We investigate the groundstates of weakly interacting bosons in a rotating trap as a function of the number of bosons, N , and the average number of vortices, NV . We identify the filling fraction ν ≡ N/NV as the parameter controlling the nature of these states. We present results indicating that, as a function of ν, there is a zero temperature phase transition between a triangular vortex lattice phase, and strongly-correlated vortex liquid phases. The vortex liquid phases appear to be the Read-Rezayi parafermion states.PACS Numbers: 03.75.Fi, 73.40.Hm, 67.57.Fg A fundamental characteristic of condensed Bose systems is their response to rotation [1]. A transition to a "normal" phase might be expected at sufficiently high angular velocities, ω, of the container (or trap) by loose analogy with a superconductor in a magnetic field. At zero temperature this phase would constitute a novel uncondensed ground state. Such a regime is entered when the vortex cores start to overlap. The corresponding value of ω is unattainable with bulk 4 He, but may be achievable in the very dilute degenerate atomic gases initially explored in Ref.[2], and studied extensively in Refs. [3][4][5][6][7][8][9], Apart from the identification [5] of the Laughlin state as the ground state at sufficiently high ω, work on the most interesting regime of large numbers of vortices has been restricted to either mean field theory [3] or exact diagonalisation [5][6][7][8]. These two approaches have exhibited apparently contradictory pictures. Within Gross-Pitaevskii (GP) mean-field theory, the groundstates are vortex lattices (distorted by the confinement), with broken rotational symmetry [3]. On the other hand, exact diagonalisations have identified groundstates which do not have crystalline correlations of vortex locations [5]; they are strongly correlated vortexliquids, closely related to incompressible liquid states responsible for the fractional quantum Hall effect [5][6][7].Here we present results of extensive exact diagonalisations (EDs) that elucidate the relationship between these two pictures. By using a periodic geometry, we have been able to study systems containing many vortices up to boson densities far in excess of previous EDs. Our results indicate that both crystalline and liquid phases of vortices exist. A clean distinction between these phases can only be made for a large number of vortices. In this limit, we argue that there is a zero-temperature phase transition as a function of the "filling fraction", ν ≡ N/N V , the ratio of the number of bosons, N , to the average number of vortices, N V . For large ν, the groundstate is a vortex lattice (characterised by broken translational/rotational symmetry). For small ν the groundstates are strongly-correlated vortex liquids. We find that the vortex-liquid groundstates are related to the ReadRezayi "parafermion" states [10] that were introduced in the context of fractional quantum Hall systems.In a frame of reference rotating with angular velocity ωẑ, the Hamiltonian for a particle of m...

We provide evidence for several novel phases in the dilute limit of rotating Bose-Einstein condensates. By exact calculation of wave functions and energies for small numbers of particles, we show that the states near integer angular momentum per particle are best considered condensates of composite entities, involving vortices and atoms. We are led to this result by explicit comparison with a description purely in terms of vortices. Several parallels with the fractional quantum Hall effect emerge, including the presence of the Pfaffian state.

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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 ...