Vortex lattices in rapidly rotating Bose-Einstein condensates: Modes and correlation functions

Abstract

After delineating the physical regimes which vortex lattices encounter in rotating Bose-Einstein condensates as the rotation rate, Ω, increases, we derive the normal modes of the vortex lattice in two dimensions at zero temperature. Taking into account effects of the finite compressibility, we find an inertial mode of frequency ≥ 2Ω, and a primarily transverse Tkachenko mode, whose frequency goes from being linear in the wave vector in the slowly rotating regime, where Ω is small compared with the lowest compressional mode frequency, to quadratic in the wave vector in the opposite limit. We calculate the correlation functions of vortex displacements and phase, density and superfluid velocities, and find that the zero-point excitations of the soft quadratic Tkachenko modes lead in a large system to a loss of long range phase correlations, growing logarithmically with distance, and hence lead to a fragmented state at zero temperature. The vortex positional ordering is preserved at zero temperature, but the thermally excited Tkachenko modes cause the relative positional fluctuations to grow logarithmically with separation at finite temperature. The superfluid density, defined in terms of the transverse velocity autocorrelation function, vanishes at all temperatures. Finally we construct the long wavelength single particle Green's function in the rotating system and calculate the condensate depletion as a function of temperature.

Stability of Ultracold Atomic Bose Condensates with Rashba Spin-Orbit Coupling against Quantum and Thermal Fluctuations

“…The integral converges in the infrared, and thus a BEC can form at finite temperature. This result is consistent with Jian and Zhai's effective field theory approach to calculate phase fluctuations [17], applied in three dimensions, through the direct relation between the condensate depletion and the phase fluctuations [23]: n 0 ∼ n e − (φ(r)−φ(r ′ )) 2 /2 , |r − r ′ | → ∞. (21) Normal state: So far, we have assumed the existence of condensate, and proved that the condensate is not destroyed by thermal fluctuations.…”

Quantum fluctuations of vortex lattices in ultracold gases

“…We follow Refs. [10,14] and take this to be ∼0.145. We also calculate the vortex core density from the estimate of Ref.…”

“…In particular, for a large uniform system, it has been shown that these fluctuations drive a phase transition out of the triangular vortex lattice to a set of strongly correlated bosonic quantum liquids (analogous to fractional quantum Hall states) when the filling factor is reduced below a critical value ν c [10]. Estimates of ν c vary between ν c 2-6 from exact diagonalization results [10][11][12] and ν c 8-14 from a Lindemann criterion [10,13,14].…”

“…[13][14][15][16] do provide important information concerning the effects of quantum fluctuations within the vortex lattice phase. These results are based on the application of mean-field theory, with quantum fluctuations described by the Bogoliubov approximation.…”

Effective field theory of a vortex lattice in a bosonic superfluid

“…[16]. The hydrodynamics of such lattices were investigated by Baym [15,17] and later, in Ref. [18], Watanabe and Murayama proposed a low-energy effective field theory of this quantum state.…”