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Rotating ground states of trapped Bose atoms with arbitrary two-body interactions

Oleg Vorov, Mahir Saleh Hussein, P. Van Isacker

Published Jul 22, 2002• 0 citations
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Abstract

In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.

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In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.
In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.
Trapped bosonic atoms with manipulated interaction strength near the sign flip of the scattering length are considered. Analytic solutions for the rotating ground states are obtained within algebraic approach, using permutation symmetry and group-theoretical properties of Bose operators. It is shown that the form of the ground states at any angular momentum is universal, and it does not depend on the details of the interaction. The ground states are either “collective rotations” or “condensed vortex states”, depending on the sign of the “modified Born scattering length”.
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.
We study the lowest energy states of a weakly interacting trapped atomic Bose-Einstein condensate in both quantum and classical cases. An analytic expression for the ground-state wavefunction of a rotating Gross-Pitaevskii condensate of trapped atoms describes the onset of vorticity in an accelerated trap, starting from the vortex entry followed by formation of growing symmetric Wigner molecules. Within a unified picture, it explains the staircase of the angular momentum jumps and the behavior of the bosonic occupancies observed in numerical and variational studies. The similarity of this behavior and mesoscopic superconductors is discussed.
Trapped Bose-Einstein condensates (BECs) differ considerably from the standard textbook example of a uniform Bose gas. In an isotropic harmonic potential V(r) = ½Mω2 r 2, the single-particle ground state introduces a new intrinsic scale of length [the ground-state size d = √ℏ/(Mω)] and energy [the ground-state energy E 0 = 32 \frac{3} {2} 23​ ℏω]. When the trap rotates at a low angular velocity, the behavior of a single vortex illustrates the crucial role of discrete quantized vorticity. For more rapid rotation, the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus, the condensate becomes effectively two dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest-Landau-level” regime, and a simple description again becomes possible. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field.
For a system of N bosons in a two-dimensional harmonic trap with frequency $\ensuremath{\omega},$ interacting via forces $V\ensuremath{\ll}\ensuremath{\Elzxh}\ensuremath{\omega},$ we develop a method to find the ground states of rotating Bose condensate as a function of two quantum numbers, the total angular momentum and the angular momentum of internal excitations (generalized yrast states). The energies of these condensed vortex states are expressed through the single two-body matrix element of interaction, V.V.V. A broad universality class of predominantly repulsive interactions for which these results hold is described by a simple integral condition on V.V.V. It includes Gaussian, $\ensuremath{\delta}$ function, and log-Coulomb forces.