Dimensional Phase Transition from an Array of 1D Luttinger Liquids to a 3D Bose-Einstein Condensate

We study the thermodynamic properties of a 2D array of coupled one-dimensional Bose gases. The system is realized with ultracold bosonic atoms loaded in the potential tubes of a two-dimensional optical lattice. For negligible coupling strength, each tube is an independent weakly interacting 1D Bose gas featuring Tomonaga Luttinger liquid behavior. By decreasing the lattice depth, we increase the coupling strength between the 1D gases and allow for the phase transition into a 3D condensate. We extract the phase diagram for such a system and compare our results with theoretical predictions. Because of the high effective mass across the periodic potential and the increased 1D interaction strength, the phase transition is shifted to large positive values of the chemical potential. Our results are prototypical to a variety of low-dimensional systems, where the coupling between the subsystems is realized in a higher spatial dimension such as coupled spin chains in magnetic insulators.

Figure 1

(a) Experimental setup: 1D Bose gases (red) are trapped in a 2D optical lattice (blue) with finite tunnel coupling and imaged with an electron beam (yellow). (b) In situ image of the density distribution from which the line profiles are extracted. (c) Corresponding analogon in real materials: Coupled dimer spin triplet chains show Bose-Einstein condensation above a critical magnetic field.

Figure 2

Density profiles (blue points) for different lattice depths [(a) $s=30$, (b) $s=14$, (c) $s=10$] at a distance of $1.2\mu \mathrm{m}$ from the trap center. The red line is a fit with the 1D theory, where only the outer parts of the profile ($\gamma >2$) were included in the fit. The circles indicate the position, where the experimental profiles deviate from the pure 1D behavior by more than 1 standard deviation.

Figure 3

Condensate fraction derived from TOF images (blue points, the inset shows the corresponding TOF absorption images) and excess atoms derived from the in situ density distribution (red points). For $s>20$ no signatures of a condensate fraction or an excess density are found in the experiment.

Figure 4

Phase diagram in the $s\text{−}\mu$ plane. The points mark the critical chemical potential at which the phase transition takes place; the dashed line is a guide to the eye. The dashed-dotted black line is the prediction of Ref. [22]; the solid red line is the prediction of a 3D treatment of the system (see text).

Figure 5

Ratio between the effective interaction for the coupled system and that for an isolated tube. Blue points are the experimental data and the black curve is the theoretical prediction (see text).