Revealing the superfluid–Mott-insulator transition in an optical lattice

Abstract

We study (by an exact numerical scheme) the single-particle density matrix of ∼ 10 3 ultracold atoms in an optical lattice with a parabolic confining potential. Our simulation is directly relevant to the interpretation and further development of the recent pioneering experiment [1]. In particular, we show that restructuring of the spatial distribution of the superfluid component when a domain of Mott-insulator phase appears in the system, results in a fine structure of the particle momentum distribution. This feature may be used to locate the point of the superfluid-Mott-insulator transition.The fascinating physics of the superfluid-insulator transition in a system of interacting bosons on a lattice has been attracting constant interest of theorists during recent years [2][3][4][5][6][7][8][9]. Lattice bosons is one of the simplest many-body problems with strong competition between the potential and kinetic energy, and a typical example of the quantum phase transition system. One of its great advantages is the possibility to study it by powerful Monte Carlo methods which nowadays allow simulations of many thousands of particles at low temperature with unprecedented accuracy (see, e.g., [9]). However, until very recently the canonical Bose-Hubbard model(where a † i creates a particle on the site i, < ij > stands for the nearest-neighbor sites, n i = a † i a i , and t, U , and µ i , are the hopping amplitude, the on-site interaction, and the on-site external field, respectively) was not particularly useful in the analysis of realistic systems. The situation has changed with the exciting success of the experiment by Greiner et al.[1] (originally proposed by Jaksch et al. [10]) in which a gas of ultracold 87 Rb atoms was trapped in a three-dimensional, simple-cubic optical lattice potential. The uniqueness of the new system is that it is adequately described by the Bose-Hubbard Hamiltonian [10,1], and allows virtually an unlimited control over the strength of the effective interparticle interaction U/t and particle density.The characteristic feature of the experimental setup of Ref.[1] is the presence of the overall parabolic potential V (r) which confines the sample. This feature could be of great advantage if one was able to directly measure the spatial density distribution in the trap. We recall the structure of the µ − U/t phase diagram for the BoseHubbard system [2] which predicts commensurate particle density distribution for the insulating phase whenever the chemical potential lies within the Mott-Hubbard gap. The slowly varying (at the length scale of the lattice period) trapping potential effectively provides a scan over µ of this phase diagram at a fixed value of U/t.Unfortunately, what is measured in the experiment is not the original spatial density distribution in the trap, but the absorption image of the free evolving atomic cloud, after the trapping/optical potential is removed; i.e. the quantity which is directly related to the the singleparticle density matrix in momentum space, ρ kk = n k .[This st...

Bosons confined in optical lattices: The numerical renormalization group revisited

“…There are no satellite peaks when the central density is noninteger, despite a broad Mott n = 1 region for an on-site repulsion U that is strong enough. This Mott region is reflected in the tail of the momentum distribution [24]. For a system with local densities varying between n = 2 and n = 3.2 (all noninteger densities), we nevertheless found satellite peaks in Fig.…”

“…It was argued in Ref. [24] that satellite peaks in the momentum distribution are related to the appearance of a Mott region in the center of the trap. Once the Mott region spanned almost the entire lattice, the peaks disappeared into the typical broad, low-peaked Mott distribution.…”

“…It was argued in Ref. [24] that the appearance of satellite peaks in the momentum distribution signals the appearance of a Mott region in the *Electronic address: Lode.Pollet@UGent.be center of the trap. We will especially focus on the momentum distribution and on the issue of the satellite peaks.…”

“…Disorder and impurities can have dramatic effects [6,11,17,[20][21][22] and lead to even other phases. The model with a quadratic confining potential [2] has been addressed in one dimension [23] and for a small lattice in three dimensions [24], using quantum Monte Carlo methods.…”

Occupation number and fluctuations in the finite-temperature Bose-Hubbard model

“…The zero-temperature BH model has been extensively investigated in the past by methods such as the Gutzwiller projection ansatz [9], the strong-coupling expansion [10][11][12][13], and Quantum Monte-Carlo [14][15][16][17][18], as well as with various mean-field approaches (see, e.g., [1,19,20]). A very powerful numerical technique in the case of one spatial dimension is the density matrix renormalization group (DMRG) [21], which is used to calculate first-and second-order (i.e., amplitude and number-number) correlations [22].…”