A Comment on the Letter by Boris Levitan, Phys. Rev. Lett. 72, 4057 (1994).

A Reply to the Comment by D. Weaire.

The sum of the information gains corresponding to measurements of position and momentum is bounded by $\log 2\Delta X\Delta P/\hslash$ for a quantum ensemble with position and momentum uncertainties $\Delta X$ and $\Delta P$. The bound implies the Heisenberg uncertainty principle and that the gain of position information can be maximized only at the expense of momentum information, and vice versa. This exclusion principle for the information contents of complementary observables is extended to finite Hilbert spaces, and to the quadrature, number, and phase observables of bosonic fields degraded by Gaussian noise.

We calculate the probability to find exactly $n$ eigenvalues in a spectral interval of a large random $N\times N$ matrix when this interval contains $s\ll N$ eigenvalues on average. The calculations exploit an analogy to the problem of finding a two-dimensional charge distribution on the interface of a semiconductor heterostructure under the influence of a split gate.

We present the exact solution of the $\mathrm{Osp}\left(1|2\right)$ invariant magnet by the Bethe ansatz approach. The associated Bethe ansatz equation exhibits a new feature by presenting an explicit and distinct phase behavior in even and odd sectors of the theory. The ground state, the low-lying excitations, and the critical properties are discussed by exploiting the Bethe ansatz solution.

The process of condensation in the system of scalar bosons with weak $\lambda {\phi }^4$ interaction is considered. The Boltzmann kinetic equation is solved numerically. We consider two kinetic stages. At the first stage condensate is still absent but there is nonzero inflow of particles towards $\mathbf{p}=0$ and the distribution function at $\mathbf{p}=0$ grows from finite values to infinity. At the second stage there are two components, condensate and particles, reaching their equilibrium values. We show that the evolution in both stages proceeds in a self-similar way and find the time needed for condensation.

Recent neutron interferometry experiments have been interpreted as demonstrating a new topological phenomenon similar in principle to the usual Aharonov-Bohm (AB) effect, but with the neutron's magnetic moment replacing the electron's charge. We show that he new phenomenon, called scalar AB (SAB) effect, follows from an ordinary local interaction, contrary to the usual AB effect, and we argue that the SAB effect is not a topological effect by any useful definition. We find that SAB actually measures an apparently novel spin autocorrelation whose operator equations of motion contain the local torque in the magnetic field. We note that the same remarks apply to the Aharonov-Casher effect.

Quantum billiards are much studied as perhaps the simplest case which presents the central difficulty that the quasiclassical approximation is expressed as a divergent series. We find here, using the Fredholm method, an exact Green's function for billiards expressed as a ratio of absolutely convergent series. We make the quasiclassical approximation to this ratio. The method provides a convergence argument for previous results and an extension of results obtained for the spectrum to the full Green's function.

We have demonstrated that, analogous to the use of quantum mechanical squeezing to beat the quantum shot noise limit, classical squeezing can be used to reduce the effect of thermal amplitude noise in precision measurements. The classical oscillator under study is a single ion in a Penning trap whose motion is cooled by coupling to a thermal bath. The thermal noise distribution is quadrature squeezed by parametric amplification. We observe 6 dB of noise reduction below the equilibrium value when the squeezing axis is 90° out of phase with the coherent signal drive.

We report the deflection of neutrons by a Coriolis force inside a slowly rotating Si crystal $\left(L\approx 10\mathrm{cm}\right)$. At an angular velocity of $5.4\times {10}^{-3\mathrm{}}$ rad/s we observe a deflection as high as $\pm$5.5 mm due to the minute effective mass of a neutron propagating under diffraction conditions in a crystal. This deflection is enhanced by more than 5 orders of magnitude over the expected deflection in free space. Remarkable features of the effect are that the trajectories of the neutrons are curved even in a nonrotating frame and that the neutrons with negative effective mass are deflected away from the force.

