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We establish, within the second quantization method, the general dipole-dipole Hamiltonian interaction of a system of $n$-level atoms. The variational energy surface of the $n$-level atoms interacting with $\ell$-mode fields and under the Van Der Waals forces is calculated with respect the tensorial product of matter and electromagnetic field coherent states. This is used to determine the quantum phase diagram associated to the ground state of the system and quantify the effect of the dipole-dipole Hamiltonian interaction. By considering real induced electric dipole moments, we find the quantum phase transitions for $2$- and $3$-level atomic systems interacting with $1$- and $2$- modes of the electromagnetic field, respectively. The corresponding order of the transitions is established by means of Ehrenfest classification; for some undetermined cases, we propose two procedures: the difference of the expectation value of the Casimir operators of the $2$-level subsystems, and by maximizing the Bures distance between neighbor variational solutions., Comment: 17 pages, 9 figures

Non-classical states are of practical interest in quantum computing and quantum metrology. These states can be detected through their Wigner function negativity in some regions. In this paper, we calculate the ground state of the three-level generalised Dicke model for a single atom and determine the structure of its phase diagram using a fidelity criterion. We also calculate the Wigner function of the electromagnetic modes of the ground state through the corresponding reduced density matrix, and show in the phase diagram the regions where entanglement is present. A finer classification for the continuous phase transitions is obtained through the computation of the surface of maximum Bures distance., Comment: 22 pages, 11 figures

A general procedure is established to calculate the quantum phase diagrams for finite matter-field Hamiltonian models. The minimum energy surface associated to the different symmetries of the model is calculated as a function of the matter-field coupling strengths. By means of the ground state wave functions, one looks for minimal fidelity or maximal Bures distance surfaces in terms of the parameters, and from them the critical regions of those surfaces characterize the finite quantum phase transitions. Following this procedure for $N_a=1$ and $N_a=4$ particles, the quantum phase diagrams are calculated for the generalised Tavis-Cummings and Dicke models of 3-level systems interacting dipolarly with $2$ modes of electromagnetic field. For $N_a=1$, the reduced density matrix of the matter allows us to determine the phase regions in a $2$-simplex (associated to a general three dimensional density matrix), on the different $3$-level atomic configurations, together with a measurement of the quantum correlations between the matter and field sectors. As the occupation probabilities can be measured experimentally, the existence of a quantum phase diagram for a finite system can be established., Comment: 30 pages, 13 figures

We present a study of the structure of phase diagrams for matter-radiation systems, based on the use of coherent states and the catastrophe formalism, that compares very well with the exact quantum solutions as well as providing analytical expressions. Emphasis is made on $2$- and $3$-level systems, but in general $n$-level systems in the presence of $\ell$ electromagnetic modes are described. Due to the infinite-dimensional nature of the Hilbert space, and using the results of the analyses and the behaviour of the solutions, we construct a sequence of ever-approximating reduced bases, which make possible the study of larger systems both, in the number of atoms and in the number of excitations. These studies are of importance in fundamental quantum optics, quantum information, and quantum cryptography scenarios., Comment: 21 pages, 16 figures

A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres--Horodecki positive partial transpose (ppt)-criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich's squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.

An algebraic procedure to find extremal density matrices for any Hamiltonian of a qudit system is established. The extremal density matrices for pure states provide a complete description of the system, that is, the energy spectra of the Hamiltonian and their corresponding projectors. For extremal density matrices representing mixed states, one gets mean values of the energy in between the maximum and minimum energies associated to the pure case. These extremal densities give also the corresponding mixture of eigenstates that yields the corresponding mean value of the energy. We enhance that the method can be extended to any hermitian operator., Comment: 31 pages, 4 figures

The discretization of the density matrix is proposed as a nonlinear positive map for systems with continuous variables. This procedure is used to calculate the entanglement between two modes through different criteria, such as Tsallis entropy, von Neumann entropy and linear entropy and the logarithmic negativity. As an example, we study the dynamics of entanglement for the two-mode squeezed vacuum state in the parametric amplifier and show good agreement with the analytic results. The loss of information on the system state due to the discretization of the density matrix is also addressed.

