Witnesses of Quantum Chaos and Nonlinear Kerr-Like Oscillator Model

Here, we present a brief insight into some current methods allowing for the detection of quantum chaos phenomena. In particular, we show examples of proposals of the parameters which could be applied as indicators of quantum-chaotic behavior and already were presented in the literature. We concentrate here on the quantum fidelity and the fidelity-like functions, defined for the wave functions describing system's evolution. The definition of the fidelity-like parameter also involves the operator of the mean number of photons/phonons. Discussing such parameter, we show here how it is possible to take into account in the discussion of quantum-chaotic systems simultaneously the behavior of the divergence of wave functions and the energy of the system represented by the mean number of photons/phonons. Next, we discuss entropy-type parameter which can also be a good candidate for the indicators of quantum chaos' phenomena. We show the ability of all considered here parameters to be witnesses of quantum-chaotic behavior for the systems of the quantum nonlinear Kerr-like oscillator--the classical counterpart of such system can exhibit chaotic evolution in its canonical form.