Figure 1

The intensity plot for the column-integrated momentum distribution near the first Brillouin zone with different trapping frequency $\omega ∕2\pi$ and temperature $T$. (a) and (c) are for the normal phase, and (b) and (d) are for the superfluid phase with the corresponding parameters: (b) $T∕T_c\sim 0.7$ and (d) $T∕T_c\sim 0.94$. The total particle number is fixed to be $N_t={10}^5$ for all cases, and the lattice barrier $V_0=10E_R$. The unit for the axes is $2\pi ∕d$.Reuse & Permissions

Figure 2

(Color online) Column-integrated momentum density profile (shown along the $x$ axis) for (a),(c),(e) noninteracting lattice bosons and (b),(d),(f) interacting lattice bosons. The solid curves are the total momentum density profile, the thick dashed curves are the momentum density profile of the normal component, and the dotted curves are the momentum density profile taking into account of the interaction broadening during the time-of-flight evolution. (a) $N_0∕N_t\sim 0.46$, $T∕T_c\sim 0.75$ ($N_0$ is the total number of atoms in the condensate and $N_t$ is the total number of atoms in the trap); (b) $N_0∕N_t\sim 0.52$, $T∕T_c\sim 0.34$; (c) $N_0∕N_t\sim 0.10$, $T∕T_c\sim 0.95$; (d) $N_0∕N_t\sim 0.10$, $T∕T_c\sim 0.65$ (inset: enlarged density profile below the dotted line); (e) $T\sim 1.1T_c$; (f) $T\sim 1.1T_c$. For the noninteracting calculations, the total particle number is fixed to be $N_t={10}^5$ and the trapping frequency is $\omega ∕2\pi =20\mathrm{Hz}$; for the calculations with interactions, the trapping potential is $\omega ∕2\pi =20\mathrm{Hz}$, the number density per site at the center of the trap is fixed to be $n\left(0\right)=1$, with the total number of particles on the order of $N_t={10}^5–{10}^6$. The lattice barrier in all cases is $V_0=10E_R$. The unit for the $x$ axes is $2\pi ∕d$.Reuse & Permissions