On the Third Critical Speed for Rotating Bose-Einstein Condensates

We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevksii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in [M. Correggi {\it et al}, {\it J. Math. Phys. \textbf{53}(2012)] that such a transition occurs when the angular velocity is of order $\varepsilon ^{-4}$, with $\varepsilon ^{-2}$ denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and $\varepsilon \ll 1$ (Thomas-Fermi regime). In this paper we identify a finite value $\Omega_{\mathrm{c}}$ such that, if $\Omega = \Omega_0/\varepsilon ^4$ with $\Omega_0 > \Omega_{\mathrm{c}}$, the condensate is in the giant vortex phase. Under the same condition we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.Comment: pdfLaTeX, 39 pages, minor changes, to appear in J. Math. Phy

Coherent properties of nano-electromechanical systems

We study the properties of a nano-electromechanical system in the coherent regime, where the electronic and vibrational time scales are of the same order. Employing a master equation approach, we obtain the stationary reduced density matrix retaining the coherences between vibrational states. Depending on the system parameters, two regimes are identified, characterized by either ($i$) an {\em effective} thermal state with a temperature {\em lower} than that of the environment or ($ii$) strong coherent effects. A marked cooling of the vibrational degree of freedom is observed with a suppression of the vibron Fano factor down to sub-Poissonian values and a reduction of the position and momentum quadratures.Comment: 12 pages, 11 figure

Theory of Nonlinear Matter Waves in Optical Lattices

We consider several effects of the matter wave dynamics which can be observed in Bose-Einstein condensates embedded into optical lattices. For low-density condensates we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross-Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force, and lattice defects affect the nonlinear matter waves.Comment: Modern Physics Letters B (invited brief review), 25 pages, 9 figure

Nexus between quantum criticality and the chemical potential pinning in high-TcT_c cuprates

For strongly correlated electrons the relation between total number of charge carriers $n_e$ and the chemical potential $\mu$ reveals for large Coulomb energy the apparently paradoxical pinning of $\mu$ within the Mott gap, as observed in high-$T_c$ cuprates. By unravelling consequences of the non-trivial topology of the charge gauge U(1) group and the associated ground state degeneracy we found a close kinship between the pinning of $\mu$ and the zero-temperature divergence of the charge compressibility $\kappa\sim\partial n_e/\partial\mu$, which marks a novel quantum criticality governed by topological charges rather than Landau principle of the symmetry breaking.Comment: 4+ pages, 2 figures, typos corrected, version as publishe

Double-Slit Interferometry with a Bose-Einstein Condensate

A Bose-Einstein "double-slit" interferometer has been recently realized experimentally by (Y. Shin et. al., Phys. Rev. Lett. 92 50405 (2004)). We analyze the interferometric steps by solving numerically the time-dependent Gross-Pitaevski equation in three-dimensional space. We focus on the adiabaticity time scales of the problem and on the creation of spurious collective excitations as a possible source of the strong dephasing observed experimentally. The role of quantum fluctuations is discussed.Comment: 4 pages, 3 figure

Strong-coupling approach to the Mott--Hubbard insulator on a Bethe lattice in Dynamical Mean-Field Theory

We calculate the Hubbard bands for the half-filled Hubbard model on a Bethe lattice with infinite coordination number up to and including third order in the inverse Hubbard interaction. We employ the Kato--Takahashi perturbation theory to solve the self-consistency equation of the Dynamical Mean-Field Theory analytically for the single-impurity Anderson model in multi-chain geometry. The weight of the secondary Hubbard sub-bands is of fourth order so that the two-chain geometry is sufficient for our study. Even close to the Mott--Hubbard transition, our results for the Mott--Hubbard gap agree very well with those from numerical Dynamical Density-Matrix Renormalization Group (DDMRG) calculations. The density of states of the lower Hubbard band also agrees very well with DDMRG data, apart from a resonance contribution at the upper band edge which cannot be reproduced in low-order perturbation theory.Comment: 40 pages, 7 figure

On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state

We show that a pair of conjectures raised in [11] concerning the construction of normal solutions to the relativistic Boltzmann equation are valid. This ensures that the results in [11] hold for any range of positive temperatures and that the relativistic Euler system under the kinetic equation of state is hyperbolic and the speed of sound cannot overcome $c/\sqrt{3}$.Comment: 6 pages. Abridged version; full version to appear in Commun. Pure Appl. Ana

Two-electron photoionization of endohedral atoms

Using $He@C_{60}$ as an example, we demonstrate that static potential of the fullerene core essentially alters the cross section of the two-electron ionization differential in one-electron energy $d\sigma ^{++}(\omega )/d\epsilon$. We found that at high photon energy prominent oscillations appear in it due to reflection of the second, slow electron wave on the $C_{60}$ shell, which "dies out" at relatively high $\epsilon$ values, of about 2$\div$3 two-electron ionization potentials. The results were presented for ratios $R_{C_{60}}(\omega ,\epsilon)\equiv d\sigma ^{++}(\omega ,\epsilon)/d\sigma ^{a++}(\omega,\epsilon)$, where $d\sigma ^{a++}(\omega,\epsilon)/d\epsilon$ is the two-electron differential photoionization cross section. We have calculated the ratio $R_{i,ful}= \sigma_{i} ^{++}(\omega)/\sigma_{i}^{a++}(\omega)$, that accounts for reflection of both photoelectrons by the $C_{60}$ shell. We have calculated also the value of two-electron photoionization cross section $\sigma ^{++}(\omega)$ and found that this value is close to that of an isolated $He$ atom.Comment: 13 pages, 4 figure