Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0509559 · Physical Review A

Vortex lattices in Bose-Einstein condensates: From the Thomas-Fermi regime to the lowest-Landau-level regime

Marco Cozzini, S. Stringari, C. Tozzo

Published Feb 22, 2006• 19 citations
arXivHTMLPDF

Abstract

We consider a periodic vortex lattice in a rotating Bose-Einstein-condensed gas, where the centrifugal potential is exactly compensated by the external harmonic trap. By introducing a gauge transformation which makes the Hamiltonian periodic, we numerically solve the two-dimensional (2D) Gross-Pitaevskii equation finding the exact mean field ground state. In particular, we explore the crossover between the Thomas-Fermi regime, holding for large values of the coupling constant, and the lowest Landau level limit, corresponding to the weakly interacting case. Explicit results are given for the equation of state, the vortex core size, as well as the elastic shear modulus, which is crucial for the calculation of the Tkachenko frequencies.

DOI
10.1103/physreva.73.023615

Related papers

Vortex lattices in fast rotating Bose-Einstein condensates between the Thomas-Fermi and the lowest-Landau-level regimes

Zhongzhou Chen, Changru Ma, Yong-li Ma

We present an explicit analytical analysis of the ground state of vortex lattice structures, based on a minimization of the Gross-Pitaevskii energy functional in a fast rotating Bose-Einstein condensate between the Thomas-Fermi and the lowest-Landau-level regimes. By a Bogoliubov-like transformation we find that the coarse-grained average of the atomic density varies as ${[1\ensuremath{-}({x}^{2}+{y}^{2})∕{R}_{\ensuremath{\perp}}^{2}]}^{3∕2}$ in the $xy$ plane, and that the vortices form an almost regular triangular lattice. Our variational calculation is quantitatively and qualitatively consistent with recent analytical, numerical, and experimental data.

Physical Review A•Nov 20, 2006•2 citations

Rotating anisotropic Bose gas with large number of vortices

Ahmet Keleş

Rapidly rotating atomic gases provide a platform for studying phenomena akin to type-II superconductors and quantum Hall systems. Recently, these systems have attracted renewed interest due to technological advances in the trap anisotropy control, in-situ observation capabilities, and cooling and rotating complex atomic species such as dipolar gases. Understanding the vortex lattice formation and quantum melting is crucial for exploring quantum Hall physics in these systems. In this paper, we theoretically investigate the vortex lattices in anisotropic quantum gases. We formulate the rotating gas Hamiltonian in the Landau gauge, and consider the effects of additional perturbations such as the trap potential in the lowest Landau level (LLL). Focusing on the gases with short-range interactions, we obtain the many-body Hamiltonian projected to lowest Landau level. We consider the limit of full Bose-Einstein condensation and obtain the governing Gross-Pitaevskii equation to identify the possible vortex phases. We numerically solve the Gross-Pitaevskii equation using the imaginary time evolution, and demonstrate the possible vortex lattices as a function of anisotropy, rotation speed and interaction strength. We show that the number of states with a support in the LLL, which determines the number of vortices, follows a Thomas-Fermi type scaling albeit with slightly different coefficients from the usual condensates.

arXiv:2503.21982•TURKISH JOURNAL OF PHYSICS•Apr 15, 2025•1 citation

Vortex structures in rotating Bose-Einstein condensates

Sergey I. Matveenko, Dmitry L. Kovrizhin, Stéphane Ouvry +1 more

We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancellation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.

arXiv:0908.2172•Physical Review A•Dec 10, 2009•30 citations

Vortex core structure and global properties of rapidly rotating Bose-Einstein condensates

Gordon Baym, C. J. Pethick

We develop an approach for calculating stationary states of rotating Bose-Einstein condensates in harmonic traps which is applicable for arbitrary ratios of the rotation frequency to the transverse frequency of the trap ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. Assuming the number of vortices to be large, we write the condensate wave function as the product of a function that describes the structure of individual vortices times an envelope function varying slowly on the scale of the vortex spacing. By minimizing the energy, we derive Gross-Pitaevskii equations that determine the properties of individual vortices and the global structure of the cloud. For low rotation rates, the structure of a vortex is that of an isolated vortex in a uniform medium, while for rotation rates approaching the frequency of the trap (the mean-field lowest-Landau-level regime), the structure is that of the lowest $p$-wave state of a particle in a harmonic trap with frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. The global structure of the cloud is determined by minimizing the energy with respect to variations of the envelope function; for conditions appropriate to most experimental investigations to date, we predict that the transverse density profile of the cloud will be of the Thomas-Fermi form, rather than the Gaussian structure predicted on the assumption that the wave function consists only of components in the lowest Landau level for a regular array of vortices.

arXiv:cond-mat/0308325•Physical Review A•Apr 27, 2004•71 citations

Thomas–Fermi profile of a fast rotating Bose–Einstein condensate

Dinh-Thi Nguyen, Nicolas Rougerie

We study the minimizers of a magnetic 2D non-linear Schrödinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is linked to the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit.

