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 adjustment of two different selfocs is considered using both exact formulas for the mode-connection coefficients expressed in terms of Hermite polynomials of several variables and a qualitative approach based on the Frank–Condon principle. Several examples of the refractive-index dependence are studied and illustrative plots for these examples are presented. The connection with the tomographic approach to quantum states of a two-dimensional oscillator and the Frank–Condon factors is established.

Frank–Condon principle and adjustment of optical waveguides with nonhomogeneous refractive indices

We discuss the known construction of two interacting superconducting circuits, based on Josephson junctions, that can be precisely engineered and easily controlled. In particular, we use the parametric excitation of two circuits, realized by an instant change of the qubit coupling, to study entropic and information properties of the density matrix of the composite system. The density matrix is obtained from the initial thermal state and is then analyzed in the approximation of small perturbation parameter and sufficiently low temperature. We also check the subadditivity condition for this system both for von Neumann and deformed entropies and look at the dependance of mutual information on the temperature of the system. Finally, we discuss the applicability of this approach to describe such system of two coupled superconducting qubits as harmonic oscillators with limited Hilbert space.

Deriving Entropic Inequalities for Two Coupled Superconducting Circuits

Using the entropic inequalities for Shannon and Tsallis entropies new inequalities for some classical polynomials are obtained. To this end, an invertible mapping for the irreducible unitary representation of groups SU(2) and SU(1, 1) like Jacoby polynomials and Gauss’ hypergeometric functions, respectively, are used.

Entropic inequalities for matrix elements of rotation group irreducible representations

Even and Odd Schrödinger Cat States in the Probability Representation of Quantum Mechanics

Review of the probability representation of qubit states and observables is presented as well as the picture of states of two-level systems in terms of Triada of Malevich’s squares. A new relation of introduced probability parameters is obtained. Also, it is offered a method to visualize the quantum channel’s maps of qubit states. Evolution of the two-level system is considered in terms of Triada of Malevich’s squares in case of Rabi and Demkov models.

Vladimir I. Man’ko

Papers

Frank–Condon principle and adjustment of optical waveguides with nonhomogeneous refractive indices

Deriving Entropic Inequalities for Two Coupled Superconducting Circuits

Entropic inequalities for matrix elements of rotation group irreducible representations

Even and Odd Schrödinger Cat States in the Probability Representation of Quantum Mechanics

Unitary and Nonunitary Evolution of Qubit States in Probability Representation of Quantum Mechanics