We propose a universal transformation from a many-boson state to a corresponding many-fermion state in the lowest Landau level approximation of rotating many-body systems, inspired by the Laughlin wave function and by the Jain composite-fermion construction. We employ the exact-diagonalization technique for finding the many-body states. The overlap between the transformed boson ground state and the true fermion ground state is calculated in order to measure the quality of the transformation. For very small and high angular momenta, the overlap is typically above 90%. For intermediate angular momenta, mixing between states complicates the picture and leads to small ground-state overlaps at some angular momenta.
Using the Kubo formalism, we demonstrate fractional quantum Hall features in a rotating Bose-Einstein condensate in a co-rotating two-dimensional optical lattice. The co-rotating lattice and trap potential allow for an effective magnetic field and compensation of the centrifugal potential. Fractional quantum Hall features are seen for the single-particle system and for few strongly interacting many-particle systems.
We calculate the structure of individual vortices in rotating Bose-Einstein condensates in a transverse harmonic trap. Making a Wigner-Seitz approximation for the unit cell of the vortex lattice, we derive the Gross-Pitaevskii equation for the condensate wave function in each cell of the lattice, including effects of varying coarse grained density. We calculate the Abrikosov parameter, the fractional core area, and the energy of individual cells.
A fast rotating Bose-Einstein condensate confined in a quadratic-plus-quartic potential is found to dynamically generate a "giant vortex" that absorbs all phase singularities into a central low density hole, thereby sustaining a quasi-one-dimensional circular superflow at a supersonic speed.
We consider several models of interacting bosons in a one dimensional lattice. Some of them are not integrable like the Bose-Hubbard others are integrable. At low density all of these models can be described by the Bose gas with delta interaction. The lattice corrections corresponding to the different models are contrasted.
We study the groundstates of rotating Bose gases when interactions are affected by a nearby Feshbach resonance. We show that exact groundstates at high angular momentum can be found analytically for a general and realistic model for the resonant interactions. We identify parameter regimes where the exact groundstates are exotic fractional quantum Hall states, the excitations of which obey non-abelian exchange statistics.
We examine a rapidly rotating Bose-Einstein condensate in an anharmonic confinement and find that many properties such as the critical rotating frequency and phase diagram are quite different from those in a harmonic trap. We investigate the phase transitions by means of average-vortex-approximation. We find that the vortex lattice consists of a vortex array with a hole in the center of the cloud as the rotating frequency $Ω$ increases and the vortex becomes invisible when $Ω$ reaches some value.
We present a calculation of the low lying spectrum of a rotating Bose-Einstein condensate. We show that in a cylindrical geometry, there exist two linear branches, one associated to usual acoustic excitation, the other corresponding to a twisting mode of the vortex lattice. Using a hydrodynamical approach we derive the elasticity coefficient of the vortex lattice and calculate the spectrum of condensate in a three dimensionnal harmonic trap with cylindrical symmetry.
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.
Recently a novel phase transition has been observed \cite{jalin} in a rotating Bose-Einstein condensate when the rotating frequency $Ω$ reaches the transverse trap frequency $ω_{\bot}$ and eventually crosses it. We study certain aspects of this experiment in terms of the condensation of composite bosons and the corresponding vortex formation using a Chern-Simon Gross-Pitaevskii theory. %
We study the signatures of rotational and phase symmetry breaking in small rotating clouds of trapped ultracold Bose atoms by looking at rigorously defined condensate wave function. Rotational symmetry breaking occurs in narrow frequency windows, where the ground state of the system has degenerated with respect to the total angular momentum, and it leads to a complex wave function that exhibits vortices clearly seen as holes in the density, as well as characteristic vorticity. Phase symmetry (or gauge symmetry) breaking, on the other hand, is clearly manifested in the interference of two indep…
Exact diagonalization study of a rotating dilute Bose-Einstein condensate reveals that as the first vortex enters the system the degeneracy of the low-energy yrast spectrum is lifted and a large energy gap emerges. As more vortices enter with faster rotation, the energy gap decreases towards zero, but eventually the spectrum exhibits a rotonlike structure above the nu=1/2 Laughlin state without having a phonon branch despite the short-range nature of the interaction.
