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A parametrization of density matrices of $d$ dimensions in terms of the raising $J_+$ and lowering $J_-$ angular momentum operators is established together with an implicit connection with the generalized Bloch-GellMann parameters. A general expression for the density matrix of the composite system of angular momenta $j_1$ and $j_2$ is obtained. In this matrix representation violations of the Bell-Clauser-Horne-Shimony-Holt inequalities are established for the $X$-states of a qubit-qubit, pure and mixed, composite system, as well as for a qubit-qutrit density matrix. In both cases maximal violation of the Bell inequalities can be reached, i.e., the Cirel'son limit. A correlation between the entanglement measure and a strong violation of the Bell factor is also given. For the qubit-qutrit composite system a time-dependent convex combination of the density matrix of the eigenstates of a two-particle Hamiltonian system is used to determine periodic maximal violations of the Bell's inequality., Comment: 17 pages, 4 figures, 1 table

We discuss in general how to geometrically visualize a qudit system, with a particular interest in thermal states. The principle of maximum entropy is used to study the geometric properties of an ensemble of finite dimensional Hamiltonian systems with known average energy. These geometric characterizations are given in terms of the generalized diagonal Bloch vectors and the invariants of the special unitary group in $n$ dimensions. As examples, Hamiltonians written in terms of linear and quadratic generators of the angular momentum algebra are considered with $J= 1$ and $J=3/2$. For these cases, paths as functions of the temperature are established in the corresponding simplex representations, which show first- and second-order quantum phase transitions, as well as the adiabatic evolution of the interaction strengths (control parameters) of the Hamiltonian models. For the Lipkin-Meshkov-Glick Hamiltonian the quantum phase diagram is explicitly shown for different temperature values in parameter space., Comment: 32 pages, 12 figures

A method for storing quantum information is presented for $3$-level atomic systems interacting dipolarly with a single radiation field. The method involves performing simple local SU(2) rotations on the Hamiltonian. Under equal detuning, these transformations decouple one of the atomic levels from the electromagnetic field for the $\Lambda$- and $V$-configurations, yielding two effective $2$-level systems (qubits) plus an isolated atomic level; this allows for the exchange of information between the qubits. This rotation preserves the quantum phase diagram of the system. The method could possibly be used as a means to manipulate quantum information, such as storage and retrieval, or communication via a transmission line., Comment: 17 pages, 6 figures

We show that the behaviour in phase space of the Wigner function associated to the electromagnetic modes carries the information of both, the entanglement properties between matter and field, and the regions in parameter space where quantum phase transitions take place. A finer classification for the continuous phase transitions is obtained through the computation of the surface of minimum fidelity., Comment: 12 pages and 6 figures. arXiv admin note: text overlap with arXiv:2009.13663

The quantum phase diagram for a finite $3$-level system in the $\Lambda$ configuration, interacting with a two-mode electromagnetic field in a cavity, is determined by means of information measures such as fidelity, fidelity susceptibility and entanglement, applied to the reduced density matrix of the matter sector of the system. The quantum phases are explained by emphasizing the spontaneous symmetry breaking along the separatrix. Additionally, a description of the reduced density matrix of one atom in terms of a simplex allows a geometric representation of the entanglement and purity properties of the system. These concepts are calculated for both, the symmetry-adapted variational coherent states and the numerical diagonalisation of the Hamiltonian, and compared. The differences in purity and entanglement obtained in both calculations can be explained and visualised by means of this simplex representation., Comment: 20 pages, 7 figures

In periodically-driven quantum systems, resonances can induce exotic nonequilibrium behavior and new phases of matter without static analog. We report on the emergence of fractional and integer resonances in a broad class of many-body Hamiltonians with a modulated hopping with a frequency that is either a fraction or an integer of the on-site interaction. We contend that there is a fundamental difference between these resonances when interactions bring the system to a Floquet prethermal state. Second-order processes dominate the dynamics in the fractional resonance case, leading to less entanglement and more localized quantum states than in the integer resonance case dominated by first-order processes. We demonstrate the dominating emergence of fractional resonances using the Magnus expansion of the effective Hamiltonian and quantify their effects on the many-body dynamics via quantum states' von Neumann entropy and Loschmidt echo. Our findings reveal novel features of the nonequilibrium quantum many-body system, such as the coexistence of Floquet prethermalization and localization, that may allow to development of quantum memories for quantum technologies and quantum information processing., Comment: 17 pages, 16 figures. We have improved the presentation of figure 6. The manuscript was published in Physical Review B

