This repository contains several codes for analysing the chaotic dynamics of a non-linear system inspired from nuclear physics. More precisely, the model describes the quadrupole vibrations of the surface of a heavy nuclei.

The code is organized in two parts, corresponding to the simulations for classical and quantum chaos.

• The classical mechanics code simulates the dynamics of the nuclear surface modeled by the following Hamiltonian:

$$ H_{cl} = \frac{A}{2}(p_0^2+p_2^2)+\frac{A}{2}(q_0^2+q_2^2) +\frac{B}{\sqrt{2}}q_0(3q_2^2-q_0^2) +\frac{D}{4}(q_0^2+q_2^2)^2 $$

The equations of motion are numerically integrated in order to build Poincare maps. In order to measure the degree of chaos for a given set of parameters at a given energy, the maximal Lyapunov exponent is computed for each trajectory.

• The quantum mechanics code simulates the quantized version of the system by diagonalizing the Hamiltonian expressed in the basis of a double harmonic oscillator

$$ \begin{split} H_B &= A \left( a_1^\dagger a_1 + a_2^\dagger a_2 \right) + \frac{B}{4} \bigg[ \left( 3 a_1^\dagger {a_2^\dagger}^2 + 3 a_1 a_2^2 - {a_1^\dagger}^3 - a_1^3 \right) \\\ &\quad + 3 \left( a_1 {a_2^\dagger}^2 + a_1^\dagger a_2^2 - a_1^\dagger a_1^2 - {a_1^\dagger}^2 a_1 + 2 a_1 a_2^\dagger a_2 + 2 a_1^\dagger a_2^\dagger a_2 \right) \bigg] \\\ &\quad + \frac{D}{16} \bigg[ 6 \left( {a_1^\dagger}^2 a_1^2 + {a_2^\dagger}^2 a_2^2 \right) + 2 \left( a_1^2 {a_2^\dagger}^2 + {a_1^\dagger}^2 a_2^2 \right) + 8 a_1^\dagger a_1 a_2^\dagger a_2 \\\ &\quad + 4 \left(a_1^\dagger a_1^3 + {a_1^\dagger}^3 a_1 + a_2^\dagger a_2^3 + {a_2^\dagger}^3 a_2 + a_1^2 a_2^\dagger a_2 + {a_1^\dagger}^2 a_2^\dagger a_2 + a_1^\dagger a_1 a_2^2 + a_1^\dagger a_1 {a_2^\dagger}^2 \right) \\\ &\quad + \left( {a_1^\dagger}^4 + a_1^4 + {a_2^\dagger}^4 + a_2^4 + 2 {a_1^\dagger}^2 {a_2^\dagger}^2 + 2 a_1^2 a_2^2 \right) \bigg]. \end{split} $$

In order to verify the accuracy of the computed eigenvalues, two different dimensions of the basis are taken and only the eigenvalues that are within $\delta$ of each other are considered. The selected eigenvalues are then separated in the 3 irreducible representations of the $C_{3v}$ symmetry group, corresponding to the symmetry of the system. The parameter $\epsilon$ is used as the maximum numerical error for the degenerated energy levels. The next step is to create the nearest neighbor spacing distributions. The contributions of the 3 irreducible representations are summed. The resulting histograms are fitted with a probability distribution corresponding to a superposition of a Poisson distribution and a Wigner one

$$ P(s) = \alpha P_P(s) + (1-\alpha) P_W(s) $$

and the $\alpha$ coefficient is obtained.

If you found the code here useful, please cite: Fingerprints of global classical phase-space structure in quantum spectra, S. Micluta-Campeanu, M.C. Raportaru, A.I. Nicolin, V. Baran, Rom. Rep. Phys. 70, 105 (2018)