A Comment on the Letter by Werner Vogel, Phys. Rev. Lett. 84, 1849 (2000). The authors of the Letter offer a Reply.

A Reply to the Comment by Lajos Diosi.

Two ${}^{87}\mathrm{Rb}$ condensates ( $F=2\mathrm{}$, $m_f=2\mathrm{}$, and $m_f=1\mathrm{}$) are produced in highly displaced harmonic traps and the collective dynamical behavior is investigated. The mutual interaction between the two condensates is evidenced in the center-of-mass oscillations as a frequency shift of $6.4\left(3\right)\%$. Calculations based on a mean-field theory well describe the observed effects of periodical collisions both on the center-of-mass motion and on the shape oscillations.

We calculate the effects of induced interactions on the transition temperature to the BCS state in dilute Fermi gases. For a pure Fermi system with two species having equal densities, the transition temperature is suppressed by a factor $(4e{)}^{1/3}\approx 2.2$, and for $\nu$ fermion species, the transition temperature is increased by a factor $(4e{)}^{\nu /3-1}\approx {2.2}^{\nu -3\mathrm{}}$. For mixtures of fermions and bosons the exchange of boson density fluctuations gives rise to an attractive interaction, and we estimate the increase of the transition temperature due to this effect.

Fick's law is important in transport theory and nonequilibrium statistical mechanics. The Heisenberg equation of motion for density is examined to see how it could be reduced to the diffusion equation, which is exactly equivalent to Fick's law. Conditions that are required have been noted and their implications explored.

We study the quadrupole oscillation of a Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ atoms confined in an axisymmetric magnetic trap, after it has been stirred by an auxiliary laser beam. The stirring may lead to the nucleation of one or more vortices, whose presence is revealed unambiguously by the precession of the axes of the quadrupolar mode. For a stirring frequency $\Omega$ below the single vortex nucleation threshold ${\Omega }_c$, no measurable precession occurs. Just above ${\Omega }_c$, the angular momentum deduced from the precession is $\sim ħ$. For stirring frequencies above ${\Omega }_c$ the angular momentum is a smooth and increasing function of $\Omega$, until an angular frequency is reached at which the vortex lattice disappears.

We have studied the hydrodynamic flow in a Bose-Einstein condensate stirred by a macroscopic object, a blue-detuned laser beam, using nondestructive in situ phase contrast imaging. A critical velocity for the onset of a pressure gradient has been observed, and shown to be density dependent. The technique has been compared to a calorimetric method used previously to measure the heating induced by the motion of the laser beam.

We solve the generalized Langevin equation driven by a stochastic force with a power-law autocorrelation function. A stationary Markov process has been applied as a model of the noise. However, the resulting velocity variance does not stabilize but diminishes with time. It is shown that algebraic distributions can induce such effects. Results are compared to those obtained with a deterministic random force. Consequences for the diffusion process are also discussed.

We show theoretically that Bell-type correlations can be observed between continuous variable measurements performed on a parametric source. An auxiliary measurement, performed on the detection environment, negates the possibility of constructing a local realistic description of these correlations.

We demonstrate a spatially resolved autocorrelation measurement with a Bose-Einstein condensate and measure the evolution of the spatial profile of its quantum mechanical phase. Upon release of the condensate from the magnetic trap, its phase develops a form that we measure to be quadratic in the spatial coordinate. Our experiments also reveal the effects of the repulsive interaction between two overlapping condensate wave packets and we measure the small momentum they impart to each other.

We present a simple experimental scheme which can be used to demonstrate an all-or-nothing–type contradiction between noncontextual hidden variables and quantum mechanics. The scheme, which is inspired by recent ideas by Cabello and García-Alcaine, shows that even for a single particle, path and spin information cannot be predetermined in a noncontextual way.

It is shown that a Dirac particle of mass $m$ and arbitrarily small momentum will tunnel without reflection through a potential barrier ${V=U}_c\left(x\right)$ of finite range provided that the potential well ${V=-U}_c\left(x\right)$ supports a bound state of energy $E=-m$. This is called a supercritical potential well.

We discuss Bose-Einstein condensation in a trapped gas of bosonic particles interacting dominantly via dipole-dipole forces. We find that in this case the mean-field interparticle interaction and, hence, the stability diagram are governed by the trapping geometry. Possible physical realizations include ultracold heteronuclear molecules, or atoms with laser induced electric dipole moments.

Bose-Einstein condensation has been achieved in a magnetically trapped sample of ${}^{85}\mathrm{Rb}$ atoms. Long-lived condensates of up to ${10}^4$ atoms have been produced by using a magnetic-field-induced Feshbach resonance to reverse the sign of the scattering length. This system provides new opportunities for the study of condensate physics. The variation of the scattering length near the resonance has been used to magnetically tune the condensate self-interaction energy over a wide range, extending from strong repulsive to large attractive interactions. When the interactions were switched from repulsive to attractive, the condensate shrank to below our resolution limit, and after $\sim 5\mathrm{ms}$ emitted a burst of high-energy atoms.

The stationary state of a quantum particle strongly coupled to a quantum thermal bath is known to be non-Gibbsian, due to entanglement with the bath. For harmonic potentials, where the system can be described by effective temperatures, thermodynamic relations are shown to take a generalized Gibbsian form that may violate the Clausius inequality. For the weakly anharmonic case, a Fokker-Planck-type description is constructed. It is shown that then work can be extracted from the bath by cyclic variation of a parameter. These apparent violations of the second law are the consequence of quantum coherence in the presence of the slightly off-equilibrium nature of the bath.

We investigate the critical behavior of systems exhibiting a continuous absorbing phase transition in the presence of a conserved field coupled to the order parameter. The results obtained point out the existence of a new universality class of nonequilibrium phase transitions that characterizes a vast set of systems including conserved threshold transfer processes and stochastic sandpile models.

In a recent experiment Lauber et al. have deformed cyclically a microwave resonator and have measured the adiabatic normal-mode wave functions for each shape along the path of deformation. The nontrivial observed cyclic phases around a threefold degeneracy were accounted for by Manolopoulos and Child within an approximate theory. However, open-path geometrical phases disagree with experiment. By solving exactly the problem, we find unsuspected extra degeneracies around the multiple one that account for the measured phase changes throughout the path. It turns out that proliferation of additional degeneracies around a multiple one is a common feature of quantum mechanics.

The quantum-mechanical $D$-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent $s$-wave scattering matrix element.

Continuous quantum nondemolition monitoring of a collective atomic spin with an off-resonant laser beam has been performed. Squeezed atomic spin states have thereby been produced with spin noise reduction to $70\%$ below the standard quantum limit expected for a coherent spin state.

We present an exact analytic solution of a nonlinear Schrödinger field interacting with a moving potential (obstacle) at supersonic speed. We discover conditions under which the field can form a stable shape-invariant structure localized around the obstacle—a dressing effect that protects the field against excitations by the obstacle. Such an effect demonstrates the existence of frictionless motion beyond the conventionally defined critical velocity.