Phase Diagram for a Bose-Einstein Condensate Moving in an Optical Lattice

The stability of superfluid currents in a system of ultracold bosons was studied using a moving optical lattice. Superfluid currents in a very weak lattice become unstable when their momentum exceeds 0.5 recoil momentum. Superfluidity vanishes already for zero momentum as the lattice deep reaches the Mott insulator (MI) phase transition. We study the phase diagram for the disappearance of superfluidity as a function of momentum and lattice depth between these two limits. Our phase boundary extrapolates to the critical lattice depth for the superfluid-to-MI transition with 2% precision. When a one-dimensional gas was loaded into a moving optical lattice a sudden broadening of the transition between stable and unstable phases was observed.

Figure 1

Phase diagram showing the stability of superfluid flow in an optical lattice and the experimental procedure. The gray curve shows the predicted boundary between superfluid flow and dissipative flow phases for a three-dimensional gas with a commensurate filling of $N=1$ atom per site [9]. The solid (dashed) arrows illustrate the experimental trajectory used for small (large) lattice depths (see text for details).

Figure 2

Determination of the critical momentum of superfluid flow. Shown is the condensate fraction as a function of a momentum $p$. (a) Condensate fraction with $u/u_c=0.61$ for a variable number of cycles of the momentum modulation (one cycle: × and blue line, two cycles: ▪ and purple line, three cycles: ▴ and red line). A dashed vertical line indicates the critical momentum where instability begins to occur. The two and three-cycle data are offset vertically for clarity. These data were fitted with an error function to guide the eye. (b) Images of interference patterns released from an optical lattice at $u/u_c=0.61$ moving with variable momentum. Instability occurred between $p=0.31p_r$ and $0.32p_r$. Some of the triangular data points in (a) were obtained from these images. (c) Condensate fraction on a log-log scale for two different interaction strengths.

Figure 3

Critical momentum for a condensate in a 3D lattice. The solid line shows the theoretical prediction for the superfluid region. The horizontal solid line is a fit to the data points in the MI phase. (Inset) Fit of critical momenta near the SF-MI phase transition.

Figure 4

Critical momentum for a 1D gas in an optical lattice. (a) The gray line indicates the mean-field theory prediction. The interaction strengths are normalized by the mean-field prediction for $u_c=5.8\times 2$ [1, 4]. Squares (crosses) represent the measured critical momentum (the center of the transition). Measurements were taken at lattice depths of 0.25, 0.50, 0.75, 1.0, 2.0 $E_R$. The lines between crosses and squares indicate the width of the transition region. (b) Condensate fraction measured at 0.25 $E_R$ and 0.75 $E_R$. The data were fitted with an error function. Squares (the critical momentum) and crosses (the center of the transition) are indicated on the plots.