Ideal Quantum Gases in D-dimensional Space and Power-law Potentials

We investigate ideal quantum gases in D-dimensional space and confined in a generic external potential by using the semiclassical approximation. In particular, we derive density of states, density profiles and critical temperatures for Fermions and Bosons trapped in isotropic power-law potentials. Form such results, one can easily obtain those of quantum gases in a rigid box and in a harmonic trap. Finally, we show that the Bose-Einstein condensation can set up in a confining power-law potential if and only if ${D/2}+{D/n}>1$, where $D$ is the space dimension and $n$ is the power-law exponent.Comment: 18 pages, Latex, to be published in Journal of Mathematical Physic

Instability and Chaos in Spatially Homogeneous Field Theories

Spatially homogeneous field theories are studied in the framework of dynamical system theory. In particular we consider a model of inflationary cosmology and a Yang-Mills-Higgs system. We discuss also the role of quantum chaos and its application to field theories.Comment: 28 pages, 4 figures, to be published in J. Math. Phy

Collapse of triaxial bright solitons in atomic Bose-Einstein condensates

We study triaxial bright solitons made of attractive Bose-condensed atoms characterized by the absence of confinement in the longitudinal axial direction but trapped by an anisotropic harmonic potential in the transverse plane. By numerically solving the three-dimensional Gross-Pitaevskii equation we investigate the effect of the transverse trap anisotropy on the critical interaction strength above which there is the collapse of the condensate. The comparison with previous predictions [Phys. Rev. A {\bf 66}, 043619 (2002)] shows significant differences for large anisotropies.Comment: Accepted for the publication in Phys. Lett.

Dynamics of a BEC bright soliton in an expulsive potential

We theoretically investigate the dynamics of a matter-wave soliton created in a harmonic potential, which is attractive in the transverse direction but expulsive in the longitudinal direction. This Bose-Einstein-condensate (BEC) bright soliton made of $^7$Li atoms has been observed in a recent experiment (Science {\bf 296}, 1290 (2002)). We show that the non-polynomial Schr\"odinger equation, an effective one-dimensional equation we derived from the three-dimensional Gross-Pitaevskii equation, is able to reproduce the main experimental features of this BEC soliton in an expulsive potential.Comment: 5 pages, 4 figures (2 of them with colors

Condensate fraction in metallic superconductors and ultracold atomic vapors

We investigate the condensate density and the condensate fraction of conduction electrons in weak-coupling superconductors by using the BCS theory and the concept of off-diagonal-long-range-order. We discuss the analytical formula of the zero-temperature condensate density of Cooper pairs as a function of Debye frequency and energy gap, and calculate the condensate fraction for some metals. We study the density of Cooper pairs also at finite temperature showing its connection with the gap order parameter and the effects of the electron-phonon coupling. Finally, we analyze similarities and differences between superconductors and ultracold Fermi atoms in the determination of their condensate density by using the BCS theory.Comment: 14 pages, 1 figure, 1 table, to be published in 'Fermions: Flavors, Properties, and Types' (Nova Science Publishers, New York)

Pulsed Quantum Tunneling with Matter Waves

In this report we investigate the macroscopic quantum tunneling of a Bose condensate falling under gravity and scattering on a Gaussian barrier that could model a mirror of far-detuned sheet of light. We analyze the effect of the inter-atomic interaction and that of a transverse confining potential. We show that the quantum tunneling can be quasi-periodic and in this way one could generate coherent Bose condensed atomic pulses. In the second part of the report, we discuss an effective 1D time-dependent non-polynomial nonlinear Schrodinger equation (NPSE), which describes cigar-shaped condensates. NPSE is obtained from the 3D Gross-Pitaevskii equation by using a variational approach. We find that NPSE gives much more accurate results than all other effective 1D equations recently proposed.Comment: 9 pages, 5 figures, report for the X International Laser Physics Workshop, Seminar on Bose-Einstein Condensation of Trapped Atoms, Moscow, July 3-7, 200

Classical and Quantum Perturbation Theory for two Non--Resonant Oscillators with Quartic Interaction

We study the classical and quantum perturbation theory for two non--resonant oscillators coupled by a nonlinear quartic interaction. In particular we analyze the question of quantum corrections to the torus quantization of the classical perturbation theory (semiclassical mechanics). We obtain up to the second order of perturbation theory an explicit analytical formula for the quantum energy levels, which is the semiclassical one plus quantum corrections. We compare the "exact" quantum levels obtained numerically to the semiclassical levels studying also the effects of quantum corrections.Comment: 11 pages, Latex, no figures, to be published in Meccanic

Chaos Suppression in the SU(2) Yang--Mills--Higgs System

We study the classical chaos--order transition in the spatially homogenous SU(2) Yang--Mills--Higgs system by using a quantal analog of Chirikov's resonance overlap criterion. We obtain an analytical estimation of the range of parameters for which there is chaos suppression.Comment: LaTex, 10 pages, to be published in Phys. Rev.

3D BEC Bright Solitons under Transverse Confinement: Analytical Results with the Nonpolynomial Schrodinger Equation

The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrodinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrodinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE.Comment: 4 pages, presented to the 5th International School/Conference 'Let's Face Chaos through Nonlinear Dynamics', Maribor, July 2002, to be published in Progress in Theoretical Physics Supplemen

Parametric Resonance Phenomena in Bose-Einstein Condensates: Breaking of Macroscopic Quantum Self-Trapping

We analyze the periodic tunneling of a Bose-Einstein condensate in a double-well potential which has an oscillating energy barrier. We show that the dynamics of the Bose condensate critically depends on the frequency $\omega$ of the oscillating energy barrier. In the regime of periodic macroscopic quantum tunneling (PMQT) with frequency $\omega_J$, the population imbalance of the condensate in the two wells can be enhanced under the condition of parametric resonance $\omega = 2 \omega_J$. Instead, in the regime of macroscopic quantum self-trapping (MQST), we find that MQST can be reduced or suppressed under the condition of parametric resonance between the frequency $\omega$ of the energy barrier and the frequency $\omega_{ST}$ of oscillation through the barrier of the very small fraction of particles which remain untrapped during MQST.Comment: 9 pages, 3 figures, prepared for the 'Laser Physics Workshop 2000', seminar on 'Bose-Einstein Condensation of Trapped Atoms', Bratislava, to be published in Laser Physic