Time-of-flight observables and the formation of Mott domains of fermions and bosons on optical lattices

We study, using quantum Monte Carlo simulations, the energetics of the formation of Mott domains of fermions and bosons trapped on one-dimensional lattices. We show that, in both cases, the sum of kinetic and interaction energies exhibits minima when Mott domains appear in the trap. In addition, we examine the derivatives of the kinetic and interaction energies, and of their sum, which display clear signatures of the Mott transition. We discuss the relevance of these findings to time-of-flight experiments that could allow the detection of the metal–Mott-insulator transition in confined fermions on optical lattices, and support established results on the superfluid–Mott-insulator transition in confined bosons on optical lattices.

Figure 1

(Color online) Intensity plot of the density fluctuations ${\Delta }_i=⟨n_i^2⟩-⟨n_i{⟩}^2$ vs position and interaction strength, in fermionic (a) and bosonic (b) systems with 30 atoms on 48 lattice sites and $V_2{a}^2=0.015t$ ($a$ is the lattice constant). The lattice size was chosen large enough so that the particle density is zero at the boundaries. For large $U$, the constant small value of ${\Delta }_i$ in the center of the trap signals the formation of the MI core. In (b), for low values of $U∕t$, we have truncated ${\Delta }_i$ at 0.5, i.e., for low on-site repulsive interactions the bosonic fluctuations of the density exceed 0.5.

Figure 2

(Color online) (a) and (d) Kinetic energy $\left(E_K\right)$; (b) and (e) interaction energy $\left(E_I\right)$; and (c) and (f) the sum $E_K+E_I$, for fermions [(a)–(c)] and bosons [(d)–(f)] in the systems of Fig. 1 (30 atoms and $V_2{a}^2=0.015t$). Energies are given in units of $t$. Continuous lines depict numerical derivatives of the energies. The arrows signal the emergence of a MI core. The QMC errors are much smaller than our symbol sizes throughout this work.

Figure 3

(Color online) Ground-state on-site interaction energy per lattice site $\left({ϵ}_I\right)$ of fermions as a function of the density [obtained from a periodic system (no trap) with 102 lattice sites], for different values of $U∕t$. At $n=1$, fermions are in the MI phase; and for any other value of $n$ they are in a metallic phase.

Figure 4

(Color online) (a) Trap $\left(E_{\mathrm{Trap}}\right)$ and total $\left(E_{\mathrm{Total}}\right)$ energies corresponding to the systems of Figs. 2d, 2e, 2f. (b) $E_K+E_I$ for systems with the same trap parameters of Fig. 2 and different inverse temperatures $\beta =1∕k_{B}T$. $\beta t=64$ corresponds to the ground-state results of Figs. 2d, 2e, 2f.

Figure 5

(Color online) Intensity plot of the density fluctuations ${\Delta }_i=⟨n_i^2⟩-⟨n_i{⟩}^2$ vs position and interaction strength in fermionic (a) and bosonic (b) systems with 40 atoms on 60 lattice sites and $V_2{a}^2=0.01t$. With increasing $U$, two Mott domains appear at the sides of the central metallic phase before a full Mott core develops in the center of the trap. In (b), for low values of $U∕t$, ${\Delta }_i$ has been truncated as in Fig. 1.

Figure 6

(Color online) (a) and (d) Kinetic energy $\left(E_K\right)$; (b) and (e) interaction energy $\left(E_I\right)$; and (c) and (f) the sum $E_K+E_I$, for fermions [(a)–(c)] and bosons [(d)–(f)] in the systems of Figs. 5 (40 atoms and $V_2{a}^2=0.01t$). Continuous lines depict the numerical derivatives of the energies. The arrows indicate when two MI domains (lower value of $U∕t$) and the MI core (larger value of $U∕t$) emerge in each system.