On mathematical models for Bose-Einstein condensates in optical lattices (expanded version)

Our aim is to analyze the various energy functionals appearing in the physics literature and describing the behavior of a Bose-Einstein condensate in an optical lattice. We want to justify the use of some reduced models. For that purpose, we will use the semi-classical analysis developed for linear problems related to the Schr\"odinger operator with periodic potential or multiple wells potentials. We justify, in some asymptotic regimes, the reduction to low dimensional problems and analyze the reduced problems

Asymptotic analysis of a secondary bifurcation of the one-dimensional Ginzburg-Landau equations of superconductivity

The bifurcation of asymmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg--Landau equations by the methods of formal asymptotics. The behavior of the bifurcating branch depends on the parameters d, the size of the superconducting slab, and $\kappa$, the Ginzburg--Landau parameter. The secondary bifurcation in which the asymmetric solution branches reconnect with the symmetric solution branch is studied for values of $(\kappa,d)$ for which it is close to the primary bifurcation from the normal state. These values of $(\kappa,d)$ form a curve in the $\kappa d$-plane, which is determined. At one point on this curve, called the quintuple point, the primary bifurcations switch from being subcritical to supercritical, requiring a separate analysis. The results answer some of the conjectures of [A. Aftalion and W. C. Troy, Phys. D, 132 (1999), pp. 214--232]

Vortex density models for superconductivity and superfluidity

We study some functionals that describe the density of vortex lines in superconductors subject to an applied magnetic field, and in Bose-Einstein condensates subject to rotational forcing, in quite general domains in 3 dimensions. These functionals are derived from more basic models via Gamma-convergence, here and in a companion paper. In our main results, we use these functionals to obtain descriptions of the critical applied magnetic field (for superconductors) and forcing (for Bose-Einstein), above which ground states exhibit nontrivial vorticity, as well as a characterization of the vortex density in terms of a non local vector-valued generalization of the classical obstacle problem.Comment: 34 page

Vortex Rings in Fast Rotating Bose-Einstein Condensates

When Bose-Eintein condensates are rotated sufficiently fast, a giant vortex phase appears, that is the condensate becomes annular with no vortices in the bulk but a macroscopic phase circulation around the central hole. In a former paper [M. Correggi, N. Rougerie, J. Yngvason, {\it arXiv:1005.0686}] we have studied this phenomenon by minimizing the two dimensional Gross-Pitaevskii energy on the unit disc. In particular we computed an upper bound to the critical speed for the transition to the giant vortex phase. In this paper we confirm that this upper bound is optimal by proving that if the rotation speed is taken slightly below the threshold there are vortices in the condensate. We prove that they gather along a particular circle on which they are evenly distributed. This is done by providing new upper and lower bounds to the GP energy.Comment: to appear in Archive of Rational Mechanics and Analysi

Inhomogeneous Vortex Patterns in Rotating Bose-Einstein Condensates

We consider a 2D rotating Bose gas described by the Gross-Pitaevskii (GP) theory and investigate the properties of the ground state of the theory for rotational speeds close to the critical speed for vortex nucleation. While one could expect that the vortex distribution should be homogeneous within the condensate we prove by means of an asymptotic analysis in the strongly interacting (Thomas-Fermi) regime that it is not. More precisely we rigorously derive a formula due to Sheehy and Radzihovsky [Phys. Rev. A 70, 063620(R) (2004)] for the vortex distribution, a consequence of which is that the vortex distribution is strongly inhomogeneous close to the critical speed and gradually homogenizes when the rotation speed is increased. From the mathematical point of view, a novelty of our approach is that we do not use any compactness argument in the proof, but instead provide explicit estimates on the difference between the vorticity measure of the GP ground state and the minimizer of a certain renormalized energy functional.Comment: 41 pages, journal ref: Communications in Mathematical Physics: Volume 321, Issue 3 (2013), Page 817-860, DOI : 10.1007/s00220-013-1697-

