-order perturbation.