Showing papers in "Reviews of Modern Physics in 1999"

Theory of Bose-Einstein condensation in trapped gases

Abstract: The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground-state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.

Quantum Field Theory

Abstract: Quantum field theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the standard model, are formulated. Quantum electrodynamics (QED), besides providing a complete foundation for atomic physics and chemistry, has supported calculations of physical quantities with unparalleled precision. The experimentally measured value of the magnetic dipole moment of the muon, $${\left({{g_\mu } - 2} \right)_{\exp }} = 233\,184\,600\,\left({1680} \right) \times {10^{ - 11}},$$ for example, should be compared with the theoretical prediction $${\left({{g_\mu } - 2} \right)_{{\rm{theor}}}} = 233\,183\,478\,\left( {308} \right) \times {10^{ - 11}}$$ (see the chapter by Hughes and Kinoshita on pp. 223-233 in this book).

Nobel Lecture: Electronic structure of matter-wave functions and density functionals

Abstract: In the intervening more than six decades enormous progress has been made in finding approximate solutions of Schrodinger's wave equation for systems with several electrons, decisively aided by modern electronic com- puters. The outstanding contributions of my Nobel Prize co-winner John Pople are in this area. The main objec- tive of the present account is to explicate DFT, which is an alternative approach to the theory of electronic struc- ture, in which the electron density distribution n(r), rather than the many-electron wave function, plays a central role. I felt that it would be useful to do this in a comparative context; hence the wording ''Wave Func- tions and Density Functionals'' in the title. In my view DFT makes two kinds of contribution to the science of multiparticle quantum systems, including problems of electronic structure of molecules and of condensed matter. The first is in the area of fundamental understanding. Theoretical chemists and physicists, following the path of the Schrodinger equation, have become accustomed to think in terms of a truncated Hilbert space of single-

Structure and phase transitions in Langmuir monolayers

Abstract: Lipid monolayers on the surface of water have been studied for over a hundred years, but in the last decade there has been a dramatic evolution in our understanding of the structures and phase transitions of these systems, driven by new experimental techniques and theoretical advances. In this review, dense monolayers of simple lipids are described in detail, including structures revealed by x-ray-diffraction experiments, computer simulations, molecular models, and a phenomenological theory of phase transitions. The effects of chirality and the structures of phospholipid monolayers are considered. Open questions and possible approaches to finding answers are discussed.

Linear scaling electronic structure methods

Abstract: Methods exhibiting linear scaling with respect to the size of the system, the so-called O(N) methods, are an essential tool for the calculation of the electronic structure of large systems containing many atoms. They are based on algorithms that take advantage of the decay properties of the density matrix. In this article the physical decay properties of the density matrix will first be studied for both metals and insulators. Several strategies for constructing O(N) algorithms will then be presented and critically examined. Some issues that are relevant only for self-consistent O(N) methods, such as the calculation of the Hartree potential and mixing issues, will also be discussed. Finally some typical applications of O(N) methods are briefly described.

Drift waves and transport

Abstract: Drift waves occur universally in magnetized plasmas producing the dominant mechanism for the transport of particles, energy and momentum across magnetic field lines. A wealth of information obtained from quasistationary laboratory experiments for plasma confinement is reviewed for drift waves driven unstable by density gradients, temperature gradients and trapped particle effects. The modern understanding of Bohm transport and the role of sheared flows and magnetic shear in reducing the transport to the gyro-Bohm rate are explained and illustrated with large scale computer simulations. The types of mixed wave and vortex turbulence spontaneously generated in nonuniform plasmas are derived with reduced magnetized fluid descriptions. The types of theoretical descriptions reviewed include weak turbulence theory, Kolmogorov anisotropic spectral indices, and the mixing length. A number of standard turbulent diffusivity formulas are given for the various space-time scales of the drift-wave turbulent mixing.

