A Critical Point

We read with great interest the illuminating review article by Barbara Goss Levi ( Physics Today, March 2002, page 18), in which she discusses the recent pioneering experiment on superfluid-Mott insulator transition in a system of ultracold atoms in an optical lattice. However, we disagree with the description of the data shown in Figure 1, in particular that "the phase transition occurs somewhere between (f) and (g)." The Garching-Munich group, which performed the experiment Levi describes, associates the quantum critical point with the case (e). We also disagree with the interpretation given by the Garching-Munich group of the interference peaks seen in their experiment. They associate the disappearance of Bragg reflection peaks with the phase transition of the Bose-Einstein condensate from a coherent, superfluid phase to a Mott insulating phase. We contend, however, that one cannot interpret the fading of Bragg peaks as a signature of a phase transition. The appearance of a Mott phase will be seen only in the fine structure of peaks.

The interference pattern of Bragg peaks results from the periodic lattice structure of the system and the phase coherence between lattice sites. As long as phase coherence exists on length scales of several lattice sites, one should see a clear picture of narrow interference peaks, with peak positions being in one-to-one correspondence with the reciprocal lattice space--that is, the Bragg peaks simply reflect the underlying periodic lattice. In a Mott phase, such a coherence is still present near the critical point (when insulating gaps are small) due to quantum hopping of loosely localized atoms to the neighboring sites; the coherence extends over a large correlation length. In this case, the interference signals only the essential quantum nature of the ground state.

The coherence disappears gradually as the tunneling strength grows weaker, as seen between (f) and (g), but this fading of Bragg peaks occurs beyond the critical point. For quantitative details and an extended discussion, see our precise numeric simulation of the experimental situation at http://arXiv.org/abs/cond-mat/0202510.