A Comment on the Letter by Ralph H. Lewis, Walter E. Lawrence, and Joseph D. Harris, Phys. Rev. Lett. 77, 5157 (1996). The authors of the Letter offer a Reply.

A Reply to the Comment by T. B. Smith and John A. Vaccaro.

The influence of frozen-in topological defects in a crystal on the long-wavelength quantum states of a particle is considered. In the continuum limit of a conveniently defined tight-binding model one is led to a covariant Schrödinger equation on a Riemann-Cartan manifold. When the tight-binding transfer energies are assumed to depend on the local lattice deformations caused by the defects, additional noncovariant terms are generated in the Hamiltonian. These terms generate bound states of the particle to edge dislocations and enhance the scattering of particles on screw dislocations.

We provide a constructive algorithm to find the best separable approximation to an arbitrary density matrix of a composite quantum system of finite dimensions. The method leads to a condition of separability and to a measure of entanglement.

Motivated by experiments on Bose atoms in traps which have attractive interactions (e.g., ${}^7\mathrm{Li}$), we consider two models which may be solved exactly. We construct the ground states subject to the constraint that the system is rotating with angular momentum proportional to the number of atoms. In a conventional system this would lead to quantized vortices; here, for attractive interactions, we find that the angular momentum is absorbed by the center of mass motion. Moreover, the state is uncondensed and is an example of a “fragmented” condensate discussed by Nozières and Saint James. The same models with repulsive interactions are fully condensed in the thermodynamic limit.

We present the theory of damping of low-energy excitations of a trapped Bose-Einstein condensate at finite temperatures, where the damping is provided by the interaction of these excitations with the thermal excitations. We emphasize the key role of stochastization in the behavior of the thermal excitations for damping in nonspherical traps. The damping rates of the lowest excitations, following from our theory, are in fair agreement with the data of recent JILA and MIT experiments. The damping of quasiclassical excitations is determined by the condensate boundary region, and the result for the damping rate is drastically different from that in a spatially homogeneous gas.

The dispersion parameter of the prehistory distribution for a potential system driven by white noise is analyzed theoretically and by means of analog and digital experiments. Nonmonotonic evolution of the dispersion with time is shown to arise provided that the potential fulfills a certain condition. It does not necessitate the existence of an unstable point, but can occur in single-minimum potentials, both symmetric and asymmetric.

It is shown that a topological vector potential (Berry phase) is induced by the act of measuring angular momentum in a direction defined by a reference particle. This vector potential appears as a consequence of the backreaction due to the quantum measurement.

Bose-Einstein condensates of sodium atoms have been confined in an optical dipole trap using a single focused infrared laser beam. This eliminates the restrictions of magnetic traps for further studies of atom lasers and Bose-Einstein condensates. More than $5\times {10}^6$ condensed atoms were transferred into the optical trap. Densities of up to $3\times {10}^{15}{\mathrm{cm}}^{-3\mathrm{}}$ of Bose condensed atoms were obtained, allowing for a measurement of the three-body loss rate constant for sodium condensates as $K_3=1.1\mathrm{}\left(3\right)\times {10}^{-30\mathrm{}}{\mathrm{cm}}^6{\mathrm{s}}^{-1\mathrm{}}$. At lower densities, the observed $1/e$ lifetime was longer than 10 s. Simultaneous confinement of Bose-Einstein condensates in several hyperfine states was demonstrated.

We consider the dynamics of a quantum degenerate trapped gas of ${}^7\mathrm{Li}$ atoms. Because the atoms have a negative $s$-wave scattering length, a Bose condensate of ${}^7\mathrm{Li}$ becomes mechanically unstable when the number of condensate atoms approaches a maximum value. We calculate the dynamics of the collapse that occurs when the unstable point is reached. In addition, we use the quantum Boltzmann equation to investigate the nonequilibrium kinetics of the atomic distribution during and after evaporative cooling. The condensate is found to undergo many cycles of growth and collapse before a stationary state is reached.