Spontaneous ordering of nanostructures on crystal surfaces

Abstract: A review is given of theoretical concepts and experimental results on spontaneous formation of periodically ordered nanometer-scale structures on crystal surfaces. Thermodynamic theory is reviewed for various classes of spontaneously ordered nanostructures, namely, for periodically faceted surfaces, for periodic surface structures of planar domains, and for ordered arrays of three-dimensional coherently strained islands. All these structures are described as equilibrium structures of elastic domains. Despite the fact that driving forces of the instability of a homogeneous phase are different in each case, the common driving force for the long-range ordering of the inhomogeneous phase is the elastic interaction. The theory of the formation of multisheet structures of islands is reviewed, which is governed by both equilibrium ordering and kinetic-controlled ordering. For the islands of the first sheet, an equilibrium structure is formed, and for the next sheets, the structure of the surface islands meets the equilibrium under the constraint of the fixed structures of the buried islands. The experimental situation for the fabrication technology of ordered arrays of semiconductor quantum dots is analyzed, including a discussion of both single-sheet and multiple-sheet ordered arrays.

Scaling, Universality, and Renormalization: Three Pillars of Modern Critical Phenomena

Abstract: Suppose we have a simple bar magnet. We know it is a ferromagnet because it is capable of picking up thumbtacks, the number of which is called the order parameter M. As we heat this system, M decreases and eventually, at a certain critical temperature T c , it reaches zero: no more thumbtacks remain! In fact, the transition is remarkably sharp, since M approaches zero at T c with infinite slope. Such singular behavior is an example of a “critical phenomenon.”

Experiments and theory in cold and ultracold collisions

Abstract: The authors review progress in understanding the nature of atomic collisions occurring at temperatures ranging from the millidegrees Kelvin to the nanodegrees Kelvin regime. The review includes advances in experiments with atom beams, light traps, and purely magnetic traps. Semiclassical and fully quantal theories are described and their appropriate applicability assessed. The review divides the subject into two principal categories: collisions in the presence of one or more light fields and ground-state collisions in the dark.

Photodissociation regions in the interstellar medium of galaxies

Abstract: The interstellar medium of galaxies is the reservoir out of which stars are born and into which stars inject newly created elements as they age. The physical properties of the interstellar medium are governed in part by the radiation emitted by these stars. Far-ultraviolet (6 eV less than h(nu) less than 13.6 eV) photons from massive stars dominate the heating and influence the chemistry of the neutral atomic gas and much of the molecular gas in galaxies. Predominantly neutral regions of the interstellar medium in which the heating and chemistry are regulated by far ultraviolet photons are termed Photo-Dissociation Regions (PDRs). These regions are the origin of most of the non-stellar infrared (IR) and the millimeter and submillimeter CO emission from galaxies. The importance of PDRs has become increasingly apparent with advances in IR and submillimeter astronomy. The IR emission from PDRs includes fine structure lines of C, C+, and O; rovibrational lines of H2, rotational lines of CO; broad middle features of polycyclic aromatic hydrocarbons; and a luminous underlying IR continuum from interstellar dust. The transition of H to H2 and C+ to CO occurs within PDRs. Comparison of observations with theoretical models of PDRs enables one to determine the density and temperature structure, the elemental abundances, the level of ionization, and the radiation field. PDR models have been applied to interstellar clouds near massive stars, planetary nebulae, red giant outflows, photoevaporating planetary disks around newly formed stars, diffuse clouds, the neutral intercloud medium, and molecular clouds in the interstellar radiation field-in summary, much of the interstellar medium in galaxies. Theoretical PDR models explain the observed correlations of the [CII] 158 microns with the COJ = 1-0 emission, the COJ = 1-0 luminosity with the interstellar molecular mass, and the [CII] 158 microns plus [OI] 63 microns luminosity with the IR continuum luminosity. On a more global scale, MR models predict the existence of two stable neutral phases of the interstellar medium, elucidate the formation and destruction of star-forming molecular clouds, and suggest radiation-induced feedback mechanisms that may regulate star formation rates and the column density of gas through giant molecular clouds.

Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion

Abstract: important corrections are presented. These corrections are calculated with the radiative transfer or Schwarzschild-Milne equation, which describes intensity transport at the ‘‘mesoscopic’’ level and is derived from the ‘‘microscopic’’ wave equation. A precise treatment of the diffuse intensity is derived which automatically includes the effects of boundary layers. Effects such as the enhanced backscatter cone and imaging of objects in opaque media are also discussed within this framework. This approach is extended to mesoscopic correlations between multiple scattered intensities that arise when scattering is strong. These correlations arise from the underlying wave character. The derivation of correlation functions and intensity distribution functions is given and experimental data are discussed. Although the focus is on light scattering, the theory is also applicable to microwaves, sound waves, and noninteracting electrons. [S0034-6861(99)00601-7]

Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states)

Abstract: Plasmas consisting exclusively of particles with a single sign of charge (e.g., pure electron plasmas and pure ion plasmas) can be confined by static electric and magnetic fields (in a Penning trap) and also be in a state of global thermal equilibrium. This important property distinguishes these totally unneutralized plasmas from neutral and quasineutral plasmas. This paper reviews the conditions for, and the structure of, the thermal equilibrium states. Both theory and experiment are discussed, but the emphasis is decidedly on theory. It is a huge advantage to be able to use thermal equilibrium statistical mechanics to describe the plasma state. Such a description is easily obtained and complete, including for example the details of the plasma shape and microscopic order. Pure electron and pure ion plasmas are routinely confined for hours and even days, and thermal equilibrium states are observed. These plasmas can be cooled to the cryogenic temperature range, where liquid and crystal-like states are realized. The authors discuss the structure of the correlated states separately for three plasma sizes: large plasmas, in which the free energy is dominated by the bulk plasma; mesoscale plasmas, in which the free energy is strongly influenced by the surface; and Coulomb clusters, in which the number of particles is so small that the canonical ensemble is not a good approximation for the microcanonical ensemble. All three cases have been studied through numerical simulations, analytic theory, and experiment. In addition to describing the structure of the thermal equilibrium states, the authors develop a thermodynamic theory of the trapped plasma system. Thermodynamic inequalities and Maxwell relations provide useful bounds on and general relationships between partial derivatives of the various thermodynamic variables.

Brane dynamics and gauge theory

Abstract: The authors review some aspects of the interplay between the dynamics of branes in string theory and the classical and quantum physics of gauge theories with different numbers of supersymmetries in various dimensions.

Conductance viewed as transmission

Abstract: Early quantum theories of electrical conduction were semiclassical. Electrons were accelerated according to Bloch’s theorem; this was balanced by back scattering due to phonons and lattice defects. Cross sections for scattering, and band structures, were calculated quantum-mechanically, but the balancing process allowed only for occupation probabilities, not permitting a totally coherent process. Also, in most instances, scatterers at separate locations were presumed to act incoherently. Totally quantum-mechanical theories stem from the 1950s, and have diverse sources. Particularly intense concern with the need for more quantum mechanical approaches was manifested in Japan, and Kubo’s formulation became the most widely accepted version. Quantum theory, as described by the Schrodinger equation, is a theory of conservative systems, and does not allow for dissipation. The Schrodinger equation readily allows us to calculate polarizability for atoms, molecules, or other isolated systems that do not permit electrons to enter or leave. Kubo’s linear-response theory is essentially an extended theory of polarizability. Some supplementary handwaving is needed to calculate a dissipative effect such as conductance, for a sample with boundaries where electrons enter and leave (Anderson, 1997). After all, no theory that ignores the interfaces of a sample to the rest of its circuit can possibly calculate the resistance of such a sample of limited extent. Modern microelectronics has provided the techniques for fabricating very small samples. These permit us to study conductance in cases where the carriers have a totally quantum mechanically coherent history within the sample, making it essential to take the interfaces into account. Mesoscopic physics, concerned with samples that are intermediate in size between the atomic scale and the macroscopic one, can now demonstrate in manufactured structures much of the quantum mechanics we associate with atoms and molecules.

