Nicolas Rougerie

Critical rotational speeds for superfluids in homogeneous traps

We present an asymptotic analysis of the effects of rapid rotation on the ground state properties of a superfluid confined in a two-dimensional trap. The trapping potential is assumed to be radial and homogeneous of degree larger than two in addition to a quadratic term. Three critical rotational velocities are identified, marking, respectively, the first appearance of vortices, the creation of a “hole” of low density within a vortex lattice, and the emergence of a giant vortex state free of vortices in the bulk. These phenomena have previously been established rigorously for a “flat” trap with fixed boundary but the “soft” traps considered in the present paper exhibit some significant differences, in particular the giant vortex regime, that necessitate a new approach. These differences concern both the shape of the bulk profile and the size of vortices relative to the width of the annulus where the bulk of the superfluid resides. Close to the giant vortex transition the profile is of Thomas-Fermi type in...

Quantum Hall states of bosons in rotating anharmonic traps

Faculty of Physics, University of Vienna, Boltzmanngasse 5 andErwin Schr¨odinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria.(Dated: January/30/2013)We study a model of bosons in the lowest Landau level in a rotating trap where the confinementpotential is a sum of a quadratic and a quartic term. The quartic term improves the stability ofthe system against centrifugal deconfinement and allows to consider rotation frequencies beyondthe frequency of the quadratic part. The interactions between particles are modeled by a Diracdelta potential. We derive rigorously conditions for ground states of the system to be stronglycorrelated in the sense that they are confined to the kernel of the interaction operator, and thuscontain the correlations of the Bose-Laughlin state. Rigorous angular momentum estimates and trialstate arguments indicate a transition from a pure Laughlin state to a state containing in addition agiant vortex at the center of the trap (Laughlin quasi-hole). There are also indications of a secondtransition where the density changes from a flat profile in a disc or an annulus to a radial Gaussianconfined to a thin annulus

Rotating superfluids in anharmonic traps: From vortex lattices to giant vortices

We study a superfluid in a rotating anharmonic trap and explicate a rigorous proof of a transition from a vortex lattice to a giant vortex state as the rotation is increased beyond a limiting speed determined by the interaction strength. The transition is characterized by the disappearance of the vortices from the annulus where the bulk of the superfluid is concentrated due to centrifugal forces while a macroscopic phase circulation remains. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and reveals significant differences between ‘soft’ anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary: In the latter case the transition takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus whereas in ‘soft’ traps the vortex lattice persists until the width of the annulus becomes comparable to the vortex cores. Moreover, the density profile in the annulus where the bulk is concentrated is, in the ‘soft’ case, approximately gaussian with long tails and not of the Thomas-Fermi type like in a trap with a fixed boundary.

Vortex Rings in Fast Rotating Bose–Einstein Condensates

When Bose–Einstein 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 (Correggi et al. in Commun Math Phys 303:451–308, 2011) we have studied this phenomenon by minimizing the two-dimensional Gross–Pitaevskii (GP) 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 uniformly distributed. This is done by providing new upper and lower bounds to the GP energy.

Critical Rotational Speeds in the Gross-Pitaevskii Theory on a Disc with Dirichlet Boundary Conditions

We study the two-dimensional Gross-Pitaevskii theory of a rotating Bose gas in a disc-shaped trap with Dirichlet boundary conditions, generalizing and extending previous results that were obtained under Neumann boundary conditions. The focus is on the energy asymptotics, vorticity and qualitative properties of the minimizers in the parameter range |log ε|≪Ω≲ε−2|log ε|−1 where Ω is the rotational velocity and the coupling parameter is written as ε−2 with ε≪1. Three critical speeds can be identified. At

Quantum Hall Phases and Plasma Analogy in Rotating Trapped Bose Gases

\varOmega=\varOmega_{\mathrm{c_{1}}}\sim |\log\varepsilon|

On the Ginzburg–Landau Functional in the Surface Superconductivity Regime

vortices start to appear and for

Emergence of Fractional Statistics for Tracer Particles in a Laughlin Liquid.

|\log\varepsilon|\ll\varOmega< \varOmega_{\mathrm{c_{2}}}\sim \varepsilon^{-1}

Derivation of Pekar's Polarons from a Microscopic Model of Quantum Crystal

the vorticity is uniformly distributed over the disc. For

Boundary Behavior of the Ginzburg–Landau Order Parameter in the Surface Superconductivity Regime

\varOmega\geq\varOmega _{\mathrm{c_{2}}}