A Reply to the Comment by J. Florencio, F. C. Sá Barreto, and O. F. de Alcantara Bonfim.

We provide evidence for several novel phases in the dilute limit of rotating Bose-Einstein condensates. By exact calculation of wave functions and energies for small numbers of particles, we show that the states near integer angular momentum per particle are best considered condensates of composite entities, involving vortices and atoms. We are led to this result by explicit comparison with a description purely in terms of vortices. Several parallels with the fractional quantum Hall effect emerge, including the presence of the Pfaffian state.

We analyze and experimentally demonstrate a new (temporal) Talbot effect, where a pulsed phase grating is applied to a cloud of cold atoms (Bose-Einstein condensate). In contrast to the usual (spatial) Talbot effect, our atoms gain kinetic energy when diffracted by the pulsed grating. The energy gain and the matter-wave-dispersion relation result in an exact (no paraxial approximation) Talbot effect where the initial wave front is reconstructed at multiples of a “Talbot time” $T_{\mathrm{T}}$. We demonstrate the Talbot effect by applying a second pulsed phase grating after a variable delay. When this second grating is applied at even (odd) multiples of $T_{\mathrm{T}}/2\mathrm{}$, it doubles (cancels) the effect of the first pulsed grating.

We compute numerically the yrast line for harmonically trapped $N$-boson systems with a weak repulsive contact interaction, studying the transition to a vortex state as the angular momentum $L$ increases and approaches $N$. There is a smooth crossover between low and high $L$ with no indication of a quantum phase transition; at $L=N$ there is a kink in the slope of the spectrum. Most strikingly, the energy and wave function appear to be analytical functions of $L$ over the entire range $2\le L\le N$. We confirm the structure of low- $L$ states proposed by Mottelson as mainly single-particle excitations with two or three units of angular momentum.

We discuss collective excitations of a trapped dilute Fermi gas within a hydrodynamic approximation. Analytical results are derived for both high- and low-temperature limits and are applied to ${}^{40}\mathrm{K}$ and ${}^6\mathrm{Li}$ systems of current experimental interest. We identify spectral signatures which can be used to detect the onset of Fermi degeneracy. Also, we find an interesting class of internal excitations with an unusual spectrum. Some of our results are relevant to the case of trapped bosons as well. Our analysis suggests several experiments which address fundamental problems of collective motion in quantum fluids.

A model in which two weakly coupled systems maintain opposite running thermodynamic arrows of time is exhibited. Each experiences its own retarded electromagnetic interaction and can be seen by the other. The possibility of opposite-arrow systems at stellar distances is explored and a relation to dark matter suggested.

The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method which we call reduced Hamiltonian interpolation (RHI). In this interpolation the $N$-particle Hamiltonian is represented through a sequence of $p$-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. A synthesis of ideas from $N$-representability and dimensional interpolation, the RHI interpolates over the number $p$ of quasiparticles (equivalent to spatial dimension) to calculate the $N$-particle energy as the mean of close upper and lower bounds. Application to bosons with harmonic interactions yields more than $99\%$ of the correlation energy.

We present a simple formula that expresses energy eigenfunctions for the Coulomb problem exactly and entirely in terms of the corresponding classical orbits. Since both energy eigenvalues and eigenfunctions can be obtained exactly by a semiclassical procedure, the full quantum dynamics of the electron in a Coulomb field can be expressed completely in terms of its classical motion.

Laser-cooled neutral atoms from a low-velocity atomic source are guided via a magnetic field generated between two parallel wires on a glass substrate. The atoms bend around three curves, each with a 15-cm radius of curvature, while traveling along a 10-cm-long track. A maximum flux of $2\times {10}^6\mathrm{atoms}/\sec$ is achieved with a current density of $3\times {10}^4\mathrm{A}/{\mathrm{cm}}^2$ in the $100\times 100$- $\mu \mathrm{m}$-cross-section wires. The kinetic energy of the guided atoms in one transverse dimension is measured to be $42\mu \mathrm{K}$.

