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We present an integral of diffraction based on particular eigenfunctions of the Laplacian in two dimensions. We show how to propagate some fields, in particular a Bessel field, a superposition of Airy beams, both over the square root of the radial coordinate, and show how to construct a field that reproduces itself periodically in propagation, i.e., a field that renders the Talbot effect. Additionally, it is shown that the superposition of Airy beams produces self-focusing., Comment: 5 pages, 4 figures

We study the eigenvalue problem for a linear potential Hamiltonian and, by writing Airy equation in terms of momentum and position operators define Airy states. We give a solution of the Schr\"odinger equation for the symmetrical linear potential in terms of the squeeze and displacement operators. Finally, we write the unit operator in terms of Airy states and find a relation between them and position and momentum eigenstates., Comment: 9 pages

We study the Jaynes-Cummings interaction when initial mixed states, for the atom or the field, are considered. The evolved mixed field density matrix is purified to a wavefunction that describes the interaction between a quantised field and an artificial four-level atom. This allows us to use the Araki-Lieb inequality to calculate the field entropy from the atomic entropy. We then generate {\it artificial} Hamiltonians that reproduce the field entropy dynamics. We finally show that more realistic Hamiltonians may be considered by using two entangled two-level atoms., Comment: 10 pages, 3 figures

We analyse the paraxial field propagation in the realm of classical optics, showing that it can be written as the action of the fractional Fourier transform, followed by the squeeze operator applied to the initial field. Secondly, we show that a wavelet transform may be viewed as the application of a displacement and squeeze operator onto the mother wavelet function., Comment: 10 pages, no figures

We propose a mixedness quantifier based on entropy fluctuations. It provides information about the degree of mixedness either for finite dimensional and infinite dimensional Hilbert spaces. It may be used to determine the reduction of the Hilbert space as it becomes maximum when, either the state is maximally mixed, or, when the Hilbert space effectively reduces its dimensions, such as in the atom field interaction where the two-level atom dictates the final dimension of the field., Comment: 13 pages, 6 figures

The Araki-Lieb inequality is commonly used to calculate the entropy of subsystems when they are initially in pure states as this forces the entropy of the two subsystems to be equal after the complete system evolves. Then, it is easy to calculate the entropy of a large subsystem by finding the entropy of the small one. To the best of our knowledge, there does not exist a way of calculating the entropy when one of the subsystems is initially in a mixed state. We show here that it is possible to use the Araki-Lieb inequality in this case and find the von Neumann entropy for the large (infinite) system. We show this in the two-level atom-field interaction., Comment: 10 pages, 4 figures

We show that it is possible to add or subtract many photons from a cavity field by interacting it resonantly with a two-level atom. The atom, after entangling with the field inside the cavity and exiting it, may be measured in one of the Schmidt states, producing a multiphoton process (in the sense that can add or annihilate more photons than a single transition allows), i.e., adding or subtracting several photons from the cavity field., Comment: 6 pages, 5 figures

Trapped ions are considered one of the best candidates to perform quantum information processing. By interacting them with laser beams they are, somehow, easy to manipulate, which makes them an excellent choice for the production of nonclassical states of their vibrational motion, the reconstruction of quasiprobability distribution functions, the production of quantum gates, etc. However, most of these effects have been produced in the so-called low intensity regime, this is, when the Rabi frequency is much smaller than the trap frequency. Because of the possibility to produce faster quantum gates in other regimes it is of importance to study this system in a more complete manner, which is the motivation for this contribution. We start by studying the way ions are trapped in Paul traps in and review the basic mechanisms of trapping. Then we show how the problem may be completely solved for trapping states; i.e., we find eigenstates of the full Hamiltonian. We show how in the low intensity regime Jaynes-Cummings and anti-Jaynes-Cummings interactions may be obtained, without using the rotating wave approximation and analyze the medium and high intensity regimes were dispersive Hamiltonians are produced. The traditional approach is also studied and used for the generation of non-classical states of the vibrational of the vibrational wavefunction. In particular, we show how to add and subtract vibrational quanta to an initial state, how to produce specific superpositions of number states and how to generate NOON states for the two-dimensional vibration of the ion. It is also shown how squeezing may be measured. The time dependent problem is studied by using Lewis-Ermakov methods, we give a solution to the problem when the time dependence of the trap is considered and also analyze an specific time dependence that produces squeezing of the initial vibrational wave function., Comment: 42 pages, 15 figures