Strongly correlated phases in rapidly rotating Bose gases

We consider a system of trapped spinless bosons interacting with a repulsive potential and subject to rotation. In the limit of rapid rotation and small scattering length, we rigorously show that the ground state energy converges to that of a simplified model Hamiltonian with contact interaction projected onto the Lowest Landau Level. This effective Hamiltonian models the bosonic analogue of the Fractional Quantum Hall Effect (FQHE). For a fixed number of particles, we also prove convergence of states; in particular, in a certain regime we show convergence towards the bosonic Laughlin wavefunction. This is the first rigorous justification of the effective FQHE Hamiltonian for rapidly rotating Bose gases. We review previous results on this effective Hamiltonian and outline open problems.Comment: AMSLaTeX, 23 page

Shape oscillation of a rotating Bose-Einstein condensate

We present a theoretical and experimental analysis of the transverse monopole mode of a fast rotating Bose-Einstein condensate. The condensate's rotation frequency is similar to the trapping frequency and the effective confinement is only ensured by a weak quartic potential. We show that the non-harmonic character of the potential has a clear influence on the mode frequency, thus making the monopole mode a precise tool for the investigation of the fast rotation regime

Nonlinear dynamics for vortex lattice formation in a rotating Bose-Einstein condensate

We study the response of a trapped Bose-Einstein condensate to a sudden turn-on of a rotating drive by solving the two-dimensional Gross-Pitaevskii equation. A weakly anisotropic rotating potential excites a quadrupole shape oscillation and its time evolution is analyzed by the quasiparticle projection method. A simple recurrence oscillation of surface mode populations is broken in the quadrupole resonance region that depends on the trap anisotropy, causing stochastization of the dynamics. In the presence of the phenomenological dissipation, an initially irrotational condensate is found to undergo damped elliptic deformation followed by unstable surface ripple excitations, some of which develop into quantized vortices that eventually form a lattice. Recent experimental results on the vortex nucleation should be explained not only by the dynamical instability but also by the Landau instability; the latter is necessary for the vortices to penetrate into the condensate.Comment: RevTex4, This preprint includes no figures. You can download the complete article and figures at http://matter.sci.osaka-cu.ac.jp/bsr/cond-mat.htm

Rapidly Rotating Fermions in an Anisotropic Trap

We consider a cold gas of non-interacting fermions in a two dimensional harmonic trap with two different trapping frequencies $\omega_x \leq \omega_y$, and discuss the effect of rotation on the density profile. Depending on the rotation frequency $\Omega$ and the trap anisotropy $\omega_y/\omega_x$, the density profile assumes two qualitatively different shapes. For small anisotropy ($\omega_y/\omega_x \ll \sqrt{1+4 \Omega^2/\omega_x^2}$), the density consists of elliptical plateaus of constant density, corresponding to Landau levels and is well described by a two dimensional local density approximation. For large anisotropy ($\omega_y/\omega_x \gg \sqrt{1+4 \Omega^2/\omega_x^2}$), the density profile is Gaussian in the strong confining direction and semicircular with prominent Friedel oscillations in the weak direction. In this regime, a one dimensional local density approximation is well suited to describe the system. The crossover between the two regimes is smooth where the step structure between the Landau level edges turn into Friedel oscillations. Increasing the temperature causes the step structure or the Friedel oscillations to wash out leaving a Boltzmann gas density profile.Comment: 14 pages, 7 figure

Vortex nucleation in Bose-Einstein condensates in time-dependent traps

Vortex nucleation in a Bose-Einstein condensate subject to a stirring potential is studied numerically using the zero-temperature, two-dimensional Gross-Pitaevskii equation. It is found that this theory is able to describe the creation of vortices, but not the crystallization of a vortex lattice. In the case of a rotating, slightly anisotropic harmonic potential, the numerical results reproduce experimental findings, thereby showing that finite temperatures are not necessary for vortex excitation below the quadrupole frequency. In the case of a condensate subject to stirring by a narrow rotating potential, the process of vortex excitation is described by a classical model that treats the multitude of vortices created by the stirrer as a continuously distributed vorticity at the center of the cloud, but retains a potential flow pattern at large distances from the center.Comment: 22 pages, 7 figures. Changes after referee report: one new figure, new refs. No conclusions altere