We discuss the collective excitation spectrum and condensate depletion of a dilute, interacting Bose condensate confined to a harmonic potential, within the Bogoliubov approximation. The spectrum is calculated explicitly for weak coupling based on a variational condensate. We discuss the qualitative nature of this spectrum and make quantitative predictions for future experiments.

We apply linear-response analysis of the Gross-Pitaevskii equation to obtain the excitation frequencies of a Bose-Einstein condensate confined in a time-averaged orbiting potential trap. Our calculated values are in excellent agreement with those observed in a recent experiment.

A new stochastic discrete space-time model of multiple random walkers is proposed to simulate one-and two-dimensional many body interacting lattice gas systems. A very simple rule set which prohibits the multiple occupancy of lattice sites by the random walkers while conserving their number is used to define the dynamics of the walkers. We found a linear time dependence for the mean squared end-to-end distances of the two-dimensional random walkers system. Extensions of the model to a wide variety of diffusive systems are noted.

The motion of an isolated vortex filament with small curvature is governed by the local induction equation. The present work is an attempt to study such an integrable system based on the Riemannian geometry of the solution manifold, by reformulating the governing equation as the geodesic equation on an infinite dimensional Lie group $C^{\infty }(S^1, \mathrm{SO}\mathrm{}(3\left)\right)$. The sectional curvatures for the vector fields along the geodesic curves and their perturbations are calculated to show their tendency to be positive for some typical filaments such as a straight vortex, a helical vortex, and a vortex soliton.

A quantum system consisting of two subsystems is separable if its density matrix can be written as $\rho =\Sigma A^{}{w}_A{\rho }_A^{\prime }\otimes {\rho }_A^{\prime \prime },$ where ${\rho }_A^{\prime }$ and ${\rho }_A^{\prime \prime }$ are density matrices for the two subsystems, and the positive weights $w_A$ satisfy $\Sigma w_A=1\mathrm{}$. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of ρ, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.

We have determined the rate of loss of atoms from a Bose-Einstein condensed gas due to binary processes in the presence of a far-detuned laser field. In this limit, the binary loss rate spectrum is markedly different from, and can greatly exceed, the basic atomic loss rate. We suggest that measurements of the loss rate spectrum can be used to determine the nature of atom interactions in a condensate.

We study the decay of the probability for a non-Markovian stationary Gaussian walker not to cross the origin up to time $t$. This result is then used to evaluate the fraction of spins that do not flip up to time $t$ in the zero temperature Monte Carlo spin flip dynamics of the Ising model. Our results are compared to extensive numerical simulations.

A Comment on the Letter by Haiqing Wei, Rushan Han, and Xiuqing Wei, Phys. Rev. Let. 75, 2071 (1995). The authors of the Letter offer a Reply.

A Reply to the Comment by C. R. Hagen.

We prove that the inequality $\partial P/\partial v<\mathrm{}0\mathrm{}$ is the stability criterion for dark solitons of the one-dimensional nonlinear Schrödinger equation with general nonlinearity. Here $v$ is the soliton's velocity, and $P$ renormalized momentum: $P=\left(i/2\right)\int ({\psi }_x^{*}\psi -{\psi }_x{\psi }^{*})(1-{\rho }_0/|\psi {|}^2)\mathrm{dx}$.

We study a quantum particle in a periodic potential and subject to slowly varying electromagnetic potentials. It is proved that, in the Heisenberg picture, the scaled position operator $\epsilon \mathbf{x}\left({\epsilon }^{-1\mathrm{}}t\right)$ has a limit as $\epsilon \to 0$. The limit operator is determined by the semiclassical equations of motion, which implies that for long times the wave packet is well approximated by the semiclassical evolution. From our result we infer the hydrodynamic limit, $\mathbf{q}\to 0,t\to \infty ,\mathbf{q}t=\mathrm{const}$, of the structure function $S(\mathbf{q},t)$ of a fluid of noninteracting fermions in a crystal potential.

