Mean-field description of ultracold Bosons on disordered two-dimensional optical lattices

In the present paper we describe the properties induced by disorder on an ultracold gas of Bosonic atoms loaded into a two-dimensional optical lattice with global confinement ensured by a parabolic potential. Our analysis is centered on the spatial distribution of the various phases, focusing particularly on the superfluid properties of the system as a function of external parameters and disorder amplitude. In particular, it is shown how disorder can suppress superfluidity, while partially preserving the system coherence. PACS numbers: 03.75.Lm, 05.30.Jp, 64.60.Cn Submitted to: J. Phys. B: At. Mol. Phys. FTC Disordered 2D optical lattices 2 In the last years, the experimental results concerning confined ultracold atoms in optical lattices have attracted much theoretical interest from many different fields, ranging from condensed-matter physics to quantum information theory (see e.g. [1, 2]). The ability of tuning the fundamental physical parameters has provided an unrivaled tool to devise specific physical situations which, beyond their intrinsic physical interest, can be employed as important simulation tools [1]. In particular, the possibility to engineer a defect-free periodic potential appears as one of the most intriguing characteristics of these systems. Such “cleanness” has allowed the experimental realization of the Bose-Hubbard (BH) model and the observation of the superfluid-Mott quantum phase transition discussed by Fisher in his seminal paper [3], where, in addition, the issue of the effect of disorder on the physical properties of the BH Hamiltonian was first addressed. Furthermore many experimental techniques such as laser speckle field [4] and the superposition of different optical lattices with incommensurate lattice constants [5, 6, 7] have recently substantiated the theoretical investigation of ultracold atoms in disordered optical lattices [8, 9], revealing a rich scenario of new physical situations such as the appearance of new phases (e.g. Boseglass phase [6, 7] ) and superfluid (SF) percolation in d-dimensional (d > 1) lattices [10, 11]. In this paper we will deal with the properties of a two-dimensional (2D) optical lattice where bosons are confined by an overall parabolic potential and subject to a random potential distribution. The experimental realization of 2D lattices is discussed in [12], [13] and [14]. In particular the latter represents a direct experimental realization of the Hamiltonian discussed in the present paper, when disorder is absent. A valuable feature of the setup here considered is the possibility to investigate a wide range of physical situations and geometries according to different external parameter choices. While the parabolic potential confines bosons in a disk-like domain, the interplay of the other external parameters (hopping amplitude and chemical potential), allows one to realize, for example, Mott phases distributed in concentric shells, each with different filling, intercalated with SF shells. The 1D counterpart of this scenario has been thoroughly studied in [15]. Here we will focus particularly on the superfluid-phase spatial distribution in the presence of a random potential. We analyze first the situation where a (single) central disk-like Mott region is surrounded by a thin SF shell, and then the case when the system is fully SF. We show that, in the first case, the quasi-1D SF domain is strongly influenced by the presence of “impurities”, leading to the drop of the SF fraction for a small increase of the disorder amplitude. In the second case, the genuine 2D geometry of the SF domain leads to the formation of circulating streams which do not disappear abruptly when the disorder amplitude is increased. In this respect our setup therefore appears to be an effective tool to investigate the combined effect of disorder and dimensionality on the SF properties of a bosonic system. The system considered is modeled by a BH Hamiltonian H = HI +HT , with