What configuration of $N$ point charges on a conducting sphere minimizes the Coulombic energy? J. J. Thomson posed this question in 1904. For $N\le 112$, numerical methods have found apparent global minimum-energy configurations; but the number of local minima appears to grow exponentially with $N$, making many such methods impractical. Here we describe a topological/numerical procedure that we believe gives the global energy minimum lattice configuration for $N$ of the form $N=10\mathrm{}(m^2{+n}^2+mn)+2\mathrm{}$ ( $m$, $n$ positive integers). For those $N$ with more than one lattice, we give a rule to choose the minimum one.

We study the ground state occupation number and its fluctuations for the Bose-Einstein condensate of trapped atoms using the microcanonical ensemble. The analytic formulas are obtained with the help of the saddle point method. We show that microcanonical fluctuations below the critical temperature tend to zero and scale with the number of particles as $1/\sqrt{N}$.

An expression is derived for the equilibrium free energy difference between two configurations of a system, in terms of an ensemble of finite-time measurements of the work performed in parametrically switching from one configuration to the other. Two well-known identities emerge as limiting cases of this result.

We are studying bistable quantum systems coupled to a thermal environment in the presence of an external controlling force. The impact of thermal and quantum effects on the switching hysteresis is studied by evaluating a time-dependent real-time double path integral with the recently developed iterative tensor multiplication scheme for the quasiadiabatic path integral propagator method for temperatures ranging from well above to well below quantum-classical crossover temperature. The switching dynamics and its intimate connection to quantum stochastic resonance is studied here for the first time without limitation to a two-state description and/or small external forces.

The Gaussian wave packet solution to the Schrödinger equation is studied for time-dependent Hamiltonians. The geometrical phase is obtained for a cyclic wave packet solution of the generalized harmonic oscillator with a nonadiabatic time-periodic Hamiltonian. It is found that the geometrical phase is independent of $ħ$, and is equal to one-half of the classical nonadiabatic Hannay angle. The Hannay angle is shown to be independent of the classical action and does not involve averaging.

We analyze the dependence of the collapse and revival of many-atom coherence of a trapped Bose-Einstein condensate on the trap potential, dimensionality of the gas, and atom number fluctuations. We show that in a class of experimentally relevant systems the collapse time vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase.

We discuss the intimate connection between the chaotic dynamics of a classical field theory and the instability of the one-loop effective action of the associated quantum field theory. Using massless scalar electrodynamics as an example, we show how the radiatively induced spontaneous symmetry breaking stabilizes the vacuum state against chaos.

Numerical study of order parameter evolution in the course of symmetry breaking transitions with Landau-Ginzburg–like dynamics shows that the density of topological defects, kinks which form during the quench, is proportional to the fourth root of its rate. This is a limited (1D) test of the more general theory of domain-size evolution in the course of symmetry breaking transformations proposed by one of us. Using these ideas, it is possible to compute the density of topological defects from the quench time scale and from the equilibrium scaling of the correlation length and relaxation time near the critical point.

We present conditions every measure of entanglement has to satisfy, and construct a whole class of “good” entanglement measures. The generalization of our class of entanglement measures to more than two particles is straightforward. We present a measure which has a statistical operational basis that might enable experimental determination of the quantitative degree of entanglement.

The form factor, $k\left(t\right)$, is the spectral statistic which best displays nonuniversal quasiclassical deviations from random matrix theory. Recent estimations of $k\left(t\right)$ for a single spectrum found interesting new effects of this type. It was supposed that $k\left(t\right)$ is self-averaging and thus did not require an ensemble average. We here argue that this supposition sometimes fails and that for many important systems an ensemble average is essential to see detailed properties of $k\left(t\right)$. In other systems, notably the nontrivial zeros of Riemann zeta function, it will be possible to see the nonuniversal properties by an analysis of a single spectrum.

With an interpretation of quantum mechanics one can not only predict probabilities for outcomes of measurements performed on quantum systems but also ascribe properties to these systems themselves. I consider interpretations (notably modal interpretations) that ascribe at least those properties to a system which are represented by the eigenprojections of the reduced state. Such interpretations solve the measurement problem for ideal measurements. Here, however, I prove that these interpretations cannot define joint property ascriptions to three or more systems.