The linear time-dependent constants of motion of the parametric amplifier are obtained and used to determine in the tomographic-probability representation the evolution of a general two-mode Gaussian state. By means of the discretization of the continuous variable density matrix, the von Neumann and linear entropies are calculated to measure the entanglement properties between the modes of the amplifier. The obtained results for the nonlocal correlations are compared with those associated to a linear map of discretized symplectic Gaussian-state tomogram onto a qubit tomogram. This qubit portrait procedure is used to establish Bell-type's inequalities, which provide a necessary condition to determine the separability of quantum states, which can be evaluated through homodyne detection. Other no-signaling nonlocal correlations are defined through the portrait procedure for noncomposite systems.

A system of $N_a$ atoms of $n$-levels interacting dipolarly with $\ell$ modes of electromagnetic field is considered. The energy surface of the system is constructed from the direct product of the coherent states of U$(n)$ in the totally symmetric representation for the matter times the $\ell$ coherent states of the electromagnetic field. A variational analysis shows that the collective region is divided into $\ell$ zones, inside each of which only one mode of the electromagnetic field contributes to the ground state. In consequence, the polychromatic phase diagram for the ground state naturally divides itself into monochromatic regions. For the case of $3$-level atoms in the $\Xi$-configuration in the presence of $2$ modes, the variational calculation is compared with the exact quantum solution showing that both are in agreement., Comment: 16 pages, 9 figures

We study the structure of the phase diagram for systems consisting of 2- and 3- level particles dipolarly interacting with a 1-mode electromagnetic field, inside a cavity, paying particular attention to the case of a finite number of particles, and showing that the divergences that appear in other treatments are a consequence of the mathematical approximations employed and can be avoided by studying the system in an exact manner quantum-mechanically or via a catastrophe formalism with variational trial states that satisfy the symmetries of the appropriate Hamiltonians. These variational states give an excellent approximation not only to the exact quantum phase space, but also to the energy spectrum and the expectation values of the atomic and field operators. Furthermore, they allow for analytic expressions in many of the cases studied. We find the loci of the transitions in phase space from one phase to the other, and the order of the quantum phase transitions are determined explicitly for each of the configurations, with and without detuning. We also derive the critical exponents for the various systems, and the phase structure at the triple point present in the {\Xi}-configuration of 3-level systems is studied., Comment: 30 pages, 31 figures, 4 tables

We establish a procedure to find the extremal density matrices for any finite Hamiltonian of a qudit system. These extremal density matrices provide an approximate description of the energy spectra of the Hamiltonian. In the case of restricting the extremal density matrices by pure states, we show that the energy spectra of the Hamiltonian is recovered for $d=2$ and $3$. We conjecture that by means of this approach the energy spectra can be recovered for the Hamiltonian of an arbitrary finite qudit system. For a given qudit system Hamiltonian, we find new inequalities connecting the mean value of the Hamiltonian and the entropy of an arbitrary state. We demonstrate that these inequalities take place for both the considered extremal density matrices and generic ones., Comment: 12 pages, 4 figures Accepted for publication in Journal of Physics A: Mathematical and Theoretical

We apply the energy surface method to study a system of Na three-level atoms interacting with a one-mode radiation field in the \Xi, \Lambda and V configurations. We obtain an estimation of the ground-state energy, the expectation value of the total number of excitations, and the separatrix of the model in the interaction parameter space, and compare the results with the exact solutions. We have first and second-order phase transitions, except for the V configuration which only presents second-order phase-transitions., Comment: 5 pp., 8 figures