The dynamical behaviour of complex quantum systems can be harnessed for information processing. With this aim, quantum reservoir computing (QRC) with Ising spin networks was recently introduced as a quantum version of classical reservoir computing. In turn, reservoir computing is a neuro-inspired machine learning technique that consists in exploiting dynamical systems to solve nonlinear and temporal tasks. We characterize the performance of the spin-based QRC model with the Information Processing Capacity (IPC), which allows to quantify the computational capabilities of a dynamical system beyond specific tasks. The influence on the IPC of the input injection frequency, time multiplexing, and different measured observables encompassing local spin measurements as well as correlations, is addressed. We find conditions for an optimum input driving and provide different alternatives for the choice of the output variables used for the readout. This work establishes a clear picture of the computational capabilities of a quantum network of spins for reservoir computing. Our results pave the way to future research on QRC both from the theoretical and experimental points of view., Comment: 12 pages, 9 figures. To be published in Cognitive Computation

A methodology is devised for building optimal bases for the generalized Dicke model based on the symmetry adapted variational solution to the problem. At order zero, the matter sector is constructed by distributing $N_a$ particles in all the possible two-level subsystems connected with electromagnetic radiation; the next order is obtained when the states of $N_a-1$ particles are added and distributed again into the two-level subsystems; and so on. In the electromagnetic sector, the order zero for each mode is the direct sum of the Fock spaces, truncated to a value of the corresponding constants of motion of each two-level subsystem; by including contributions of the other modes, the next orders are obtained. As an example of the procedure we consider $4$ atoms in the $\Xi$ configuration interacting dipolarly with two modes of electromagnetic radiation. The results may be applied to situations in quantum optics, quantum information, and quantum computing., Comment: 12 pages and 7 figures

We establish a novel procedure to analyze the entanglement properties of extremal density matrices depending on the parameters of a finite dimensional Hamiltonian. It was applied to a general 2-qubit Hamiltonian which could exhibit Kramers degeneracy. This is done through the extremal density matrix formalism, which allows to extend the conventional variational principle to mixed states. By applying the positive partial transpose criterion in terms of the Correlation and Schlienz-Mahler matrices on the extremal density matrices, we demonstrate that it is possible to reach both pure and mixed entangled states, changing properly the parameters of the Hamiltonian. For time-reversal invariant Hamiltonians, the extremal pure states can be entangled or not and we prove that they are not time-reversal invariants. For extremal mixed states we have in general 5 possible cases: three of them are entangled and the other two separable., Comment: Extremal density matrices; 2-qubit system; Entanglement; Kramers degeneracy

A dynamic procedure is established within the generalised Tavis-Cummings model to generate light states with discrete point symmetries, given by the cyclic group ${\cal C}_n$. We consider arbitrary dipolar coupling strengths of the atoms with a one-mode electromagnetic field in a cavity. The method uses mainly the matter-field entanglement properties of the system, which can be extended to any number of $3$-level atoms. An initial state constituted by the superposition of two states with definite total excitation numbers, $\vert \psi \rangle_{M_1}$, and $\vert \psi \rangle_{M_2}$, is considered. It can be generated by the proper selection of the time-of-flight of an atom passing through the cavity. We demonstrate that the resulting Husimi function of the light is invariant under cyclic point transformations of order $n=\vert M_1-M_2\vert$., Comment: 24 pages, 7 figures

