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:1107.5160 · Physical Review A

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

Michele Correggi, Florian Pinsker, Nicolas Rougerie, Jakob Yngvason

Published Nov 14, 2011• 41 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.84.053614

Related papers

Vortex Phases of Rotating Superfluids

M. Correggi, F. Pinsker, Nicolas Rougerie +1 more

We report on the first mathematically rigorous proofs of a transition to a giant vortex state of a superfluid in rotating anharmonic traps. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and large rotational velocity and is based on precise asymptotic estimates on the ground state energy. An interesting aspect is a significant difference between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary. In the former case vortices persist in the bulk until the width of the annulus becomes comparable to the size of the vortex cores. In the second case the transition already takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus. Moreover, the density profiles in the annulus are different in the two cases. In both cases rotational symmetry of the density in a true ground state is broken, even though a symmetric variational ansatz gives an excellent approximation to the energy.

•Aug 16, 2016•4 citations

Vortex Phases of Rotating Superfluids

Michele Correggi, Florian Pinsker, Nicolas Rougerie +1 more

We report on the first mathematically rigorous proofs of a transition to a giant vortex state of a superfluid in rotating anharmonic traps. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and large rotational velocity and is based on precise asymptotic estimates on the ground state energy. An interesting aspect is a significant difference between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary. In the former case vortices persist in the bulk until the width of the annulus becomes comparable to the size of the vortex cores. In the second case the transition already takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus. Moreover, the density profiles in the annulus are different in the two cases. In both cases rotational symmetry of the density in a true ground state is broken, even though a symmetric variational ansatz gives an excellent approximation to the energy.

arXiv:1212.3680•Journal of Physics Conference Series•Feb 8, 2013•7 citations

Critical rotational speeds for superfluids in homogeneous traps

Michele Correggi, F. Pinsker, Nicolas Rougerie +1 more

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 “flat” traps, whereas it is gaussian for soft traps, and the “last” vortices to survive in the bulk before the giant vortex transition are small relative to the width of the annulus in the former case but of comparable size in the latter.

arXiv:1108.5429•Journal of Mathematical Physics•May 14, 2012•36 citations

Transition to the giant vortex state in a harmonic-plus-quartic trap

Han Fu, E. Zaremba

We consider a rapidly rotating Bose-condensed gas in a harmonic-plus-quartic trap. At sufficiently high rotation rates, the condensate acquires an annular geometry with the superposition of a vortex lattice. With increasing rotation rate, the lattice evolves into a single ring of vortices. Of interest is the transition from this state to the giant vortex state in which the circulation is carried by only a central vortex. By analyzing the Gross-Pitaevskii energy functional variationally, we have been able to map out the phase boundary between these two states as a function of the rotation rate and the various trapped gas parameters. For strong interactions, the transition is first order. Our variational results are in good qualitative agreement with those obtained by means of a direct numerical solution of the Gross-Pitaevskii equation.

arXiv:cond-mat/0508515•Physical Review A•Jan 12, 2006•47 citations

Rapidly rotating Bose-Einstein condensates in strongly anharmonic traps

Michele Correggi, Tanja Rindler-Daller, Jakob Yngvason

We study a rotating Bose-Einstein condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of two-dimensional Gross-Pitaevskii theory. We write the coupling constant for the interactions between the gas atoms as 1∕ε2 and we are interested in the limit ε→0 (Thomas-Fermi limit) with the angular velocity Ω depending on ε. We derive rigorously the leading asymptotics of the ground state energy and the density profile when Ω tends to infinity as a power of 1∕ε. If Ω(ε)=Ω0∕ε a “hole” (i.e., a region where the density becomes exponentially small as 1∕ε→∞) develops for Ω0 above a certain critical value. If Ω(ε)⪢1∕ε the hole essentially exhausts the container and a “giant vortex” develops with the density concentrated in a thin layer at the boundary. While we do not analyze the detailed vortex structure we prove that rotational symmetry is broken in the ground state for const∣logε∣<Ω(ε)≲const∕ε.

arXiv:math-ph/0606058•Journal of Mathematical Physics•Apr 1, 2007•37 citations

Dynamics of rapidly rotating Bose-Einstein condensates in a harmonic plus quartic trap

