Exact Vortex Nucleation and Cooperative Vortex Tunneling in Dilute Bose-Einstein Condensates

Abstract

With the imminent advent of mesoscopic rotating BECs in the lowest Landau level (LLL) regime, we explore LLL vortex nucleation. An exact many-body analysis is presented in a weakly elliptical trap for up to 400 particles. Striking non-mean field features are exposed at filling factors ≫ 1. E. g. near the critical rotation frequency pairs of energy levels approach each other with exponential accuracy. A physical interpretation is provided by requantising a mean field (MF) theory, where 1/N plays the role of Planck's constant, revealing two vortices cooperatively tunneling between classically degenerate energy minima. The tunnel splitting variation is described in terms of frequency, particle number and ellipticity.PACS numbers: 03.75.Hh, 03.75.LmThe physics of vortices in slowly rotating degenerate gases [1] has reached the level of maturity where it is now used as a tool to study other phenomena, such as polarised fermi gases [2]. However achieving rapid rotation -to explore thoroughly the MF quantum Hall (QH) regime [3,4,5,6] in the lowest Landau level (LLL) [7] and to reach correlated QH states [7,8,9] -remains a challenge.A promising approach to accessing the QH regime is to have very dilute BECs, perhaps constructed by slicing up a condensate with an optical lattice [10]. In this Letter we show that even well away from the correlated regime there are pronounced quantum effects which become increasingly striking as the particle number decreases. We will show that the exact many-body ground states may be interpreted as exhibiting vortex tunneling leading to superpositions of mean-field states with vortices residing at different locations. This mesoscopic limit is consistent with the thrust of experimental effort in the near future. (In terms of ν = N/N v , where N is the number of particles, and N v the number of vortices, ν = 1/2 corresponds to the Laughlin state, and we will study 10 ν 400.) Vortex nucleation [11] has been studied in the ThomasFermi regime, both experimentally [12,13,14] and theoretically [15,16]. The conclusion is that under adiabatic ramping of the rotation frequency [13,14,15] the process is determined by an hydrodynamic instability. Under those conditions, the thermodynamic instability to vortex entry is apparently unobservable, occurring at lower rotation frequencies.It is known [17] that in a BEC in the LLL in an axisymmetric(AS) trap that there is a first-order thermodynamic instability to vortex entry (with no hydrodynamic instability needed). In this Letter, we will show that the situation is very different in a non-AS trap. The equilibrium of vortices in a non-AS trap has already been analysed at a MF level [18,19,20] within the LLL[21] and at the Bogoliubov level[22]).Our starting point is the standard model Hamiltonian, H, for a cold gas of N particles residing in a plane:Units of length, a ⊥ , and energy, ω ⊥ , are those provided by the harmonic trap; angular momenta, L z n are scaled by . There are two remaining dimensionless parameters. Firstly, Ω, is the angular velocity of ...

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