Ground-state and finite-temperature properties of correlated ultracold bosons on optical lattices

We investigate the ground-state and finite-temperature properties of the Bose-Hubbard model using the standard basis operator method applied to the equation of motion for Green's functions. We show that by introducing the self-consistent approach, the theory goes significantly beyond the mean-field approximation, allowing more-precise description of quantum and thermal fluctuations. For optical lattice systems with ultracold bosons, we find relevant quantitative agreement with numerical and experimental data. In particular, the presented method reveals the importance of thermal fluctuations in time-of-flight experiments. For the most reliable fitting of the visibility analytical curve to the experimental data, the value of temperature is determined and proximity to the quantum regime is discussed. The problem of identification of the superfluid–Mott-insulator phase boundary from time-of-flight experiments is also considered.

Figure 1

Zero-temperature phase diagram with average density of one boson per lattice site $\left(n=1\right)$ for two-dimensional square (a) and three-dimensional simple cubic (b) lattice. The dashed red line shows the numerical MC results [20, 49], the dashed orange line represents the mean-field calculations, and the solid black line is obtained by the SC-SBO method proposed in this work.

Figure 2

Dependence of critical temperature $T_c/J$ on $U/J$ for the N-SF transition in the cubic lattice. The solid and dashed lines represent the same theories as in Fig. 1. Additionally, white circles represent the experimental data from Ref. [18] and the green dashed line corresponds to the three-states MF approximation in comparison to the orange dashed line in MF approximation with infinite number of states at site [22].

Figure 3

Dependence of quasiparticle gap $\mathrm{\Delta }/U$ on the hopping energy $J$ in the MI phase. (a) Two-dimensional lattice $\left(T=0\right)$ and (b) three-dimensional lattice [arrows indicate different temperatures used in SC-SBO calculation; for MF-RPA (red dashed line) the curve is plotted for $T=0$]. In panel (a) we plot experimental data from Ref. [13] as white circles with error bars.

Figure 4

Simulation of the TOF experiment taken at finite temperatures within quantum critical region where ${(U/J)}_c\approx 29.80$. The first plot of each (a)–(c) row is the density of column-integrated momentum distribution function $N_{\perp }(k_x,k_y)$ and the second one is the cut $N_{\perp }(k_x,0)$. In the following rows the temperatures are $\left(\text{a}\right)T/J=13,\left(\text{b}\right)T/J=6,\left(\text{c}\right)T/J=0.1$.

Figure 5

Dependence of visibility $\nu$ on the interaction strength $U/J$. In both plots we compare the experimental data with theoretical curves: (a) SC-SBO and MF-RPA in $T=0\text{K}$ and (b) SC-SBO in $T/J=13.7$. Experimental data were collected for ${\text{Rb}}^{\text{87}}$ atoms at the $(F=2,m_F=2)$ state loaded on an optical lattice [31]. The inset in panel (b) enlarges the visibility in the proximity of low interaction regime $U/J<50$.

Figure 6

Zero-temperature phase diagram between SF and MI phase for a (a) two-dimensional system and (b) three-dimensional system. SC-SBO, black solid line; SC-SBO ver. 2, black dashed line; and MC, red dashed line.

Figure 7

Average particle density for a (a) two-dimensional system and (b) three-dimensional system when including fluctuations in the MI phase for first lobe $n=1$. SC-SBO, black solid line; SC-SBO ver. 2, black dashed line; and MF-RPA, orange dashed line (see also Secs. 3c and 3b). To better compare different theories, we take the critical value of the hopping parameter $J_c$ (at the tip of the lobe), which corresponds to the given approach.