Dimensionally induced one-dimensional to three-dimensional phase transition of the weakly interacting ultracold Bose gas

We investigate the dimensionally induced phase transition from the normal to the Bose-Einstein-condensed phase for a weakly interacting Bose gas in an optical lattice. To this end we make use of the Hartree-Fock-Bogoliubov-Popov theory, where we include numerically exact hopping energies and effective interaction strengths. At first we determine the critical chemical potential, where we find a much better agreement with recent experimental data than a pure Hartree-Fock treatment. This finding originates from an important correction due to quantum fluctuations, which are enhanced in lower dimensions. Furthermore, we determine for the one-dimensional to three-dimensional transition the power-law exponent of the critical temperature for two different noninteracting Bose gas models yielding the same value of 1/2, which indicates that they belong to the same universality class. For the weakly interacting Bose gas we find for both models that this exponent is robust with respect to finite interaction strengths.

Figure 1

Schematic setup of 2D lattice model for two dimensionless lattice depths $s$: bosonic clouds in longitudinal direction are depicted in blue and optical lattice potential in $xy$ plane is represented by black lines.

Figure 2

Hopping energy $J$ as function of dimensionless lattice depth $s$: lines depict from the top to the bottom solutions of three different approximation methods described in the text, respectively. Since the top blue curve defined by Eq. (10) vanishes at $s=0$, it is plotted only up to its maximal value.

Figure 3

One-dimensional effective interaction strength $g_{\mathrm{eff}}^{1\mathrm{D}}$ as function of dimensionless lattice depth $s$. Top curve corresponds to tight-binding approximation (11), bottom curve stems from Eq. (13) and numerically determined Wannier function, as used in Ref. [21].

Figure 4

Phase diagram in $s\text{−}\mu$ plane: red data points are reproduced from Ref. [21] and describe phase boundary between decoupled 1D tubes and 3D condensate, red bottom curve is HF result from Ref. [21], solid lines represent HF, and dashed lines HFBP results.

Figure 5

Power-law exponent $\alpha$ as functions of gas parameter ${\gamma }^{1/3}=n^{1/3}{a}_s$ for (a) 2D lattice model and (b) 3D lattice model. Dashed lines represent corresponding value $\alpha =1/2$ for noninteracting case.