© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:cond-mat/0405182 · Physical Review A

Vortex lattice stability and phase coherence in three-dimensional rapidly rotating Bose-Einstein condensates

S. Andrew Gifford, Gordon Baym

Published Sep 2, 2004• 22 citations

Abstract

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.

DOI: 10.1103/physreva.70.033602

Related papers

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

Gordon Baym

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.

arXiv:cond-mat/0308342•Physical Review A•Apr 27, 2004•51 citations

Tkachenko Modes of Vortex Lattices in Rapidly Rotating Bose-Einstein Condensates

Gordon Baym

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)].

arXiv:cond-mat/0305294•Physical Review Letters•Sep 9, 2003•123 citations

Tkachenko modes and structural phase transitions of the vortex lattice of a two-component Bose-Einstein condensate

Murat Keçeli, Mehmet Özgür Oktel

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.

arXiv:cond-mat/0511512•Physical Review A•Feb 17, 2006•21 citations

Observation of Tkachenko Oscillations in Rapidly Rotating Bose-Einstein Condensates

Ian R. Coddington, Peter Engels, Volker Schweikhard +1 more

We directly image Tkachenko waves in a vortex lattice in a dilute-gas Bose-Einstein condensate. The low (sub-Hz) resonant frequencies are a consequence of the small but nonvanishing elastic shear modulus of the vortex-filled superfluid. The frequencies are measured for rotation rates as high as 98% of the centrifugal limit for the harmonically confined gas. Agreement with a hydrodynamic theory worsens with increasing rotation rate, perhaps due to the increasing fraction of the volume displaced by the vortex cores. We also observe two low-lying m=0 longitudinal modes at about 20 times higher frequency.

arXiv:cond-mat/0305008•Physical Review Letters•Sep 5, 2003•210 citations

Tkachenko Oscillations and the Compressibility of a Rotating Bose-Einstein Condensate

Marco Cozzini, Лев П. Питаевский, S. Stringari

The elastic oscillations of the vortex lattice of a cold Bose gas (Tkachenko modes) are shown to play a crucial role in the saturation of the compressibility sum rule, as a consequence of the hybridization with the longitudinal degrees of freedom. The presence of the vortex lattice is responsible for a q2 behavior of the static structure factor at small wave vectors q, which implies the absence of long range order in 2D configurations at zero temperature. Sum rules are used to calculate the Tkachenko frequency in the presence of harmonic trapping. Results are derived in the Thomas-Fermi regime and compared with experiments as well as with previous theoretical estimates.

arXiv:cond-mat/0401516•Physical Review Letters•Jun 3, 2004•28 citations

Stability of the vortex lattice in a rotating superfluid

Gordon Baym

We analyze the stability of the vortex lattice in a rotating superfluid against thermal fluctuations associated with the long-wavelength Tkachenko modes of the lattice. Inclusion of only the two-dimensional modes leads formally to instability in infinite lattices; however, when the full three-dimensional spectrum of modes is taken into account, the thermally-induced lattice displacements are indeed finite.

arXiv:cond-mat/9412066•Physical review. B, Condensed matter•May 1, 1995•13 citations

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.

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)].

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.

We directly image Tkachenko waves in a vortex lattice in a dilute-gas Bose-Einstein condensate. The low (sub-Hz) resonant frequencies are a consequence of the small but nonvanishing elastic shear modulus of the vortex-filled superfluid. The frequencies are measured for rotation rates as high as 98% of the centrifugal limit for the harmonically confined gas. Agreement with a hydrodynamic theory worsens with increasing rotation rate, perhaps due to the increasing fraction of the volume displaced by the vortex cores. We also observe two low-lying m=0 longitudinal modes at about 20 times higher frequency.

The elastic oscillations of the vortex lattice of a cold Bose gas (Tkachenko modes) are shown to play a crucial role in the saturation of the compressibility sum rule, as a consequence of the hybridization with the longitudinal degrees of freedom. The presence of the vortex lattice is responsible for a q2 behavior of the static structure factor at small wave vectors q, which implies the absence of long range order in 2D configurations at zero temperature. Sum rules are used to calculate the Tkachenko frequency in the presence of harmonic trapping. Results are derived in the Thomas-Fermi regime and compared with experiments as well as with previous theoretical estimates.