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

TL;DR

This paper analyzes the normal modes and correlation functions of vortex lattices in rapidly rotating Bose-Einstein condensates, revealing how quantum and thermal fluctuations affect long-range order and superfluid properties.

Contribution

It derives the vortex lattice normal modes considering finite compressibility, and explores the effects of quantum and thermal fluctuations on correlations and superfluid density.

Findings

Identification of inertial and Tkachenko modes in vortex lattices.

Zero-point fluctuations cause loss of long-range phase coherence.

Thermal excitations lead to logarithmic growth of fluctuations and vanishing superfluid density.

Abstract

After delineating the physical regimes which vortex lattices encounter in rotating Bose-Einstein condensates as the rotation rate, $\Omega$, 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 $\ge 2\Omega$, and a primarily transverse Tkachenko mode, whose frequency goes from being linear in the wave vector in the slowly rotating regime, where $\Omega$ 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.