Nuclear energy level complexity: Fano factor signature of chaotic behavior of nearest-neighbor time-series analysis

Background: The Fano factor is used to characterize statistical noises, in quantum optics to determine correlations and anticorrelations, in transport theory to characterize the limit of quantum chaos experienced by electrons in a mesoscopic cavity. However, to our knowledge, it has not been used in nuclear physics to study the spectral fluctuations.Purpose: In the present contribution, we applied the square root of the Fano factor ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{f}$) to random matrix theory and to the energy level statistics of the nuclear excitation spectrum to determine if the Fano factor can be used to study nuclear energy spectra distributions.Methods: We studied the fluctuation of the excited states of $^{48}\mathrm{Ca}, ^{48}\mathrm{Ti}$, and $^{46}\mathrm{Ti}$ with symmetry ${J}^{P}={3}^{+}$ as well as the Wigner distribution and the Fourier power spectrum of the energy level spacings for different quadrupole interactions so we can determine when we have a chaotic distribution. Later on we compared with the Fano factor of the same distributions.Results: The Fano factor agrees with that obtained by the other methods. We show that $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{f}=0.5$ corresponds to quantum chaos for the energy levels in the nuclear spectra.Conclusions: The Fano factor can be used in nuclear physics (and other areas of science) to determine quantum chaos.