Growth of nanostructures by cluster deposition: Experiments and simple models

Abstract: This paper presents simple models useful in analyzing the growth of nanostructures obtained by cluster deposition. After a brief survey of applications and experimental methods, the author describes the Monte Carlo techniques for simulating nanostructure growth. Simulations of the first stages, the submonolayer regime, are reported for a wide variety of experimental situations: complete condensation, growth with reevaporation, nucleation on defects, and total or null cluster-cluster coalescence. [Note: Software for all these simulation programs, which are also useful for analyzing growth from atomic beams, is available on request from the author.] The aim of the paper is to help experimentalists, in analyzing their data, to determine which processes are important and to quantify them. Experiments on growth from cluster beams are discussed, as is the measurement of cluster mobility on the surface. Surprisingly high mobility values are found. An important issue for future technological applications of cluster deposition is the relation between the size of the incident clusters and the size of the islands obtained on the substrate, which is described by an approximate formula depending on the melting temperature of the deposited material. Finally, the author examines the atomic mechanisms that can explain the diffusion of clusters on a substrate and their mutual interaction, to aggregate keeping their integrity or to coalesce. [S0034-6861(99)00405-5]

The `Friction' of Vacuum, and other Fluctuation-Induced Forces

Abstract: The static Casimir effect describes an attractive force between two conducting plates, due to quantum fluctuations of the electromagnetic (EM) field in the intervening space. Thermal fluctuations of correlated fluids (such as critical mixtures, super-fluids, liquid crystals, or electrolytes) are also modified by the boundaries, resulting in finite-size corrections at criticality, and additional forces that affect wetting and layering phenomena. Modified fluctuations of the EM field can also account for the ``van der Waals'' interaction between conducting spheres, and have analogs in the fluctuation-induced interactions between inclusions on a membrane. We employ a path integral formalism to study these phenomena for boundaries of arbitrary shape. This allows us to examine the many unexpected phenomena of the dynamic Casimir effect due to moving boundaries. With the inclusion of quantum fluctuations, the EM vacuum behaves essentially as a complex fluid, and modifies the motion of objects through it. In particular, from the mechanical response function of the EM vacuum, we extract a plethora of interesting results, the most notable being: (i) The effective mass of a plate depends on its shape, and becomes anisotropic. (ii) There is dissipation and damping of the motion, again dependent upon shape and direction of motion, due to emission of photons. (iii) There is a continuous spectrum of resonant cavity modes that can be excited by the motion of the (neutral) boundaries.

Nonlinear optics of normal-mode-coupling semiconductor microcavities

Abstract: The authors review the nonlinear optical properties of semiconductor quantum wells that are grown inside high-Q Bragg-mirror microcavities. Light-matter coupling in this system is particularly pronounced, leading in the linear regime to a polaritonic mixing of the excitonic quantum well resonance and the single longitudinal cavity mode. The resulting normal-mode splitting of the optical resonance is observed in reflection, transmission, and luminescence experiments. In the nonlinear regime the strong light-matter coupling influences the excitation-dependent bleaching of the normal-mode resonances for nonresonant excitation, leads to transient saturation and normal-mode oscillations for resonant pulsed excitation and is responsible for the density-dependent signatures in the luminescence characteristics. These and many more experimental observations are summarized and explained in this review using a microscopic theory for the Coulomb interacting electron-hole system in the quantum well that is nonperturbatively coupled to the cavity light field.

Granular Matter: A Tentative View

Abstract: Granular matter refers to particle systems in which the size d is larger than one micron. Below one micron, thermal agitation is important, and Brownian motion can be seen. Above one micron, thermal agitation is negligible. We are interested here in many-particle systems, at zero temperature, occupying a large variety of metastable states: if we pour sand on a table, it would like to go to a ground state, with a monolayer of grains giving the lowest gravitational energy. But in reality the sand remains as a heap; the shape of the heap and the stress distribution inside depend critically on how the heap was made. Hence come many difficulties.

