Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. St\"ackel systems are studied in particular details. Examples in conformally and non-conformally flat manifolds are given.Comment: 18 page

Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds

Given a $n$-dimensional Riemannian manifold of arbitrary signature, we illustrate an algebraic method for constructing the coordinate webs separating the geodesic Hamilton-Jacobi equation by means of the eigenvalues of $m \leq n$ Killing two-tensors. Moreover, from the analysis of the eigenvalues, information about the possible symmetries of the web foliations arises. Three cases are examined: the orthogonal separation, the general separation, including non-orthogonal and isotropic coordinates, and the conformal separation, where Killing tensors are replaced by conformal Killing tensors. The method is illustrated by several examples and an application to the L-systems is provided.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Extended Hamiltonians and shift, ladder functions and operators

In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended manifolds, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians admits factorized constants of motion and we determine their expression. The classical constants may be non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension.Comment: 25 page

Polynomial constants of motion for Calogero-type systems in three dimensions

We give an explicit and concise formula for higher-degree polynomial first integrals of a family of Calogero-type Hamiltonian systems in dimension three. These first integrals, together with the already known ones, prove the maximal superintegrability of the systems.Comment: A small flaw in the proof of Theorem 3 has been amended. Some remarks about dihedral symmetries and superintegrability have been adde

First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator

We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians $H$ obtained as one-dimensional extensions of natural (geodesic) $n$-dimensional Hamiltonians $L$. The Liouville integrability of $L$ implies the (minimal) superintegrability of $H$. We prove that, as a consequence of natural integrability conditions, it is necessary for the construction that the curvature of the metric tensor associated with $L$ is constant. As examples, the procedure is applied to one-dimensional $L$, including and improving earlier results, and to two and three-dimensional $L$, providing new superintegrable systems.Comment: Theorem 1, Lemmas 1 and 2, Example 2 are correcte

Conversion between electromagnetically induced transparency and absorption in a three-level lambda system

We show that it is possible to change from a {\it subnatural} electromagnetically induced transparency (EIT) feature to a {\it subnatural} electromagnetically induced absorption (EIA) feature in a (degenerate) three-level $\Lambda$ system. The change is effected by turning on a second control beam counter-propagating with respect to the first beam. We observe this change in the $D_2$ line of Rb in a room-temperature vapor cell. The observations are supported by density-matrix analysis of the complete sublevel structure including the effect of Doppler averaging, but can be understood qualitatively as arising due to the formation of $N$-type systems with the two control beams. Since many of the applications of EIT and EIA rely on the anomalous dispersion near the resonances, this introduces a new ability to control the sign of the dispersion.Comment: 6 pages, 7 figure

Polarization-rotation resonances with subnatural widths using a control laser

We demonstrate extremely narrow resonances for polarization rotation in an atomic vapor. The resonances are created using a strong control laser on the same transition, which polarizes the atoms due to optical pumping among the magnetic sublevels. As the power in the control laser is increased, successively higher-order nested polarization rotation resonances are created, with progressively narrower linewidths. We study these resonances in the $D_2$ line of Rb in a room-temperature vapor cell, and demonstrate a width of $0.14 \, \Gamma$ for the third-order rotation. The explanation based on a simplified $\Lambda$V-type level structure is borne out by a density-matrix analysis of the system. The dispersive lineshape and subnatural width of the resonance lends itself naturally to applications such as laser locking to atomic transitions and precision measurements.Comment: 5 pages, 6 figure