We show how various mathematical formalisms, specifically the catastrophe formalism and group theory, aid in the study of relevant systems in quantum optics. We describe the phase transition of the Dicke model for a finite number N of atoms, via 3 different methods, which lead to universal parametric curves for the expectation value of the first quadrature of the electromagnetic field and the expectation value of the number operator, as functions of the atomic relative population. These are valid for all values of the matter-field coupling parameter, and valid for both the ground and first-excited states. Using these mathematical tools, the critical value of the atom-field coupling parameter is found as a function of the number of atoms, from which its critical exponent is derived., Comment: 14 pp, 9 figures

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 operators. 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., Comment: 11 pages, 5 figures, match published version. Use of the discontinuity in the critical exponents of the fidelity susceptibility to obtain analytically the class of universality associated with the Lipkin model. The mention to spurious divergences has been replaced with "have no counterpart in the exact diagonalization"

The critical value of the atom-field coupling strength for a finite number of atoms is deter- mined by means of both, semiclassical and exact solutions. In the semiclassical approach we use a variational procedure with coherent and symmetry-adapted states, while for the exact quantum solution the concept of fidelity is employed. These procedures allow for the determination of the phase transitions in the model, and coincide in the thermodynamic limit. For the three cases men- tioned above, universal parametric curves are obtained for the expectation values of both the first quadrature of the electromagnetic field, and the atomic relative population, as implicit functions of the atom-field coupling parameter, valid for the ground- and first-excited states., Comment: 10 pages, 7 figures

A mean field description of the Dicke model is presented, employing the Holstein-Primakoff realization of the angular momentum algebra. It is shown that, in the thermodynamic limit, when the number of atoms interacting with the photons goes to infinity the energy surface takes a simple form, allowing for a direct description of many observables., Comment: To appear in the Proc. of FPP6 - Foundations of Probability and Physics-6 conference at Linnaeus University in V\"axj\"o, Sweden, June 13-16, 2011, American Institute of Physics Conference Proceedings

We show that an excellent approximation to the exact quantum solution of the ground state of the Tavis-Cummings model is obtained by means of a semi-classical projected state. This state has an analytical form in terms of the model parameters and, in contrast to the exact quantum state, it allows for an analytical calculation of the expectation values of field and matter observables, entanglement entropy between field and matter, squeezing parameter, and population probability distributions. The fidelity between this projected state and the exact quantum ground state is very close to 1, except for the region of classical phase transitions. We compare the analytical results with those of the exact solution obtained through the direct Hamiltonian diagonalization as a function of the atomic separation energy and the matter-field coupling., Comment: 22 pages, 13 figures, accepted for publication in Physics Scripta

Quantum phase transitions and observables of interest of the ground state in the Tavis-Cummings model are analyzed, for any number of atoms, by using a tensorial product of coherent states. It is found that this "trial" state constitutes a very good approximation to the exact quantum solution, in that it globally reproduces the expectation values of the matter and field observables. These include the population and dipole moments of the two-level atoms and the squeezing parameter. Agreement in the field-matter entanglement and in the fidelity measures, of interest in quantum information theory, is also found.The analysis is carried out in all three regions defined by the separatrix which gives rise to the quantum phase transitions. It is argued that this agreement is due to the gaussian structure of the probability distributions of the constant of motion and the number of photons. The expectation values of the ground state observables are given in analytic form, and the change of the ground state structure of the system when the separatrix is crossed is also studied., Comment: 38 pages, 16 figures

Some properties of Plebanski squeezing operator and squeezed states created with time-dependent quadratic in position and momentum Hamiltonians are reviewed. New type of tomography of quantum states called squeeze tomography is discussed., Comment: 17 pages LaTeX, 2 figures PostScript To appear in the Proccedings of the Conference ``Topics in Mathematical Physics, General Relativity and Cosmology on the Ocassion of the 75th Birthday of Professor Jerzy F. Plebanski'' Mexico City, August 2002 (World Scientific, Singapore)