Experiment and the foundations of quantum physics

Abstract: Quantum physics, a child of the early 20th century, is probably the most successful description of nature ever invented by man. The range of phenomena it has been applied to is enormous. It covers phenomena from the elementary-particle level all me way to the physics of the early universe. Many modern technologies would be impossible without quantum physics—witness, for example, that all information technologies are based on a quantum understanding of solids, particularly of semiconductors, or that the operation of lasers is based on a quantum understanding of atomic and molecular phenomena.

Built upon sand: Theoretical ideas inspired by granular flows

Abstract: Granulated materials, like sand and sugar and salt, are composed of many pieces that can move independently. The study of collisions and flow in these materials requires new theoretical ideas beyond those in the standard statistical mechanics or hydrodynamics or traditional solid mechanics. Granular materials differ from standard molecular materials in that frictional forces among grains can dissipate energy and drive the system toward frozen or glassy configurations. In experimental studies of these materials, one sees complex flow patterns similar to those of ordinary liquids, but also freezing, plasticity, and hysteresis. To explain these results, theorists have looked at models based upon inelastic collisions among particles. With the aid of computer simulations of these models they have tried to build a ``statistical dynamics'' of inelastic collisions. One effect seen, called inelastic collapse, is a freezing of some of the degrees of freedom induced by an infinity of inelastic collisions. More often some degrees of freedom are partially frozen, so that there can be a rather cold clump of material in correlated motion. Conversely, thin layers of material may be mobile, while all the material around them is frozen. In these and other ways, granular motion looks different from movement in other kinds of materials. Simulations in simple geometries may also be used to ask questions like When does the usual Boltzmann-Gibbs-Maxwell statistical mechanics arise?, What are the nature of the probability distributions for forces between the grains?, and Might the system possibly be described by uniform partial differential equations? One might even say that the study of granular materials gives one a chance to reinvent statistical mechanics in a new context.

Nobel Lecture: Quantum chemical models

Abstract: The fundamental underpinnings of theoretical chemistry were uncovered in a relatively short period at the beginning of the present century. Rutherford’s discovery of the nucleus in 1910 completed the identification of the constituent subparticles of atoms and molecules and was followed shortly thereafter by the Bohr treatment of electronic orbits in atoms, the ‘‘old quantum theory.’’ The relation between the positive nuclear charge, atomic number and position of an atom in the periodic table was uncovered by 1913. It proved difficult to extend Bohr’s orbits to a polyatomic situation and the next advance had to await the development of the wave theory of matter and the associated quantum mechanics in the early 1920s. By 1926, Heisenberg had developed matrix mechanics and Schrödinger had proposed the basic nonrelativistic wave equation governing the motion of nuclei and electrons in molecules. The latter,

Dynamic transitions and hysteresis

Abstract: When an interacting many-body system, such as a magnet, is driven in time by an external perturbation, such as a magnetic field, the system cannot respond instantaneously due to relaxational delay. The response of such a system under a time-dependent field leads to many novel physical phenomena with intriguing physics and important technological applications. For oscillating fields, one obtains hysteresis that would not occur under quasistatic conditions in the presence of thermal fluctuations. Under some extreme conditions of the driving field, one can also obtain a nonzero average value of the variable undergoing such ``dynamic hysteresis.'' This nonzero value indicates a breaking of the symmetry of the hysteresis loop about the origin. Such a transition to the ``spontaneously broken symmetric phase'' occurs dynamically when the driving frequency of the field increases beyond its threshold value, which depends on the field amplitude and the temperature. Similar dynamic transitions also occur for pulsed and stochastically varying fields. We present an overview of the ongoing research in this not-so-old field of dynamic hysteresis and transitions.

The Fractional Quantum Hall Effect

Abstract: Two-dimensional electron systems in a high magnetic field behave very strangely. They exhibit rational fractional quantum numbers and contain exactly fractionally charged particles. Electrons seem to absorb magnetic flux quanta, altering their statistics and consuming the magnetic field. They condense into a manifold of novel ground states of boson and fermion character. These fascinating properties are not characteristic of any individual electron but rather emerge from the highly correlated motion of many.