In this paper we show that Onsager-Machlup time-reversal properties of thermodynamic fluctuations and Onsager reciprocity relations for transport coefficients can hold also if the microscopic dynamics is not reversible. This result is based on the explicit construction of models which can be analyzed rigorously.

We consider layered two-dimensional Ising and directed walk models and show that the two problems are inherently related. The information about the zero-field thermodynamical properties of the Ising model is contained in the transfer matrix of the directed walk. For several hierarchical and aperiodic distributions of the couplings, critical exponents for the two problems are obtained exactly through renormalization.

Systems evolving according to effective non-Hermitian Hamiltonians are considered. To each eigenvalue of the effective Hamiltonian is associated two eigenstates $〈\psi |$ and $|\phi 〉$ which evolve backward and forward in time, respectively. Adiabatic measurements on such systems are analyzed. The outcome of the adiabatic measurement of an observable $A$ is the weak value $〈\psi |A|\phi 〉/〈\psi |\phi 〉$ associated with the two-state vector $〈\psi ||\phi 〉$. The possibility of performing such measurements in a laboratory is discussed.

Collective excitations of a dilute Bose condensate have been observed. These excitations are analogous to phonons in superfluid helium. Bose condensates were created by evaporatively cooling magnetically trapped sodium atoms. Excitations were induced by a modulation of the trapping potential, and detected as shape oscillations in the freely expanding condensates. The frequencies of the lowest modes agreed well with theoretical predictions based on mean-field theory. Before the onset of Bose-Einstein condensation, we observed sound waves in a dense ultracold gas.

The average number $〈I\left(t\right)〉$ of islands (connected sets of unvisited sites) enclosed by the support of a 2D simple random walk is studied for a large number $t$ of steps. On an infinite square lattice the result $〈I\left(t\right)〉\simeq \frac{1}{2}\pi (\pi -2\mathrm{})t/\ln {}^{2}8t$ is the difference between a creation and a destruction term that are both $\propto t/\ln 8t$. On a finite lattice of $N$ sites, for $N$ large, $〈I\left(t\right)〉$ attains a maximum value $\left[\right(\pi -2\mathrm{})/2e]N/\ln N$ for $t\simeq {\pi }^{-1\mathrm{}}N\ln N$, and goes down to $O\left(1\right)$ for $t\simeq {\pi }^{-1\mathrm{}}N{\ln }^{2}N$. The scaling function describing this behavior is given and compared with simulations by Coutinho et al.

A self-consistent approximation for the structure factor of three-dimensional lattice gases yields remarkably accurate predictions (less than 3% error over most of the temperature range) for the correlation length, isothermal compressibility, specific heat, and the coexistence curve. Critical temperatures agree to within 0.2%, and other critical properties to within (1–2)%, of the best numerical estimates. Until temperature and density are within 1% of their critical values, the approximate effective critical exponents do not differ appreciably from their estimated exact form; they attain their limiting spherical-model values only much closer to critical. The method should prove useful for a variety of three-dimensional lattice-gas and fluid problems; it is inappropriate to two dimensions, where it predicts criticality at zero temperature.

The Onsager regression hypothesis states that the regression of fluctuations is governed by macroscopic equations describing the approach to equilibrium. It is here asserted that this hypothesis fails in the quantum case. This is shown first by explicit calculation for the example of quantum Brownian motion of an oscillator and then in general from the fluctuation-dissipation theorem. It is asserted that the correct generalization of the Onsager hypothesis is the fluctuation-dissipation theorem.

A method of generating an arbitrary pattern of atoms by using an interferometric technique is reported. A laser-cooled metastable Ne atomic beam in the ${1s}_3$ state is diffracted by a computer generated binary hologram. A black-and-white pattern is drawn on a microchannel detector with a spatial resolution exceeding $65\mu \mathrm{m}$ and a resolving power better than $30$.