Gross-Pitaevskii Theory of the

We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state; more precisely, for any value of the angular velocity and for large enough values of the interaction strength, the ground state of the functional is not an eigenfunction of the angular momentum. This has interesting consequences on the Bose gas with spin; in particular, the ground state energy depends non-trivially on the number of spin components, and the dierent components do not have the same wave function. For the special case of a harmonic trap potential, we give explicit upper and lower bounds on the critical coupling constant for symmetry breaking. 1We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state; more precisely, for any value of the angular velocity and for large enough values of the interaction strength, the ground state of the functional is not an eigenfunction of the angular momentum. This has interesting consequences on the Bose gas with spin; in particular, the ground state energy depends non-trivially on the number of spin components, and the dierent components do not have the same wave function. For the special case of a harmonic trap potential, we give explicit upper and lower bounds on the critical coupling constant for symmetry breaking. 1

Jan 1, 200159 citations

Yrast line of a rapidly rotating Bose gas: Gross-Pitaevskii regime

Élliott H. Lieb, Robert Seiringer, Jakob Yngvason

We consider an ultracold rotating Bose gas in a harmonic trap close to the critical angular velocity, so that the system can be considered to be confined to the lowest Landau level. With this assumption we prove that the Gross-Pitaevskii energy functional accurately describes the ground-state energy of the corresponding $N$ -body Hamiltonian with contact interaction provided the total angular momentum $L$ is much less than ${N}^{2}$ . While the Gross-Pitaevskii energy is always an obvious variational upper bound to the ground-state energy, a more refined analysis is needed to establish it as an exact lower bound. We also discuss the question of Bose-Einstein condensation in the parameter range considered. Coherent states together with inequalities in spaces of analytic functions are the main technical tools.We consider an ultracold rotating Bose gas in a harmonic trap close to the critical angular velocity, so that the system can be considered to be confined to the lowest Landau level. With this assumption we prove that the Gross-Pitaevskii energy functional accurately describes the ground-state energy of the corresponding $N$-body Hamiltonian with contact interaction provided the total angular momentum $L$ is much less than ${N}^{2}$. While the Gross-Pitaevskii energy is always an obvious variational upper bound to the ground-state energy, a more refined analysis is needed to establish it as an exact lower bound. We also discuss the question of Bose-Einstein condensation in the parameter range considered. Coherent states together with inequalities in spaces of analytic functions are the main technical tools.

arXiv:0904.1750Physical Review AJun 24, 200931 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.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/0606058Journal of Mathematical PhysicsApr 1, 200737 citations

Ground state properties of rotating spin-orbital-angular-momentum coupled Bose-Einstein condensates

Lei Zhao, Xu Qiu, Yi LIANG +2 more

By numerically solving the single-particle stationary Schrödinger equation and the Gross-Pitaevskii equation with mean-field interactions at zero temperature, the ground state properties of the rotating spin-orbital-angular-momentum coupled Bose-Einstein condensates in a harmonic trapping potential are investigated in this work. The results show that the rotation lifts the double degeneracy of the single-particle energy spectrum in the angular momentum space, and leads to the vortex state. The angular momentum of the vortex depends on the rotating frequency, the intensity of the laser beam, and the spin-orbital-angular-momentum coupling. In particular, if the rotating frequency is below a critical value, the angular momentum of the ground state vortex remains unaffected by the rotating frequency. When the rotating frequency exceeds the critical value, the angular momentum of the ground state vortex will increase with the rotating frequency increasing. By assuming that the system is confined in a ring trap, the expression of the single-particle energy spectrum in the angular momentum space can be obtained, which clarifies how the rotation frequency affects the angular momentum of the ground state. In the presence of atomic interactions, similar phenomena can also be observed in the mean-field ground state at zero temperature.By numerically solving the single-particle stationary Schrödinger equation and the Gross-Pitaevskii equation with mean-field interactions at zero temperature, the ground state properties of the rotating spin-orbital-angular-momentum coupled Bose-Einstein condensates in a harmonic trapping potential are investigated in this work. The results show that the rotation lifts the double degeneracy of the single-particle energy spectrum in the angular momentum space, and leads to the vortex state. The angular momentum of the vortex depends on the rotating frequency, the intensity of the laser beam, and the spin-orbital-angular-momentum coupling. In particular, if the rotating frequency is below a critical value, the angular momentum of the ground state vortex remains unaffected by the rotating frequency. When the rotating frequency exceeds the critical value, the angular momentum of the ground state vortex will increase with the rotating frequency increasing. By assuming that the system is confined in a ring trap, the expression of the single-particle energy spectrum in the angular momentum space can be obtained, which clarifies how the rotation frequency affects the angular momentum of the ground state. In the presence of atomic interactions, similar phenomena can also be observed in the mean-field ground state at zero temperature.

