Anderson localization of Bogoliubov excitations on quasi‐1D strips

Abstract

Anderson localization of Bogoliubov excitations is studied for disordered lattice Bose gases in planar quasi-one-dimensional geometries. The inverse localization length is computed as function of energy by a numerical transfer-matrix scheme, for strips of different widths. These results are described accurately by analytical formulas based on a weak-disorder expansion of backscattering mean free paths. I. SETTING, OBJECTIVES, AND SCOPESingle-particle Anderson localization in quasi-1D geometries with correlated disorder is already a challenging problem [1][2][3][4]. With interactions, the situation becomes even more interesting. Here, we study the localization properties of the Bogoliubov quasi-particles (BQP) of weakly disordered Bose gases at zero temperature on quasi-1D lattices. While interactions tend to screen the disorder and thus stabilize extended (quasi-)condensates [5] for weak disorder [6,7], BQPs are expected to be localized irrespective of their energy or disorder strength in low dimensions [8][9][10], thus emulating noninteracting particles in the orthogonal Wigner-Dyson universality class [11]. Yet, BQPs differ qualitatively from noninteracting particles because of their collective, phononlike behavior at low energy. Moreover, they experience a randomness mediated by the inhomogeneous condensate background, which responds nonlinearly and nonlocally to the bare disorder [12][13][14]. This interplay has been extensively examined in 1D [15][16][17][18], where direct backscattering provides the only pathway to localization.In this Letter, we discuss the localization length of BQPs on quasi-1D planar lattices of transverse width N × 1 (shown in Fig. 1a). While the techniques developed below also apply to N x ×N y bars, strips provide the simplest realization of a multi-channel geometry, where phase-coherent scattering between modes is the rule. The bosons are described by the Bose-Hubbard (BH) model with on-site interaction U > 0, hopping t (with periodic boundary conditions across the strip) and on-site disorder V x . The random potential V x is drawn from a Gaussian distribution and has no spatial correlation on the lattice scale, V x V x = δ xx V 2 , where the bar denotes disorder averaging and V 2 is the on-site variance. For each realization of disorder, the mean-field (MF) ground-state density n x is determined by the condensate amplitude Φ x = √ n x that solves the discrete Gross-Pitaevskii (GP)where the sum runs over the nearest neighbors of x. The chemical potential µ controls the average occupation n = n x . In the limit of large occupations [19], an expansion of the BH Hamiltonian around the MF solution yields the Bogoliubov-de Gennes (BdG) equationswhere u x and v x are the BQP particle and hole components at energy E. These Bogoliubov excitations determine ground-state properties as well as the (thermo-) dynamical response of the system. In the remainder of this paper, we compute the dependence of the BQP localization length on the excitation energy. We first present a numerical transfer-m...

Thermodynamics of a two-dimensional dipolar Bose gas with correlated disorder in the roton regime

“…This is attributed to the fact that the particles are localized in the respective minima of the external random potential and thus form distributed randomly obstacles for the motion of the superfluid. However this localization is different from Anderson localization of Bogoliubov quasiparticles observed by Lugan et al [65,67]. The Bogoliubov quasiparticles experience a randomness mediated by the inhomogeneous condensate background, which responds nonlinearly and nonlocally to an effective potential that is different from the usual bare disorder [53,[65][66][67].…”

“…The Bogoliubov quasiparticles experience a randomness mediated by the inhomogeneous condensate background, which responds nonlinearly and nonlocally to an effective potential that is different from the usual bare disorder [53,[65][66][67]. Therefore, the localization properties are changed compared to bare particles although the general symmetry class is the same [66,67].…”