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

Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

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Entanglement in many-body systems

Complex (dusty) plasmas: current status, open issues, perspectives

: The NRL Plasma Formulary originated over twenty five years ago and has been revised several times during this period

NRL: Plasma Formulary

Complex (dusty) plasmas are composed of a weakly ionized gas and charged microparticles and represent the plasma state of soft matter. Complex plasmas have several remarkable features: Dynamical time scales associated with microparticles are ``stretched'' to tens of milliseconds, yet the microparticles themselves can be easily visualized individually. Furthermore, since the background gas is dilute, the particle dynamics in strongly coupled complex plasmas is virtually undamped, which provides a direct analogy to regular liquids and solids in terms of the atomistic dynamics. Finally, complex plasmas can be easily manipulated in different ways---also at the level of individual particles. Altogether, this gives us a unique opportunity to go beyond the limits of continuous media and study---at the kinetic level---various generic processes occurring in liquids or solids, in regimes ranging from the onset of cooperative phenomena to large strongly coupled systems. In the first part of the review some of the basic and new physics are highlighted which complex plasmas enable us to study, and in the second (major) part strong coupling phenomena in an interdisciplinary context are examined. The connections with complex fluids are emphasized and a number of generic liquid and solid-state issues are addressed. In summary, application oriented research is discussed.

Complex plasmas: An interdisciplinary research field

Low-frequency modes that develop as a result of an instability in a dusty rf discharge plasma were studied experimentally, leading to an empirical explanation for the instability. In the experiment, particle diameter grew with time. Two instability modes appeared after growth to a sufficient size. A filamentary mode appeared abruptly, and later a great void mode developed as a dust-free region with an intense glow inside and a sharp boundary outside. These modes were characterized by two-dimensional laser light scattering, video imaging, optical emission spectroscopy, Langmuir probe measurements, and Fourier analysis of the fluctuation spectrum. Dust growth was measured by electron microscopy and optical extinction, yielding the dust particle size and dust number density. The electron density was found to be enhanced inside the great void, due to an absence of electron depletion on the dust grains. The great void was explained by the ion drag force, which becomes stronger than the opposing Coulomb force once the particle size reaches a critical diameter. When a dust-free region develops, its electron density is enhanced, the ionization rate increases, and the ion flow that pushes particles outward is further augmented. The plasma used in the experiment grew particles by sputtering of the electrodes, although the same instabilities are expected to occur in other types of dusty plasma discharges as well.

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Instabilities in a dusty plasma with ion drag and ionization

Observations show that plasma crystals, suspended in the sheath of a radio-frequency discharge, rotate under the influence of a vertical magnetic field Depending on the discharge conditions, two different cases are observed: a rigid-body rotation (all the particles move with a constant angular velocity) and sheared rotation (the angular velocity of particles has a radial distribution) When the discharge voltage is increased sufficiently, the particles may even reverse their direction of motion A simple analytical model is used to explain qualitatively the mechanism of the observed particle motion and its dependence on the confining potential and discharge conditions The model takes into account electrostatic, ion drag, neutral drag, and effective interparticle interaction forces For the special case of rigid-body rotation, the confining potential is reconstructed Using data for the radial dependence of particle rotation velocity, the shear stresses are estimated The critical shear stress at which shear-induced melting occurs is used to roughly estimate the shear elastic modulus of the plasma crystal The latter is also used to estimate the viscosity contribution due to elasticity in the plasma liquid Further development is suggested in order to quantitatively implement these ideas

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Rigid and differential plasma crystal rotation induced by magnetic fields

Transverse shear waves were observed experimentally in a two-dimensional screened Coulomb crystal. They were excited by applying a chopped laser beam to a 2D dusty plasma, i.e., a monolayer of charged microspheres levitated in a plasma. Measurements of the dispersion relation reveal an acoustic, i.e., nondispersive, character over the entire range of wave numbers measured, 0.2<k(r)a/pi<0.7, where a is the interparticle spacing. Comparison to theory provides a measurement of the particles' charge.

/pdf/transverse-waves-in-a-two-dimensional-screened-coulomb-34h0n5hihw.pdf

Transverse waves in a two-dimensional screened-coulomb crystal (Dusty plasma)

Solitary waves are experimentally studied in a monolayer hexagonal dust lattice which is formed from monodisperse plastic microspheres and levitated in the sheath of an rf discharge. It is found that the product of the soliton amplitude and the square of the soliton width is constant as the soliton propagates. The analytical theory describing the experiment is based on the equations of motion written for a linear chain. It takes into account damping, dispersion, and nonlinearity. The numerical simulation of a linear chain produces double solitons like those observed in the experiment.

Dissipative longitudinal solitons in a two-dimensional strongly coupled complex (dusty) plasma.

Shock waves with a linear front were experimentally studied in a monolayer hexagonal Yukawa lattice which was formed from charged monodisperse plastic microspheres and levitated in the sheath of a radio-frequency discharge. It was found that the shock can cause phase transitions from a crystalline to gaslike and liquidlike states. Melting occurred in two stages. First, the lattice was compressed in the direction of shock propagation and second, the particle velocities were randomized a few lattice lines downstream. The Mach number of the shock reached 2.7.

D. Samsonov

Papers

Instabilities in a dusty plasma with ion drag and ionization

Rigid and differential plasma crystal rotation induced by magnetic fields

Transverse waves in a two-dimensional screened-coulomb crystal (Dusty plasma)

Dissipative longitudinal solitons in a two-dimensional strongly coupled complex (dusty) plasma.

Shock Melting of a Two-Dimensional Complex (Dusty) Plasma