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Physical Review A

Vortex stability in Bose-Einstein condensates in the presence of disorder

M. Ll. Pons, Anna Sanpera

Published Mar 24, 2017• 2 citations
Publisher

Abstract

We study the time evolution of a Bose-Einstein condensate with a vortex on it when it is released from a trap and expands freely in a spatially uncorrelated disordered media. As customary in such cases, we perform the evolution over different disorder realizations and average over the disorder to obtain the quantities of interest. The propagation of noninteracting quantum systems in disordered media is strongly linked to Anderson localization which can be understood as formation of islands of constant phase due to coherent backscattering. We find that the vortex superfluid localizes in such media and, moreover, the vortex is resilient to disorder effects. This is a single-particle effect. In the presence of interactions, no matter how small they are, the vortex rapidly decays into phase discontinuities although localization is still present. The study of dispersion of a bosonic condensate with vorticity in a disordered media bears similarities with the stability of topological excitations in two-dimensional $p$-wave fermionic superfluids where the ground state is a Majorana mode that arises in the form of a vortex in the order parameter.

DOI
10.1103/physreva.95.033626

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We study the time evolution of a Bose-Einstein condensate with a vortex on it when it is released from a trap and expands freely in a spatially uncorrelated disordered media. As customary in such cases, we perform the evolution over different disorder realizations and average over the disorder to obtain the quantities of interest. The propagation of noninteracting quantum systems in disordered media is strongly linked to Anderson localization which can be understood as formation of islands of constant phase due to coherent backscattering. We find that the vortex superfluid localizes in such media and, moreover, the vortex is resilient to disorder effects. This is a single-particle effect. In the presence of interactions, no matter how small they are, the vortex rapidly decays into phase discontinuities although localization is still present. The study of dispersion of a bosonic condensate with vorticity in a disordered media bears similarities with the stability of topological excitations in two-dimensional ppp-wave fermionic superfluids where the ground state is a Majorana mode that arises in the form of a vortex in the order parameter.
We study the onset of localization from excited states of trapped Bose- Einstein Condensates expanding in presence of Gaussian uncorrelated random disorder. In 1D systems, we observe that for a fixed ratio between the disorder strength and the initial energy, excited states localize exponentially with a localization length that decreases as the energy of the initial state increases. Moreover, the localized state keeps the shape of the initial state wave function with an exponential tail. In 2D, we analyze the interplay between vorticiy and localization by examining the dispersion of a state containing a vortex on it in a disordered media. Despite localization can be associated to islands of constant phase, the presence of a vortex in the initial state leads to dislocations and phase jumps in the localized state. The study of dispersion of a bosonic condensate with vorticity bears similarities to the stability of topological excitations in 2D p-wave fermionic superfluids.
We numerically explore the long-time expansion of a one-dimensional Bose-Einstein condensate in a disorder potential employing the Gross-Pitaevskii equation. The goal is to search for unique signatures of Anderson localization in the presence of particle-particle interactions. Using typical experimental parameters, we show that the time scale for which the nonequilibrium dynamics of the interacting system begins to diverge from the noninteracting system exceeds the observation times up to now accessible in the experiment. We find evidence that the long-time evolution of the wave packet is characterized by (sub)diffusive spreading and a growing effective localization length, suggesting that interactions destroy Anderson localization.
We theoretically investigate the localization of an expanding Bose–Einstein condensate (BEC) with repulsive atom–atom interactions in a disordered potential. We focus on the regime where the initial inter-atomic interactions dominate over the kinetic energy and the disorder. At equilibrium in a trapping potential and for the considered small disorder, the condensate shows a Thomas–Fermi shape modified by the disorder. When the condensate is released from the trap, a strong suppression of the expansion is obtained in contrast to the situation in a periodic potential with similar characteristics. This effect crucially depends on both the momentum distribution of the expanding BEC and the strength of the disorder. For strong disorder as in the experiments reported by Clément et al 2005 Phys. Rev. Lett. 95 170409 and Fort et al 2005 Phys. Rev. Lett. 95 170410, the suppression of the expansion results from the fragmentation of the core of the condensate and from classical reflections from large modulations of the disordered potential in the tails of the condensate. We identify the corresponding disorder-induced trapping scenario for which large atom–atom interactions and strong reflections from single modulations of the disordered potential play central roles. For weak disorder, the suppression of the expansion signals the onset of Anderson localization, which is due to multiple scattering from the modulations of the disordered potential. We compute analytically the localized density profile of the condensate and show that the localization crucially depends on the correlation function of the disorder. In particular, for speckle potentials the long-range correlations induce an effective mobility edge in 1D finite systems. Numerical calculations performed in the mean-field approximation support our analysis for both strong and weak disorder.
We investigate a two-dimensional (2D) Bose system with the long range interactions in the presence of disorder. Formation of the bound states at strong impurity sites gives rise to a depletion of the superfluid density. We predict the intermediate superfluid state where the condensate and localized bosons are present simultaneously. We find that interactions suppress localization and that with the increase of the boson density the system experiences a sharp delocalization crossover into a state where all bosons are delocalized. We map our results onto a 3D system of vortices in type II superconductors in the presence of columnar defects; the intermediate superfluid state maps to an intermediate vortex liquid where vortex liquid neighbors pinned vortices. We predict the depinning crossover within the vortex liquid and depinning induced vortex lattice-Bose glass melting.
We study the interplay of disorder localization and strong local interactions within the Anderson-Hubbard model. Taking into account local Mott-Hubbard physics and static screening of the disorder potential, the system is mapped onto an effective single-particle Anderson model, which is studied within the self-consistent theory of electron localization. For fermions, we find rich nonmonotonic behavior of the localization length ξ, particularly in two-dimensional systems, including an interaction-induced exponential enhancement of ξ for small and intermediate disorders and a strong reduction of ξ due to hopping suppression by strong interactions. In three dimensions, we identify for half filling a Mott-Hubbard-assisted Anderson localized phase existing between the metallic and the Mott-Hubbardgapped phases. For small U there is re-entrant behavior from the Anderson localized phase to the metallic phase. For bosons, the unrestricted particle occupation number per lattice site yields a monotonic enhancement of ξ as a function of decreasing interaction, which we assume to persist until the superfluid Bose- Einstein condensate phase is entered. Besides, we study cold atomic gases expanding, by a diffusion process, in a weak random potential. We show that the density-density correlation function of the expanding gas is strongly affected by disorder and we estimate the typical size of a speckle spot, i.e., a region of enhanced or depleted density. Both a Fermi gas and a Bose-Einstein condensate (in a mean-field approach) are considered.
We study the stability of vortices in a binary system of Bose-Einstein condensat es, with their wave functions modeled by a set of coupled, time-dependent Gross-Pitaevskii equations. Beginning with an effective two-dimensional system, we identify miscible and immiscible regimes characterized by the inter- and intra-atomic interactions and the initial configuration of the system. We then consider a binary system of Bose-Einstein condensates placed in a rotating harmonic trap and study the single united vortex state in this system. We derive an approximate form for the energy of a single united vortex in the binary system and the critical angular velocity for the global stability of a vortex at the center of the trap. We also compute the metastability onset angular velocity for the local stability of a vortex at the center of the trap. Numerical solutions to the Gross-Pitaevskii equations support these expressions. These rotational results inform us of a novel bimodal formation within the miscible regime of the binary condensate system. We thus demonstrate the nontrivial aspects of vortex stability in interacting binary Bose-Einstein condensates as a result of their nonlinear interactions.