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arXiv:cond-mat/0004307 · Physical Review A

Weakly interacting Bose-Einstein condensates under rotation

G. M. Kavoulakis, Ben R. Mottelson, C. J. Pethick

Published Nov 14, 2000• 104 citations
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Abstract

We investigate the ground and low excited states of a rotating, weakly interacting Bose-Einstein condensed gas in a harmonic trap for a given angular momentum. Analytical results in various limits as well as numerical results are presented, and these are compared with those of previous studies. Within the mean-field approximation and for repulsive interaction between the atoms, we find that for very low values of the total angular momentum per particle, $L/\stackrel{\ensuremath{\rightarrow}}{N}0,$ where $L\ensuremath{\Elzxh}$ is the angular momentum and N is the total number of particles, the angular momentum is carried by quadrupole $(|m|=2)$ surface modes. For $L/N=1,$ a vortexlike state is formed and all the atoms occupy the $m=1$ state. For small negative values of $L/N\ensuremath{-}1,$ the states with $m=0$ and $m=2$ become populated, and for small positive values of $L/N\ensuremath{-}1,$ atoms in the states with $m=5$ and $m=6$ carry the additional angular momentum. In the whole region $0<~L/N<~1,$ we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of $L/N.$ We have also found that an array of singly quantized vortices is formed as $L/N$ increases. Finally, we have gone beyond the mean-field approximation and have calculated the energy of the lowest state up to order N for small negative values of $L/N\ensuremath{-}1,$ as well as the energy of the low-lying excited states.

