Ultracold bosons in random lattice potentials

Abstract

The Bose-Hubbard model describes short-range interacting spinless bosons on a random lattice. This model is realized in ultracold atomic systems where quantum phase transitions can be observed. When loaded into a deep potential depth perfect lattice, a cloud of cold bosonic atoms undergoes a quantum phase transition from a superfluid to a gapped Mott insulating phase. In the presence of additional random external potential, rare condensate regions can emerge inside an insulating background, forming a gapless Bose-glass state. Despite the well-understood characteristics of the Bose-Hubbard model, a precise analytic investigation of the effects of disorder on the energy spectrum remains unavailable, and a concrete characterization of the Bose-glass energy spectra, as well as its phase boundary with the Mott phase, is lacking. Here, we investigate how disorder affects the elementary excitations of the Bose-Hubbard model in the strongly interacting limit, and study the Mott-to-Bose glass quantum phase transition for zero and finite temperatures. We develop a perturbation method in the functional integral formalism to obtain a strong-coupling expansion for the single-particle Green's function in terms of the tunneling energy. By applying the partial summation method, we calculate the influence of an infinite subset of terms in such an expansion to the spectral function. Using the Poincaré-Lindstedt method, we compute the renormalized expression to the local density of states. We demonstrate that the spectrum is composed of stable-delocalized excitations at low energies and damped-localized excitations at slightly higher energies. When disorder becomes of the order of interactions, the stable-delocalized excitations become dispersionless due to their increased effective mass. In such a limit, the lifetime of the damped-localized excitations increases, and they dominate the whole energy band. We argue that such damped-localized excitations correspond to the low-energy excited states of the Bose-glass phase. Furthermore, by analyzing the local density of states, we show that this spectral information serves as a reliable parameter to unambiguously distinguish the Mott from the Bose-glass states both at zero and finite temperatures. Our results go beyond the mean-field theory predictions for the characterization of these ground states. Finally, we suggest an effective-action approach and a weak disorder expansion for the analysis of the superfluid excitation spectra.