We construct an effective field theory describing the collective Tkachenko oscillation mode of a vortex lattice in a two-dimensional rotating Bose-Einstein condensate in the long-wavelength regime. The theory has the form of a noncommutative field theory of a Nambu-Goldstone boson, which exhibits a noncommutative version of dipole symmetry. From the effective field theory, we show that, at zero temperature, the decay width Γ of the Tkachenko mode scales with its energy E as Γ∼E3 in the low-energy limit. We also discuss the width of the Tkachenko mode at a small temperature. Published by the American Physical Society 2024
In a rapidly rotating Bose-Einstein condensate (BEC) the condensate wave function is analogous to that of a type II superconductor close to the critical magnetic field
Hc2. This wave function is a superposition of the lowest Landau levels (LLL) of a charged particle in a magnetic field and corresponds to a vortex lattice. This leads to the inverted-parabola density profile of the BEC in the parabolic trap similar to slower rotation outside the LLL regime. Then one can apply the continuum theory of the Tkachenko modes (transverse sound modes in the vortex lattice) suggested for slower rotation, but with another shear modulus also known for type II superconductors at
Hc2. The theory yields a simple expression for the Tkachenko eigenfrequencies of the BEC cloud. The comparison with experiment shows that the experimental conditions have not yet reached the LLL regime.
We develop a low-energy effective field theory of a two-dimensional bosonic superfluid on the lowest Landau level at zero temperature and identify a Berry term that governs the dynamics of coarse-grained superfluid degrees of freedom. For an infinite vortex crystal we compute how the Berry term affects the low-energy spectrum of soft collective Tkachenko oscillations and nondissipative Hall responses of the particle number current and stress tensor. This term gives rise to a quadratic in momentum term in the Hall conductivity, but does not generate a nondissipative Hall viscosity.
A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional harmonic trap can be described with the lowest Landau-level set of single-particle states. The condensate wave function $\ensuremath{\psi}(x,y)$ is a Gaussian $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}{r}^{2}/2)$, multiplied by an analytic function
f(z) of the complex variable
z=x+iy. The criterion for a quantum phase transition to a non-superfluid correlated many-body state is usually expressed in terms of the ratio of the number of particles to the number of vortices. Here a similar description applies to a rapidly rotating nonsymmetric two-dimensional trap with arbitrary quadratic anisotropy (${\ensuremath{\omega}}_{x}^{2}<{\ensuremath{\omega}}_{y}^{2}$). The corresponding condensate wave function $\ensuremath{\psi}(x,y)$ is a complex anisotropic Gaussian with a phase proportional to
xy, multiplied by an analytic function
f(z), where $z=x+i{\ensuremath{\beta}}_{\ensuremath{-}}y$ is a stretched complex variable and $0\ensuremath{\leqslant}{\ensuremath{\beta}}_{\ensuremath{-}}\ensuremath{\leqslant}1$ is a real parameter that depends on the trap anisotropy and the rotation frequency. Both in the mean-field Thomas-Fermi approximation and in the mean-field lowest Landau level approximation with many visible vortices, an anisotropic parabolic density profile minimizes the energy. An elongated condensate grows along the soft trap direction yet ultimately shrinks along the tight trap direction. The criterion for the quantum phase transition to a correlated state is generalized (
1) in terms of
N/Lz, which suggests that a nonsymmetric trap should make it easier to observe this transition, or (
2) in terms of a ``fragmented'' correlated state, which suggests that a nonsymmetric trap should make it harder to observe this transition. An alternative scenario involves a crossover to a quasi one-dimensional condensate without visible vortices, as suggested by Aftalion et al., Phys. Rev. A 79, 011603(R) (2009).
After reviewing the ideal Bose-Einstein gas in a box and in a harmonic trap, the effect of interactions on the formation of a Bose-Einstein condensate are discussed, along with the dynamics of small-amplitude perturbations (the Bogoliubov equations). When the condensate rotates with angular velocity $\ensuremath{\Omega}$, one or several vortices nucleate, leading to many observable consequences. With more rapid rotation, the vortices form a dense triangular array, and the collective behavior of these vortices has additional experimental implications. For $\ensuremath{\Omega}$ near the radial trap frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$, the lowest-Landau-level approximation becomes applicable, providing a simple picture of such rapidly rotating condensates. Eventually, as $\ensuremath{\Omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{\ensuremath{\perp}}$, the rotating dilute gas is expected to undergo a quantum phase transition from a superfluid to various highly correlated (nonsuperfluid) states analogous to those familiar from the fractional quantum Hall effect for electrons in a strong perpendicular magnetic field.