High-resolution Elemental Mapping of the Lunar Surface

New instruments and missions are being proposed to study the lunar surface as a result of the resurgence of interest in returning to the Moon. One instrument recently proposed is similar in concept to the x-ray fluorescence detectors flown on Apollo, but utilizes fluorescence from the L- and M-shells rather than the K-shell. This soft X-Ray Flourescence Imager (XRFI) is discussed

Local Spin-Gauge Symmetry of the Bose-Einstein Condensates in Atomic Gases

The Bose-Einstein condensates of alkali atomic gases are spinor fields with local ``spin-gauge" symmetry. This symmetry is manifested by a superfluid velocity ${\bf u}_{s}$ (or gauge field) generated by the Berry phase of the spin field. In ``static" traps, ${\bf u}_{s}$ splits the degeneracy of the harmonic energy levels, breaks the inversion symmetry of the vortex nucleation frequency ${\bf \Omega}_{c1}$, and can lead to {\em vortex ground states}. The inversion symmetry of ${\bf \Omega}_{c1}$, however, is not broken in ``dynamic" traps. Rotations of the atom cloud can be generated by adiabatic effects without physically rotating the entire trap.Comment: Typos in the previous version corrected, thanks to the careful reading of Daniel L. Cox. 13 pages + 2 Figures in uuencode + gzip for

Instantons and radial excitations in attractive Bose-Einstein condensates

Imaginary- and real-time versions of an equation for the condensate density are presented which describe dynamics and decay of any spherical Bose-Einstein condensate (BEC) within the mean field appraoch. We obtain quantized energies of collective finite amplitude radial oscillations and exact numerical instanton solutions which describe quantum tunneling from both the metastable and radially excited states of the BEC of 7Li atoms. The mass parameter for the radial motion is found different from the gaussian value assumed hitherto, but the effect of this difference on decay exponents is small. The collective breathing states form slightly compressed harmonic spectrum, n=4 state lying lower than the second Bogolyubov (small amplitude) mode. The decay of these states, if excited, may simulate a shorter than true lifetime of the metastable state. By scaling arguments, results extend to other attractive BEC-s.Comment: 6 pages, 3 figure

Variational study of a dilute Bose condensate in a harmonic trap

A two-parameter trial condensate wave function is used to find an approximate variational solution to the Gross-Pitaevskii equation for $N_0$ condensed bosons in an isotropic harmonic trap with oscillator length $d_0$ and interacting through a repulsive two-body scattering length $a>0$. The dimensionless parameter ${\cal N}_0 \equiv N_0a/d_0$ characterizes the effect of the interparticle interactions, with ${\cal N}_0 \ll 1$ for an ideal gas and ${\cal N}_0 \gg 1$ for a strongly interacting system (the Thomas-Fermi limit). The trial function interpolates smoothly between these two limits, and the three separate contributions (kinetic energy, trap potential energy, and two-body interaction energy) to the variational condensate energy and the condensate chemical potential are determined parametrically for any value of ${\cal N}_0$, along with illustrative numerical values. The straightforward generalization to an anisotropic harmonic trap is considered briefly.Comment: 14 pages, RevTeX, submitted to Journal of Low Temperature Physic

Stabilizing an Attractive Bose-Einstein Condensate by Driving a Surface Collective Mode

Bose-Einstein condensates of $^7$Li have been limited in number due to attractive interatomic interactions. Beyond this number, the condensate undergoes collective collapse. We study theoretically the effect of driving low-lying collective modes of the condensate by a weak asymmetric sinusoidally time-dependent field. We find that driving the radial breathing mode further destabilizes the condensate, while excitation of the quadrupolar surface mode causes the condensate to become more stable by imparting quasi-angular momentum to it. We show that a significantly larger number of atoms may occupy the condensate, which can then be sustained almost indefinitely. All effects are predicted to be clearly visible in experiments and efforts are under way for their experimental realization.Comment: 4 ReVTeX pages + 2 postscript figure

Microscopic Treatment of Binary Interactions in the Non-Equilibrium Dynamics of Partially Bose-condensed Trapped Gases

In this paper we use microscopic arguments to derive a nonlinear Schr\"{o}dinger equation for trapped Bose-condensed gases. This is made possible by considering the equations of motion of various anomalous averages. The resulting equation explicitly includes the effect of repeated binary interactions (in particular ladders) between the atoms. Moreover, under the conditions that dressing of the intermediate states of a collision can be ignored, this equation is shown to reduce to the conventional Gross-Pitaevskii equation in the pseudopotential limit. Extending the treatment, we show first how the occupation of excited (bare particle) states affects the collisions, and thus obtain the many-body T-matrix approximation in a trap. In addition, we discuss how the bare particle many-body T-matrix gets dressed by mean fields due to condensed and excited atoms. We conclude that the most commonly used version of the Gross-Pitaevskii equation can only be put on a microscopic basis for a restrictive range of conditions. For partial condensation, we need to take account of interactions between condensed and excited atoms, which, in a consistent formulation, should also be expressed in terms of the many-body T-matrix. This can be achieved by considering fluctuations around the condensate mean field beyond those included in the conventional finite temperature mean field, i.e. Hartree-Fock-Bogoliubov (HFB), theory.Comment: Resolved some problems with printing of figure

Collective excitations of Bose-Einstein condensed gases at finite temperatures

We have applied the Popov version of the Hartree-Fock-Bogoliubov (HFB) approximation to calculate the finite-temperature excitation spectrum of a Bose-Einstein condensate (BEC) of $^{87}$Rb atoms. For lower values of the temperature, we find excellent agreement with recently-published experimental data for the JILA TOP trap. In contrast to recent comparison of the results of HFB--Popov theory with experimental condensate fractions and specific heats, there is disagreement of the theoretical and recent experimental results near the BEC phase transition temperature.Comment: 4 pages, Latex, with 4 figures. More info at http://amo.phy.gasou.edu/bec.htm

A particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas

The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number $N$ of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator $A$ for the total number of particles and the sum of a ``condensate wavefunction'' $\xi(x)$ and a phonon field operator $\chi(x)$ in the form $\psi(x) \approx A\{\xi(x) + \chi(x)/\sqrt{N}\}$ when the field operator acts on the N particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of $\sqrt{N}$, an thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different N.Comment: RevTeX, 11 page

Vortex Waves in a Cloud of Bose Einstein - Condensed, Trapped Alkali - Metal Atoms

We consider the vortex state solution for a rotating cloud of trapped, Bose Einstein - condensed alkali atoms and study finite temperature effects. We find that thermally excited vortex waves can distort the vortex state significantly, even at the very low temperatures relevant to the experiments.Comment: to appear in Phys. Rev.

Beyond the Thomas-Fermi approximation for a trapped condensed Bose-Einstein gas

Corrections to the zero-temperature Thomas-Fermi description of a dilute interacting condensed Bose-Einstein gas confined in an isotropic harmonic trap arise due to the presence of a boundary layer near the condensate surface. Within the Bogoliubov approximation, the various contributions to the ground-state condensate energy all have terms of order R^{-4}ln R and R^{-4}, where R is the number-dependent dimensionless condensate radius in units of the oscillator length. The zero-order hydrodynamic density-fluctuation amplitudes are extended beyond the Thomas-Fermi radius through the boundary layer to provide a uniform description throughout all space. The first-order correction to the excitation frequencies is shown to be of order R^{-4}.Comment: 12 pages, 2 figures, revtex. Completely revised discussion of the boundary-layer corrections to collective excitations, and two new figures added. To appear in Phys. Rev. A (October, 1998