A study of the $\lambda$- and $N$-atomic configurations under dipolar interaction with $2$ modes of electromagnetic radiation is presented. The corresponding quantum phase diagrams are obtained by means of a variational procedure. Both configurations exhibit normal and collective (super-radiant) regimes. While the latter in the $\lambda$-configuration divides itself into $2$ subregions, corresponding to each of the modes, that in the $N$-configuration may be divided into $2$ or $3$ subregions depending on whether the field modes divide the atomic system into $2$ separate subsystems or not. Our variational procedure compares well with the exact quantum solution. The properties of the relevant field and matter observables are obtained., Comment: 8 pages, 7 figures, 4 tables

We introduce a combination of coherent states as variational test functions for the atomic and radiation sectors to describe a system of Na three- level atoms interacting with a one-mode quantised electromagnetic field, with and without the rotating wave approximation, which preserves the symmetry presented by the Hamiltonian. These provide us with the possibility of finding analytical solutions for the ground and first excited states. We study the properties of these solutions for the V-configuration in the double resonance condition, and calculate the expectation values of the number of photons, the atomic populations, the total number of excitations, and their corresponding fluctuations. We also calculate the photon number distribution and the linear entropy of the reduced density matrix to estimate the entanglement between matter and radiation. For the first time, we exhibit analytical expressions for all of these quantities, as well as an analytical description for the phase diagram in parameter space, which distinguishes the normal and collective regions, and which gives us all the quantum phase transitions of the ground state from one region to the other as we vary the interaction parameters (the matter-field coupling constants) of the model, in functional form., Comment: 18 pages, 9 figures

The energy spectrum of a system of $N_a$ atoms of $n$ levels interacting with a one-mode electromagnetic field is studied in the dipole and rotating wave approximations. We find that, under the resonant condition, it exhibits a mirror symmetry with respect to the energy $E=M$ where $M$ the total number of excitations. Thus, for any eigenstate $|\psi_M^{+}\rangle$ with energy $E=M+{\cal E}$ there exists a related eigenstate $|\psi_M^{-}\rangle$ with energy $E=M-{\cal E}$ via the unitary parity operator in the number of photons . This is independent of the dipolar coupling between the levels. We give explicit examples for $3$-level systems., Comment: 5 pages, 4 figures

The energy spectrum of a 3-level atomic system in the $\Xi$-configuration is studied. This configuration presents a triple point independently of the number of atoms, which remains in the thermo- dynamic limit. This means that in a vicinity of this point any quantum fluctuation will drastically change the composition of the ground state of the system. We study the expectation values of the atomic population of each level, the number of photons, and the probability distribution of photons at the triple point., Comment: 5 pages, 8 figures

We calculate numerically the fidelity and its susceptibility for the ground state of the Dicke model. A minimum in the fidelity identifies the critical value of the interaction where a quantum phase crossover, the precursor of a phase transition for finite number of atoms N, takes place. The evolution of these observables is studied as a function of N, and their critical exponents evaluated. Using the critical exponents the universal curve for the specific susceptibility is recovered. An estimate to the precision to which the ground state wave function is numerically calculated is given, and found to have its lowest value, for a fixed truncation, in a vicinity of the critical coupling., Comment: 9 Pages, 9 Figures, Proceedings of Nuclear Physics Symposium (Cocoyoc), IOP Conference Series in press

We consider $N_a$ three-level atoms (or systems) interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations. The order of the quantum phase transitions is determined explicitly for each of the configurations $\Xi$, $\Lambda$ and $V$, with and without detuning. The semi-classical and exact quantum calculations for both the expectation values of the total number of excitations $\cal{M}=\langle \bm{M} \rangle$ and photon number $n=\langle \bm{n} \rangle$ have an excellent correspondence as functions of the control parameters. We prove that the ground state of the collective regime obeys sub-Poissonian statistics for the ${\cal M}$ and $n$ distribution functions. Therefore, their corresponding fluctuations are not well described by the semiclassical approximation. We show that this can be corrected by projecting the variational state to a definite value of ${\cal M}$., Comment: 28 pages

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., Comment: 8 pages, 10 figures, Conference Proceedings of CEWQO-2012, to be published as a Topical Issue of the journal Physica Scripta

