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

Some inequalities for probability vector are discussed. The probability representation of quantum mechanics where the states are mapped onto probability vectors (either finite or infinite dimensional) called the state tomograms is used. Examples of inequalities for qudit tomograms and a state extended uncertainly relation are considered. Tomographic cumulant related to photon state tomographic probability distributions is introduced and it is used as parameter of the state nongaussianity.

State-extended uncertainly relations and tomographic inequalities as quantum system state characteristics

Out of any unitary representation of a group, positive-type functions on the group can be obtained. These functions allow one to construct positive semi-definite matrices that may be used to define new inequalities for higher moments of observables associated with classical probability distribution functions and density states of quantum systems. The inequalities stemming from the Heisenberg–Weyl group representations are considered in connection with Gaussian distributions. We obtain new inequalities for multi-variable Hermite polynomials.

Positive-type functions on groups and new inequalities in classical and quantum mechanics

We analyze a recently found inequality for eigenvalues of the density matrix and purity parameter describing either a bipartite system state or a single qudit state. The Minkowski type trace inequality for the density matrices of the qudit states is rewritten in terms of the purities. The properties of the obtained inequality are discussed. The $X$-states of the two qubits and the single qudit are considered in detail. A study of the relation of the obtained purity inequalities with the entanglement is presented.

Inequalities for purity parameters for multipartite and single qudit states

The role of the Wigner function in the dynamics of charged particle beams in high-energy accelerating machines is discussed. This is done within the quantum-like de- scription of the thermal wave model (TWM). A brief review of the numerical experiments showing satisfactory agreement between TWM and the particle tracking simulations is presented. A simple analysis in phase space in terms of the Wigner quasidistribution, showing that TWM is capable of reproducing the beam dynamics in the presence of the space charge effects, is put froward. The branch of electron optics describing the transport of non-laminar paraxial high-energy charged-particle beams involves the dynamics of an extremely large number of charged particles that is mainly dominated by the electromagnetic interactions, while the effects of the temperature cannot be neglected. The behavior of such a system is collective and affected by the thermal spreading among the particles (thermal regime). This discipline has been strongly developed in particle accelerator physics (1). A peculiar aspect of the electron optics of non-laminar beams is the mixing of the particle tra- jectories, i.e., mixing among the electron rays (1). In paraxial beams, this effect causes a slight random deviation of the electron rays slopes (with respect to the propagation direction). In vacuo, the electron ray slopes are Boltzmann-distributed, hence the instantaneous transverse momentum spread (normalized to the longitudinal relativistic particle momentum) is σp ∼ vT /c � 1( vT and c being the transverse thermal velocity and the speed of light, respectively). However, it is a matter of fact, both theoretically predicted and experimentally observed, that during the beam motion the instantaneous spread of the electron rays positions in the transverse plane, say σ, is related to σp by the following inequality:

/pdf/the-role-of-the-wigner-function-in-charged-particle-beam-1sg6pr9uma.pdf

The role of the Wigner function in charged-particle beam transport

The gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated. Explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained. Gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions of states of quantum system are introduced, and their evolution equations having the Liouville equation in corresponding representations as the classical limit are found.

Vladimir I. Man’ko

Papers

State-extended uncertainly relations and tomographic inequalities as quantum system state characteristics

Positive-type functions on groups and new inequalities in classical and quantum mechanics

Inequalities for purity parameters for multipartite and single qudit states

The role of the Wigner function in charged-particle beam transport

Gauge transformation of quantum states in probability representation