Page conjectured very recently [Phys. Rev. Lett. 71, 1291 (1993)] that if a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension m≤n is given by ${\mathit{S}}_{\mathit{m},}$n=${\mathit{tsum}}_{\mathit{k}=\mathit{n}+1\mathrm{}}^{\mathit{m}\mathit{n}}$1/k-(m-1)/2n. We outline a proof of this elegant formula.

We show that, in a narrow gap in the spectrum of two linearly coupled Korteweg–de Vries equations with opposite signs of the dispersion coefficient, a two-parameter family of solitons of a novel type may exist. These are envelope solitons with decaying oscillating tails, which are radically different from the gap solitons previously known in nonlinear optics. In particular, they may become singular at some value of the velocity, and degenerate into algebraic solitons in another special case. It is demonstrated that gap solitons of the same type may also exist in a nonlinear optical system consisting of focusing and defocusing tunnel-coupled planar lightguides.

We consider the geometric phase factor in quantum mechanics for the case of the continuous spectrum, and we find that it is nothing but the S matrix; for certain specific states the geometric phase is simply twice the usual phase shift.

The nonlocality responsible for violations of Bell’s inequalities is not equivalent to that used in teleportation, although they probably are two aspects of the same physical property. There are mixed states which do not violate any Bell type inequality, but still can be used for teleportation.

A new 1D discrete nonlinear Schrödinger (NLS) Hamiltonian is introduced which includes the integrable Ablowitz-Ladik system as a limit. The symmetry properties of the system are studied. The relationship between intrinsic localized states and the soliton of the Ablowitz-Ladik NLS is discussed, including the role of discretization as a mechanism controlling collapse. It is pointed out that a staggered localized state can be viewed as a particle of a negative effective mass. It is shown that staggered localized states can exist in the discrete dark NLS. The motion of localized states and Peierls-Nabarro pinning are also studied.

Let A and B be two atoms or, more generally, a ‘‘source’’ and a ‘‘detector’’ separated by some distance R. At t=0A is in an excited state, B in its ground state, and no photons are present. A theorem is proved that in contrast to Einstein causality and finite signal velocity the excitation probability of B is nonzero immediately after t=0. Implications are discussed.

The statistical mechanics of an anyon gas in a magnetic field is addressed. A harmonic regulator is used to define a proper thermodynamic limit. When the magnetic field is sufficiently strong, only exact N-anyon ground states, where anyons occupy the lowest Landau level, contribute to the equation of state. Particular attention is paid to the interval of definition of the statistical parameter α∈[-1,0] where a gap exists. Interestingly enough, one finds that at the critical filling ν=-1/α where the pressure diverges, the external magnetic field is entirely screened by the flux tubes carried by the anyons.

By numerical and analytic study of the exact solution to the six-vertex model in applied fields, we explore in detail an unusual coexistence point which occurs in the low-temperature ferroelectric regime. This point, whose existence had been previously conjectured, corresponds to additional singularities in the free energy which were overlooked in the discussion of the original solution of the model. Furthermore, we find interesting scaling properties in the vicinity of this point. Implications for crystal shapes and the phase separation of unstable crystal surfaces are also discussed.

We consider the focusing and deflection of a nonresonant atomic beam propagating through a spatially inhomogeneous quantized electromagnetic field. Different Fock states of the field deflect the atoms in different angles and focus them at different points. We find a regime in which individual foci corresponding to neighboring Fock states can be reoslved even for large average number of photons. In this sense this quantum lens for atomic waves leaves intact the discreteness of the photons even in the classical limit.

A new continuum model for hard-core fermions with ν colors is presented and solved exactly in arbitrary dimensions. The model has the SU(ν,1) supersymmetry in an exchange interaction and a kinetic-energy part. Through a renormalization to regularize operator products at short distances, free SU(ν) fermions emerge as eigenstates of the model. This corresponds to a nonrelativistic and supersymmetric version of the equivalence of SU(ν)×SU(ν) Wess-Zumino-Witten Lagrangian to free relativistic fermions.