Standard Bell inequalities apply to correlations arising when two or more macroscopically separated systems are each subjected to a single ideal measurement. While these inequalities demonstrate that quantum mechanics and local hidden variable theories yield incompatible predictions for pure states, they fail for mixed states: For mixed states, correlations arising from a single ideal measurement on each system may obey standard Bell inequalities, yet when each system is subjected to a sequence of ideal measurements the correlations are nonlocal. For some situations even this last procedure fails, and we must consider more complex (“nonideal”) measurements.

The question in the title may be answered by considering the outcome of a “weak measurement” in the sense of Aharonov, Albert, and Vaidman. Various properties of the resulting time are discussed, including its close relation to the Larmor times. It is a universal description of a broad class of measurement interactions, and its physical implications are unambiguous.

We use a one-dimensional random walk on $D$-dimensional hyperspheres to determine the critical behavior of statistical systems in hyperspherical geometries. First, we demonstrate the properties of such a walk by studying the phase diagram of a percolation problem. We find a line of second and first order phase transitions separated by a tricritical point. Then, we analyze the adsorption-desorption transition for a polymer growing near the attractive boundary of a cylindrical cell membrane. We find that the fraction of adsorbed monomers on the boundary vanishes exponentially when the adsorption energy decreases towards its critical value.

A Comment on the Letter by Paolo Sibani and Peter B. Littlewood, Phys. Rev. Lett. 71, 1482 (1993).

We propose a new ensemble for Monte Carlo simulations, in which each state is assigned a statistical weight $1/k$, where $k$ is the number of states with smaller or equal energy. This ensemble has robust ergodicity properties and gives significant weight to the ground state, making it effective for hard optimization problems. It can be used to find free energies at all temperatures and picks up aspects of critical behavior (if present) without any parameter tuning. We test it on the traveling salesperson problem, the Edwards-Anderson spin glass, and the triangular antiferromagnet.

We derive the first nontrivial rigorous bounds on the mean distance between nearest neighbors $\lambda$ in ergodic, isotropic packings of hard $D$-dimensional spheres that depend on the packing fraction and nearest-neighbor distribution function. Several interesting implications of these bounds for equilibrium as well as nonequilibrium ensembles are explored. For an equilibrium ensemble, we find accurate analytical approximations for $\lambda$ for $D=2\mathrm{}$ and 3 that apply up to random close packing. Our theoretical results are in excellent agreement with available computer-simulation data.

We study theoretically the long time asymptotic of a quantum particle moving in a random time-dependent potential with finite correlation time, in $d=1\mathrm{}$. By applying a new unitary numerical scheme we first show the minor importance of quantum interference and then derive an effective Langevin-type equation for the corresponding clasical problem in the limit of weak potential. We find that on intermediate time scales $\overline{E_{\mathrm{kin}}\left(t\right)}\sim t^{2/5}$, while the true long time asymptotic is determined by a new friction term, which gives rise to a stationary power law velocity distribution, multifractality of the velocity moments, and a slowing down of the superdiffusive behavior.

An analog of Kolmogorov's superconvergent perturbation theory in classical mechanics is constructed for self-adjoint operators. It is different from the usual Rayleigh-Schrödinger perturbation theory and yields expansions for eigenvalues and eigenvectors in terms of functions of the perturbation parameter.

We introduce two quantum Monte Carlo algorithms and employ them to study the superfluid-insulator transition in a two-dimensional system of hard-core bosons. One of the algorithms is appropriate for zero temperature and is based upon the Green's function Monte Carlo method; the other is an extension of the finite-temperature world-line cluster method. In each case we find that the dynamical exponent is consistent with the theoretical prediction of $z=2\mathrm{}$ by Fisher and co-workers.

We present an optical realization of a thermal ratchet. Directed motion of Brownian particles in water is induced by modulating in time a spatially periodic but asymmetric optical potential. The net drift shows a maximum as a function of the modulation period. The experimental results agree with a simple theoretical model based on diffusion.