Jong-kwan Kim, Alexander L. Fetter

A two-dimensional rapidly rotating Bose-Einstein condensate in a harmonic plus quartic trap is expected to have unusual vortex states that do not occur in a pure harmonic trap. At a critical rotation speed ${\ensuremath{\Omega}}_{h}$, a central hole appears in the condensate, and at some faster rotation speed ${\ensuremath{\Omega}}_{g}$, the system undergoes a transition to a giant vortex state with pure irrotational flow. Using a time-dependent variational analysis, we study the behavior of an annular condensate with a single concentric ring of vortices. The transition to a giant vortex state is investigated by comparing the energy of the two equilibrium states (the ring of vortices and the giant vortex) and also by studying the dynamical stability of small excitation modes of the ring of vortices.

arXiv:cond-mat/0505078•Physical Review A•Aug 29, 2005•38 citations
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.
We report on the first mathematically rigorous proofs of a transition to a giant vortex state of a superfluid in rotating anharmonic traps. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and large rotational velocity and is based on precise asymptotic estimates on the ground state energy. An interesting aspect is a significant difference between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary. In the former case vortices persist in the bulk until the width of the annulus becomes comparable to the size of the vortex cores. In the second case the transition already takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus. Moreover, the density profiles in the annulus are different in the two cases. In both cases rotational symmetry of the density in a true ground state is broken, even though a symmetric variational ansatz gives an excellent approximation to the energy.
We report on the first mathematically rigorous proofs of a transition to a giant vortex state of a superfluid in rotating anharmonic traps. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and large rotational velocity and is based on precise asymptotic estimates on the ground state energy. An interesting aspect is a significant difference between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary. In the former case vortices persist in the bulk until the width of the annulus becomes comparable to the size of the vortex cores. In the second case the transition already takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus. Moreover, the density profiles in the annulus are different in the two cases. In both cases rotational symmetry of the density in a true ground state is broken, even though a symmetric variational ansatz gives an excellent approximation to the energy.
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 “flat” traps, whereas it is gaussian for soft traps, and the “last” vortices to survive in the bulk before the giant vortex transition are small relative to the width of the annulus in the former case but of comparable size in the latter.
We consider a rapidly rotating Bose-condensed gas in a harmonic-plus-quartic trap. At sufficiently high rotation rates, the condensate acquires an annular geometry with the superposition of a vortex lattice. With increasing rotation rate, the lattice evolves into a single ring of vortices. Of interest is the transition from this state to the giant vortex state in which the circulation is carried by only a central vortex. By analyzing the Gross-Pitaevskii energy functional variationally, we have been able to map out the phase boundary between these two states as a function of the rotation rate and the various trapped gas parameters. For strong interactions, the transition is first order. Our variational results are in good qualitative agreement with those obtained by means of a direct numerical solution of the Gross-Pitaevskii equation.
We study a rotating Bose-Einstein condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of two-dimensional Gross-Pitaevskii theory. We write the coupling constant for the interactions between the gas atoms as 1∕ε2 and we are interested in the limit ε→0 (Thomas-Fermi limit) with the angular velocity Ω depending on ε. We derive rigorously the leading asymptotics of the ground state energy and the density profile when Ω tends to infinity as a power of 1∕ε. If Ω(ε)=Ω0∕ε a “hole” (i.e., a region where the density becomes exponentially small as 1∕ε→∞) develops for Ω0 above a certain critical value. If Ω(ε)⪢1∕ε the hole essentially exhausts the container and a “giant vortex” develops with the density concentrated in a thin layer at the boundary. While we do not analyze the detailed vortex structure we prove that rotational symmetry is broken in the ground state for const∣logε∣<Ω(ε)≲const∕ε.
A two-dimensional rapidly rotating Bose-Einstein condensate in a harmonic plus quartic trap is expected to have unusual vortex states that do not occur in a pure harmonic trap. At a critical rotation speed ${\ensuremath{\Omega}}_{h}$, a central hole appears in the condensate, and at some faster rotation speed ${\ensuremath{\Omega}}_{g}$, the system undergoes a transition to a giant vortex state with pure irrotational flow. Using a time-dependent variational analysis, we study the behavior of an annular condensate with a single concentric ring of vortices. The transition to a giant vortex state is investigated by comparing the energy of the two equilibrium states (the ring of vortices and the giant vortex) and also by studying the dynamical stability of small excitation modes of the ring of vortices.