A fast time-evolution method is developed for systems for which the dynamical behavior can be reduced to the eigenvector/eigenvalue problem. The method does not use the eigenvectors/eigenvalues themselves and is based on a polynominal expansion of the formal operator solution in the eigenfrequency domain. It is complementary to the standard time-integration approaches and allows one to calculate or simulate the state of a system at arbitrary times. The time evolution of, e.g., classical harmonic atomic systems and quantum systems described by linear Hamiltonians can be treated by this method.

Nonlinear effects in the interference of Bose-Einstein condensates are studied using exact solutions of the one-dimensional nonlinear Schrödinger equation, which is applicable when the lateral motion is confined or negligible. With the inverse scattering method, the interference pattern is studied as a scattering problem with the linear Schrödinger equation, whose potential is profiled by the initial density distribution of the condensates. Our theory not only provides an analytical framework for quantitative predictions for the one-dimensional case, it also gives an intuitive understanding of some mysterious features of the interference patterns observed in experiments and numerical simulations.

We use a multiple time scale boundary layer theory to derive the equation of motion for a dark (or grey) soliton propagating through an effectively one-dimensional cloud of Bose-Einstein condensate, assuming only that the background density and velocity vary slowly on the soliton scale. We show that solitons can exhibit viscous or radiative acceleration (antidamping), which we estimate as slow but observable on experimental time scales.

Motivated by the recent discoveries of spin-1 and pseudo-spin-1/2 Bose gas, we have studied the general structure of the Bose gases with arbitrary spin. A general method is developed to uncover the elementary building blocks of the angular momentum eigenstates, as well as the relations (or interactions) between them. Applications of this method to Bose gas with integer spins $(f=1,2,3\mathrm{})$ and half integer spins $(f=1\mathrm{}/2,3/2)$ reveal many surprising structures.

We obtain analytical formulas for the statistics, in particular, for the characteristic function and all cumulants, of the Bose-Einstein condensate in dilute weakly interacting and ideal equilibrium gases in the canonical ensemble via the particle-number-conserving operator formalism of Girardeau and Arnowitt. We prove that the ground-state occupation statistics is not Gaussian even in the thermodynamic limit. We calculate the effect of Bogoliubov coupling on suppression of ground-state occupation fluctuations and show that they are governed by a pair-correlation, squeezing mechanism.

A new form of stochastic resonance (SR) that occurs in multilevel threshold signal detectors is reported. In contrast to classical SR, which extends the dynamic range of threshold detectors to subthreshold signal levels, this new form of SR extends the dynamic range to suprathreshold signal strengths. The effect is most dominant, and can outperform networks based on standard engineering design, when all thresholds adapt to the dc level of the signal. This has an interesting analogy to dc adaptation in neurons. The possible connection between these two effects is discussed.

We investigated the electron tunneling out of a quantum dot in the presence of continuous monitoring by a detector. It is shown that the Schrödinger equation for the whole system can be reduced to new Bloch-type rate equations describing the time development of the detector and the measured system at once. Using these equations we find that the continuous measurement of the unstable system does not affect its exponential decay, $\exp (-\Gamma t)$, contrary to expectations based on the quantum Zeno effect. However, the width of the energy distribution of the tunneling electron is no longer $\Gamma$, but increases due to the decoherence, generated by the detector.

We apply random-matrix-theory (RMT) to the analysis of evolution of wave packets in energy space. We study the crossover from ballistic behavior to saturation, the possibility of having an intermediate diffusive behavior, and the feasibility of strong localization effect. Both theoretical considerations and numerical results are presented. Using quantal-classical correspondence considerations we question the validity of the emerging dynamical picture. In particular, we claim that the appearance of the intermediate diffusive behavior is possibly an artifact of the RMT strategy.

We report the observation of the scissors mode of a Bose-Einstein condensed gas of ${}^{87}\mathrm{Rb}$ atoms in a magnetic trap, which gives direct evidence of superfluidity in this system. The scissors mode of oscillation is excited by a sudden rotation of the anisotropic trapping potential. For a gas above $T_c$ (normal fluid) we detect the occurrence of oscillations at two frequencies, with the lower frequency corresponding to the rigid body value of the moment of inertia. Well below $T_c$ the condensate oscillates at a single frequency, without damping, as expected for a superfluid.

