Randomness is fundamental in quantum theory, with many philosophical and practical implications. In this paper we discuss the concept of algorithmic randomness, which provides a quantitative method to assess the Borel normality of a given sequence of
numbers, a necessary condition for it to be considered random. We use Borel normality as a tool to investigate the randomness of ten sequences of bits generated from the differences between detection times of photon pairs generated by spontaneous parametric downconversion. These sequences are shown to fulfil the randomness criteria without difficulties. As deviations from Borel normality for photon-generated random number sequences have been reported in previous work, a strategy to understand these diverging findings is outlined.
A Hamiltonian describing the collective behaviour of N interacting spins can be mapped to a bosonic one employing the Holstein-Primakoff realisation, at the expense of having an infinite series in powers of the boson creation and annihilation operato
rs. Truncating this series up to quadratic terms allows for the obtention of analytic solutions through a Bogoliubov transformation, which becomes exact in the limit N -> infinit. The Hamiltonian exhibits a phase transition from single spin excitations to a collective mode. In a vicinity of this phase transition the truncated solutions predict the existence of singularities for finite number of spins, which have no counterpart in the exact diagonalization. Renormalisation allows to extract from these divergences the exact behaviour of relevant observables with the number of spins around the phase transition, and relate it with the class of universality to which the model belongs. In the present work a detailed analysis of these aspects is presented for the Lipkin model.
Two-level atoms interacting with a one mode cavity field at zero temperature have order parameters which reflect the presence of a quantum phase transition at a critical value of the atom-cavity coupling strength. Two popular examples are the number
of photons inside the cavity and the number of excited atoms. Coherent states provide a mean field description, which becomes exact in the thermodynamic limit. Employing symmetry adapted (SA) SU(2) coherent states (SACS) the critical behavior can be described for a finite number of atoms. A variation after projection treatment, involving a numerical minimization of the SA energy surface, associates the finite number phase transition with a discontinuity in the order parameters, which originates from a competition between two local minima in the SA energy surface.
We analyze the ability of the three different Liquid Drop Mass (LDM) formulas to describe nuclear masses for nuclei in various deformation regions. Separating the 2149 measured nuclear species in eight sets with similar quadrupole deformations, we sh
ow that the masses of prolate deformed nuclei are better described than those of spherical ones. In fact, the prolate deformed nuclei are fitted with an RMS smaller than 750 keV, while for spherical and semi-magic species the RMS is always larger than 2000 keV. These results are found to be independent of pairing. The macroscopic sector of the Duflo-Zuker (DZ) mass model reproduces shell effects, while most of the deformation dependence is lost and the RMS is larger than in any LDM. Adding to the LDM the microscopically motivated DZ master terms introduces the shell effects, allowing for a significant reduction in the RMS of the fit but still exhibiting a better description of prolate deformed nuclei. The inclusion of shell effects following the Interacting Boson Models ideas produces similar results.
We show that the Liquid Drop Model is best suited to describe the masses of prolate deformed nuclei than of spherical nuclei. To this end three Liquid Drop Mass formulas are employed to describe nuclear masses of eight sets of nuclei with similar qua
drupole deformations. It is shown that they are able to fit the measured masses of prolate deformed nuclei with an RMS smaller than 750 keV, while for the spherical nuclei the RMS is, in the three cases, larger than 2000 keV. The RMS of the best fit of the masses of semi-magic nuclei is also larger than 2000 keV. The parameters of the three models are studied, showing that the surface symmetry term is the one which varies the most from one group of nuclei to another. In one model, isospin dependent terms are also found to exhibit strong changes. The inclusion of shell effects allows for better fits, which continue to be better in the prolate deformed nuclei region