The role of anyonic excitations in fast rotating harmonically trapped Bose gases in a fractional Quantum Hall State is examined. Standard Chern-Simons anyons as well as "non standard" anyons obtained from a statistical interaction having Maxwell-Chern-Simons dynamics and suitable non minimal coupling to matter are considered. Their respective ability to stabilize attractive Bose gases under fast rotation in the thermodynamical limit is studied. Stability can be obtained for standard anyons while for non standard anyons, stability requires that the range of the corresponding statistical interac…
We study the composite fermion construction at and below the single vortex ($L=N$) state of weakly interacting rotating Bose gases, presenting a new method for handling the large number of derivatives typically occurring via the Slater determinant. Remarkably, the CF wave function at $L=N$ becomes {\em exact} in the large $N$ limit, even though this construction is not, {\em a priori}, expected to work in the low angular momentum regime. This implies an interesting mathematical identity which may be useful in other contexts.
Through exact numerical diagonalization, the von Neumann entropy is calculated for the Laughlin and Pfaffian quantum Hall states in rotating interacting Bose gases at zero temperature in the lowest Landau level limit. The particles comprising the states are indistinguishable, so the required spatial bipartitioning is effected by tracing over a subset of single-particle orbitals. The topological entropy is then extracted through a finite-size scaling analysis. The results for the Laughlin and the Pfaffian states agree with the expected values of $\ln\sqrt{2}$ and $\ln\sqrt{4}$, respectively.
We numerically study the system of rapidly rotating Bose atoms at the filling factor (ratio of particle number to vortex number) $ν=1$ with the dipolar interaction. A moderate dipolar interaction stabilizes the incompressible quantum liquid at $ν=1$. Further addition induces a collapse of it. The state after the collapse is a compressible state which has phases with stripes and bubbles. There are two types of bubbles with a different array. We also investigate models constructed from truncated interactions and the models with the three-body contact interaction. They also have phases with strip…
We show that dipolar interactions have dramatic effects on the groundstates of rotating atomic Bose gases in the weak interaction limit. With increasing dipolar interaction (relative to the net contact interaction), the mean-field, or high filling fraction, groundstate undergoes a series of transitions between vortex lattices of different symmetries: triangular, square, ``stripe'', and ``bubble'' phases. We also study the effects of dipolar interactions on the quantum fluids at low filling fractions. We show that the incompressible Laughlin state at filling fraction $ν=1/2$ is replaced by comp…
We use a Chern Simons Landau-Ginzburg (CSLG) framework related to hierarchies of composite bosons to describe 2D harmonically trapped fast rotating Bose gases in Fractional Quantum Hall Effect (FQHE) states. The predicted values for $ν$ (ratio of particle to vortex numbers) are $ν$$=$${{p}\over{q}}$ ($p$, $q$ are any integers) with even product $pq$, including numerically favored values previously found and predicting a richer set of values. We show that those values can be understood from a bosonic analog of the law of the corresponding states relevant to the electronic FQHE. A tentative glob…
We study the Quantum Hall phases that appear in the dilute limit of rotating Bose-Einstein condensates. By exact diagonalization in a spherical geometry we obtain the ground-state and low-lying excited states of a small number of bosons as a function of the filling fraction nu, ratio of the number of bosons to the number of vortices. We show the occurrence of the Jain principal sequence of incompressible liquids for nu = 1/2, 2/3, 3/4, 4/3, 5/4 as well as the Pfaffian state for nu =1. The collective excitations for the Jain sequence are well described by a composite-fermion scheme.
The role of anyonic excitations in fast rotating harmonically trapped Bose gases in a fractional Quantum Hall state is examined. Standard Chern-Simons anyons as well as "non standard" anyons obtained from a statistical interaction having Maxwell-Chern-Simons dynamics and suitable non minimal coupling to matter are considered. Their respective ability to stabilize attractive Bose gases under fast rotation in the thermodynamical limit is studied. Stability can be obtained for standard anyons while for non standard anyons, stability requires that the range of the corresponding statistical interac…
We observed the expansion of vortex-free Bose-condensates after their sudden release from a slowly rotating anisotropic trap. Our results show clear experimental evidence of the irrotational flow expected for a superfluid. The expansion from a rotating trap has strong features associated with the superfluid nature of a Bose-condensate, namely that the condensate cannot at any point be cylindrically symmetric with respect to the axis of rotation since such a wavefunction cannot possess angular momentum. Consequently, an initially rotating condensate expands in a distinctively different way to o…