The spatial behavior of nonclassical light

Abstract: Nonclassical effects such as squeezing, antibunching, and sub-Poissonian statistics of photons have been attracting attention in quantum optics over the last decade. Up to now most theoretical and experimental investigations have been carried out exclusively in the time domain while neglecting the spatial aspects by considering only one spatial mode of the electromagnetic field. In many situations such an approximation is well justified. There are, however, problems that do not allow in principle a single-mode consideration. This is the case when one wants to investigate the quantum fluctuations of light at different spatial points in the plane perpendicular to the direction of propagation of the light beam. Such an investigation requires a complete description of quantum fluctuations of light in both time and space and cannot be done within a single-mode theory. This space-time description brings about a natural generalization into the spatial domain of such notions as the standard quantum limit, squeezing, antibunching, etc. It predicts, for example, the possibility of generating a light beam with sub-Poissonian statistics of photons not only in time but also in the beam's transverse plane. Of particular relevance to the applications is a situation in which the cross section of the light beam contains several nonoverlapping areas with sub-Poissonian statistics of photons in each. Photodetection of such a beam produces several sub-shot-noise photocurrents depending on the number of independent areas with sub-Poissonian statistics. This is in marked contrast to the case of a single-mode sub-Poissonian light beam in which any attempt to collect light from only a part of the beam deteriorates the degree of shot-noise reduction. This property of multimode squeezed light opens a range of interesting new applications in optical imaging, optical parallel processing of information, parallel computing, and many other areas in which it is desirable to have a light beam with regular photon statistics across its transverse area. The aim of this review is to describe the recent development in this branch of quantum optics.

History of the search for continuous melting

Abstract: The melting of crystals has resisted efforts to understand the microscopic process for more than a century. The course of the struggle has stimulated the development of quantum mechanics, concepts of long-range order, the role of dimensionality in condensed matter, surface physics, and more sensitive experimental techniques to test the theories. After years of probing the mechanism within the bulk material, we learn that the answer has been lying on the surface.

Nobel Lecture: Fractional quantization

Abstract: One of my favorite times in the academic year occurs in early spring when I give my class of extremely bright graduate students, who have mastered quantum mechanics but are otherwise unsuspecting and innocent, a take-home exam in which they are asked to deduce superfluidity from first principles. There is no doubt a special place in hell being reserved for me at this very moment for this mean trick, for the task is impossible. Superfluidity, like the fractional quantum Hall effect, is an emergent phenomenon—a low-energy collective effect of huge numbers of particles that cannot be deduced from the microscopic equations of motion in a rigorous way and that disappears completely when the system is taken apart (Anderson, 1972). There are prototypes for superfluids, of course, and students who memorize them have taken the first step down the long road to understanding the phenomenon, but these are all approximate and in the end not deductive at all, but fits to experiment. The students feel betrayed and hurt by this experience because they have been trained to think in reductionist terms and thus to believe that everything not amenable to such thinking is unimportant. But nature is much more heartless than I am, and those students who stay in physics long enough to seriously confront the experimental record eventually come to understand that the reductionist idea is wrong a great deal of the time, and perhaps always. One common response in the early stages of learning is that superconductivity and the quantum Hall effect are not fundamental and therefore not worth taking seriously. When this happens I just open up the AIP Handbook and show the disbeliever that the accepted values of e and h are defined by these effects, and that ends that. The world is full of things for which one’s understanding, i.e., one’s ability to predict what will happen in an experiment, is degraded by taking the system apart, including most delightfully the standard model of elementary particles itself. I myself have come to suspect most of the important outstanding problems in physics are emergent in nature, including particularly quantum gravity. One of the things an emergent phenomenon can do is create new particles. When a large number of atoms condense into a crystal, the phonon, the elementary quantum of sound, becomes a perfectly legitimate particle at low energy scales. It propagates freely, does not decay, carries momentum and energy related to wave-

