There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Table Of Integrals Series And Products

We review the field of cavity optomechanics, which explores the interaction between electromagnetic radiation and nano- or micromechanical motion This review covers the basics of optical cavities and mechanical resonators, their mutual optomechanical interaction mediated by the radiation pressure force, the large variety of experimental systems which exhibit this interaction, optical measurements of mechanical motion, dynamical backaction amplification and cooling, nonlinear dynamics, multimode optomechanics, and proposals for future cavity quantum optomechanics experiments In addition, we describe the perspectives for fundamental quantum physics and for possible applications of optomechanical devices

Cavity Optomechanics

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

http://cds.cern.ch/record/262851/files/9405029.pdf

Supersymmetry and quantum mechanics

Rotation of such objects as an atomic nucleus or a chromodynamical string can result in specific effects in scattering processes and multiparticle production. Secondary fragments of the rotating nucleus or of the decaying string can move like stones thrown from a sling. That would be detected as the azimuthal asymmetry of particle distributions in individual events. Non-classical states of the created particles like the Schr"\{o}dinger cats are produced. Some classical and quantum-mechanical estimates of possible effects are given. Experimental facts which can be used for their verification are discussed.

Particles and Nuclei as Quantum Slings

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.

Lyapunov exponent in quantum mechanics. A phase-space approach

Gaussian beams include a number of field or wavefunctions which have a clear quantum mechanical analogue: coherent states, correlated coherent states and discrete modes for quantum oscillators. These are used to model optical fibers and to describe the output of laser devices. When these beams leave their source and travel freely through space, they loose their coherence and aberrate. The source of this aberration is purely geometrical, and is termed spherical aberration. We describe this process in the framework of the Fermat-Hamilton formulation of optics, studying the behaviour of the center of the beam, its width, and the way in which the initial uncorrelation of position and momentum is lost.

The influence of spherical aberration on gaussian beam propagation

The problem of classical particle in linear potential is studied by using the formalism of Hilbert space and tomographic probability distribution. The Liouville equation for this problem is solved by finding the density matrix satisfying von Newmann-like equation in the form of product of wave functions. The relation to quantum mechanics is discussed.

Wave function of classical particle in linear potential

New entropic and information inequalities for density matrices and vector tomographic portraits of spin-1 quantum particle states are obtained

Vladimir I. Man’ko

Papers

Particles and Nuclei as Quantum Slings

Lyapunov exponent in quantum mechanics. A phase-space approach

The influence of spherical aberration on gaussian beam propagation

Wave function of classical particle in linear potential

Entropic and information inequalities in the tomographic probability description of spin-1 particles