We report on an experiment with polarized atoms passing a standing light wave, where the atom's position is encoded (entangled) with the excitation amplitude of two long lived electronic states. Position information can be obtained a posteriori by measuring the electronic state. If this measurement is performed, the atom is localized in an array of virtual slits. This gives rise to diffraction of the atomic de Broglie waves from a measurement induced amplitude grating, thus showing the back action onto the momentum due to the localization. We further demonstrate experimentally that the encoding can be totally erased.

The rate of change of the entropy S and free energy A with respect to time for a nonequilibrium system in a steady state is considered. A modified Liouville equation is derived to account for the phase space Jacobian arising from nontrivial phase space compression. With due allowance for this factor, the relations $\dot{S}=0\mathrm{}$ and $\dot{A}=0\mathrm{}$ are obtained, results that are consistent with the assumption of a smooth, differentiable phase space distribution function.

We report the demonstration of a Sagnac-effect atom interferometer gyroscope which uses stimulated Raman transitions to coherently manipulate atomic wave packets. We have measured the Earth's rotation rate, and demonstrated a short-term sensitivity to rotations of $2\times {10}^{-8\mathrm{}}\left(\mathrm{rad}/\mathrm{s}\right)/\sqrt{\mathrm{Hz}}$.

Quantum open systems are described in the Markovian limit by master equations in Lindblad form. I argue that common “quantum jump” techniques, which solve the master equation by unraveling its evolution into stochastic trajectories in Hilbert space, correspond closely to a particular set of decoherent histories. This is illustrated by a simple model of a photon counting experiment.

We discuss the collective modes of a trapped Bose gas in the hydrodynamic regime where atomic collisions ensure local thermal equilibrium for the distribution function. Starting from the conservation laws, in the linearized limit we derive a closed equation for the velocity fluctuations in a trapped Bose gas above the Bose-Einstein transition temperature. Explicit solutions for a parabolic trap are given. We find that the surface modes above the transition have the same dispersion relation as the one recently obtained by Stringari for the oscillations of the condensate at $T=0\mathrm{}$ within the Thomas-Fermi approximation. Results are also given for the monopole “breathing” mode as well as for the $m=0\mathrm{}$ excitations which result from the coupling of the monopole and quadrupole modes in an anisotropic parabolic well.

We present a detailed study of the temperature dependence of the condensate and noncondensate density profiles of a Bose-condensed gas in a parabolic trap. These quantities are calculated self-consistently using the Hartree-Fock-Bogoliubov equations within the Popov approximation. Below the Bose-Einstein transition the excitation frequencies have a relatively weak temperature dependence even though the condensate is strongly depleted. As the condensate density goes to zero through the transition, the excitation frequencies are strongly affected and approach the frequencies of a noninteracting trapped gas in the high temperature limit.

We propose a variational wave function which uses coherent states to represent the spin state of particles, and describes systems with spin-dependent correlations. Spin states are parametrized by a continuous variable that looks like a classical spin vector. Correlation operators are represented as simple integrals, without commutator terms, which can be evaluated using standard or Monte Carlo techniques. Explicit calculations on liquid ${}^3$He using our wave function give good agreement with standard methods. Applications to systems in condensed matter and nuclear physics are discussed.

Directed percolation is one of the generic dynamic universality classes. We study the crossover from isotropic to directed percolation by representing the combined problem as a random cluster model. A parameter $r$ controls the spontaneous birth of new forest fires. We obtain the exact crossover exponent $y_{\mathrm{DP}}{=y}_T-1\mathrm{}$ at $r=1\mathrm{}$ using Coulomb gas methods. Isotropic percolation is stable in 2D. Our numerical finite-size scaling results confirm this. An intuitive argument suggests that directed percolation at $r=0\mathrm{}$ does not change stability for $D\ge 3$. It remains unstable, such that forest fires at $0<r<\mathrm{}1\mathrm{}$ have the same scaling properties as isotropic percolation all dimensions.