Molecular chirality and chiral parameters

Abstract: The fundamental issues of symmetry related to chirality are discussed and applied to simple situations relevant to liquid crystals The authors show that any chiral measure of a geometric object is a pseudoscalar (invariant under proper rotations but changing sign under improper rotations) and must involve three-point correlations that only come into play when the molecule has at least four atoms In general, a molecule is characterized by an infinite set of chiral parameters The authors illustrate the fact that these parameters can have differing signs and can vanish at different points as a molecule is continuously deformed into its mirror image From this it is concluded that handedness is not an absolute concept but depends on the property being observed Within a simplified model of classical interactions, the chiral parameter of the constituent molecules that determines the macroscopic pitch of cholesterics is identified

Nobel Lecture: The fractional quantum Hall effect

Abstract: The fractional quantum Hall effect is a very counterintuitive physical phenomenon. It implies that many electrons, acting in concert, can create new particles having a charge smaller than the charge of any individual electron. This is not the way things are supposed to be. A collection of objects may assemble to form a bigger object, or the parts may remain their size, but they don’t create anything smaller. If the new particles were doubly charged, it wouldn’t be so paradoxical— electrons could ‘‘just stick together’’ and form pairs. But fractional charges are very bizarre indeed. Not only are they smaller than the charge of any constituent electron, but they are exactly 1/3 or 1/5 or 1/7 etc. of an electronic charge, depending on the conditions under which they have been prepared. And yet we know with certainty that none of these electrons has split up into pieces. Fractional charge is the most puzzling of the observations, but there are others. Quantum numbers—usually integers or half-integers—turn out to be also fractional, such as 2/5, 4/9, and 11/7, or even 5/23. Moreover, bits of magnetic field can get attached to each electron, creating yet other objects. Such composite particles have properties very different from those of the electrons. They sometimes seem to be oblivious to huge magnetic fields and move in straight lines, although any bare electron would orbit on a very tight circle. Their mass is unrelated to the mass of the original electron but arises solely from interactions with their neighbors. More so, the attached magnetic field changes drastically the characteristics of the particles, from fermions to bosons and back to fermions, depending on the field strength. And finally, some of these composites are conjectured to coalesce and form pairs, vaguely similar to the formation of electron pairs in superconductivity. This would provide yet another astounding new state with weird properties. All of these strange phenomena occur in twodimensional electron systems at low temperatures exposed to a high magnetic field—only electrons and a magnetic field. The electrons reside within a solid, at the interface between two slightly different semiconductors. This is presently the smoothest plane we can fabricate to restrict the electrons’ motion to two dimensions. Quantum mechanics does the rest.

Quantum effects in one-photon and two-photon interference

Abstract: Although interference is intrinsically a classical wave phenomenon, the superposition principle which underlies all interference is also at the heart of quantum mechanics. Feynman has referred to interference as really “the only mystery” of quantum mechanics. Furthermore, in some interference experiments we encounter the idea of quantum entanglement, which has also been described as really the only quantum mystery. Clearly interference confronts us with some quite basic questions of interpretation. Despite its long history, going back to Thomas Young at the beginning of the 19th century, optical interference still challenges our understanding, and the last word on the subject probably has not yet been written. With the development of experimental techniques for fast and sensitive measurements of light, it has become possible to carry out many of the Gedanken experiments whose interpretation was widely debated in the 1920s and 1930s in the course of the development of quantum mechanics. Although this article focuses entirely on experiments with light, interference has also been observed with many kinds of material particles like electrons, neutrons, and atoms. We particularly draw the reader’s attention to the beautiful experiments with neutron beams by Rauch and co-workers and others (see, for example, Badurek et al.,1988). Quantum optical interference effects are key topics of a recent book (Greenstein and Zajonc, 1997), an extended rather thorough review (Buzek and Knight, 1995) and an article in Physics Today(Greenberger et al.,1993).