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arXiv:cond-mat/0201020 · Physical Review Letters

Quantum Melting and Absence of Bose-Einstein Condensation in Two-Dimensional Vortex Matter

Jairo Sinova, Charles B. Hanna, Allan H. MacDonald

Published Jun 28, 2002• 147 citations
arXivHTMLPDF

Abstract

We demonstrate that quantum fluctuations suppress Bose-Einstein condensation of quasi-two-dimensional bosons in a rapidly rotating trap. Our conclusions rest in part on the derivation of an exact expression for the boson action in terms of vortex position coordinates, and in part on a solution of the weakly interacting boson Bogoliubov equations, which simplify in the rapid-rotation limit. We obtain analytic expressions for the collective-excitation dispersion, which is quadratic rather than linear. Our estimates for the boson filling factor at which the vortex lattice melts are consistent with recent exact-diagonalization calculations.

DOI
10.1103/physrevlett.89.030403

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We demonstrate that quantum fluctuations suppress Bose-Einstein condensation of quasi-two-dimensional bosons in a rapidly rotating trap. Our conclusions rest in part on the derivation of an exact expression for the boson action in terms of vortex position coordinates, and in part on a solution of the weakly interacting boson Bogoliubov equations, which simplify in the rapid-rotation limit. We obtain analytic expressions for the collective-excitation dispersion, which is quadratic rather than linear. Our estimates for the boson filling factor at which the vortex lattice melts are consistent with recent exact-diagonalization calculations.
We investigate vortex states and quantum fluctuations of bilayer atomic gases in rapid rotation. Among mean-field solutions of the Gross-Pitaevskii equation, we consider two types of vortex configurations: the vortex core positions of the layers are coincident or staggered. It is found that the coincident type is energetically preferred in the practical parameter regime. We also calculate the dispersion relations of collective modes and the filling factor for bosons outside the condensates. In the double layer system, quantum depletion is found to be suppressed due to interlayer tunneling. This means that Bose-Einstein condensation is stabilized compared to the single layer case.
We investigate a shift in the critical temperature of rotating Bose-Einstein condensates mediated by the melting of the vortex lattice. Numerical simulations reveal that this temperature exhibits contrasting behavior depending on the system configuration: a negative shift occurs for fixed trap potentials due to the expansion of the condensate, while a positive shift is observed for fixed volumes, where vortex-lattice rigidity suppresses thermal fluctuations. We introduce a vortex-energy model that captures the role of vortex interactions, the positional energy of the vortex lattice, as well as the phase transition and how the vortex lattice disappears. The findings provide insights into the thermodynamic properties of rotating condensates and the dynamics of vortex-lattice melting, offering potential parallels with other quantum systems such as type-II superconductors.
We study the thermal fluctuations of vortex positions in small vortex clusters in a harmonically trapped rotating Bose-Einstein condensate. It is shown that the order-disorder transition of two-shell clusters occurs via the decoupling of shells with respect to each other. The corresponding ``melting'' temperature depends strongly on the commensurability between numbers of vortices in shells. We show that melting can be achieved at experimentally attainable parameters and very low temperatures. Also studied is the effect of thermal fluctuations on vortices in an anisotropic trap with small quadrupole deformation. We show that thermal fluctuations lead to the decoupling of a vortex cluster from the pinning potential produced by this deformation. The decoupling temperatures are estimated and strong commensurability effects are revealed.
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 present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancellation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.