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arXiv:1101.0269 · Physical Review A

Tkachenko modes and their damping in the vortex lattice regime of rapidly rotating bosons

Sergey I. Matveenko, Gora V. Shlyapnikov

Published Mar 7, 2011• 18 citations
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Abstract

We have found an exact analytical solution of the Bogoliubov--de Gennes equations for the Tkachenko modes of the vortex lattice in the lowest Landau level (LLL) in the thermodynamic limit (geometry of an infinite plane) at any momenta and calculated their damping rates. At finite temperatures both Beliaev and Landau damping leads to momentum-independent damping rates in the low-energy limit, which shows that, at sufficiently low energies, Tkachenko modes become strongly damped. We then found that the mean-square fluctuations of the density grow logarithmically at large distances, which indicates that the state is ordered in the vortex lattice only on a finite (although exponentially large) distance scale and introduces a low-momentum cutoff. Using this circumstance we showed that at finite temperatures the one-body density matrix undergoes an exponential decay at large distances.

DOI
10.1103/physreva.83.033604

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We have found an exact analytical solution of the Bogoliubov--de Gennes equations for the Tkachenko modes of the vortex lattice in the lowest Landau level (LLL) in the thermodynamic limit (geometry of an infinite plane) at any momenta and calculated their damping rates. At finite temperatures both Beliaev and Landau damping leads to momentum-independent damping rates in the low-energy limit, which shows that, at sufficiently low energies, Tkachenko modes become strongly damped. We then found that the mean-square fluctuations of the density grow logarithmically at large distances, which indicates that the state is ordered in the vortex lattice only on a finite (although exponentially large) distance scale and introduces a low-momentum cutoff. Using this circumstance we showed that at finite temperatures the one-body density matrix undergoes an exponential decay at large distances.
After delineating the physical regimes that vortex lattices encounter in rotating Bose-Einstein condensates as the rotation rate $\ensuremath{\Omega}$ 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 $\ensuremath{\geqslant}2\ensuremath{\Omega}$, and a primarily transverse Tkachenko mode, whose frequency goes from being linear in the wave vector in the slowly rotating regime, where $\ensuremath{\Omega}$ 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.
We calculate the in-plane modes of the vortex lattice in a rotating Bose condensate, from the slowly rotating to mean-field quantum Hall limits. The Tkachenko mode frequency, linear in wave vector k for lattice rotational velocities Omega much smaller than the lowest sound wave frequency in a finite system, becomes quadratic in k in the opposite limit. The system also supports an inertial mode of frequency >or=2omega. The calculated frequencies are in excellent agreement with recent observations of Tkachenko modes by Phys. Rev. Lett., 91, 100402 (2003)].
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{H}_{c2}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{H}_{c2}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 establish the general equations of motion for the modes of a vortex lattice in a rapidly rotating Bose-Einstein condensate in three dimensions, taking into account the elastic energy of the lattice and the vortex line bending energy. As in two dimensions, the vortex lattice supports Tkachenko and gapped sound modes. In contrast, in three dimensions the Tkachenko mode frequency at long wavelengths becomes linear in the wave vector for any propagation direction out of the transverse plane. We compute the correlation functions of the vortex displacements and the superfluid order parameter for a homogeneous Bose gas of bounded extent in the axial direction. At zero temperature the vortex displacement correlations are convergent at large separation, but at finite temperatures, they grow with separation. The growth of the vortex displacements should lead to observable melting of vortex lattices at higher temperatures and somewhat lower particle number and faster rotation than in current experiments. At zero temperature a system of large extent in the axial direction maintains long range order-parameter correlations for large separation, but at finite temperatures the correlations decay with separation.
The complete low-energy collective-excitation spectrum of vortex lattices is discussed for rotating Bose-Einstein condensates by solving the Bogoliubov-de Gennes equation, yielding, e.g., the Tkachenko mode recently observed at JILA. The totally symmetric subset of these modes includes the transverse shear, common longitudinal, and differential longitudinal modes. We also solve the time-dependent Gross-Pitaevskii equation to simulate the actual JILA experiment, obtaining the Tkachenko mode and identifying a pair of breathing modes. Combining both approaches allows one to unambiguously identify every observed mode.
We consider a rapidly rotating two-component Bose-Einstein condensate (BEC) containing a vortex lattice. We calculate the dispersion relation for small oscillations of vortex positions (Tkachenko modes) in the mean-field quantum Hall regime, taking into account the coupling of these modes with density excitations. Using an analytic form for the density of the vortex lattice, we numerically calculate the elastic constants for different lattice geometries. We also apply this method to calculate the elastic constant for the single-component triangular lattice. For a two-component BEC, there are two kinds of Tkachenko modes, which we call acoustic and optical in analogy with phonons. For all lattice types, acoustic Tkachenko mode frequencies have quadratic wave-number dependence at long wavelengths, while the optical Tkachenko modes have linear dependence. For triangular lattices the dispersion of the Tkachenko modes are isotropic, while for other lattice types the dispersion relations show directional dependence consistent with the symmetry of the lattice. Depending on the intercomponent interaction there are five distinct lattice types, and four structural phase transitions between them. Two of these transitions are second order and are accompanied by the softening of an acoustic Tkachenko mode. The remaining two transitions are first order and while one of them is accompanied by the softening of an optical mode, the other does not have any dramatic effect on the Tkachenko spectrum. We also find an instability of the vortex lattice when the intercomponent repulsion becomes stronger than the repulsion within components.