DOI
10.1103/physreva.62.063605

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Työväentutkimus Vuosikirja•Mar 1, 2006•0 citations
We investigate the ground and low excited states of a rotating, weakly interacting Bose-Einstein condensed gas in a harmonic trap for a given angular momentum. Analytical results in various limits as well as numerical results are presented, and these are compared with those of previous studies. Within the mean-field approximation and for repulsive interaction between the atoms, we find that for very low values of the total angular momentum per particle, L/N\ensuremath→0,L/\stackrel{\ensuremath{\rightarrow}}{N}0,L/N\ensuremath→​0, where L\ensuremath\ElzxhL\ensuremath{\Elzxh}L\ensuremath\Elzxh is the angular momentum and N is the total number of particles, the angular momentum is carried by quadrupole (∣m∣=2)(|m|=2)(∣m∣=2) surface modes. For L/N=1,L/N=1,L/N=1, a vortexlike state is formed and all the atoms occupy the m=1m=1m=1 state. For small negative values of L/N\ensuremath−1,L/N\ensuremath{-}1,L/N\ensuremath−1, the states with m=0m=0m=0 and m=2m=2m=2 become populated, and for small positive values of L/N\ensuremath−1,L/N\ensuremath{-}1,L/N\ensuremath−1, atoms in the states with m=5m=5m=5 and m=6m=6m=6 carry the additional angular momentum. In the whole region 0< L/N< 1,0<~L/N<~1,0< L/N< 1, we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of L/N.L/N.L/N. We have also found that an array of singly quantized vortices is formed as L/NL/NL/N increases. Finally, we have gone beyond the mean-field approximation and have calculated the energy of the lowest state up to order N for small negative values of L/N\ensuremath−1,L/N\ensuremath{-}1,L/N\ensuremath−1, as well as the energy of the low-lying excited states.
We consider a weakly interacting Bose-Einstein condensate that rotates in either a harmonic, or a weakly anharmonic trapping potential. Performing numerical calculations, we investigate the behavior of the gas in these two cases as the angular momentum, or equivalently as the rotational frequency of the trap increases. While in the case of a purely harmonic potential the gas makes a transition from the mean-field regime to the correlated Laughlin regime, in the case of anharmonic confinement the mean-field approximation remains always valid. We compare our derived results in these two cases, using both the mean-field approximation, as well as the diagonalization of the many-body Hamiltonian considering a small atom number.
We study the spectrum of rotating Bose-Einstein condensates in the limit of weak repulsive interactions and present analytical and numerical results for energies and wave functions. While the low-lying states are of collective nature, the high-lying states are dominated by single-particle excitations and can be accurately approximated by simple polynomial expressions. In the limit that the number of particles is large compared to the number of excited quanta, the single-particle states become excellent approximations to the eigenstates, and a rather simple ordering scheme is obtained.
The excitation spectrum of a highly condensed two-dimensional trapped Bose-Einstein condensate (BEC) is investigated within the rotating frame of reference. The rotation is used to transfer high-lying excited states to the low-energy spectrum of the BEC. We employ many-body linear-response theory and show that, once the rotation leads to a quantized vortex in the ground state, already the low-energy part of the excitation spectrum shows substantial many-body effects beyond the realm of mean-field theory. We demonstrate numerically that the many-body effects grow with the vorticity of the ground state, meaning that the rotation enhances them even for very weak repulsion. Furthermore, we explore the impact of the number of bosons NNN in the condensate on a low-lying single-particle excitation, which is describable within mean-field theory. Our analysis shows deviations between the many-body and mean-field results which clearly persist when NNN is increased up to the experimentally relevant regime, typically ranging from several thousand up to a million bosons in size. Implications are briefly discussed.
We consider a gas of N(< 15)N(<~15)N(< 15) Bose particles with hard-core repulsion, contained in a quasi-two-dimensional harmonic trap and subjected to an overall angular velocity $\ensuremath{\Omega}$ about the z axis. Exact diagonalization of the $n\ifmmode\times\else\texttimes\fi{}n$ many-body Hamiltonian matrix in given subspaces of the total (quantized) angular momentum Lz,{L}_{z},Lz​, with $n\ensuremath{\sim}{10}^{5}$ (e.g., for Lz=N=15,n=240782){L}_{z}=N=15, n=240782)Lz​=N=15,n=240782) was carried out using Davidson's algorithm. The many-body variational ground-state wave function, as also the corresponding energy and the reduced one-particle density-matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})={\ensuremath{\sum}}_{\ensuremath{\mu}}{\ensuremath{\lambda}}_{\ensuremath{\mu}}{\ensuremath{\chi}}_{\ensuremath{\mu}}^{*}(\mathbf{r}){\ensuremath{\chi}}_{\ensuremath{\mu}}({\mathbf{r}}^{\ensuremath{'}})$ were determined. With the usual identification of $\ensuremath{\Omega}$ as the Lagrange multiplier associated with Lz{L}_{z}Lz​ for a rotating system, the ${L}_{z}\ensuremath{-}\ensuremath{\Omega}$ phase diagram (or the stability line) was determined that gave a number of critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{},$ at which the ground-state angular momentum and the associated condensate fraction, given by the largest eigenvalue of the reduced one-particle density matrix, undergo abrupt jumps. For a given N, a number of (total) angular momentum states were found to be stable at successively higher critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{}.$ All the states in the regime N>Lz>0N>{L}_{z}>0N>Lz​>0 are metastable. For Lz>N,{L}_{z}>N,Lz​>N, the Lz{L}_{z}Lz​ values for the stable ground states generally increased with increasing critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}},$ and the condensate was strongly depleted. The critical ${\ensuremath{\Omega}}_{\mathrm{ci}}$ values, however, decreased with increasing interaction strength as well as the particle number, and were systematically greater than the nonvariational yrast-state values for the Lz=N{L}_{z}=NLz​=N single vortex state. We have also observed that the condensate fraction for the single vortex state (as also for the higher vortex states) did not change significantly even as the two-body interaction strength was varied over several $(\ensuremath{\sim}4)$ orders of magnitude in the moderately to the weakly interacting regime.
This thesis contains theoretical studies of weakly interacting, trapped Bose-Einstein condensed gases. The main emphasis is on the rotational properties of these systems. A typical Bose-Einstein condensate reacts to external rotation by forming singly quantized vortices. By the word ”typical” we refer to a condensate with repulsive interparticle interactions and with a frozen spin degree of freedom. However, there are some atomic species for which the low-energy interactions are attractive. Moreover, the tuning of different condensate properties is relatively easy. For example, there are ways to experimentally change the effective interaction from repulsive to attractive. For an attractively interacting condensate the vortex is not the only possible rotational state but the sample may also experience rotation through its center of mass. In the first part of the thesis, we analyze some properties of a rotating attractive Bose gas at the zero temperature limit. First, we note that in order to maintain energetical stability, the gas has to be rotated in an external trap which is steeper than a harmonic trap. We study the system in an anharmonic quadratic + quartic potential. The rotating ground state phase-space is determined with analytical and numerical methods. If the interactions are strong enough, the rotating ground state appears to be a state with the rotating center of mass. On the other hand, at the limit of weak interactions the system experiences rotation by forming a multiply quantized vortex. We also determine the spectrum of elementary excitations for these quantum many-particle states. We find that the broken symmetry of the center of mass rotating state induces an additional zero-energy (Goldstone) mode and that the small oscillations of the ground state exhibit the decoupling between the center of mass and relative motions. If the spin of the atoms is allowed to change freely, a rich variety of topological excitations in condensates can be studied. External rotation is one of the tools for examining the excitations in spinor condensates. In a spinless scalar condensate the appearance of a quantized vortex is manifested by the vortex line in which the condensate density vanishes. In the plane perpendicular to the vortex line, one speaks of the vortex core for the density minimum. We study the properties of slowly rotated F = 1 spinor condensates and show that for ferromagnetic interactions the ground state is a coreless vortex. While the individual spin components may carry ordinary vortices the overall density of the system is coreless. We also study a method to create a point-like monopole structure in a ferromagnetic condensate by suitably imprinting vortices and solitons for different spin components of the condensate.