The largest Lyapunov exponent ${\lambda }^{+}$ for a dilute gas with short range interactions in equilibrium is studied by a mapping to a clock model, in which every particle carries a watch, with a discrete time that is advanced at collisions. This model has a propagating front solution with a speed that determines ${\lambda }^{+}$, for which we find a density dependence as predicted by Krylov, but with a larger prefactor. Simulations for the clock model and for hard sphere and hard disk systems confirm these results and are in excellent mutual agreement. They show a slow convergence of ${\lambda }^{+}$ with increasing particle number, in good agreement with a prediction by Brunet and Derrida.

For a two-particle two-state system, sets of compatible propositions exist for which quantum mechanics and noncontextual hidden-variable theories make conflicting predictions for every individual system whatever its quantum state. This permits a simple all-or-nothing state-independent experimental verification of the Bell-Kochen-Specker theorem.

The position operator (defined within the Schrödinger representation in the standard way) becomes meaningless when periodic boundary conditions are adopted for the wave function, as usual in condensed matter physics. I show how to define the position expectation value by means of a simple many-body operator acting on the wave function of the extended system. The relationships of the present findings to the Berry-phase theory of polarization are discussed.

We consider the $T=0\mathrm{}$ ground state of mixtures of bosons and fermions. Applying the Thomas-Fermi approximation we show that, depending on the strength of boson-boson and boson-fermion interactions, the Fermi gas may constitute a “shell” around or a “core” inside the Bose condensate, or even both.

Numerical evidence is given for nonergodic (nonmixing) behavior, exhibiting ideal transport, of a simple nonintegrable many-body quantum system in the thermodynamic limit, namely, the kicked $t$- $V$ model of spinless fermions on a ring. However, for sufficiently large kick parameters $t$ and $V$ we recover quantum ergodicity, and normal transport, which can be described by random matrix theory.

We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable (i.e., finite) positive operator valued measurement on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of $N$ independent and identically prepared two-level systems (qubits).

A Bose-Einstein condensate with attractive interaction can be metastable if it is spatially confined and if the number of condensate bosons $N_0$ is below a certain critical value $N_{\mathrm{c}}$. By applying a variational method and the instanton technique to the Gross-Pitaevskii energy functional, we find analytically the frequency of the collective excitation and the rate of macroscopic quantum tunneling. We show that near the critical point the tunneling exponent vanishes according to $({1-N}_0/N_{\mathrm{c}}{)}^{5/4}$ and that macroscopic quantum tunneling can be a dominant decay mechanism of the condensate for $N_0$ very close to $N_{\mathrm{c}}$.

Theoretical design of global optimization algorithms can profitably utilize recent statistical mechanical treatments of potential energy surfaces (PES's). Here we analyze a particular method to explain its success in locating global minima on surfaces with a multiple-funnel structure, where trapping in local minima with different morphologies is expected. We find that a key factor in overcoming trapping is the transformation applied to the PES which broadens the thermodynamic transitions. The global minimum then has a significant probability of occupation at temperatures where the free energy barriers between funnels are surmountable.

We investigate the dynamics of a soliton that behaves as an extended particle. The soliton motion in an effective bistable potential can be chaotic in a similar way as the Duffing oscillator. We generalize the concept of geometrical resonance to spatiotemporal systems and apply it to design a nonfeedback mechanism of chaos control using localized perturbations. We show the existence of solitonic stochastic resonance.

A “polydisperse” system has an infinite number of conserved densities. We give a rational procedure for projecting its infinite-dimensional free energy surface onto a subspace comprising a finite number of linear combinations of densities (“moments”), in which the phase behavior is then found as usual. If the excess free energy of the system depends only on the moments used, exact cloud, shadow, and spinodal curves result; two-phase and multiphase regions are approximate, but refinable indefinitely by adding extra moments. The approach is computationally robust and gives new geometrical insights into the thermodynamics of polydispersity.