Dark solitons in cigar-shaped Bose-Einstein condensates of ${}^{87}\mathrm{Rb}$ are created by a phase imprinting method. Coherent and dissipative dynamics of the solitons has been observed.

In a recent MIT experiment, a new form of superradiant Rayleigh scattering was observed in Bose-Einstein condensates. We present a detailed theory of this phenomenon in which condensate depletion leads to mode competition, which, together with the directional dependence of the scattering rate, is ultimately responsible for the observed phenomena. The nonlinear response of the system is shown to be highly sensitive to initial quantum fluctuations which cause large run to run variations in the observed superradiant pulses.

We study a nonlinear response of the underdamped generalized Frenkel-Kontorova model (a system of interacting atoms placed into an external periodic potential) on the dc driving force. Both studied variants of the model, one with a transverse degree of freedom, and another with two-dimensional isotropic substrate potential, exhibit an interesting “fuse”-like behavior: at small forces the system is in the regime of running kinks with a nonzero conductivity, while at higher forces the system is trapped in an immobile state. This behavior is explained in terms of a dynamical transition between two states of kinks.

As a nonlinear optical system consisting of a Kerr medium inserted in a feedback loop is exposed to a light intensity growing linearly from below to above the threshold for pattern formation, the critical slowing down around threshold freezes the defect population. The measured number of defects immediately after the transition scales with the quench time as predicted by Zurek for a two-dimensional Ginzburg-Landau model. The further temporal evolution of the defect number is in agreement with a simple annihilation model, once the drift of defects specific for our system is taken into account.

We propose a simple method that allows, in one dimension, to solve exactly a wide class of classical stochastic many-body systems far from equilibrium. For the sake of illustration and without loss of generality, we focus on a model that describes the asymmetric diffusion of hard core particles in the presence of an external source and instantaneous annihilation. Starting from a master equation formulation of the problem, we show that the density and multipoint correlation functions obey a closed set of integro-differential equations which in turn can be solved numerically and/or analytically.

We present the stochastic Schrödinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it to the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our powerful stochastic approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger as $ħ\to 0$.

A continuous sequence of infinitesimal unitary transformations, combined with an operator product expansion for vertex operators, is used to diagonalize the quantum sine-Gordon model for ${\beta }^2\in (2\pi ,\infty )$. The leading order of this approximation already gives very accurate results for the single-particle gap in the strong-coupling phase. This approach can be understood as an extension of perturbative scaling theory since it links weak- to strong-coupling behavior in a systematic expansion. The method should also be useful for other strong-coupling problems that can be formulated in terms of vertex operators.

The internal vortex structure of a trapped spin-1 Bose-Einstein condensate is investigated. It is shown that it has a variety of configurations depending on, in particular, the ratio of the relevant scattering lengths and the total magnetization.

We present a general approach to greatly increase at little cost the efficiency of Monte Carlo algorithms. To each observable to be computed we associate a renormalized observable (improved estimator) having the same average but a different variance. By writing down the zero-variance condition a fundamental equation determining the optimal choice for the renormalized observable is derived (zero-variance principle for each observable separately). We show, with several examples including classical and quantum Monte Carlo calculations, that the method can be very powerful.

We have realized a quantum accelerator mode using a system consisting of ultracold cesium atoms falling through a pulsed standing wave of off resonant light. We present a picture of this system based on diffraction and show that the effect arises from the application of blazed matter wave diffraction gratings. The implications of our results for quantum chaos and the prospect of constructing a large angular separation matter wave beam splitter are discussed.

We investigate the oscillation of a dilute atomic gas generated by a sudden rotation of the confining trap (scissors mode). This oscillation reveals the effects of superfluidity exhibited by a Bose-Einstein condensate. The scissors mode is also investigated in a classical gas above $T_c$ in various collisional regimes. The crucial difference with respect to the superfluid case arises from the occurrence of low frequency components, which are responsible for the rigid value of the moment of inertia. Different experimental procedures to excite the scissors mode are discussed.