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arXiv:1501.07278 · Physical Review A

Fluctuation effects in rotating Bose-Einstein condensates with broken SU(2) and U(1)×U(1) symmetries in the presence of intercomponent density-density interactions

Peder Notto Galteland, Egor S. Babaev, Asle Sudbø

Published Jan 6, 2015• 8 citations
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Abstract

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 $\mathrm{SU}(2)$ 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 $\mathrm{SU}(2)$ 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.

DOI
10.1103/physreva.91.013605

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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 U(1)\ifmmode×\else\texttimes\fiU(1)\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)U(1)\ifmmode×\else\texttimes\fiU(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 [U(1)\ifmmode×\else\texttimes\fiU(1)][\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)][U(1)\ifmmode×\else\texttimes\fiU(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.
We consider a two-component immiscible Bose-Einstein condensate with dominating intra-species repulsive density-density interactions. In the ground-state phase of such a system only one condensates is present. This can be viewed as a spontaneous breakdown of Z2\mathbb{Z}_2Z2​ symmetry. We study the phase diagram of the system at finite temperature beyond mean-field approximation. In the absence of rotation, we show that the system undergoes a first order phase transition from this ground state to a miscible two-component normal fluid as temperature is increased. In the presence of rotation, the system features a competition between vortex-vortex interaction and short range density-density interactions. This leads to a rotation-driven "mixing" phase transition in a spatially inhomogeneous state with additional broken U(1)\mathrm{U}(1)U(1) symmetry. Thermal fluctuations in this state lead to nematic two-component sheets of vortex liquids. At sufficiently strong inter-component interaction, we find that the superfluid and Z2\mathbb{Z}_2Z2​ phase transitions split. This results in the formation of an intermediate state which breaks only Z2\mathbb{Z}_2Z2​ symmetry. It represents two phase separated normal fluids with density imbalance.
We consider a two-component interacting bosonic condensate with dominating intra-species repulsive density–density interactions. We study the phase diagram of the system at finite temperature with rotation, using large-scale Monte Carlo simulations of a two-component Ginzburg–Landau model of the system. In the presence of rotation, the system features a competition between long-range vortex–vortex interactions and short-range density–density interactions. This leads to a rotation-driven 'mixing' phase transition in a spatially inhomogeneous state with a broken symmetry. Thermal fluctuations in this state lead to nematic two-component sheets of vortex liquids. At sufficiently strong inter-component interaction, we find that the superfluid and phase transitions split. This results in the formation of an intermediate state which breaks only symmetry. It represents two phase separated normal fluids with a difference in their densities.
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 vortex lattices in three-component Bose-Einstein condensates (BECs) under rotation, where three kinds of fractional vortices winding one of three components are present. Unlike the cases of two-component BECs where the phases of square and triangular lattices are present depending on the intercomponent coupling constant and the rotation speed, we find triangular ordered ``colorful'' vortex lattices where three kinds of fractional vortices are placed in order without defects, in all parameter regions where the intercomponent coupling ${g}^{\ensuremath{'}}$ is less than the intracomponent coupling ggg. When $g>{g}^{\ensuremath{'}}$ on the other hand, we find the phase separation: In a region where one component is present, the other two components must vanish, where we find ghost vortices in these two components whose positions are separated. In the boundary $g={g}^{\ensuremath{'}}$, the accidental U(3) symmetry is present, in which case two vortices in different components are close to each other in some regions.
We review the topic of quantized vortices in multicomponent Bose–Einstein condensates of dilute atomic gases, with an emphasis on the two-component condensates. First, we review the fundamental structure, stability and dynamics of a single vortex state in a slowly rotating two-component condensates. To understand recent experimental results, we use the coupled Gross–Pitaevskii equations and the generalized nonlinear sigma model. An axisymmetric vortex state, which was observed by the JILA group, can be regarded as a topologically trivial skyrmion in the pseudospin representation. The internal, coherent coupling between the two components breaks the axisymmetry of the vortex state, resulting in a stable vortex molecule (a meron pair). We also mention unconventional vortex states and monopole excitations in a spin-1 Bose–Einstein condensate. Next, we discuss a rich variety of vortex states realized in rapidly rotating two-component Bose–Einstein condensates. We introduce a phase diagram with axes of rotation frequency and the intercomponent coupling strength. This phase diagram reveals unconventional vortex states such as a square lattice, a double-core lattice, vortex stripes and vortex sheets, all of which are in an experimentally accessible parameter regime. The coherent coupling leads to an effective attractive interaction between two components, providing not only a promising candidate to tune the intercomponent interaction to study the rich vortex phases but also a new regime to explore vortex states consisting of vortex molecules characterized by anisotropic vorticity. A recent experiment by the JILA group vindicated the formation of a square vortex lattice in this system.
In a single-component superfluid under rotation a broken symmetry in the order parameter space results in a broken translational symmetry in real space: a vortex lattice. If translational symmetry is restored, the phase of the order parameter disorders and thus the broken symmetry in the order parameter space is also restored. We show that for Bose-Einstein condensate mixtures in optical lattices with negative dissipationless drag, a new situation arises. This state is a modulated vortex liquid which breaks translational symmetry in the direction transverse to the rotation vector.