Quantum control and the Strocchi map

Identifying the real and imaginary parts of wave functions with coordinates and momenta, quantum evolution may be mapped onto a classical Hamiltonian system. In addition to the symplectic form, quantum mechanics also has a positive-definite real inner product which provides a geometrical interpretation of the measurement process. Together they endow the quantum Hilbert space with the structure of a K\"{a}ller manifold. Quantum control is discussed in this setting. Quantum time-evolution corresponds to smooth Hamiltonian dynamics and measurements to jumps in the phase space. This adds additional power to quantum control, non unitarily controllable systems becoming controllable by ``measurement plus evolution''. A picture of quantum evolution as Hamiltonian dynamics in a classical-like phase-space is the appropriate setting to carry over techniques from classical to quantum control. This is illustrated by a discussion of optimal control and sliding mode techniques.Comment: 16 pages Late

Lyapunov exponent in quantum mechanics. A phase-space approach

Using the symplectic tomography map, both for the probability distributions in classical phase space and for the Wigner functions of its quantum counterpart, we discuss a notion of Lyapunov exponent for quantum dynamics. Because the marginal distributions, obtained by the tomography map, are always well defined probabilities, the correspondence between classical and quantum notions is very clear. Then we also obtain the corresponding expressions in Hilbert space. Some examples are worked out. Classical and quantum exponents are seen to coincide for local and non-local time-dependent quadratic potentials. For non-quadratic potentials classical and quantum exponents are different and some insight is obtained on the taming effect of quantum mechanics on classical chaos. A detailed analysis is made for the standard map. Providing an unambiguous extension of the notion of Lyapunov exponent to quantum mechnics, the method that is developed is also computationally efficient in obtaining analytical results for the Lyapunov exponent, both classical and quantum.Comment: 30 pages Late

Tomographic Representation of Minisuperspace Quantum Cosmology and Noether Symmetries

The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select "observable" universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.Comment: 15 page

Tomographic entropy and cosmology

The probability representation of quantum mechanics including propagators and tomograms of quantum states of the universe and its application to quantum gravity and cosmology are reviewed. The minisuperspaces modeled by oscillator, free pointlike particle and repulsive oscillator are considered. The notion of tomographic entropy and its properties are used to find some inequalities for the tomographic probability determining the quantum state of the universe. The sense of the inequality as a lower bound for the entropy is clarified.Comment: 19 page

Coherent states for exactly solvable potentials

A general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials e.g., Morse and P{\"o}schl-Teller, is given. The method, {\it a priori}, is potential independent and connects with earlier developed ones, including the oscillator based approaches for coherent states and their generalizations. This approach can be straightforwardly extended to construct more general coherent states for the quantum mechanical potential problems, like the nonlinear coherent states for the oscillators. The time evolution properties of some of these coherent states, show revival and fractional revival, as manifested in the autocorrelation functions, as well as, in the quantum carpet structures.Comment: 11 pages, 4 eps figures, uses graphicx packag

Quantum Characterization of a Werner-like Mixture

We introduce a Werner-like mixture [R. F. Werner, Phys. Rev. A {\bf 40}, 4277 (1989)] by considering two correlated but different degrees of freedom, one with discrete variables and the other with continuous variables. We evaluate the mixedness of this state, and its degree of entanglement establishing its usefulness for quantum information processing like quantum teleportation. Then, we provide its tomographic characterization. Finally, we show how such a mixture can be generated and measured in a trapped system like one electron in a Penning trap.Comment: 8 pages ReVTeX, 8 eps figure

Contractions, deformations and curvature

The role of curvature in relation with Lie algebra contractions of the pseudo-ortogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley-Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such "quantum" spaces.Comment: 17 pages. Based on the talk given in the Oberwolfach workshop: Deformations and Contractions in Mathematics and Physics (Germany, january 2006) organized by M. de Montigny, A. Fialowski, S. Novikov and M. Schlichenmaie