arXiv:1007.2416v1 [cond-mat.quant-gas] 14 Jul 2010 Quantum localization without disorder in interacting Bose-Einstein condensates Roberto Franzosi, Salvatore M. Giampaolo, and Fabrizio Illuminati Dipartimento di Matematica e Informatica, Universit` a degli Studi di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy, CNR-SPIN, and INFN Sezione di Napoli, Gruppo collegato di Salerno, I-84084 Fisciano (SA), Italy (Dated: July 14, 2010) We discuss the possibility of exponential quantum localization in systems of ultracold bosonic atoms with repulsive interactions in open optical lattices without disorder. We show that exponential localization occur in the maximally excited state of the lowest energy band. We establish the conditions under which the presence of the upper energy bands can be neglected, determine the successive stages and the quantum phase boundaries at which localization occurs, and discuss how to detect it experimentally by visibility measurements. The discussed mechanism is a bona fide type of quantum localization, solely due to the interplay between nonlinearity and a bounded energy spectrum. In particular, it does not require the presence of random disorder or other local sources of noise, in striking contrast with Anderson localization. PACS numbers: 03.75.Lm, 05.30.Jp, 03.65.Sq The phenomenon of Anderson localization in disor- dered quantum systems [1] was originally discovered in the context of the study of electrons in a crystal with imperfections [2]. In fact, it is much more general [3] and has been observed in a variety of systems, including light waves in random media [4, 5]. Despite remarkable efforts, Anderson localization has not been observed directly in crystals, owing to the high electron-electron and electron- phonon interactions. It has finally been observed in non- interacting Bose-Einstein condensates in one-dimensional quasi-periodic optical lattices [6], that feature a crossover between extended and exponentially localized states, as in the case of purely random disorder in higher dimen- sions; moreover, the effects of random disorder in optical lattices can also be simulated manipulating the interac- tions in multi-species mixtures [7]. Indeed, ultracold degenerate gases in optical lattices provide an unprecedented toolbox for the experimental realization of what were once just toy models sketch- ing the key features of complex condensed matter sys- tems. One prominent example is the Bose-Hubbard model [8, 9], whose suggested realization in optical lat- tices loaded with ultracold bosonic atoms [10] was soon achieved in a spectacular breakthrough experiment [11]. Driven by this brilliant result, a growing number of in- vestigations has focused on the possibility to use optical lattices to realize various phenomena of considerable in- terest in condensed matter physics [12, 13]. Amongst these, in the last years much attention has been devoted to the study of localized quantum phases in many-body systems. For instance, it has been showed that it is possi- ble to use boundary dissipation [14] or the control of the sign of the local interactions, exploiting Feshbach reso- nances, to switch from the repulsive Hubbard model to the attractive one, whose ground state may feature a col- lapse of all the atoms of the system into a single site of the lattice [15–17]. The transition to collapse is essentially due to the combination of the nonlinear dependence of the local Hamiltonian on the site occupation that makes energetically favorable those states that are characterized by a concentration of all atoms in a single site. In the present work we describe a route to quantum localization in interacting many-boson systems, that has one important feature in common with the transition to collapse discussed in Refs.[15–17] because it does not re- quire the presence of any source of disorder. However, it differs crucially from the former in that it is not realized as an on-site concentration in the ground state of a sys- tem with attractive interactions, but rather as a proper exponential localization in the maximally excited state of the lowest energy band in systems with repulsive in- teractions. It is a novel bona fide mechanism of quantum localization that is solely due to the interplay between nonlinearity and a bounded energy spectrum and does not require the presence of random disorder, i.e. the key element in Anderson localization. Let us consider a system of N ultracold atoms with re- pulsive on-site interactions described by a Bose-Hubbard model on a one-dimensional lattice of M sites: H = U 2 d  j=−d ˆ n j (ˆ n j − 1) − T d−1  j=−d  ˆ a † j ˆ a j+1 + h.c.  . (1) One needs to consider open chains to look, even in prin- ciple, for the possibility of true localization. Indeed, in a translationally-invariant geometry the atoms would be unable to localize on a definite site. Namely, even in the presence of strong repulsive on-site interactions, the max- imally excited state would be essentially a Schr¨ odinger- cat state, i.e. a superposition of localized states charac- terized by a flat distribution of the atomic density over the entire lattice [17]. In Eq.(1) d =(M − 1)/2, ˆ a j (ˆ a † j ) are the bosonic annihilation (creation) operators on the j -th site, ˆ n j =ˆ a † j ˆ a j are the occupation number opera- tors, U> 0 is the strength of the repulsive nonlinear on-site interaction, and T is the hopping amplitude be- tween neighboring sites. In order to determine an optimized analytic approxi-