Regular and Chaotic Collective Modes in Nuclei

Abstract

IntroductionAtomic nuclei constitute an exemplary realization of chaotic dynamics in the quantum domain. These dense clouds of strongly interacting particles were at the dawn of the field of physics called quantum chaos [1,2]. In the 1980s, when its fundamentals were formulated, the field might be seen just as an exotic branch of quantum mechanics, but the present rapid growth of "quantum technologies" gives it a more practical potential.Paradoxically, the most difficult task of "quantum chaos" is to find what this term actually means. It is clear that the classical definition of chaos based on an extreme sensitivity of the motion to initial conditions (exponential divergence of trajectories, or the "butterflywing effect") cannot be applied in quantum mechanics due to linearity of the Schrödinger equation. Instead, distinct signatures of chaos in bound quantum systems lie in statistical properties of discrete energy spectra. More specifically, the systems having chaotic classical counterparts were found [3] to exhibit strong correlations between energy levels across the whole spectrum-correlations that are consistent with the theory of random matrices, originally proposed in the 1950s for the description of slow neutron resonances in nuclei [4]. This connection, which also leads to important dynamical consequences, has been tested in several schematic models, particularly in two-dimensional cavities, so-called "billiards," with shapes generating different proportions of regular and chaotic classical motions [1,2].Nuclei also exhibit rather well-pronounced collective motions-coherent rotations or vibrations of a large number of nucleons. These motions are well described by models far simpler than those considering all the microscopic degrees of freedom. Both generally accepted representatives of such

Understanding chaos via nuclei

“…In particular, in the subsequent region up to B ≈ 0.7 we observe a series of fine-structured peaks in the f reg dependence. 6 The increased regularity in this parameter domain is connected with rather complex Poincaré maps, such as those shown in the middle column of Fig. 6.…”

“…The complete dependence of f reg on the GCM Hamiltonian parameters and energy can be found in Ref. [6].…”

“…Two examples of trajectories (chaotic and regular) represent some high-energy vibrations generated by the Hamiltonian (2) with the same choice of parameters. Adapted from Ref [6]…”

Quantum Chaos in the Extended Dicke Model

“…In practice, we need to cut off the bosonic number states at a larger but finite value . Moreover, the dependence of the chaoticity in the Dicke model on the energy [ 63 , 72 , 73 ] further implies that it is also necessary to cut off the energy in order to obtain the finite number of considered states. In our numerical simulations, we set and restrict our analysis to the eigenstates with energies ; the convergence of our results is carefully examined.…”

Quantum Chaos in the Extended Dicke Model

“…In practice, we need to cut off the bosonic number states at a larger but finite value N c . Moreover, the dependence of the chaoticity in the Dicke model on the energy [63,72,73] further implies that it is also necessary to cut off the energy in order to get the finite number of considered states. In our numerical simulations, we set N c = 320 and restrict our analysis on the eigenstates with energies E/N ∈ [0.4, 4], the convergence of our results has been carefully examined.…”