The Quantum Arnold Transformation and its applications

The Quantum Arnold Transformation, a unitary operator mapping the solutions of the Schr¨odinger equation for time-dependent quadratic Hamiltonians into free-particle solutions, is revisited. Possible applications and extensions are also outlined: the analytic construction of harmonic states for the free particle, the Quantum Arnold-Ermakov-Pinney transformation and the description of the Release & Recapture method

Coherent States for infinite homogeneous waveguide arrays

Perelomov coherent states for equally spaced, infinite homogeneous waveguide arrays with Euclidean E(2) symmetry are defined, and new resolutions of the identity are constructed in Cartesian and polar coordinates. The key point to construct these resolutions of the identity is the fact that coherent states satisfy Helmholtz equation (in coherent states labels) an thus a non-local scalar product with a convolution kernel can be introduced which is invariant under the Euclidean group. It is also shown that these coherent states for the Eucliean E(2) group have a simple and natural physical realization in these waveguide arrays.Spanish MICINN through the project PGC2018-097831-B-I00 and Junta de Andalucı́a through the project FEDER/UJA-1381026

The triple-shell structure and collimated outflows of the planetary nebula NGC 6891

Narrow-band H$\alpha$ and [N {\sc ii}] images and high-dispersion spatially-resolved echelle spectroscopy of the planetary nebula NGC 6891 are presented. These observations show a great wealth of structures. The bright central nebula is surrounded by an attached shell and a detached outer halo. Both the inner and intermediate shells can be described as ellipsoids with similar major to minor axial ratios, but different spatial orientations. The kinematical ages of the intermediate shell and halo are 4,800 and 28,000 years, respectively. The inter-shell time lapse is in good agreement with the evolutionary inter-pulse time lapse. A highly collimated outflow is observed to protrude from the tips of the major axis of the inner nebula and impact on the outer edge of the intermediate shell. Kinematics and excitation of this outflow provide conclusive evidence that it is deflected during the interaction with the outer edge of the intermediate shell. At the same time, both the kinematics and the morphology of the intermediate shell appear to be affected by this interaction.Comment: To be published in MNRAS, 9 pages, 5 figure

Harmonic states for the free particle

Different families of states, which are solutions of the time-dependent free Schr\"odinger equation, are imported from the harmonic oscillator using the Quantum Arnold Transformation introduced in a previous paper. Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.Comment: 21 pages, 3 figures. Title changed, content added, references adde