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

The coupled parametric quantum oscillators are discussed. The ground, coherent states and integrals of motion for the parametric chain with time-dependent frequencies are constructed. The possibility of controlling in principle the dispersions of coordinates and momenta is suggested. The squeezing coefficients in coherent states are found.

Squeezing in Quantum Parametric Chain (

A simple system of two particles in a bidimensional configurational space $S$ is studied. The possibility of breaking in $S$ the time independent Schr\"{o}dinger equation of the system into two separated one-dimensional one-body Schr\"{o}dinger equations is assumed. In this paper, we focus on how the latter property is countered by imposing such boundary conditions as confinement in a limited region of $S$ and/or restrictions on the joint coordinate probability density stemming from the sign-invariance condition of the relative coordinate (an impenetrability condition). Our investigation demonstrates the reducibility of the problem under scrutiny into that of a single particle living in a limited domain of its bidimensional configurational space. These general ideas are illustrated introducing the coordinates $X_c$ and $x$ of the center of mass of two particles and of the associated relative motion, respectively. The effects of the confinement and the impenetrability are then analyzed by studying with the help of an appropriate Green's function and the time evolution of the covariance of $X_c$ and $x$. Moreover, to calculate the state of the single particle constrained within a square, a rhombus, a triangle and a rectangle the Green's function expression in terms of Jacobi $\theta_3$-function is applied. All the results are illustrated by examples.

Breakdown of separability due to confinement, submitted to Report on mathematical physics

We extend the Bell inequality known for two qubits to the four-level atom, including an artificial atom realized by the superconducting circuit, and qudit with j=3/2. We formulate the extended inequality as the inequality valid for an arbitrary Hermitian nonnegative 4x4-matrix with unit trace for both separable and entangled matrices.

Hidden Bell correlations in the four-level atom

Systems with multimode nonstationary Hamiltonians (quadratic in position and momentum operators) are reviewed. The tomographic probability distributions (tomograms) for the Fock states and Gaussian states of the quadratic systems are discussed. The tomograms for the Fock states are expressed in terms of multivariable Hermite polynomials. In view of the obvious physical relations, some new formulas for multivariable Hermite polynomials are found. Examples of a driven parametric oscillator and a charged particle moving in the electromagnetic field are presented.

Vladimir I. Man’ko

Papers

Squeezing in Quantum Parametric Chain (

Breakdown of separability due to confinement, submitted to Report on mathematical physics

Hidden Bell correlations in the four-level atom

Tomography of Multimode Quantum Systems with Quadratic Hamiltonians and Multivariable Hermite Polynomials

Tomography of quantum states of a symmetric rotor