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

We introduce examples of three- and four-mode entangled Gaussian mixed states that are not detected by the scaling and Peres–Horodecki separability criteria. The presented modification of the scaling criterion resolves this problem. Also it is shown that the new criterion reproduces the main features of the scaling pictures for different cases of entangled states, while the previous versions lead to completely different outcomes. This property of the presented scheme is evidence of its higher generality.

Scaling separability criterion: application to Gaussian states

The symplectic tomograms of 2D Hermite--Gauss beams are found and expressed in terms of the Hermite polynomials squared. It is shown that measurements of optical-field intensities may be used to determine the tomograms of electromagnetic-radiation modes. Furthermore, entropic uncertainty relations associated with these tomograms are found and applied to establish the compatibility conditions of the the field profile properties with Hermite--Gauss beam description. Numerical evaluations for some Hermite--Gauss modes illustrating the corresponding entropic uncertainty relations are finally given.

Entropic uncertainty relations for electromagnetic beams

We construct the positive invertible map of the mixed states of a single qutrit onto the antisymmetrized bipartite qutrit states (quasifermions). It is shown that using this one-to-one correspondence between qutrit states and states of two three-dimensional quasifermions one may attribute hidden entanglement to a single mixed state of qutrit.

/pdf/virtual-correlations-in-single-qutrit-3a11wio5nj.pdf

Virtual Correlations in Single Qutrit

We study the evolution of the driven harmonic oscillator in the probability representation of quantum mechanics. We use the photon-number tomographic-probability-distribution function to describe the quantum states of the system. We give a general review of the photon-number tomographic framework, including a discussion on the connection with other representations of quantum mechanics. We find tomograms of coherent states as well as excited states of the harmonic oscillator in an explicit form. We discuss the time evolution of the photon-number tomograms and transforms of the propagators for different representations of quantum mechanics. We obtain the propagator for the photon-number tomographic-distribution function for the case of the driven oscillator in an explicit form.

The driven oscillator in the photon-number probability representation of quantum mechanics

The invertable map of spin state density operator onto quasiprobability distribution of three continuous variables is constructed. The connection with two-mode electromagnetic field oscillators is discussed. The inversion formula for spin-density matrix is given in terms of Hermite polynomials. The connection with the spin-tomographic representation is examined. The problem of the magnetic moment rotation in the magnetic field is considered in the suggested representation of spin states.

Vladimir I. Man’ko

Papers

Scaling separability criterion: application to Gaussian states

Entropic uncertainty relations for electromagnetic beams

Virtual Correlations in Single Qutrit

The driven oscillator in the photon-number probability representation of quantum mechanics

Hermite Polynomial Representation of the Spin States