We show that semi-classical states adapted to the symmetry of the Hamiltonian are an excellent approximation to the exact quantum solution of the ground and first excited states of the Dicke model. Their overlap to the exact quantum states is very close to 1 except in a close vicinity of the quantum phase transition. Furthermore, they have analytic forms in terms of the model parameters and allow us to calculate analytically the expectation values of field and matter observables. Some of these differ considerably from results obtained via the standard coherent states, and by means of Holstein-Primakoff series expansion of the Dicke Hamiltonian. Comparison with exact solutions obtained numerically support our results. In particular, it is shown that the expectation values of the number of photons and of the number of excited atoms have no singularities at the phase transition. We comment on why other authors have previously found otherwise.

The Dicke Hamiltonian describes the simplest quantum system with atoms interacting with photons: N two level atoms inside a perfectly reflecting cavity which allows only one electromagnetic mode. It has also been successfully employed to describe superconducting circuits which behave as artificial atoms coupled to a resonator. The system exhibits a transition to a superradiant phase at zero temperature. When the interaction strength reaches its critical value, both the number of photons and of atoms in excited states in the cavity, together with their fluctuations, exhibit a sudden increase from zero. Employing symmetry-adapted coherent states it is shown that these properties scale with the number of atoms, that their reported divergences at the critical point represent the limit when this number goes to infinity, and that in this limit they remain divergent in the superradiant phase. Analytical expressions are presented for all observables of interest, for any number of atoms. Comparisons with exact numerical solutions strongly support the results., Comment: 5 pages, 2 figures, to appear as a Rapid Communication in Physical Review A

Observing Delta Scuti stars is most important as their multi-frequency spectrum of radial pulsations provide strong constraints on the physics of the stars interior; so any new detection and observation of these stars is a valuable contribution to asteroseismology. While performing uvby-beta photoelectric photometry of some RR Lyrae stars acquired in 2005 at the Observatorio Astronomico Nacional, Mexico, we also observed several standard stars, HD115520 among them. After the reduction this star showed indications of variability. In view of this, a new observing run was carried out in 2006 during which we were able to demonstrate its variability and its nature as a Delta Scuti star. New observations in 2007 permitted us to determine its periodic content with more accuracy. This, along with the uvby-beta photoelectric photometry allowed us to deduce its physical characteristics and pulsational modes., Comment: 6 pages. To be published in Journal of Physics: Conference Series. Proc. of the workshop Helioseismology, Asteroseismology and MHD Connections held in Gottingen, Germany, 20-24 August 2007

All the new layer perovskite superconductors seem to show a phenomenon of symmetry mixing with repect to the order parameter. An analysis of the different alternative of mixing and how far could them be presented is carried out. For the particular case of s+id symmetry of the gap, the temperature dependence of the specific heat ($C_{es}$) and the thermodynamic critical magnetic field ($H_{c}$) are calculated. A double peak transition is observed on $C_{es}(T)$ in the mixed regime while the single peak behavior is recovered for a purely symmetric state (s or d). $C_{es}$ presents a quadratic law at low tempeartures for a d-wave gap and for the s-wave one the typical exponential attenuation. The temperature dependence of $H_{c}$ shows a clear phase transition of second order at temperatures where the d-wave component becomes negligible. A comparison with other results and experiments is done., Comment: Revised version and accepted to be published in Physica C

An analysis of the different approaches used within the van Hove BCS model for the high temperature superconductors has been done. How far the employment of an asymptotic expression for the density of states underestimates the thermodynamic parameters, as for example, the critical temperature, the gap at zero temperature, and the known universal ratios of the BCS model, is discussed. Analytical expressions obtained for some thermodynamic functions are compared with the numerical results for which the exact 2D density of states has been considered. Approximate analytical expressions for the temperature dependence of the reduced gap, the superconducting electronic specific heat, and the critical magnetic field have been obtained and compared with the corresponding numerical results. The validity of the different approximations and of the equations obtained for the thermodynamic functions in the framework of the van Hove model are also analyzed., Comment: LaTex,7 eps figures