For a Bose-Einstein condensate placed in a rotating trap, we study the simplified energy of a vortex line derived in Aftalion-Riviere Phys. Rev. A 64, 043611 (2001) in order to determine the shape of the vortex line according to the rotational velocity and the elongation of the condensate. The energy reflects the competition between the length of the vortex which needs to be minimized taking into account the anisotropy of the trap and the rotation term which pushes the vortex along the z axis. We prove that if the condensate has the shape of a pancake, the vortex stays straight along the z axi…
The close theoretical analogy between the physics of rapidly rotating atomic Bose condensates and the quantum Hall effect (i.e., a two dimensional electron gas in a strong magnetic field) was first pointed out ten years ago. As a consequence of this analogy, a large number of strongly correlated quantum Hall-type states have been predicted to occur in rotating Bose systems, and suggestions have been made how to manipulate and observe their fractional quasiparticle excitations. Due to a very rapid development in experimental techniques over the past years, experiments on BEC now appear to be cl…
We show that for rotating harmonically trapped Bose gases in a fractional quantum Hall state, the anyonic excitation statistics in the rotating gas can effectively play a {\em dynamical} role. For particular values of the two-dimensional coupling constant $g = -2π\hbar^2 (2k-1)/m$, where $k$ is a positive integer, the system becomes a noninteracting gas of anyons, with exactly obtainable solutions satisfying Bogomol'nyi self-dual order parameter equations. Attractive Bose gases under rapid rotation thus can be stabilized in the thermodynamic limit due to the anyonic statistics of their quasipa…
The Nambu-Goldstone mode, associated with vortex nucleation in a harmonically confined, two-dimensional dilute Bose-Einstein condensate, is identified with the lowest-lying envelope of octupole-mode branches, which are separated from each other by the admixture of quadrupolar excitations. As the vortex approaches the center of the condensate and the system's axisymmetry is restored, the Nambu-Goldstone mode becomes massive due to its coupling to higher rotational bands.
We investigate possible parafermionic states in rapidly rotating ultracold bosonic atomic gases at lowest Landau level filling factor nu=k/2. We study how the system size and interactions act upon the overlap between the true ground state and a candidate Read-Rezayi state. We also consider the quasihole states which are expected to display non-Abelian statistics. We numerically evaluate the degeneracy of these states and show agreement with a formula given by E. Ardonne. We compute the overlaps between low-lying exact eigenstates and quasihole candidate wavefunctions. We discuss the validity o…
We study the groundstates of weakly interacting atomic Bose gases under conditions of rapid rotation. We present the results of large-scale exact diagonalisation studies on a periodic geometry (a torus) which allows studies of compressible states with broken translational symmetry. Focusing on filling factor $ν=2$, we show a competition between the triangular vortex lattice, a quantum smectic state, and the incompressible $k=4$ Read-Rezayi state. We discuss the corrections arising from finite size effects, and the likely behaviour for large system sizes. The Read-Rezayi state is stabilised by …
We develop a stochastic Gross-Pitaveskii theory suitable for the study of Bose-Einstein condensation in a {\em rotating} dilute Bose gas. The theory is used to model the dynamical and equilibrium properties of a rapidly rotating Bose gas quenched through the critical point for condensation, as in the experiment of Haljan et al. [Phys. Rev. Lett., 87, 21043 (2001)]. In contrast to stirring a vortex-free condensate, where topological constraints require that vortices enter from the edge of the condensate, we find that phase defects in the initial non-condensed cloud are trapped en masse in the e…
We exploit the analogy with the quantum Hall (QH) effect for electrons to study the possible atomic QH states of a rapidly-rotating Bose-Einstein condensate. Actually, there is a nearly perfect map of the present problem in the QH regime to the QH physics for electrons. The profound map enables one to give a physically appealing definitions of the filling fraction and the "atomic Hall conductance" that is quantized for atomic Laughlin states. This quantization might imply an exotic fractionalization of atomic mass. We also briefly discuss an effective Chern-Simons theory for describing the ato…
The edge excitations and related topological orders of correlated states of a fast rotating Bose gas are studied. Using exact diagonalization of small systems, we compute the energies and number of edge excitations, as well as the boson occupancy near the edge for various states. The chiral Luttinger-liquid theory of Wen is found to be a good description of the edges of the bosonic Laughlin and other states identified as members of the principal Jain sequence for bosons. However, we find that in a harmonic trap the edge of the state identified as the Moore-Read (Pfaffian) state shows a number …
The vortex density of a rotating superfluid, divided by its particle mass, dictates the superfluid's angular velocity through the Feynman relation. To find how the Feynman relation applies to superfluid mixtures, we investigate a rotating two-component Bose-Einstein condensate, composed of bosons with different masses. We find that in the case of sufficiently strong interspecies attraction, the vortex lattices of the two condensates lock and rotate at the drive frequency, while the superfluids themselves rotate at two different velocities, whose ratio is the ratio between the particle mass of …