Transition temperature data obtained as a function of particle density in the ${}^4\mathrm{He}$-Vycor system are compared with recent theoretical calculations for 3D Bose-condensed systems. In the low density dilute Bose gas regime we find, in agreement with theory, a positive shift in the transition temperature of the form $\Delta T/T_0=\gamma ({\mathrm{na}}^3{)}^{1/3}$. At higher densities a maximum is found in the ratio of $T_c/T_0$ for a value of the interaction parameter, ${\mathrm{na}}^3$, that is in agreement with path-integral Monte Carlo calculations.

Despite its long history, there are many fundamental issues concerning random packings of spheres that remain elusive, including a precise definition of random close packing (RCP). We argue that the current picture of RCP cannot be made mathematically precise and support this conclusion via a molecular dynamics study of hard spheres using the Lubachevsky-Stillinger compression algorithm. We suggest that this impasse can be broken by introducing the new concept of a maximally random jammed state, which can be made precise.

A nonparametric Bayesian approach is developed to determine quantum potentials from empirical data for quantum systems at finite temperature. The approach combines the likelihood model of quantum mechanics with a priori information on potentials implemented in the form of stochastic processes. Its specific advantages are the possibilities to deal with heterogeneous data and to express a priori information explicitly in terms of the potential of interest. A numerical solution in maximum a posteriori approximation is obtained for one-dimensional problems. As nonparametric estimates, the results depend strongly on the implemented a priori information.

An observable criterion is derived that allows one to distinguish nonclassical states of the harmonic oscillator from those having a classical counterpart. A quantum state is shown to have no classical counterpart if and only if the characteristic functions of the quadrature distributions or the $s$-parametrized phase-space distributions exhibit a slower decay than for the ground state of the oscillator. This renders it possible to experimentally check the failure of the $P$ function to be a probability measure.

Thermally activated escape over a potential barrier in the presence of periodic driving is considered. By means of novel time-dependent path-integral methods we derive asymptotically exact weak-noise expressions for both the instantaneous and the time-averaged escape rate. The agreement with accurate numerical results is excellent over a wide range of driving strengths and driving frequencies.

We obtain numerically a scale-invariant distribution of the bandwidths $S$ for the critical Harper model, which is closely described by a semi-Poisson $P\left(S\right)=4S\exp (-2S)$ curve. After a suitable unfolding of spectra, derived from different boundary conditions, a semi-Poisson level spacing distribution and a sub-Poisson linear number variance are deduced from the bandwidth distribution. The obtained results support possible universality of the critical spectral statistics and suggest its connection to spectral multifractality.

The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical $O\left(N\right)$ loop or a $Q$-state Potts cluster, is solved in two dimensions. The dimension $\hat{f}\left(\theta \right)$ of the boundary set with local wedge angle $\theta$ is $\hat{f}\left(\theta \right)=\frac{\pi }{\theta }-\frac{25-c}{12}\frac{(\pi -\theta {)}^2}{\theta (2\pi -\theta )}$, with $c$ the central charge of the model. As a corollary, the dimensions $D_{\mathrm{EP}}$ of the external perimeter and $D_{\mathrm{H}}$ of the hull of a Potts cluster obey the duality equation $(D_{\mathrm{EP}}-1\mathrm{})(D_{\mathrm{H}}-1\mathrm{})=\frac{1}{4}$. A related covariant MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.

A gedanken experiment is proposed for “weighing” the total mass of a closed system from within the system. We prove that for an internal observer the time $\tau$, required to measure the total energy with accuracy $\Delta E$, is bounded according to $\tau \Delta E>\mathrm{}ħ$. This time-energy uncertainty principle for a closed system follows from the measurement backreaction on the system. We generally examine what other conserved observables are in principle measurable within a closed system and what are the corresponding uncertainty relations.

A cloud of laser-cooled ${}^{85}\mathrm{Rb}$ atoms is coupled through a magnetic funnel into a miniature waveguide formed by four current-carrying wires embedded in a silica fiber. The atom cloud has a $\sim 100\mu \mathrm{m}$ radius within the fiber and propagates over cm distances. We study the coupling, propagation, and transverse distribution of atoms in the fiber, and find good agreement with theory. This prototype demonstrates the feasibility of miniature guides as a tool in the new field of integrated atom optics, leading to single-mode propagation of de Broglie waves and the possible preparation of 1D atom clouds.

A completely positive master equation describing quantum dissipation for a Brownian particle is derived starting from microphysical collisions, exploiting a recently introduced approach to subdynamics of a macrosystem. The obtained equation can be cast into Lindblad form with a single generator for each Cartesian direction. Temperature dependent friction and diffusion coefficients for both position and momentum are expressed in terms of the collision cross section.