Localization and delocalization of ground states of Bose-Einstein condensates under disorder

This paper studies the localization behaviour of Bose-Einstein condensates in disorder potentials, modeled by a Gross-Pitaevskii eigenvalue problem on a bounded interval. In the regime of weak particle interaction, we are able to quantify exponential localization of the ground state, depending on statistical parameters and the strength of the potential. Numerical studies further show delocalization if we leave the identified parameter range, which is in agreement with experimental data. These mathematical and numerical findings allow the prediction of physically relevant regimes where localization of ground states may be observed experimentally.

[1]  Daniel Peterseim,et al.  Sobolev Gradient Flow for the Gross-Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency , 2018, SIAM J. Numer. Anal..

[2]  K. B. Davis,et al.  Bose-Einstein Condensation in a Gas of Sodium Atoms , 1995, EQEC'96. 1996 European Quantum Electronic Conference.

[3]  Yvon Maday,et al.  Numerical Analysis of Nonlinear Eigenvalue Problems , 2009, J. Sci. Comput..

[4]  P Zoller,et al.  Atomic Bose and Anderson glasses in optical lattices. , 2003, Physical review letters.

[5]  Elliott H. Lieb,et al.  A Rigorous Derivation¶of the Gross–Pitaevskii Energy Functional¶for a Two-dimensional Bose Gas , 2000 .

[6]  Localization of Quantum States and Landscape Functions , 2015, 1510.06353.

[7]  D. Wiersma,et al.  Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide. , 2005, Physical review letters.

[8]  J. Fröhlich,et al.  Absence of diffusion in the Anderson tight binding model for large disorder or low energy , 1983 .

[9]  S. Mayboroda,et al.  Universal mechanism for Anderson and weak localization , 2012, Proceedings of the National Academy of Sciences.

[10]  Bose.  Plancks Gesetz und Lichtquantenhypothese , 1924 .

[11]  Daniel Peterseim,et al.  Localized Computation of Eigenstates of Random Schrödinger Operators , 2019, SIAM J. Sci. Comput..

[12]  P. Anderson.  Absence of Diffusion in Certain Random Lattices , 1958 .

[13]  A. Aspect,et al.  Disorder-induced trapping versus Anderson localization in Bose–Einstein condensates expanding in disordered potentials , 2008, 0803.1774.

[14]  Wolfgang Ketterle,et al.  Bose-Einstein Condensation: Identity Crisis for Indistinguishable Particles , 2007 .

[15]  Marcel Filoche,et al.  Effective Confining Potential of Quantum States in Disordered Media. , 2015, Physical review letters.

[16]  Daniel Peterseim,et al.  The J-method for the Gross-Pitaevskii eigenvalue problem , 2019, ArXiv.

[17]  A. Aspect,et al.  Direct observation of Anderson localization of matter waves in a controlled disorder , 2008, Nature.

[18]  Ralf Kornhuber,et al.  Numerical Homogenization of Elliptic Multiscale Problems by Subspace Decomposition , 2016, Multiscale Model. Simul..

[19]  P. Mason,et al.  Vortices in Bose-Einstein Condensates , 2009 .

[20]  W. Bao,et al.  Mathematical Models and Numerical Methods for Bose-Einstein Condensation , 2012, 1212.5341.

[21]  A. Einstein.  Quantentheorie des einatomigen idealen Gases , 2006 .

[22]  Experimental study of the transport of coherent interacting matter-waves in a 1D random potential induced by laser speckle , 2006, cond-mat/0606015.

[23]  Ultracold bosonic atoms in two-colour superlattices , 2002, cond-mat/0205412.

[24]  Daniel Peterseim,et al.  Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials , 2018, Mathematical Models and Methods in Applied Sciences.

[25]  M. Aizenman.  LOCALIZATION AT WEAK DISORDER: SOME ELEMENTARY BOUNDS , 1994 .

[26]  P. Markowich,et al.  Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation , 2003, cond-mat/0303239.

[27]  R. Seiringer,et al.  Disordered Bose Einstein Condensates with Interaction , 2012, 1209.4046.

[28]  Andrew V. Knyazev,et al.  A subspace preconditioning algorithm for eigenvector/eigenvalue computation , 1995, Adv. Comput. Math..

[29]  F. Martinelli,et al.  Constructive proof of localization in the Anderson tight binding model , 1985 .

[31]  Massimo Inguscio,et al.  Anderson localization of a non-interacting Bose–Einstein condensate , 2008, Nature.

[32]  E. D'yakonov,et al.  Minimization of the computational labor in determining the first eigenvalues of differential operators , 1980 .

[33]  Analysis of localization phenomena in weakly interacting disordered lattice gases , 2006, cond-mat/0609774.

[34]  F. Dalfovo,et al.  Theory of Bose-Einstein condensation in trapped gases , 1998, cond-mat/9806038.

[35]  Ralf Kornhuber,et al.  An analysis of a class of variational multiscale methods based on subspace decomposition , 2016, Math. Comput..

[36]  S. F. Caballero-Benitez,et al.  Localisation of weakly interacting bosons in two dimensions: disorder vs lattice geometry effects , 2016, Scientific Reports.

[37]  C. Wieman,et al.  Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor , 1995, Science.

[38]  Suppression of transport of an interacting elongated Bose-Einstein condensate in a random potential. , 2005, Physical review letters.

[39]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[40]  Routes towards Anderson-like localization of Bose-Einstein condensates in disordered optical lattices. , 2005, Physical review letters.

[41]  M. Aizenman,et al.  Localization at large disorder and at extreme energies: An elementary derivations , 1993 .

[42]  A. Weinstein,et al.  Methods of intermediate problems for eigenvalues: theory and ramifications , 1972 .

[43]  L. Zalcman,et al.  Fourier Series and Integrals , 2007 .

[44]  W. Bao,et al.  MATHEMATICAL THEORY AND NUMERICAL METHODS FOR , 2012 .