arXiv:2201.04418•Pure and Applied Analysis•Dec 4, 2022•4 citations

Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates

Anxo Biasi, Piotr Bizoń, Ben Craps +1 more

The lowest Landau level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of Bose-Einstein condensates (BEC) in two-dimensional isotropic harmonic traps in the limit of weak interactions. Building on recent developments in the field of spatially confined extended Hamiltonian systems, we find a fully nonlinear solution of this equation representing periodically modulated precession of a single vortex. Motions of this type have been previously seen in numerical simulations and experiments at moderately weak coupling. Our paper provides a controlled analytic prediction for trajectories of a single vortex, suggests new targets for experiments, and opens up the prospect of finding analytic multivortex solutions.

arXiv:1705.00867•Physical Review A•Nov 10, 2017•43 citations
We consider a periodic vortex lattice in a rotating Bose-Einstein-condensed gas, where the centrifugal potential is exactly compensated by the external harmonic trap. By introducing a gauge transformation which makes the Hamiltonian periodic, we numerically solve the two-dimensional (2D) Gross-Pitaevskii equation finding the exact mean field ground state. In particular, we explore the crossover between the Thomas-Fermi regime, holding for large values of the coupling constant, and the lowest Landau level limit, corresponding to the weakly interacting case. Explicit results are given for the equation of state, the vortex core size, as well as the elastic shear modulus, which is crucial for the calculation of the Tkachenko frequencies.
We present an explicit analytical analysis of the ground state of vortex lattice structures, based on a minimization of the Gross-Pitaevskii energy functional in a fast rotating Bose-Einstein condensate between the Thomas-Fermi and the lowest-Landau-level regimes. By a Bogoliubov-like transformation we find that the coarse-grained average of the atomic density varies as ${[1\ensuremath{-}({x}^{2}+{y}^{2})∕{R}_{\ensuremath{\perp}}^{2}]}^{3∕2}$ in the xyxyxy plane, and that the vortices form an almost regular triangular lattice. Our variational calculation is quantitatively and qualitatively consistent with recent analytical, numerical, and experimental data.
Rapidly rotating atomic gases provide a platform for studying phenomena akin to type-II superconductors and quantum Hall systems. Recently, these systems have attracted renewed interest due to technological advances in the trap anisotropy control, in-situ observation capabilities, and cooling and rotating complex atomic species such as dipolar gases. Understanding the vortex lattice formation and quantum melting is crucial for exploring quantum Hall physics in these systems. In this paper, we theoretically investigate the vortex lattices in anisotropic quantum gases. We formulate the rotating gas Hamiltonian in the Landau gauge, and consider the effects of additional perturbations such as the trap potential in the lowest Landau level (LLL). Focusing on the gases with short-range interactions, we obtain the many-body Hamiltonian projected to lowest Landau level. We consider the limit of full Bose-Einstein condensation and obtain the governing Gross-Pitaevskii equation to identify the possible vortex phases. We numerically solve the Gross-Pitaevskii equation using the imaginary time evolution, and demonstrate the possible vortex lattices as a function of anisotropy, rotation speed and interaction strength. We show that the number of states with a support in the LLL, which determines the number of vortices, follows a Thomas-Fermi type scaling albeit with slightly different coefficients from the usual condensates.
We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancellation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.
We develop an approach for calculating stationary states of rotating Bose-Einstein condensates in harmonic traps which is applicable for arbitrary ratios of the rotation frequency to the transverse frequency of the trap ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. Assuming the number of vortices to be large, we write the condensate wave function as the product of a function that describes the structure of individual vortices times an envelope function varying slowly on the scale of the vortex spacing. By minimizing the energy, we derive Gross-Pitaevskii equations that determine the properties of individual vortices and the global structure of the cloud. For low rotation rates, the structure of a vortex is that of an isolated vortex in a uniform medium, while for rotation rates approaching the frequency of the trap (the mean-field lowest-Landau-level regime), the structure is that of the lowest ppp-wave state of a particle in a harmonic trap with frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. The global structure of the cloud is determined by minimizing the energy with respect to variations of the envelope function; for conditions appropriate to most experimental investigations to date, we predict that the transverse density profile of the cloud will be of the Thomas-Fermi form, rather than the Gaussian structure predicted on the assumption that the wave function consists only of components in the lowest Landau level for a regular array of vortices.
We study the minimizers of a magnetic 2D non-linear Schrödinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is linked to the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit.
The lowest Landau level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of Bose-Einstein condensates (BEC) in two-dimensional isotropic harmonic traps in the limit of weak interactions. Building on recent developments in the field of spatially confined extended Hamiltonian systems, we find a fully nonlinear solution of this equation representing periodically modulated precession of a single vortex. Motions of this type have been previously seen in numerical simulations and experiments at moderately weak coupling. Our paper provides a controlled analytic prediction for trajectories of a single vortex, suggests new targets for experiments, and opens up the prospect of finding analytic multivortex solutions.