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

Probability representation of quantum mechanics and star product quantization

Quantum probability measure for parametric oscillators

Wigner Picture and Tomographic Representation of Envelope Solitons

Entangled and separable states of a bipartite (multipartite) system are studied in the tomographic representation of quantum states. Properties of tomograms (joint probability distributions) corresponding to entangled states are discussed. The connection with star-product quantization is presented. U(N)-tomography and spin tomography as well as the relation of the tomograms to positive and completely positive maps are considered. The tomographic criterion of separability (necessary and sufficient condition) is formulated in terms of the equality of the specific function depending on unitary group parameters and positive map semigroup parameters to unity. Generalized Werner states are used as an example.

http://iopscience.iop.org/article/10.1088/1464-4266/6/2/007/pdf

Entanglement in probability representation of quantum states and tomographic criterion of separability

We present one-dimensional (1D) stability analysis of a recently
proposed method to filter and control localized states of the
Bose–Einstein condensate (BEC), based on novel trapping
techniques that allow one to conceive methods to select a
particular BEC shape by controlling and manipulating the external
potential well in the three-dimensional (3D)
Gross–Pitaevskii equation (GPE). Within the framework of this
method, under suitable conditions, the GPE can be exactly
decomposed into a pair of coupled equations: a transverse
two-dimensional (2D) linear Schrodinger equation and a
one-dimensional (1D) longitudinal nonlinear Schrodinger
equation (NLSE) with, in a general case, a time-dependent
nonlinear coupling coefficient. We review the general idea how
to filter and control localized solutions of the GPE. Then,
the 1D longitudinal NLSE is numerically solved
with suitable non-ideal controlling potentials that differ from
the ideal one so as to introduce relatively small errors
in the designed spatial profile. It is shown that a BEC with an
asymmetric initial position in the confining potential exhibits
breather-like oscillations in the longitudinal direction but,
nevertheless, the BEC state remains confined within the potential
well for a long time. In particular, while the condensate remains
essentially stable, preserving its longitudinal soliton-like
shape, only a small part is lost into “radiation”.

Vladimir I. Man’ko

Papers

Probability representation of quantum mechanics and star product quantization

Quantum probability measure for parametric oscillators

Wigner Picture and Tomographic Representation of Envelope Solitons

Entanglement in probability representation of quantum states and tomographic criterion of separability

1D stability analysis of filtering and controlling the solitons in Bose-Einstein condensates