Acta Physica SinicaJan 1, 20250 citations

A renormalized Gross–Pitaevskii theory and vortices in a strongly interacting Bose gas

Ch. Moseley, K. Ziegler

We consider a strongly interacting Bose-Einstein condensate in a spherical harmonic trap. The system is treated by applying a slave-boson representation for hard-core bosons. A renormalized Gross-Pitaevskii theory is derived for the condensate wave function that describes the dilute regime (like the conventional Gross-Pitaevskii theory) as well as the dense regime. We calculate the condensate density of a rotating condensate for both the vortex-free condensate and the condensate with a single vortex and determine the critical angular velocity for the formation of a stable vortex in a rotating trap.We consider a strongly interacting Bose-Einstein condensate in a spherical harmonic trap. The system is treated by applying a slave-boson representation for hard-core bosons. A renormalized Gross-Pitaevskii theory is derived for the condensate wave function that describes the dilute regime (like the conventional Gross-Pitaevskii theory) as well as the dense regime. We calculate the condensate density of a rotating condensate for both the vortex-free condensate and the condensate with a single vortex and determine the critical angular velocity for the formation of a stable vortex in a rotating trap.

arXiv:cond-mat/0505598Journal of Physics B Atomic Molecular and Optical PhysicsJan 24, 20076 citations

Ground state asymptotics of a dilute, rotating gas

Robert Seiringer

We investigate the ground state properties of a gas of interacting particles confined in an external potential in three dimensions and subject to rotation around an axis of symmetry. We consider the Gross–Pitaevskii (GP) limit of a dilute gas. Analysing both the absolute and the bosonic ground states of the system, we show, in particular, their different behaviour for a certain range of parameters. This parameter range is determined by the question whether the rotational symmetry in the minimizer of the GP functional is broken or not. For the absolute ground state, we prove that in the GP limit a modified GP functional depending on density matrices correctly describes the energy and reduced density matrices, independent of symmetry breaking. For the bosonic ground state this holds true if and only if the symmetry is unbroken.We investigate the ground state properties of a gas of interacting particles confined in an external potential in three dimensions and subject to rotation around an axis of symmetry. We consider the Gross–Pitaevskii (GP) limit of a dilute gas. Analysing both the absolute and the bosonic ground states of the system, we show, in particular, their different behaviour for a certain range of parameters. This parameter range is determined by the question whether the rotational symmetry in the minimizer of the GP functional is broken or not. For the absolute ground state, we prove that in the GP limit a modified GP functional depending on density matrices correctly describes the energy and reduced density matrices, independent of symmetry breaking. For the bosonic ground state this holds true if and only if the symmetry is unbroken.

arXiv:math-ph/0306022Journal of Physics A Mathematical and GeneralSep 2, 200355 citations

N

-body Hamiltonian with contact interaction provided the total angular momentum

L

is much less than

{N}^{2}

. While the Gross-Pitaevskii energy is always an obvious variational upper bound to the ground-state energy, a more refined analysis is needed to establish it as an exact lower bound. We also discuss the question of Bose-Einstein condensation in the parameter range considered. Coherent states together with inequalities in spaces of analytic functions are the main technical tools.