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arXiv:2306.11873 · New Journal of Physics

Engineering phase and density of Bose–Einstein condensates in curved waveguides with toroidal topology

Y. O. Nikolaieva, Luca Salasnich, A. I. Yakimenko

Published Sep 7, 2023• 6 citations
arXivHTMLPDF

Abstract

Abstract We investigate the effects of ellipticity-induced curvature on atomic Bose–Einstein condensates confined in quasi-one-dimensional closed-loop waveguides. Our theoretical study reveals intriguing phenomena arising from the interplay between curvature and interactions. Density modulations are observed in regions of high curvature, but these modulations are suppressed by strong repulsive interactions. Additionally, we observe phase accumulation in regions with the lowest curvature when the waveguide with superflow has high eccentricity. Furthermore, waveguides hosting vortices exhibit dynamic transformations between states with different angular momenta. These findings provide insights into the behavior of atomic condensates in curved waveguides, with implications for fundamental physics and quantum technologies. The interplay between curvature and interactions offers opportunities for exploring novel quantum phenomena and engineering quantum states in confined geometries.

DOI
10.1088/1367-2630/acf783

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Abstract We investigate the effects of ellipticity-induced curvature on atomic Bose–Einstein condensates confined in quasi-one-dimensional closed-loop waveguides. Our theoretical study reveals intriguing phenomena arising from the interplay between curvature and interactions. Density modulations are observed in regions of high curvature, but these modulations are suppressed by strong repulsive interactions. Additionally, we observe phase accumulation in regions with the lowest curvature when the waveguide with superflow has high eccentricity. Furthermore, waveguides hosting vortices exhibit dynamic transformations between states with different angular momenta. These findings provide insights into the behavior of atomic condensates in curved waveguides, with implications for fundamental physics and quantum technologies. The interplay between curvature and interactions offers opportunities for exploring novel quantum phenomena and engineering quantum states in confined geometries.
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We investigate the effects of spatial curvature for an atomic Bose-Einstein condensate confined in an elliptical waveguide. The system is well described by an effective 1D Gross-Pitaevskii equation with a quantum-curvature potential, which has the shape of a double-well but crucially depends on the eccentricity of the ellipse. The ground state of the system displays a quantum phase transition from a two-peak configuration to a one-peak configuration at a critical attractive interaction strength. In correspondence of this phase transition the superfluid fraction strongly reduces and goes to zero for a sufficiently attractive Bose-Bose interaction.
We investigate vortex solitons in models representing “pancake-shaped” Bose-Einstein condensates (BECs) with attractive interatomic interactions and an anisotropic harmonic trapping potential, or polarization-preserving waveguides, such as a photonic-crystal or radial-bandgap fiber with an elliptically deformed core. Elliptic-vortex states bifurcate from dipole modes. In spite of strongly asymmetric shapes, the elliptic vortices with topological charge 1 are stable in a wide region in their existence domain. The dependence of the stability region on the eccentricity of the elliptic trap is reported. Input circular wave functions with embedded vorticity slowly relax into stationary elliptic vortices, featuring persistent rotation in the course of the evolution. All higher-order vortices are unstable.
We investigate the nonlinear dynamics of dark solitons in a one-dimensional Bose-Einstein condensate confined to a curved geometry. Using the Gross-Pitaevskii equation in curvilinear coordinates and a perturbative expansion in the local curvature, we derive a set of coupled evolution equations for the soliton velocity and the curvature. For the case of constant curvature, such as circular geometries, the soliton dynamics is governed solely by the initial velocity and curvature. Remarkably, the soliton travels a nearly constant angular trajectory across two orders of magnitude in curvature, suggesting an emergent conserved quantity, independent of its initial velocity. We extend our analysis to elliptical trajectories with spatially varying curvature and show that soliton dynamics remain determined by the local curvature profile. In these cases, the model of effective constant curvature describes accurately the dynamics given the local curvature has smooth variation. When the soliton crosses regions of rapid curvature variation and/or non-monotonic behavior, the model fails to describe to soliton dynamics, although the overall behavior can still be fully mapped to the curvature profile. Our results provide a quantitative framework for understanding the role of geometry in soliton dynamics and pave the way for future studies of nonlinear excitations in curved quantum systems.
Modern quantum engineering techniques allow for synthesizing quantum systems in exotic lattice geometries, from self-similar fractal networks to negatively curved hyperbolic graphs. We demonstrate that these structures profoundly reshape Bose-Einstein condensation. Fractal lattices dramatically lower the condensation temperature and enhance condensation fluctuations. In a Sierpiński carpet, quasi-degeneracies in the tight-binding spectrum fragment the condensate. Hyperbolic lattices, on the other hand, exhibit condensation features similar to regular three-dimensional lattices, despite their embedding in only two dimensions: The critical temperature increases as the system grows, and the temperature-dependence of the condensate fraction follows the same power-law as for cubic lattices. We explain these similarities through the similarity of the densities of state at low energies. When strong repulsive interactions are included, the gas enters a Mott insulating state. Using a multi-site Gutzwiller approach as well as a simple strong-coupling expansion, for the Sierpiński triangle we find a smooth interpolation between the characteristic insulating lobes of one-dimensional and two-dimensional systems. Our findings establish lattice geometry as a powerful tuning knob for quantum phase phenomena and pave the way for experimental exploration in photonic waveguide arrays and Rydberg-atom tweezer arrays.
We compute the ground state of a Bose–Einstein condensate confined on a curved surface and unravel the effects of curvatures. Starting with a general formulation for any smooth surface, we apply it to a prolate ellipsoid, which is inspired by recent bubble trap experiments. Using only elementary tools, we perform a perturbative approach to the Gross–Pitaevskii equation and a general Ansatz, followed by a dimensional reduction. We derive an effective two-dimensional equation that includes a curvature-dependent geometric potential. We compute the ground state using the Thomas–Fermi approximation and, for an isotropic confinement, we find that the highest accumulation of atoms happens in the regions with the greatest difference between the principal curvatures. For a prolate ellipsoid, this accumulation happens on the equator, which is contrary to previous findings that describe accumulation on the poles of a bubble trap. Finally, we explain the reasons for this difference: the higher accumulation of atoms on the poles happens due to anisotropies in the confinement, while the higher accumulation on the equator happens exclusively due to the geometric properties of the surface.