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arXiv:cond-mat/0604416 · Physical Review Letters

Thermal Fluctuations of Vortex Matter in Trapped Bose-Einstein Condensates

Steinar Kragset, Egor S. Babaev, Asle Sudbø

Published Oct 25, 2006• 16 citations
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Abstract

We perform Monte Carlo studies of vortices in three dimensions in a cylindrical confinement, with uniform and nonuniform density. The former is relevant to rotating 4He; the latter is relevant to a rotating trapped Bose-Einstein condensate. In the former case, we find dominant angular thermal vortex fluctuations close to the cylinder wall. For the latter case, a novel effect is that at low temperatures the vortex solid close to the center of the trap crosses directly over to a tensionless vortex tangle near the edge of the trap. At higher temperatures an intermediate tensionful vortex liquid located between the vortex solid and the vortex tangle may exist.

DOI
10.1103/physrevlett.97.170403

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We perform Monte Carlo studies of vortices in three dimensions in a cylindrical confinement, with uniform and nonuniform density. The former is relevant to rotating 4He; the latter is relevant to a rotating trapped Bose-Einstein condensate. In the former case, we find dominant angular thermal vortex fluctuations close to the cylinder wall. For the latter case, a novel effect is that at low temperatures the vortex solid close to the center of the trap crosses directly over to a tensionless vortex tangle near the edge of the trap. At higher temperatures an intermediate tensionful vortex liquid located between the vortex solid and the vortex tangle may exist.
Most of the literature on quantum vortices predicting various states of vortex matter in three dimensions at finite temperatures in quantum fluids is based on the assumption of an extended and homogeneous system. This is well known not to be the case in actual Bose-Einstein condensates in traps, which are finite systems with nonuniform density. This raises the question to what extent one can speak of different aggregate states of vortex matter (vortex lattices, liquids, and tensionless vortex tangles) in these systems. To address this point, in the present work we focus on the finite-size, boundaries and density inhomogeneity effects on thermal vortex matter in a Bose-Einstein condensate. To this end we perform Monte Carlo simulations on a model system describing trapped Bose-Einstein condensates. Throughout the paper, we draw on analogies with results for vortex matter obtained for extended systems. We also consider, for comparison, the cylindrical confinement geometry with uniform density profile from the center out to the edge of the trap. The trapping potential is taken to be generically of an anharmonic form in such a way as to interpolate between a harmonic trap and a cylindrical confinement geometry. It also allows for a dip in the density profile at the center. We find distinct thermal equilibrium properties of the vortex system as the qualitative characteristics of the trapping potential are varied. For a uniform cylindrical confinement geometry, we find a vortex lattice at the center of the trap as well as ringlike thermal fluctuations of the vortex system as the trap edge is approached. For a harmonic trap, we find a distinct region at the edge of the trap where the vortex lines appear to have lost their line tension. Due to the steep density gradient, this crosses directly over to a vortex-line lattice at the center of the trap at low temperatures. At higher temperatures, an intermediate tensionful vortex liquid may exist. For an anharmonic trap where the ground state condensate density features a local minimum at the center of the trap, we find a corresponding region which appears to feature a tensionless vortex-line liquid phase. This work suggests that, finiteness and intrinsic inhomogeneity of the system notwithstanding, one nonetheless can approximately invoke the notion of distinct aggregate states of vortex matter realized at certain length scales. This might be helpful, in particular, in the search for possible new states of vortex matter in Bose-Einstein condensates with multiple components and different symmetries.
The properties of a rotating Bose-Einstein condensate confined in a prolate cylindrically symmetric trap are explored both analytically and numerically. As the rotation frequency increases, an ever greater number of vortices are energetically favored. Though the cloud anisotropy and moment of inertia approach those of a classical fluid at high frequencies, the observed vortex density is consistently lower than the solid-body estimate. Furthermore, the vortices are found to arrange themselves in highly regular triangular arrays, with little distortion even near the condensate surface. These results are shown to be a direct consequence of the inhomogeneous confining potential.
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.
We develop a stochastic Gross-Pitaveskii theory suitable for the study of Bose-Einstein condensation in a rotating dilute Bose gas. The theory is used to model the dynamical and equilibrium properties of a rapidly rotating Bose gas quenched through the critical point for condensation, as in the experiment of Haljan et al. [Phys. Rev. Lett. 87, 210403 (2001)]. In contrast to stirring a vortex-free condensate, where topological constraints require that vortices enter from the edge of the condensate, we find that phase defects in the initial noncondensed cloud are trapped en masse in the emerging condensate. Bose-stimulated condensate growth proceeds into a disordered vortex configuration. At sufficiently low temperature the vortices then order into a regular Abrikosov lattice in thermal equilibrium with the rotating cloud. We calculate the effect of thermal fluctuations on vortex ordering in the final gas at different temperatures, and find that the BEC transition is accompanied by lattice melting associated with diminishing long-range correlations between vortices across the system.
We study the dynamics of a vortex in an atomic Bose-condensed gas at finite temperature within the Zaremba-Nikuni-Griffin formalism. In a harmonically trapped pancake-shaped condensate, an off-centered vortex is known to decay by spiraling out toward the edge of the condensate. We quantify the dependence of this decay on temperature, atomic collisions, and thermal cloud rotation. Near the trap center where the density varies slowly, we show that our numerical results agree with the predictions of the Hall-Vinen phenomenological friction force model used to describe quantized vorticity in superfluid systems. Our result thus clarifies the microscopic origin of the friction and provides an ab initio determination of its value.
Thermal fluctuations and melting transitions for rotating single-component superfluids have been intensively studied and are well understood. In contrast, the thermal effects on vortex states for two-component superfluids with density-density interaction, which have a much richer variety of vortex ground states, have been much less studied. Here, we investigate the thermal effects on vortex matter in superfluids with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ broken symmetries and intercomponent density-density interactions, as well as the case with a larger SU(2)\mathrm{SU}(2)SU(2) broken symmetry obtainable from the $[\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]$-symmetric case by tuning scattering lengths. In the former case we find that, in addition to first-order melting transitions, the system exhibits thermally driven phase transitions between square and hexagonal lattices. Our main result, however, concerns the case where the condensate exhibits SU(2)\mathrm{SU}(2)SU(2) symmetry, and where vortices are not topological. At finite temperature, the system exhibits effects which do not have a counterpart in single-component systems. Namely, it has a state where thermally averaged quantities show no regular vortex lattice, yet the system retains superfluid coherence along the axis of rotation. In such a state, the thermal fluctuations result in transitions between different (nearly) degenerate vortex states without undergoing a melting transition. Our results apply to multicomponent Bose-Einstein condensates, and we suggest how to detect some of these unusual effects experimentally in such systems.