The renormalizability of the self-avoiding manifold Edwards model is established. We use a new short distance multilocal operator product expansion, which extends methods of local field theories to a large class of models with nonlocal singular interactions. This validates the direct renormalization method introduced before, as well as scaling laws. A new general hyperscaling relation is derived. Manifolds at the FTHETA point and long-range Coulomb interactions are briefly discussed.

We analyze the low energy behavior of interacting fermions in continuous dimensions d between one and two. It is shown that Fermi liquid fixed points are stable with respect to residual scattering of quasiparticles by regular interactions in any dimension above one, while the structure of corrections to quasiparticle behavior changes drastically below two dimensions. The crossover from Luttinger liquid behavior in 1D to Fermi liquid behavior in higher dimensions is described by a tomographic Luttinger model with effective interactions approaching zero with a power d-1 of the distance from the Fermi surface.

We prove that for any Monte Carlo algorithm of Metropolis type, the autocorraletion time of a suitable ‘‘energy’’-like observable is bounded below by a multiple of the corresponding ‘‘specific heat.’’ This bound does not depend on whether the proposed moves are local or nonlocal; it depends only on the distance between the desired probability distribution π and the probability distribution ${\mathrm{\pi }}^{\mathrm{}\left(0\right)\mathrm{}}$ for which the proposal matrix satisfies detailed balance. We show, with several examples, that this result is particularly powerful when applied to nonlocal algorithms.

In atomic Stern-Gerlach (polarization) interferometry with time-dependent magnetic fields both the spatial and temporal parts of the atomic phase are generally affected. This leads to a total energy shift and to an inelastic momentum transfer. Some of the related effects are studied using a beam of metastable hydrogen atoms. In particular the scalar Aharonov-Bohm effect and its nondispersivity are considered, in addition to other phase shift effects occurring where the field gradient differs from zero.

It is shown that a neutral particle with an electric dipole moment which moves in a magnetic field acquires a topological phase. This phase may be observed in atom or molecular interferometry.

We introduce a new type of boundary conditions, smooth boundary conditions, for numerical studies of quantum lattice systems. In a number of circumstances, these boundary conditions have substantially smaller finite-size effects than periodic or open boundary conditions. They can be applied to nearly any short-ranged Hamiltonian system in any dimensionality and within almost any type of numerical approach.

Using independent sources one can realize an ‘‘event-ready’’ Bell–Einstein-Podolsky-Rosen experiment in which one can measure directly the probabilities of the various outcomes including nondetection of both particles. Our proposal involves two parametric down-converters. Subcoherence-time monitoring of the idlers provides a noninteractive quantum measurement entangling and preselecting the independent signals without touching them. We give the conditions for high fringe visibility and particle collection efficiency as required for a Bell test.

After establishing a formal theory for getting solutions of one type of high-dimensional partial differential equation, two sets of generalized symmetries of the 3D Toda theory, which arises from a particular reduction of the 4D self-dual gravity equation, are obtained concretely by a simple formula. Each set of symmetries constitutes a generalized ${\mathrm{\omega }}_{\mathrm{\infty }}$ algebra which contains three types of the usual ${\mathrm{\omega }}_{\mathrm{\infty }}$ algebras as special cases. Some open questions are discussed.

A novel mathematically simple and physically transparent derivation of the Efimov effect is presented by means of a new efficient method for solving coordinate space Faddeev equations. The method, which is directly applicable on small computers, is used to obtain energies and wave functions of the Efimov states.

We report a new two-photon polarization correlation experiment for realizing the Einstein-Podolsky-Rosen-Bohm (EPRB) state and for testing Bell-type inequalities. We use the pair of orthogonally polarized light quanta generated in type-II parametric down-conversion. In a 1 nm bandwidth, we observe from the output of a 0.5 mm β-${\mathrm{BaB}}_2$${\mathrm{O}}_4$ crystal the EPRB correlations in coincidence counts, and measure an associated Bell inequality violation of 22 standard deviations.